12th Standard Maths State Board (Tamilnadu) - study material, free online tests, previous year question papers, answer keys, topper answers, centum question paper, exam tips

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 5 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Let f(x, y) = sin(xy²) + \(e^{{x^3}+5y}\)for all ∈ R². Calculate \(\frac { \partial f }{ \partial x } ,\frac { \partial f }{ \partial y } ,\frac { { \partial }^{ 2 }f }{ { \partial y\partial x } } \)and \(\frac { { \partial }^{ 2 }f }{ { \partial x\partial y } } \)
2)
Let w(x, y) = xy+\(\frac { { e }^{ y } }{ { y }^{ 2 }+1 } \) for all (x, y) ∈ R². Calculate \(\frac { { \partial }^{ 2 }w }{ { \partial y\partial x } } \) and \(\frac { { \partial }^{ 2 }w }{ { \partial x\partial y } } \)
3)
For each of the following functions find the f_x, f_y, and show that f_xy= f_yx
f(x, y) = \(\frac { 3x }{ y+sinx \ } \)
4)
If U(x, y, z) = log (x³ + y³ + z³), find \(\frac { \partial U }{ \partial x } +\frac { \partial U }{ \partial y } +\frac { \partial U }{ \partial z } \)
5)
If V(x,y) = e^x(x cos y - y siny), then prove that \(\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } =\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } } \) = 0

12th Standard English Medium Maths Subject Applications of Integration Book Back 1 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The area between y² = 4x and its latus rectum is
2)
The value of \(\int _{ 0 }^{ 1 }{ x{ (1-x) }^{ 99 }dx } \) is
3)
The value of \(\int _{ 0 }^{ \pi }{ \frac { dx }{ 1+{ 5 }^{ cos\ x } } } \) is
4)
If \(\frac{\Gamma(n+2)}{\Gamma(n)}=90\) then n is
5)
The value of \(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ { cos }^{ 3 }3x\ dx }\ is\)

12th Standard English Medium Maths Subject Applications of Integration Book Back 1 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The value of \(\int _{ 0 }^{ a }{ { (\sqrt { { a }^{ 2 }-{ x }^{ 2 } } ) }^{ 3 } } dx\) is
2)
If \(\int _{ 0 }^{ x }{ f(t)dt=x+\int _{ x }^{ 1 }{ tf } (t)dt } \), then the value of f (1) is
3)
\(\text { The value of } \int_{0}^{\frac{2}{3}} \frac{d x}{\sqrt{4-9 x^{2}}} \text { is }\)
4)
The value of \(\int _{ -1 }^{ 2 }{ |x|dx } \) is
5)
For any value of \(n \in \mathbb{Z}, \int_{0}^{\pi} e^{\cos ^{2} x} \cos ^{3}[(2 n+1) x] d x\) is

12th Standard English Medium Maths Subject Applications of Integration Book Back 2 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Evaluate: \(\int _{ 0 }^{ 3 }{ (3{ x }^{ 2 }-4x+5) } dx\)
2)
Evaluate :\(\int _{ 0 }^{ 1 }{ [2x] } dx\) where [⋅] is the greatest integer function
3)
Evaluate the following \(\int _{ 0 }^{ \pi /2 }{ { cos}^{ 7}x\quad dx } \)
4)
Evaluate the following
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }x{ cos }^{ 4 }xdx } \)
5)
Find the area bounded by y=x²+2,x-x-axis, x=1 and x=2

12th Standard English Medium Maths Subject Applications of Integration Book Back 2 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Evaluate \(\int _{ b }^{ \infty }{ \frac { 1 }{ { a }^{ 2 }+{ x }^{ 2 } } dx,a>0,b\in R } \)
2)
Evaluate the following
\(\int _{ 0 }^{ 1 }{ { x }^{ 2 }{ (1-x) }^{ 3 }dx } \)
3)
Find the area of the region bounded by the curve y = sin x and the ordinate x=0 \(x=\frac { \pi }{ 3 } \)
4)
Find the area enclosed between the parabola y²=4ax and the line x=a, x=9a.
5)
Find the area bounded by the curve y = cosax in one arc of the curve.

12th Standard English Medium Maths Subject Applications of Integration Book Back 3 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find an approximate value of \(\int _{ 1 }^{ 1.5 }{ xdx } \) by applying the left-end rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}.
2)
Evaluate :\(\int _{ 0 }^{ \frac { \pi }{ 3 } }{ \frac { sec\ x\ tan\ x }{ 1+{ sec }^{ 2 }x } dx } \)
3)
Evaluate :\(\int _{ 0 }^{ 9 }{ \frac { 1 }{ x+\sqrt { x } } dx } \)
4)
Evaluate \(\int _{ 1 }^{ 2 }{ \frac { x }{ (x+1)(x+2) } dx } \)
5)
Evaluate: \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { cos\theta }{ (1+sin\theta )(2+sin\theta ) } } d\theta \)

12th Standard English Medium Maths Subject Applications of Integration Book Back 3 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Prove that \(\int _{ 0 }^{ \infty }{ { e }^{ -x }{ x }^{ n }dx=n! } \) where n is a positive integer.
2)
Find the area of the region bounded by the line 6x + 5y = 30, x − axis and the lines x = −1 and x = 3.
3)
Find the area of the region bounded by the line 7x − 5y = 35, x−axis and the lines x = −2 and x = 3.
4)
Find the area of the region bounded between the parabola y² = 4ax and its latus rectum.
5)
Find the area of the region bounded by the y-axis and the parabola x = 5 − 4y − y².

12th Standard English Medium Maths Subject Applications of Integration Book Back 5 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Estimate the value of \(\int _{ 0 }^{ 0.5 }{ { x }^{ 2 } } dx\) using the Riemann sums corresponding to 5 subintervals of equal width and applying
(i) left-end rule
(ii) right-end rule
(iii) the mid-point rule.
2)
Evaluate\(\int _{ 1 }^{ 4 }{ ({ 2x }^{ 2 }+3) } \) dx, as the limit of a sum
3)
Show that \(\int ^{1}_{0} (tan ^{-1} x + tan ^{-1}(1-x))\) dx = \(\frac {\pi}{2}\) - log_e2
4)
Find the area of the region bounded by y = cos x, y = sin x, the lines x = \(\frac{\pi}{4}\) and x = \(\frac{5\pi}{4}\).
5)
The curve y = (x − 2)²+1 has a minimum point at P. A point Q on the curve is such that the slope of PQ is 2. Find the area bounded by the curve and the chord PQ.

12th Standard English Medium Maths Subject Applications of Integration Book Back 5 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Evaluate the following:
\(\int _{ 0 }^{ 1 }{ { x }^{ 3 }{ e }^{ -2x }dx } \)
2)
The region enclosed by the circle x² + y² = a² is divided into two segments by the line x = h. Find the area of the smaller segment.
3)
Find, by integration, the area of the region bounded by the lines 5x − 2y = 15, x + y + 4 = 0 and the x-axis
4)
Father of a family wishes to divide his square field bounded by x = 0, x = 4, y = 4 and y = 0 along the curve y² = 4x and x²= 4y into three equal parts for his wife, daughter and son. Is it possible to divide? If so, find the area to be divided among them.
5)
Find, by integration, the volume of the container which is in the shape of a right circular conical frustum.

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 1 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The order and degree of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ \left( \frac { dy }{ dx } \right) }^{ 1/3 }+{ x }^{ 1/4 }=0\) are respectively
2)
The differential equation representing the family of curves y = Acos(x + B), where A and B are parameters,is
3)
The solution of \(\frac{d y}{d x}+p(x) y=0\) is
4)
The integrating factor of the differential equation \(\frac { dy }{ dx } +y=\frac { 1+y }{ \lambda } \) is
5)
The integrating factor of the differential equation \(\frac{d y}{d x}+P(x) y=Q(x)\) is x, then P(x)

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 1 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The number of arbitrary constants in the particular solution of a differential equation of third order is
2)
Integrating factor of the differential equation \(\frac{d y}{d x}=\frac{x+y+1}{x+1}\) is
3)
The population P in any year t is such that the rate of increase in the population is proportional to the population. Then
4)
If the solution of the differential equation \(\frac{d y}{d x}=\frac{a x+3}{2 y+f}\)represents a circle, then the value of a is
5)
The slope at any point of a curve y = f (x) is given by \(\frac{dy}{dx}=3x^2\) and it passes through (-1, 1). Then the equation of the curve is

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 2 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
For each of the following differential equations, determine its order, degree (if exists)
\({ \left( \frac { { d }^{ 3 }y }{ d{ x }^{ 3 } } \right) }^{ \frac { 2 }{ 3 } }-3\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +5\frac { dy }{ dx } +4=0\)
2)
For each of the following differential equations, determine its order, degree (if exists)
\({ x }^{ 2 }\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +{ \left[ 1+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 1 }{ 2 } }=0\)
3)
For each of the following differential equations, determine its order, degree (if exists)
\({ \left( \frac { d^2y }{ dx^2 } \right) }^{ 3 }=\sqrt { 1+\left( \frac { dy }{ dx } \right) } \)
4)
For each of the following differential equations, determine its order, degree (if exists)
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +5\frac { dy }{ dx } +\int { ydx } ={ x }^{ 3 }\)
5)
For each of the following differential equations, determine its order, degree (if exists)
\(x={ e }^{ xy\left( \frac { dy }{ dx } \right) }\)

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 2 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find value of m so that the function y = e^mx is a solution of the given differential equation, y''− 5y' + 6y = 0
2)
Show that y = a cos bx is a solution of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ b }^{ 2 }y=0\).
3)
Determine the order and degree (if exists) of the following differential equations:
\(3\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ={ \left[ 4+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 3 }{ 2 } }\)
4)
Determine the order and degree (if exists) of the following differential equations:
dy + (xy − cos x)dx = 0
5)
Find the differential equation of the family of parabolas y² = 4ax, where a is an arbitrary constant.

12th Standard Maths Public Question Paper - March 2022 updated Previous Year Question Papers - by QB Admin - May 13, 2022 - View & Read

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 3 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the differential equation of the family of parabolas with vertex at (0, −1) and having axis along the y-axis.
2)
Find the differential equation corresponding to the family of curves represented by the equation y = Ae^8x + Be^-8x, where A and B are arbitrary constants.
3)
Find the differential equation of the curve represented by xy = ae^x + be^−x + x².
4)
Show that y = ae^-3x + b, where a and b are arbitary constants, is a solution of the differential equation\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3\frac { dy }{ dx } =0\)
5)
Find the differential equation of the family of all ellipses having foci on the x -axis and centre at the origin.

12th Standard Mathematics Public Question Paper Answer key - March 2022 updated Previous Year Question Papers - by QB Admin - May 13, 2022 - View & Read

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 3 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the differential equation of the family of circles passing through the origin and having their centres on the x -axis.
2)
Find the differential equation corresponding to the family of curves represented by the equation y = Ae^8x + Be^-8x, where A and B are arbitrary constants.
3)
The slope of the tangent to the curve at any point is the reciprocal of four times the ordinate at that point. The curve passes through (2, 5). Find the equation of the curve.
4)
Form the differential equation by eliminating the arbitrary constants A and B from y = A cos x + B sin x.
5)
Solve : \(\frac { dy }{ dx } =\sqrt { 4x+2y-1 } \)

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 5 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Solve \((1+{ 2e }^{ x/y })dx+2{ e }^{ x/y }\left( 1-\frac { x }{ y } \right) dy=0\)
2)
Solve the following differential equations
(x³+ y³) dy-x²ydx = 0
3)
Solve the Linear differential equation:
\(\frac { dy }{ dx } +\frac { y }{ x } =sinx\)
4)
The growth of a population is proportional to the number present. If the population of a colony doubles in 50 years, in how many years will the population become triple?
5)
A pot of boiling water at 100^oC is removed from a stove at time t = 0 and left to cool in the kitchen. After 5 minutes, the water temperature has decreased to 80^oC , and another 5 minutes later it has dropped to 65^oC. Determine the temperature of the kitchen.

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 5 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Solve the Linear differential equation:
\(\frac { dy }{ dx } +\frac { y }{ x } =sinx\)
2)
Solve the Linear differential equation:
\(({ x }^{ 2 }+1)\frac { d }{ y } dx+2xy=\sqrt { { x }^{ 2 }+4 } \)
3)
The growth of a population is proportional to the number present. If the population of a colony doubles in 50 years, in how many years will the population become triple?
4)
In a murder investigation, a corpse was found by a detective at exactly 8 p.m. Being alert, the detective also measured the body temperature and found it to be 70^oF. Two hours later, the detective measured the body temperature again and found it to be 60^oF. If the room temperature is 50^oF, and assuming that the body temperature of the person before death was 98.6^oF, at what time did the murder occur? [log(2.43) = 0.88789; log(0.5)=-0.69315]
5)
A tank initially contains 50 litres of pure water. Starting at time t = 0 a brine containing with 2 grams of dissolved salt per litre flows into the tank at the rate of 3 litres per minute. The mixture is kept uniform by stirring and the well-stirred mixture simultaneously flows out of the tank at the same rate. Find the amount of salt present in the tank at any time t > 0.

12th Standard Maths English Medium Application of Matrices and Determinants Important Questions updated Sample Question Papers - by Question Bank Software - May 13, 2022 - View & Read

1)
If \(\rho\) (A) ≠ \(\rho\) ([AIB]), then the system is _____________
2)
If A = \(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \), then adj(adj A) is
3)
If A is a matrix of order m \(\times\) n, then \(\rho\) (A) is _________
4)
If A = [2 0 1] then the rank of AA^T is ______
5)
The two lines are Parallel (non-coincident) then the solution is ___________

12th Standard English Medium Maths Subject Probability Distributions Book Back 1 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is
2)
A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is
3)
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. Then the possible values of X are
4)
If the function \(f(x)=\frac { 1 }{ 12 } \) for a < x < b, represents a probability density function of a continuous random variable X, then which of the following cannot be the value of a and b?
5)
If X is a binomial random variable with expected value 6 and variance 2.4, then P(X = 5) is

12th Standard English Medium Maths Subject Probability Distributions Book Back 1 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Suppose that X takes on one of the values 0, 1, and 2. If for some constant k, \(P(X=i)=k P(X=i-1) \text { for } i=1,2 \text { and } P(X=0)=\frac{1}{7}\) , then the value of k is
2)
The probability mass function of a random variable is defined as:

x -2 -1 0 1 2

f(x) k 2k 3k 4k 5k

Then E(X ) is equal to:
3)
Let X have a Bernoulli distribution with mean 0.4, then the variance of (2X - 3) is
4)
If in 6 trials, X is a binomial variable which follows the relation 9P(X = 4) = P(X = 2), then the probability of success is
5)
A computer salesperson knows from his past experience that he sells computers to one in every twenty customers who enter the showroom. What is the probability that he will sell a computer to exactly two of the next three customers?

12th Standard English Medium Maths Subject Probability Distributions Book Back 2 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
For each of the following differential equations, determine its order, degree (if exists)
\(\frac { dy }{ dx } +xy=cotx\)
2)
For each of the following differential equations, determine its order, degree (if exists)
\({ \left( \frac { { d }^{ 3 }y }{ d{ x }^{ 3 } } \right) }^{ \frac { 2 }{ 3 } }-3\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +5\frac { dy }{ dx } +4=0\)
3)
For each of the following differential equations, determine its order, degree (if exists)
\(\sqrt { \frac { dy }{ dx } } -4\frac { dy }{ dx } -7x=0\)
4)
For each of the following differential equations, determine its order, degree (if exists)
\(y\left( \frac { dy }{ dx } \right) =\frac { x }{ \left( \frac { dy }{ dx } \right) +{ \left( \frac { dy }{ dx } \right) }^{ 3 } } \)
5)
Express each of the following physical statements in the form of differential equation.
(i) Radium decays at a rate proportional to the amount Q present.
(ii) The population P of a city increases at a rate proportional to the product of population and to the difference between 5,00,000 and the population.
(iii) For a certain substance, the rate of change of vapor pressure P with respect to temperature T is proportional to the vapor pressure and inversely proportional to the square of the temperature.
(iv) A saving amount pays 8% interest per year, compounded continuously. In addition, the income from another investment is credited to the amount continuously at the rate of Rs. 400 per year.

12th Standard English Medium Maths Subject Probability Distributions Book Back 2 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Determine the order and degree (if exists) of the following differential equations:
\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3{ \left( \frac { dy }{ dx } \right) }^{ 2 }={ x }^{ 2 }log\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \)
2)
Determine the order and degree (if exists) of the following differential equations:
\(3\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ={ \left[ 4+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 3 }{ 2 } }\)
3)
Determine the order and degree (if exists) of the following differential equations:
dy + (xy − cos x)dx = 0
4)
Solve:\(\frac { dy }{ dx } \) = (3x+y+4)².
5)
Solve the following differential equations or show that the solution of
\(\\ \\ \\ \frac { dy }{ dx } =\sqrt { \frac { 1-{ y }^{ 2 } }{ 1-{ x }^{ 2 } } } \)

12th Standard English Medium Maths Subject Probability Distributions Book Back 3 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
In a pack of 52 playing cards, two cards are drawn at random simultaneously. If the number of black cards drawn is a random variable, find the values of the random variable and number of points in its inverse images.
2)
Two balls are chosen randomly from an urn containing 6 red and 8 black balls. Suppose that we win Rs. 15 for each red ball selected and we lose Rs. 10 for each black ball selected. X denotes the winning amount, then find the values of X and number of points in its inverse images.
3)
A six sided die is marked '2' on one face, '3' on two ofits faces, and '4' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find the values of the random variable and number of points in its inverse images.
4)
The probability density function of X is
\(f(x)=\left\{\begin{array}{cc} x & 0
find P(0.2 ≤ X< 0.6)
5)
The probability density function of X is
\(f(x)=\left\{\begin{array}{cc} x & 0
find P(0.5≤X<1.5)

12th Standard English Medium Maths Subject Probability Distributions Book Back 3 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
For the random variable X with the given probability mass function as below, find the mean and variance.
\(f(x)=\begin{cases} \begin{matrix} \cfrac { 1 }{ 2 } e^{ -\frac { x }{ 2 } } & for\quad x>0 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\)
2)
If X~ B(n, p) such that 4P(X = 4) = P(X = 2) and n = 6. Find the distribution, mean and standard deviation of X.
3)
An urn contains 2 white balls and 3 red balls. A sample of 3 balls are chosen at random from the urn. If X denotes the number of red balls chosen, find the values taken by the random variable X and its number of inverse images
4)
Two balls are chosen randomly from an urn containing 6 white and 4 black balls. Suppose that we win Rs. 30 for each black ball selected and we lose Rs. 20 for each white ball selected. If X denotes the winning amount, then find the values of X and number of points in its inverse images.
5)
Two fair coins are tossed simultaneously (equivalent to a fair coin is tossed twice). Find the probability mass function for number of heads occurred.

12th Standard English Medium Maths Subject Probability Distributions Book Back 5 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Let X be a random variable denoting the life time of an electrical equipment having probability density function
\(f(x)=\begin{cases} \begin{matrix} { ke }^{ -2x } & forx>0 \end{matrix} \\ \begin{matrix} 0 & forx\le 0 \end{matrix} \end{cases}\)
Find
(i) the value of k
(ii) Distribution function
(iii) P(X < 2)
(iv) calculate the probability that X is at least for four unit of time
(v) P(X = 3)
2)
Suppose that f (x) given below represents a probability mass function

x 1 2 3 4 5 6

f(x) c² 2c² 3c² 4c² c 2c

Find
(i) the value of c
(ii) Mean and variance.
3)
Two balls are chosen randomly from an urn containing 8 white and 4 black balls. Suppose that we win Rs. 20 for each black ball selected and we lose Rs. 10 for each white ball selected. Find the expected winning amount and variance
4)
The mean and variance of a binomial variate X are respectively 2 and 1.5. Find
(i) P(X = 0)
(ii) P(X =1)
(iii) P(X ≥1)
5)
On the average, 20% of the products manufactured by ABC Company are found to be defective. If we select 6 of these products at random and X denote the number of defective products find the probability that
(i) two products are defective
(ii) at most one product is defective
(iii) at least two products are defective.

12th Standard English Medium Maths Subject Probability Distributions Book Back 5 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 Iitres and a maximum of 600 litres with probability density function
\(\begin{cases} \begin{matrix} k & 200\le x\le 600 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\)
Find
(i) the value of k
(ii) the distribution function
(iii) the probability that daily sales will fall between 300 litres and 500 litres?
2)
If X is the random variable with probability density function f(x) given by,
\(f(x)=\begin{cases} \begin{matrix} x+1 & -1\le x<0 \end{matrix} \\ \begin{matrix} -x+1 & 0\le x<1 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\)
then find
(i) the distribution function F(x)
(ii) P( -0.5 ≤X ≤ 0.5)
3)
If X is the random variable with distribution function F(x) given by,

then find (i) the probability density function f(x)
(ii) P(0.3 ≤ X ≤ 0.6)
4)
A retailer purchases a certain kind of electronic device from a manufacturer. The manufacturer, indicates that the defective rate of the device is 5%. The inspector of the retailer randomly picks 10 items from a shipment. What is the probability that there will be
(i) at least one defective item
(ii) exactly two defective items.
5)
The mean and standard deviation of a binomial variate X are respectively 6 and 2.
Find
(i) the probability mass function
(ii) P(X = 3)
(iii) P(X\(\ge \)2).

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 1 Mark Questions with Solution Part -I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If a compound statement involves 3 simple statements, then the number of rows in the truth table is
2)
Which one is the contrapositive of the statement (pVq)⟶r?
3)
Which one of the following is incorrect? For any two propositions p and q, we have
4)
The dual of ᄀ(p V q) V [p V (p ∧ ᄀr)] is
5)
Which one of the following is not true?

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 1 Mark Questions with Solution Part -II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
In the set Q define a⊙b = a+b+ab. For what value of y, 3⊙(y⊙5) = 7?
2)
Which one is the contrapositive of the statement (pVq)⟶r?
3)
Which one of the following is incorrect? For any two propositions p and q, we have
4)
The dual of ᄀ(p V q) V [p V (p ∧ ᄀr)] is
5)
Which one of the following is not true?

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 2 Mark Questions with Solution Part -I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Determine whether ∗ is a binary operation on the sets given below.
(a*b) = a√b is binary on R
2)
On Z, define \(⊗ \mathrm{by}(m * n)\) = mn + nm: ∀m, n∈Z. Is ⊗ binary on Z?
3)
Determine the truth value of each of the following statements
(i) If 6 + 2 = 5 , then the milk is white.
(ii) China is in Europe or \(\sqrt3\) is an integer
(iii) It is not true that 5 + 5 = 9 or Earth is a planet
(iv) 11 is a prime number and all the sides of a rectangle are equal
4)
Which one of the following sentences is a proposition?
(i) 4 + 7 =12
(ii) What are you doing?
(iii) 3ⁿ ≤ 81, n ∈ N
(iv) Peacock is our national bird
(v) How tall this mountain is!
5)
Consider the binary operation ∗ defined on the set A = {a, b, c, d} by the following table:

* a b c d

a a c b d

b d a b c

c c d a a

d d b a c

Is it commutative and associative?

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 2 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Let p: Jupiter is a planet and q: India is an island be any two simple statements. Give verbal sentence describing each of the following statements.
(i) ¬p
(ii) p ∧ ¬q
(iii) ¬p ∨ q
(iv) p➝ ¬q
(v) p↔q
2)
Write each of the following sentences in symbolic form using statement variables p and q.
(i) 19 is not a prime number and all the angles of a triangle are equal.
(ii) 19 is a prime number or all the angles of a triangle are not equal
(iii) 19 is a prime number and all the angles of a triangle are equal
(iv) 19 is not a prime number
3)
Fill in the following table so that the binary operation ∗ on A = {a, b, c} is commutative.

* a b c

a b

b c b a

c a c
4)
Write the converse, inverse, and contrapositive of each of the following implication.
If x and y are numbers such that x = y, then x² = y²
5)
Construct the truth table for the following statements.
(¬p ⟶ r) ∧ ( p ↔️ q)

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 3 Mark Questions with Solution Part -I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation + on Z.
2)
Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation - on Z.
3)
Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation + on Z_e = the set of all even integers
4)
Verify
(i) closure property
(ii) commutative property, and
(iii) associative property of the following operation on the given set. (a*b) = a^b;∀a, b∈N (exponentiation property)
5)
How many rows are needed for following statement formulae?
\(p \vee \neg t \wedge(p \vee \neg s)\)

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 3 Mark Questions with Solution Part -II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Establish the equivalence property p ➝ q ≡ ㄱp ν q
2)
Let \(A=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{matrix}\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix} \right) ,B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{matrix} \right) ,C=\left( \begin{matrix} 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix} \right) \)be any three boolean matrices of the same type.
Find AΛB
3)
Verify whether the following compound propositions are tautologies or contradictions or contingency
(p ∧ q) ∧ ¬ (p ∨ q)
4)
Let \(A=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{matrix}\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix} \right) ,B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{matrix} \right) ,C=\left( \begin{matrix} 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix} \right) \)be any three boolean matrices of the same type.
Find (A∧B)∨C
5)
Verify whether the following compound propositions are tautologies or contradictions or contingency
(( p V q)∧ ¬ p) ➝ q

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 5 Mark Questions with Solution Part -I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Using the equivalence property, show that p ↔️ q ≡ ( p ∧ q) v (ㄱp ∧ ㄱq)
2)
Verify whether the following compound propositions are tautologies or contradictions or contingency
((p⟶ q) ∧ (q ⟶ r)) ⟶ (p ⟶ r)
3)
Let M = \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\} \) and let * be the matrix multiplication. Determine whether M is closed under ∗. If so, examine the commutative and associative properties satisfied by ∗ on M.
4)
Let A be Q\{1}. Define ∗ on A by x*y = x + y − xy. Is ∗ binary on A? If so, examine the commutative and associative properties satisfied by ∗ on A.
5)
Check whether the statement p➝(q➝p) is a tautology or a contradiction without using the truth table.

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 5 Mark Questions with Solution Part -II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Using the equivalence property, show that p ↔️ q ≡ ( p ∧ q) v (ㄱp ∧ ㄱq)
2)
Let M = \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\} \) and let * be the matrix multiplication. Determine whether M is closed under ∗. If so, examine the commutative and associative properties satisfied by ∗ on M.
3)
Let M = \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\} \) and let ∗ be the matrix multiplication. Determine whether M is closed under ∗ . If so, examine the existence of identity, existence of inverse properties for the operation ∗ on M.
4)
Using truth table check whether the statements ¬(p V q) V (¬p ∧ q) and ¬p are logically equivalent.
5)
Prove p⟶(q⟶r) ☰ (p ∧ q)⟶r without using truth table.

12th Standard English Medium Maths Subject Book Back 1 Mark Questions with Solution Part -I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is
3)
If A = \(\left[ \begin{matrix} \cos { \theta } & \sin { \theta } \\ -\sin { \theta } & \cos { \theta } \end{matrix} \right] \) and A(adj A) = \(\left[ \begin{matrix} k & 0 \\ 0 & k \end{matrix} \right] \), then k =
4)
If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| = _______
5)
If z = cos\(\frac { \pi }{ 4 } \) + i sin\(\frac { \pi }{ 6 } \), then ______

12th Standard English Medium Maths Subject Book Back 1 Mark Questions with Solution Part -II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If x < 0, y < 0 such that xy = 1, then tan^-1(x) + tan^-1(y) =_____
2)
The distance between the foci of a hyperbola is 16 and e = \(\sqrt { 2 } \). Its equation is ____________
3)
If B, B¹ are the ends of minor axis, F₁, F₂ are foci of the ellipse \(\frac { { x }^{ 2 } }{ 8 } +\frac { { y }^{ 2 } }{ 4 } \) = 1 then area of F₁BF₂B¹ is __________
4)
Let \(\overset { \rightarrow }{ a } \), \(\overset { \rightarrow }{ b } \)and \(\overset { \rightarrow }{ c } \)be three non- coplanar vectors and let \(\overset { \rightarrow }{ p } ,\overset { \rightarrow }{ q } ,\overset { \rightarrow }{ r } \) be the vectors defined by the relations \(\overset { \rightarrow }{ P } =\frac { \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ q } =\frac { \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ r } =\frac { \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } \) Then the value of \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) .\overset { \rightarrow }{ p } +\left( \overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) .\overset { \rightarrow }{ q } +\left( \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \right) .\overset { \rightarrow }{ r } \)= ____________
5)
The number of vectors of unit length perpendicular to the vectors \(\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) \) and \(\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right) \)is __________

12th Standard English Medium Maths Subject Book Back 2 Mark Questions with Solution Part -I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If A is a non-singular matrix of odd order, prove that |adj A| is positive
2)
Find the rank of each of the following matrices:
\(\left[ \begin{matrix} 3 & 2 & 5 \\ 1 & 1 & 2 \\ 3 & 3 & 6 \end{matrix} \right] \)
3)
If z₁= 1 - 3i, z₂= - 4i, and z₃ = 5 , show that (z₁+ z₂) + z₃= z₁+ (z₂+ z₃)
4)
Simplify \(\left( \frac { 1+i }{ 1-i } \right) ^{ 3 }-\left( \frac { 1-i }{ 1+i } \right) ^{ 3 }\) into rectangular form
5)
Construct a cubic equation with roots 1, 2 and 3

12th Standard English Medium Maths Subject Book Back 3 Mark Questions with Solution Part -I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the inverse of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \end{matrix} \right] \).
2)
If A = \(\left[ \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} \right] \), show that A^-1 = \(\frac {1}{2}\) (A² - 3I).
3)
Find the adjoint of the following:
\(\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right] \)
4)
Prove the following properties z is real if and only if z = \(\bar { z } \)
5)
Show that \(\left( 2+i\sqrt { 3 } \right) ^{ 10 }-\left( 2-i\sqrt { 3 } \right) ^{ 10 }\) is purely imaginary

12th Standard English Medium Maths Subject Book Back 2 Mark Questions with Solution Part -II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If A = \(\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \) is non-singular, find A⁻¹.
2)
Find the rank of the matrix \(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{matrix} \right] \) by reducing it to a row-echelon form.
3)
Find the following \(\left| \frac { 2+i }{ -1+2i } \right| \)
4)
If \(\omega \neq 1\) is a cube root of unity, then the show that \(\cfrac { a+b\omega +c{ \omega }^{ 2 } }{ b+c\omega +{ a\omega }^{ 2 } } +\cfrac { a+b\omega +{ c\omega }^{ 2 } }{ c+a\omega +b{ \omega }^{ 2 } } =-1\)
5)
Find the square roots of −6+8i

12th Standard English Medium Maths Subject Book Back 3 Mark Questions with Solution Part -II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Solve the equations
x⁴+ 3x³- 3x - 1 = 0
2)
Evaluate :\(\int _{ 0 }^{ 1 }{ \frac { 2x+7 }{ { 5x }^{ 2 }+9 } } dx\)
3)
Evaluate: \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { cos\theta }{ (1+sin\theta )(2+sin\theta ) } } d\theta \)
4)
Show that \(\int _{ 0 }^{ \pi }{ g(sinx)dx=2 } \int _{ 0 }^{ \frac { \pi }{ 2 } }{ g(sinx)dx, } \) where g(sin x) is a function of sin x.
5)
Evaluate \(\int _{ 0 }^{ x }{ { x }^{ 2 } } \)cos nx dx, where n is a positive integer.

12th Standard English Medium Maths Subject Book Back 5 Mark Questions with Solution Part -I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If A = \(\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right] \), find x and y such that A² + xA + yI₂ = O₂. Hence, find A^-1.
2)
Solve the following system of equations, using matrix inversion method:
2x₁ + 3x₂ + 3x₃ = 5, x₁ - 2x₂ + x₃ = -4, 3x₁ - x₂ - 2x₃ = 3.
3)
(a) If A = \(\left[ \begin{matrix} -5 & 1 & 3 \\ 7 & 1 & -5 \\ 1 & -1 & 1 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 1 & 2 \\ 3 & 2 & 1 \\ 2 & 1 & 3 \end{matrix} \right] \), find the products AB and BA and hence solve the system of equations x + y + 2z = 1, 3x + 2y + z = 7, 2x + y + 3z = 2.
4)
Investigate the values of λ and μ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y - 5z = 8, 2x + 3y + λz = μ, have
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions.
5)
Determine the values of λ for which the following system of equations (3λ − 8)x + 3y + 3z = 0, 3x + (3λ − 8)y + 3z = 0, 3x + 3y + (3λ − 8)z = 0. has a non-trivial solution.

12th Standard English Medium Maths Subject Book Back 5 Mark Questions with Solution Part -II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8 \end{matrix} \right] \)
2)
Determine k and solve the equation 2x³-6x²+3x+k = 0 if one of its roots is twice the sum of the other two roots.
3)
For the ellipse 4x²+ y²+ 24x − 2y + 21 = 0, find the centre, vertices and the foci. Also prove that the length of latus rectum is 2
4)
Parabolic cable of a 60m portion of the roadbed of a suspension bridge are positioned as shown below. Vertical Cables are to be spaced every 6m along this portion of the roadbed. Calculate the lengths of first two of these vertical cables from the vertex.
5)
Prove by vector method that the perpendiculars (attitudes) from the vertices to the opposite sides of a triangle are concurrent.

Stateboard 12th Standard Maths Subject Public Question Paper - March 2022 updated Previous Year Question Papers - by QB Admin - May 13, 2022 - View & Read

Stateboard 12th Standard Maths Subject English Medium Public Answer Key- March 2022 updated Previous Year Question Papers - by QB Admin - May 13, 2022 - View & Read

12th Standard English Medium Maths Subject Application of Matrices and Determinants Book Back 1 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =
2)
If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is
3)
If A = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right] \) then the value of a₂₃ is
4)
If A, B and C are invertible matrices of some order, then which one of the following is not true?
5)
If (AB)^-1 = \(\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right] \), then B^-1 =

12th Standard English Medium Maths Subject Application of Matrices and Determinants Book Back 2 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the rank of the matrix \(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{matrix} \right] \) by reducing it to a row-echelon form.
2)
Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 2 & -4 \\ -1 & 2 \end{matrix} \right] \)
3)
Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 1 \\ 3 \end{matrix}\begin{matrix} -2 \\ -6 \end{matrix}\begin{matrix} -1 \\ -3 \end{matrix}\begin{matrix} 0 \\ 1 \end{matrix} \right] \)
4)
Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 1 & -2 & 3 \\ 2 & 4 & -6 \\ 5 & 1 & -1 \end{matrix} \right] \)
5)
Find the rank of the following matrices by minor method or show that the rank of matrix is 3
\(\left[ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 8 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 2 \\ 1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ 0 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 3 \\ 2 \end{matrix} \end{matrix} \right] \)

12th Standard English Medium Maths Subject Application of Matrices and Determinants Book Back 2 Mark Questions with Solution Part -II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If A = \(\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \) is non-singular, find A⁻¹.
2)
If A is a non-singular matrix of odd order, prove that |adj A| is positive
3)
If A is symmetric, prove that then adj A is also symmetric.
4)
Prove that \(\left[ \begin{matrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{matrix} \right] \) is orthogonal.
5)
Find the adjoint of the following:
\(\left[ \begin{matrix} -3 & 4 \\ 6 & 2 \end{matrix} \right] \)

12th Standard English Medium Maths Subject Application of Matrices and Determinants Book Back 3 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the inverse of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \end{matrix} \right] \).
2)
Verify the property (A^T)^-1 = (A^-1)^T with A = \(\left[ \begin{matrix} 2 & 9 \\ 1 & 7 \end{matrix} \right] \).
3)
Verify (AB)^-1 = B^-1A^-1 with A = \(\left[ \begin{matrix} 0 & -3 \\ 1 & 4 \end{matrix} \right] \), B = \(\left[ \begin{matrix} -2 & -3 \\ 0 & -1 \end{matrix} \right] \).
4)
If A = \(\frac { 1 }{ 9 } \left[ \begin{matrix} -8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & -8 & 4 \end{matrix} \right] \), prove that A⁻¹ = A^T.
5)
If A = \(\left[ \begin{matrix} 8 & -4 \\ -5 & 3 \end{matrix} \right] \), verify that A(adj A) = (adj A)A = |A|I₂.

12th Standard English Medium Maths Subject Application of Matrices and Determinants Book Back 3 Mark Questions with Solution Part -II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Reduce the matrix \(\left[ \begin{matrix} 0 \\ -1 \\ 4 \end{matrix}\begin{matrix} 3 \\ 0 \\ 2 \end{matrix}\begin{matrix} 1 \\ 2 \\ 0 \end{matrix}\begin{matrix} 6 \\ 5 \\ 0 \end{matrix} \right] \) to row-echelon form.
2)
Find the rank of the matrix \(\left[ \begin{matrix} 2 \\ \begin{matrix} -3 \\ 6 \end{matrix} \end{matrix}\begin{matrix} -2 \\ \begin{matrix} 4 \\ 2 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} -2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} -1 \\ 7 \end{matrix} \end{matrix} \right] \) by reducing it to an echelon form.
3)
Solve the following system of linear equations, using matrix inversion method:
5x + 2y = 3, 3x + 2y = 5.
4)
Solve the following system of linear equations by matrix inversion method:
2x + 5y = −2, x + 2y = −3
5)
Find the adjoint of the following:
\(\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right] \)

12th Standard English Medium Maths Subject Application of Matrices and Determinants Book Back 5 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If A = \(\left[ \begin{matrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{matrix} \right] \), verify thatA(adj A) = (adj A)A = |A| I₃.
2)
If A = \(\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right] \), find x and y such that A² + xA + yI₂ = O₂. Hence, find A^-1.
3)
If A = \(\frac { 1 }{ 7 } \left[ \begin{matrix} 6 & -3 & a \\ b & -2 & 6 \\ 2 & c & 3 \end{matrix} \right] \) is orthogonal, find a, b and c , and hence A⁻¹.
4)
If F(\(\alpha\)) = \(\left[ \begin{matrix} \cos { \alpha } & 0 & \sin { \alpha } \\ 0 & 1 & 0 \\ -\sin { \alpha } & 0 & \cos { \alpha } \end{matrix} \right] \), show that [F(\(\alpha\))]^-1 = F(-\(\alpha\)).
5)
If A = \(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] \), show that A² - 3A - 7I₂ = O₂. Hence find A⁻¹.

12th Standard English Medium Maths Subject Application of Matrices and Determinants Book Back 5 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The upward speed v(t)of a rocket at time t is approximated by v(t) = at² + bt + c, 0 ≤ t ≤ 100 where a, b and c are constants. It has been found that the speed at times t = 3, t = 6, and t = 9 seconds are respectively, 64, 133, and 208 miles per second respectively. Find the speed at time t = 15 seconds. (Use Gaussian elimination method.)
2)
Investigate for what values of λ and μ the system of linear equations x + 2y + z = 7 , x + y + λz = μ , x + 3y − 5z = 5 has
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions
3)
Test for consistency and if possible, solve the following systems of equations by rank method.
2x - y + z = 2, 6x - 3y + 3z = 6, 4x - 2y + 2z = 4
4)
Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 1 & -1 & 0 \\ 1 & 0 & -1 \\ 6 & -2 & -3 \end{matrix} \right] \)
5)
Solve the following systems of linear equations by Cramer’s rule:
\(\frac { 3 }{ x } -\frac { 4 }{ y } -\frac { 2 }{ z } \) -1 = 0, \(\frac { 1 }{ x } +\frac { 2 }{ y } +\frac { 1 }{ z } \) - 2 = 0, \(\frac { 2 }{ x } -\frac { 5 }{ y } -\frac { 4 }{ z } \) + 1 = 0

12th Standard English Medium Maths Subject Complex Numbers Book Back 1 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is
2)
The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is
3)
The area of the triangle formed by the complex numbers z, iz and z+iz in the Argand’s diagram is
4)
The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is
5)
If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to

12th Standard English Medium Maths Subject Application of Matrices and Determinants Creative 1 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then _____________
2)
If A^T is the transpose of a square matrix A, then ___________
3)
If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| = _______
4)
If A is a square matrix of order n, then |adj A| = ______________
5)
If A =\(\left( \begin{matrix} cosx & sinx \\ -sinx & cosx \end{matrix} \right) \) and A(adj A) =\(\lambda \) \(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \) then \(\lambda \) is ________

12th Standard English Medium Maths Subject Complex Numbers Book Back 1 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If \(\frac { z-1 }{ z+1 } \) is purely imaginary, then |z| is
2)
If z = x + iy is a complex number such that |z+2| = |z−2|, then the locus of z is
3)
If \(\alpha \) and \(\beta \) are the roots of x²+x+1 = 0, then \({ \alpha }^{ 2020 }+{ \beta }^{ 2020 }\) is
4)
The product of all four values of \(\left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) ^{ \frac { 3 }{ 4 } }\) is
5)
If \(\omega =cis\cfrac { 2\pi }{ 3 } \), then the number of distinct roots of \(\left| \begin{matrix} z+1 & \omega & { \omega }^{ 2 } \\ \omega & z+{ \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & z+\omega \end{matrix} \right| \)=0

12th Standard English Medium Maths Subject Application of Matrices and Determinants Creative 1 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The system of linear equations x + y + z = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if _________
2)
Let A be a 3 \(\times\) 3 matrix and B its adjoint matrix If |B| = 64, then |A| = ___________
3)
The system of linear equations x + y + z = 2, 2x + y - z = 3, 3x + 2y + kz = has a unique solution if __________
4)
If A is a matrix of order m \(\times\) n, then \(\rho\) (A) is _________
5)
If \(\rho\) (A) = r then which of the following is correct?

12th Standard English Medium Maths Subject Complex Numbers Book Back 2 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Show that the following equations represent a circle, and find its centre and radius \(\left| z-2-i \right| =3\)
2)
Write in polar form of the following complex numbers
\(2+i2\sqrt { 3 } \)
3)
Simplify the following
\({ i }^{ 59 }+\frac { 1 }{ { i }^{ 59 } } \)
4)
If z = x + iy , find the following in rectangular form.
Im(3z + 4\(\bar { z } \) − 4i)
5)
Simplify the following:
i i²i³...i⁴⁰

12th Standard English Medium Maths Subject Application of Matrices and Determinants Creative 2 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
For the matrix A, if A³ = I, then find A^-1.
2)
Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.
3)
Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).
4)
Find k if the equations x + 2y + 2z = 0, x - 3y - 3z = 0, 2x + y + kz = 0 have only the trivial solution.
5)
Solve 6x - 7y = 16, 9x - 5y = 35 using (Cramer's rule).

12th Standard English Medium Maths Subject Complex Numbers Book Back 2 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Simplify the following
i i ²i³...i²⁰⁰⁰
2)
Evaluate the following if z = 5−2i and w = −1+3i
2z + 3w
3)
Find the square roots of −6+8i
4)
Find the following \(\left| \overline { (1+i) } (2+3i)(4i-3) \right| \)
5)
Find the modulus and principal argument of the following complex numbers.
\(-\sqrt { 3 } +i\)

12th Standard English Medium Maths Subject Application of Matrices and Determinants Creative 2 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
For any 2 \(\times\) 2 matrix, if A (adj A) =\(\left[ \begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix} \right] \) then find |A|.
2)
If A is a square matrix such that A³ = I, then prove that A is non-singular.
3)
Find the rank of the matrix \(\left[ \begin{matrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{matrix} \right] \).
4)
Show that the equations 3x + y + 9z = 0, 3x + 2y + 12z = 0 and 2x + y + 7z = 0 have nontrivial solutions also.
5)
Solve : 2x - y = 3, 5x + y = 4 using matrices.

12th Standard English Medium Maths Subject Complex Numbers Book Back 3 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Given the complex number z = 2 + 3i, represent the complex numbers in Argand diagram z, iz , and z+iz
2)
Find the rectangular form of the complex numbers
\(\left( cos\frac { \pi }{ 6 } +isin\frac { \pi }{ 6 } \right) \left( cos\frac { \pi }{ 12 } +isin\frac { \pi }{ 12 } \right) \)
3)
If \(cos\alpha +cos\beta +cos\gamma =sin\alpha +sin\beta +sin\gamma =0\) then show that
(i) \(cos3\alpha +cos3\beta +cos3\gamma =3cos(\alpha +\beta +\gamma )\)
(ii) \(sin3\alpha +sin3\beta +sin3\gamma +sin3\gamma =3sin\left( \alpha +\beta +\gamma \right) \)
4)
Find the quotient \(\frac { 2\left( cos\frac { 9\pi }{ 4 } +isin\frac { 9\pi }{ 4 } \right) }{ 4\left( cos\left( \frac { -3\pi }{ 2 } + \right) isin\left( \frac { -3\pi }{ 2 } \right) \right) } \) in rectangular form
5)
Evaluate the following if z = 5−2i and w = −1+3i
(z + w)²

12th Standard English Medium Maths Subject Application of Matrices and Determinants Creative 3 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
For what value of t will the system tx +3y - z = 1, x + 2y + z = 2, -tx + y + 2z = -1 fail to have unique solution?
2)
Verify (AB)^-1 = B^-1 A^-1 for A =\(\left[ \begin{matrix} 2 & 1 \\ 5 & 3 \end{matrix} \right] \) and B =\(\left[ \begin{matrix} 4 & 5 \\ 3 & 4 \end{matrix} \right] \).
3)
Find the rank of the matrix math \(\left[ \begin{matrix} 4 \\ -2 \\ 1 \end{matrix}\begin{matrix} 4 \\ 3 \\ 4 \end{matrix}\begin{matrix} 0 \\ -1 \\ 8 \end{matrix}\begin{matrix} 3 \\ 5 \\ 7 \end{matrix} \right] \).
4)
Solve 2x - 3y = 7, 4x - 6y = 14 by Gaussian Jordan method.
5)
If the rank of the matrix \(\left[ \begin{matrix} \lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda \end{matrix} \right] \) is 2, then find ⋋.

12th Standard English Medium Maths Subject Complex Numbers Book Back 3 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the value of the real numbers x and y, if the complex number (2+i)x+(1−i)y+2i −3 and x+(−1+2i)y+1+i are equal
2)
If z₁, z₂ and z₃ are complex numbers such that |z₁| = |z₂| = |z₃| = |z₁+z₂+z₃| = 1 find the value of \(\left| \frac { 1 }{ { z }_{ 1 } } +\frac { 1 }{ z_{ 2 } } +\frac { 1 }{ { z }_{ 3 } } \right| \)
3)
If |z| = 2 show that \(3\le \left| z+3+4i \right| \le 7\)
4)
Find the rectangular form of the complex numbers
\(\left( cos\frac { \pi }{ 6 } +isin\frac { \pi }{ 6 } \right) \left( cos\frac { \pi }{ 12 } +isin\frac { \pi }{ 12 } \right) \)
5)
If (x₁+ iy₁)(x₂+ iy₂)(x3 + iy₃)...(x_n+ iy_n) = a + ib, show that
(x₁²+ y₁²)(x₂²+ y₂²)(x₃²+ y₃²)...(x_n²+ y_n²) = a²+ b²

12th Standard English Medium Maths Subject Application of Matrices and Determinants Creative 3 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Solve: 2x + 3y = 10, x + 6y = 4 using Cramer's rule.
2)
Solve: 3x+ay = 4, 2x + ay = 2, a ≠ 0 by Cramer's rule.
3)
Under what conditions will the rank of the matrix \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & h-2 & 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} h+2 \\ 3 \end{matrix} \end{matrix} \right] \) be less than 3?
4)
Verify that (A^-1)^T = (A^T)^-1 for A =\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).
5)
Solve: x + y + 3z = 4, 2x + 2y + 6z = 7, 2x + y + z = 10.

12th Standard English Medium Maths Subject Complex Numbers Book Back 5 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Suppose z₁, z₂and z₃are the vertices of an equilateral triangle inscribed in the circle |z| = 2. If z₁ = 1 + i\(\sqrt { 3 } \) then find z₂ and z₃.
2)
Find all cube roots of \(\sqrt { 3 } +i\)
3)
Solve the equation z³+ 8i = 0, where \(z \in \mathbb{C}\)
4)
Simplify: (1+i)¹⁸
5)
Simplify: \(\left( -\sqrt { 3 } +3i \right) ^{ 31 }\)

12th Standard English Medium Maths Subject Application of Matrices and Determinants Creative 5 Mark Questions with Solution updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Using determinants; find the quadratic defined by f(x) = ax² + bx + c, if f(1) = 0, f(2) = -2 and f(3) = -6.
2)
Solve: \(\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z } \) = 2
3)
The sum of three numbers is 20. If we multiply the third number by 2 and add the first number to the result we get 23. By adding second and third numbers to 3 times the first number we get 46. Find the numbers using Cramer's rule.
4)
Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c = 0.
5)
Using Gaussian Jordan method, find the values of λ and μ so that the system of equations 2x - 3y + 5z = 12, 3x + y + λz =μ, x - 7y + 8z = 17 has
(i) unique solution
(ii) infinite solutions and
(iii) no solution.

12th Standard English Medium Maths Subject Complex Numbers Book Back 5 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Show that \(\left( \frac { 19+9i }{ 5-3i } \right) ^{ 15 }-\left( \frac { 8+i }{ 1+2i } \right) ^{ 15 }\) is purely imaginary.
2)
Let z₁, z₂ and z₃ be complex numbers such that \(\left| { z }_{ 1 } \right\| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =r>0\) and z₁+ z₂+ z₃ \(\neq \) 0 prove that \(\left| \frac { { z }_{ 1 }{ z }_{ 2 }+{ z }_{ 2 }{ z }_{ 3 }+{ z }_{ 3 }{ z }_{ 1 } }{ { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 } } \right| \) = r
3)
If z₁, z_2, and z₃ are three complex numbers such that |z₁| = 1, |z₂| = 2|z₃| = 3 and |z₁+ z₂+ z₃| = 1, show that |9z₁z₂+ 4z₁z₂+ z₂z₃| = 6
4)
If z = x + iy is a complex number such that Im \(\left( \frac { 2z+1 }{ iz+1 } \right) =0\) show that the locus of z is 2x²+ 2y²+ x - 2y = 0
5)
If z = x + iy and arg \(\left( \frac { z-i }{ z+2 } \right) =\frac { \pi }{ 4 } \), then show that x²+ y²+ 3x - 3y + 2 = 0

12th Standard English Medium Maths Subject Complex Numbers Creative 1 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If \(\sqrt { a+ib } \) = x + iy, then possible value of \(\sqrt { a-ib }\) is ___________
2)
If z = cos\(\frac { \pi }{ 4 } \) + i sin\(\frac { \pi }{ 6 } \), then ______
3)
.If a = 3+i and z = 2-3i, then the points on the Argand diagram representing az, 3az and - az are _________
4)
If a = 3 + i and z = 2 - 3i, then the points on the Argand diagram representing az, 3az and - az are ___________
5)
If z = 1-cos θ + i sin θ, then |z| = _____________

12th Standard English Medium Maths Subject Theory of Equations Book Back 1 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
A zero of x³ + 64 is
2)
If f and g are polynomials of degrees m and n respectively, and if h(x) = (f ^_og)(x), then the degree of h is
3)
A polynomial equation in x of degree n always has
4)
If α, β and γ are the zeros of x³+ px²+ qx + r, then \(\Sigma \frac { 1 }{ \alpha } \) is
5)
According to the rational root theorem, which number is not possible rational zero of 4x⁷+ 2x⁴- 10x³- 5?

12th Standard English Medium Maths Subject Complex Numbers Creative 1 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The value of (1+i) (1+i²) (1+i³) (1+i⁴) is ____________
2)
The amplitude of \(\frac{1}{i}\) is equal to _______
3)
The complex number z which satisfies the condition \(\left| \frac { 1+z }{ 1-z } \right| \) = 1 lies on _________
4)
\(\frac { 1+e^{ -i\theta } }{ 1+{ e }^{ i\theta } } \) =__________
5)
If ω is the cube root of unity, then the value of (1-ω) (1-ω²) (1-ω⁴) (1-ω⁸) is _________

12th Standard English Medium Maths Subject Theory of Equations Book Back 1 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The polynomial x³- kx²+ 9x has three real zeros if and only if, k satisfies
2)
The number of real numbers in [0, 2π] satisfying sin⁴x - 2sin²x + 1 is
3)
If x³+12x²+10ax+1999 definitely has a positive zero, if and only if
4)
The polynomial x³+ 2x + 3 has
5)
The number of positive zeros of the polynomial \(\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }\)(-1)^rx^r is

12th Standard English Medium Maths Subject Complex Numbers Creative 2 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find Re (z) and im (z) if z = 5i¹¹ + 7i³
2)
If z₁ and z₂ are 1-i, -2+4i then find Im\(\left( \frac { { z }_{ 1 }{ z }_{ 2 } }{ \bar { { z }_{ 1 } } } \right) \).
3)
Find the value of the complex number (i²⁵)³.
4)
Find the argument of -2
5)
Find the values of the real number x and y if 3x + (2x - 3y) i = 6 + 3i⁹.

12th Standard English Medium Maths Subject Theory of Equations Book Back 2 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If α, β, and γ are the roots of the equation x³+ px²+ qx + r = 0, find the value of \(\Sigma \frac { 1 }{ \beta \gamma } \) in terms of the coefficients.
2)
Construct a cubic equation with roots 1, 2 and 3
3)
If α, β, γ and \(\delta\) are the roots of the polynomial equation 2x⁴+ 5x³− 7x²+ 8 = 0, find a quadratic equation with integer coefficients whose roots are α + β + γ + \(\delta\) and αβ૪\(\delta\).
4)
Formulate into a mathematical problem to find a number such that when its cube root is added to it, the result is 6.
5)
A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was left standing.

12th Standard English Medium Maths Subject Complex Numbers Creative 2 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If z₁ and z₂ are two complex numbers, such that |z₁| = Iz₂|, then is it necessary that z₁ = z₂?
2)
If (cosθ + i sinθ)² = x + iy, then show that x²+y² =1
3)
If z =\(\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 107 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 107 }\), then show that Im (z) = 0
4)
If 1, ω, ω² are the cube roots of unity show that (1+ω²)³ - (1+ω)³ = 0
5)
Find the modules of (1+ 3i)³

12th Standard English Medium Maths Subject Theory of Equations Book Back 2 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the exact number of real zeros and imaginary of the polynomial x⁹+9x⁷+7x⁵+5x³+3x.
2)
Construct a cubic equation with roots 1, 1 and −2
3)
Construct a cubic equation with roots 2, −2, and 4.
4)
Examine for the rational roots of x⁸- 3x + 1 = 0
5)
Discuss the nature of the roots of the following polynomials:
x⁵-19x⁴+ 2x³+ 5x²+11

12th Standard English Medium Maths Subject Complex Numbers Creative 3 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the circle roots of -27.
2)
Show that \(\left| \frac { z-3 }{ z+3 } \right| \) = 2 represent a circle.
3)
If the complex number 2 + i and 1-2i are equidistant from x + iy then show that x+3y = 0.
4)
Find the locus of z if |3z - 5| = 3 |z + 1| where z = x + iy.
5)
If \(\frac { (a+i)^{ 2 } }{ 2a-i } \) = p + iq, show that p²+q² = \(\frac { ({ a }^{ 2 }+i)^{ 2 } }{ 4a^{ 2 }+1 } \).

12th Standard English Medium Maths Subject Theory of Equations Book Back 3 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Solve the equation 2x³+11x²−9x−18 = 0.
2)
Find the condition that the roots of ax³+ bx²+ cx + d = 0 are in geometric progression. Assume a, b, c, d ≠ 0.
3)
Solve the equation 3x³-26x²+52x - 24 = 0 if its roots form a geometric progression.
4)
Solve the equation
2x³- 9x²+ 10x = 3
5)
Find all real numbers satisfying 4^x- 3(2^x+2) + 2⁵= 0

12th Standard English Medium Maths Subject Complex Numbers Creative 3 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Explain the falacy:
2)
Find the principal value of -2i.
3)
Show that the complex numbers 3 + 2i, 5i, -3 + 2i and -i form a square.
4)
Find the locus of z if Re\(\left( \frac { z+1 }{ z-i } \right) \) = 0 where z = x+iy.
5)
Find the locus of z if Re\(\\ \left( \frac { \bar { z } +1 }{ \bar { z } -i } \right) \) = 0.

12th Standard English Medium Maths Subject Theory of Equations Book Back 3 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If α and β are the roots of the quadratic equation 17x²+43x−73 = 0 , construct a quadratic equation whose roots are α + 2 and β + 2.
2)
If α and β are the roots of the quadratic equation 2x²−7x+13 = 0 , construct a quadratic equation whose roots are α² and β².
3)
If the sides of a cubic box are increased by 1, 2, 3 units respectively to form a cuboid, then the volume is increased by 52 cubic units. Find the volume of the cuboid.
4)
If α, β and γ are the roots of the cubic equation x³+2x²+3x+4 = 0, form a cubic equation whose roots are, 2α, 2β, 2γ
5)
Find the sum of squares of roots of the equation 2x⁴- 8x³+ 6x²-3 = 0.

12th Standard English Medium Maths Subject Complex Numbers Creative 5 Mark Questions with Solution updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If 1, ω, ω² are the cube roots of unity then show that (1+5ω²+ω⁴) (1+5ω+ω²) (5+ω+ω⁵) = 64
2)
Show that \(\left( \frac { i+\sqrt { 3 } }{ -i+\sqrt { 3 } } \right) ^{ 2\omega }+\left( \frac { i-\sqrt { 3 } }{ i+\sqrt { 3 } } \right) ^{ 2\omega }\) = -1
3)
Verify that 2 arg(-1) ≠ arg(-1)²
4)
Find all the roots \((2-2i)^{ \frac { 1 }{ 3 } }\) and also find the product of its roots.
5)
Find the radius and centre of the circle \(z\bar { z } \)-(2+3i)z-(2-3i)\(\bar { z } \)+9 = 0 where z is a complex number.

12th Standard English Medium Maths Subject Theory of Equations Book Back 5 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Solve : (x - 5) (x - 7) (x + 6) (x + 4) = 504
2)
Find all zeros of the polynomial x⁶- 3x⁵- 5x⁴+ 22x³- 39x²- 39x + 135, if it is known that 1+2i and \(\sqrt{3}\) are two of its zeros.
3)
Solve the following equation: x⁴-10x³+ 26x²-10x + 1 = 0
4)
Solve the equation 6x⁴- 5x³- 38x²- 5x + 6 = 0 if it is known that \(\frac{1}{3}\) is a solution.
5)
Discuss the maximum possible number of positive and negative roots of the polynomial equations x²−5x+6 and x²−5x+16 . Also draw rough sketch of the graphs

12th Standard English Medium Maths Subject Theory of Equations Creative 1 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The quadratic equation whose roots are ∝ and β is ___________
2)
The equation \(\sqrt { x+1 } -\sqrt { x-1 } =\sqrt { 4x-1 } \) has ____________
3)
For real x, the equation \(\left| \frac { x }{ x-1 } \right| +|x|=\frac { { x }^{ 2 } }{ |x-1| } \) has ________
4)
If \((2+\sqrt{3})^{x^{2}-2 x+1}+(2-\sqrt{3})^{x^{2}-2 x-1}=\frac{2}{2-\sqrt{3}}\) then x = _________
5)
If ∝, β, ૪ are the roots of 9x³-7x+6 = 0, then ∝ β ૪ is __________

12th Standard English Medium Maths Subject Theory of Equations Book Back 5 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Solve the equation 3x³- 16x²+ 23x - 6 = 0 if the product of two roots is 1.
2)
Form the equation whose roots are the squares of the roots of the cubic equation x³+ ax²+ bx + c = 0.
3)
Solve the equation x³− 9x²+14x + 24 = 0 if it is given that two of its roots are in the ratio 3:2.
4)
Find a polynomial equation of minimum degree with rational coefficients, having \(\sqrt{5}\)−\(\sqrt{3}\) as a root.
5)
If 2+i and 3-\(\sqrt{2}\) are roots of the equation x⁶-13x⁵+ 62x⁴-126x³+ 65x²+127x-140 = 0, find all roots.

12th Standard English Medium Maths Subject Theory of Equations Creative 1 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If a, b, c ∈ Q and p +√q (p, q ∈ Q) is an irrational root of ax²+bx+c = 0 then the other root is ___________
2)
If f(x) = 0 has n roots, then f'(x) = 0 has __________ roots
3)
If ∝, β, ૪ are the roots of the equation x³-3x+11 = 0, then ∝+β+૪ is __________.
4)
If x² - hx - 21 = 0 and x² - 3hx + 35 = 0 (h > 0) have a common root, then h = ___________
5)
If p(x) = ax² + bx + c and Q(x) = -ax² + dx + c where ac ≠ 0 then p(x). Q(x) = 0 has at least _______ real roots.

12th Standard English Medium Maths Subject Theory of Equations Creative 2 Mark Questions with Solution updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If sin ∝, cos ∝ are the roots of the equation ax²+ bx + c-0 (c ≠ 0), then prove that (n + c)²- b² + c²
2)
Find value of a for which the sum of the squares of the equation x² - (a- 2) x - a -1 = 0 assumes the least value.
3)
Find the Interval for a for which 3x²+2(a²+1) x+(a²-3a+2) possesses roots of opposite sign.
4)
Find x If \(x=\sqrt { 2+\sqrt { 2+\sqrt { 2+....+upto\infty } } } \)
5)
Find the number of positive and negative roots of the equation x⁷ - 6x⁶ + 7x⁵ + 5x²+2x+2

12th Standard English Medium Maths Subject Theory of Equations Creative 3 Mark Questions with Solution updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the number of real solutions of sin (e^x) -5^x + 5^-x
2)
Find the number of positive integral solutions of (pairs of positive integers satisfying) x² - y² = 353702.
3)
Solve: 2x+2^x-1+2^x-2= 7x+7^x-1+7^x-2
4)
Solve: (x-1)⁴+(x-5)⁴= 82
5)
Solve: \({ (5+2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }+{ (5-2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }=10\)

12th Standard English Medium Maths Subject Theory of Equations Creative 5 Mark Questions with Solution updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If the sum of the roots of the quadratic equation ax²+ bx + c = 0 (abc ≠ 0) is equal to the sum of the squares of their reciprocals, then \(\frac { a }{ c } ,\frac { b }{ a } ,\frac { c }{ b } \) are H.P.
2)
If a, b, c, d and p are distinct non-zero real numbers such that (a²+b²+c²) p²-2 (ab+bc+cd) p+(b²+c²+d²)≤ 0 then prove that a, b, c, d are in G.P and ad = bc
3)
If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p.
4)
If the equation x² + bx + ca = 0 and x² + cx + ab = 0 have a comnion root and b≠c, then prove that their roots will satisfy the equation x² + ax + bc = 0.
5)
Solve: (2x² - 3x + 1) (2x² + 5x + 1) = 9x².

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Creative 1 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If \({ sin }^{ -1 }x-cos^{ -1 }x=\frac { \pi }{ 6 } \) then ___________
2)
If \(\alpha ={ tan }^{ -1 }\left( tan\frac { 5\pi }{ 4 } \right) \) and \(\beta ={ tan }^{ -1 }\left( -tan\frac { 2\pi }{ 3 } \right) \) then ___________
3)
If \(\alpha ={ tan }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2y-x } \right) ,\beta ={ tan }^{ -1 }\left( \frac { 2x-y }{ \sqrt { 3y } } \right) \) then \(\alpha -\beta \) __________
4)
If tan^-1(3) + tan^-1(x) = tan^-1(8) then x = ____________
5)
\(sin\left\{ 2{ cos }^{ -1 }\left( \frac { -3 }{ 5 } \right) \right\} =\) __________

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Creative 1 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If \({ tan }^{ -1 }\left\{ \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right\} =\alpha \) then x² = _____________
2)
\({ tan }^{ -1 }\left( \frac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \frac { 2 }{ 11 } \right) \) = ____________
3)
The value of \({ cos }^{ -1 }\left( \cos\cfrac { 5\pi }{ 3 } \right) +sin^{ -1 }\left( \sin\cfrac{5\pi }{ 3 } \right) \) is ______________
4)
If \({ cos }^{ -1 }x>x>{ sin }^{ -1 }x\) then _________
5)
The value of sin 2(tan^-1 0.75) is ___________

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Creative 2 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the principal value of sin^-1(-1).
2)
If \({ cot }^{ -1 }\left( \frac { 1 }{ 7 } \right) =\theta \) find the value of cos \(\theta \)
3)
Prove that \({ tan }^{ -1 }\left( \frac { 1 }{ 7 } \right) +{ tan }^{ -1 }\left( \frac { 1 }{ 13 } \right) ={ tan }^{ -1 }\left( \frac { 2 }{ 9 } \right) \)
4)
Evaluate \(sin\left( \frac { 1 }{ 2 } { cos }^{ -1 }\frac { 4 }{ 5 } \right) \)
5)
Evaluate \(sin\left( { cos }^{ -1 }\left( \frac { 1 }{ 2 } \right) \right) \)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Creative 2 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the principal value of \({ tan }^{ -1 }\left( \frac { -1 }{ \sqrt { 3 } } \right) \)
2)
Find the principal value of \({ cos }^{ -1 }\left( \frac { -1 }{ 2 } \right) \)
3)
If \({ sin }^{ -1 }\left( \frac { 1 }{ 2 } \right) ={ tan }^{ -1 }x\) then find the value of x,
4)
Evaluate \(sin\left( { cos }^{ -1 }\left( \frac { 3 }{ 5 } \right) \right) \)
5)
Prove that \(2{ tan }^{ -1 }\left( \frac { 2 }{ 3 } \right) ={ tan }^{ -1 }\left( \frac { 12 }{ 5 } \right) \)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Creative 3 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Evaluate \(cos\left[ { sin }^{ -1 }\frac { 3 }{ 5 } +{ sin }^{ -1 }\frac { 5 }{ 13 } \right] \)
2)
Solve: \({ tan }^{ -1 }\left( \cfrac { x-1 }{ x-2 } \right) +{ tan }^{ -1 }\left( \cfrac { x+1 }{ x+2 } \right) =\cfrac { \pi }{ 4 } \)
3)
If \(sin\left( { sin }^{ -1 }\frac { 1 }{ 5 } +{ cos }^{ -1 }x \right) =1\) then find the value ofx.
4)
Evaluate \(cos\left[ { cos }^{ -1 }\left( \frac { -\sqrt { 3 } }{ 2 } +\frac { \pi }{ 6 } \right) \right] \)
5)
Solve: cos(tan^-1x) = \(sin\left( { cot }^{ -1 }\frac { 3 }{ 4 } \right) \)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Creative 3 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Prove that \({ cos }^{ -1 }\left( \frac { 4 }{ 5 } \right) +{ tan }^{ -1 }\left( \frac { 3 }{ 5 } \right) ={ tan }^{ -1 }\left( \frac { 27 }{ 11 } \right) \)
2)
Prove that \({ tan }^{ -1 }\left( \frac { m }{ n } \right) -{ tan }^{ -1 }\left( \frac { m-n }{ m+n } \right) =\frac { \pi }{ 4 } \)
3)
Solve \({ tan }^{ -1 }\left( \frac { 2x }{ 1-{ x }^{ 2 } } \right) +{ cot }^{ -1 }\left( \frac { 1-{ x }^{ 2 } }{ 2x } \right) =\frac { \pi }{ 3 } ,x>0\)
4)
Prove that \({ tan }^{ -1 }\sqrt { x } =\frac { 1 }{ 2 } { cos }^{ -1 }={ \frac { 1 }{ 2 } { cos }^{ -1 }\left( \frac { 1-x }{ 1+x } \right) ,x\in \left| 0,1 \right| }\)
5)
Find the real solutions of the equation
\({ tan }^{ -1 }\sqrt { x(x+1) } +{ sin }^{ -1 }\sqrt { { x }^{ 2 }+x+1 } =\frac { \pi }{ 2 } \)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Creative 5 Mark Questions with Solution updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the domain of the following functions
(i) f(x) = sin^-1(2x - 3)
(ii) f(x) = sin^-1x + cos x
2)
Write the function \(f(x)=\tan ^{-1} \sqrt{\frac{a-x}{a+x}}-a<x<a \) in the simplest form
3)
Simplify \({ sin }^{ -1 }\left( \frac { sinx+cosx }{ \sqrt { 2 } } \right) ,\frac { \pi }{ 4 }\)
4)
If \({ tan }^{ -1 }\left( \frac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right) =a\) than prove that x²= sin 2a
5)
Prove that \({ tan }^{ -1 }\left( \frac { 1-x }{ 1+x } \right) -{ tan }^{ -1 }\left( \frac { 1-y }{ 1+y } \right) ={ sin }^{ -1 }\left( \frac { y-x }{ \sqrt { 1+{ x }^{ 2 } } .\sqrt { 1+{ y }^{ 2 } } } \right)\)

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Creative 1 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The equation of the directrix of the parabola y²+ 4y + 4x + 2 = 0 is ___________
2)
y²- 2x - 2y + 5 = 0 is a _________
3)
The eccentricity of the ellipse 9x²+ 5y² - 30y = 0 is __________
4)
If the distance between the foci is 2 and the distance between the direction is 5, then the equation of the ellipse is __________
5)
The equation 7x²- 6\(\sqrt { 3 } \) xy + 13y² - 4\(\sqrt { 3 } \) x - 4y - 12 = 0 represents ____________

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Creative 1 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If (0, 4) and (0, 2) are the vertex and focus of a parabola then its equation is ___________
2)
Equation of tangent at (-4, -4) on x² = -4y is _____________
3)
The length of the latus rectum of the ellipse \(\frac { { x }^{ 2 } }{ 36 } +\frac { { y }^{ 2 } }{ 49 } \) = 1 is __________
4)
In an ellipse, the distance between its foci is 6 and its minor axis is 8, then e is ________
5)
The auxiliary circle of the ellipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { { y }^{ 2 } }{ 16 } \) = 1 is __________

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Creative 2 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the length of the tangent from (2, -3) to the circle x² + y²- 8x - 9y + 12 = 0.
2)
If a parabolic reflector is 24 cm in diameter and 6 cm deep, find its locus.
3)
Find the locus of a point which divides so that the sum of its distances from (-4, 0) and (4, 0) is 10 units.
4)
Find the eccentricity of the ellipse with foci on x-axis if its latus rectum be equal to one half of its major axis.
5)
Find the equation of the hyperbola whose vertices are (0, ±7) and e = \(\frac { 4 }{ 3 } \)

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Creative 2 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the equation of tangent to the circle x² +y² + 2x - 3y - 8 = 0 at (2, 3).
2)
Find the equation of the parabola with vertex at the origin, passing through (2, -3) and symmetric about x-axis
3)
If the line y = 3x + 1, touches the parabola y² = 4ax, find the length of the latus rectum?
4)
For the ellipse x² + 3y² = a², find the length of major and minor axis.
5)
Find the eccentricity of the hyperbola with foci on the x-axis if the length of its conjugate axis is \({ \left( \frac { 3 }{ 4 } \right) }^{ th }\) of the length of its tranverse axis.

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Creative 3 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the value of p so that 3x + 4y - p = 0 is a tangent to the circle x² +y²- 64 = 0.
2)
Find the area of th triangle found by the lines Joining the vertex of the parabola x² = -36y to the ends of the latus rectum.
3)
Find the equation of the ellipse whose latus rectum is 5 and e = \(\frac { 2 }{ 3 } \)
4)
For the hyperbola 3x² - 6y² = -18, find the length of transverse and conjugate axes and eccentricity.
5)
Show that the line x + y + 1 = 0 touches the hyperbola \(\frac { { x }^{ 2 } }{ 16 } -\frac { { y }^{ 2 } }{ 15 } \) = 1 and find the co-ordinates of the point of contact

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Creative 3 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the circumference and area of the circle x² +y² - 2x + 5y + 7 = 0
2)
Find the condition for the line lx + my + n = 0 is tangent to the circle x² + y² = a²
3)
Find the equation of the ellipse whose e = \(\frac34\), foci ony-axis, centre at origin and passing through (6, 4).
4)
Find the equation of the hyperbola whose conjugate axis is 5 and the distance between the foci is 13.
5)
Find the value of c if y = x + c is a tangent to the hyperbola 9x² - 16y² = 144.

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Creative 5 Mark Questions with Solution updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?
2)
An equilateral triangle is inscribed in the parabola y² = 4ax whose vertex is at the vertex of the parabola. Find the length of its side.
3)
The girder of a railway bridge is a parabola with its vertex at the highest point 15 m above the ends. If the span is 120 m, find the height of the bridge at 24 m from the middle point.
4)
The foci of a hyperbola coincides with the foci of the ellipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { y^{ 2 } }{ 9 } =1\). Find the equation of the hyperbola if its eccentricity is 2.
5)
A kho-kho player In a practice session while running realises that the sum of tne distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him of the distance between the poles is 6m.

12th Standard English Medium Maths Subject Applications of Vector Algebra Creative 1 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Let \(\overset { \rightarrow }{ a } \), \(\overset { \rightarrow }{ b } \)and \(\overset { \rightarrow }{ c } \)be three non- coplanar vectors and let \(\overset { \rightarrow }{ p } ,\overset { \rightarrow }{ q } ,\overset { \rightarrow }{ r } \) be the vectors defined by the relations \(\overset { \rightarrow }{ P } =\frac { \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ q } =\frac { \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ r } =\frac { \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } \) Then the value of \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) .\overset { \rightarrow }{ p } +\left( \overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) .\overset { \rightarrow }{ q } +\left( \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \right) .\overset { \rightarrow }{ r } \)= ____________
2)
The volume of the parallelepiped whose sides are given by \(\overset { \rightarrow }{ OA } =2\overset { \wedge }{ i } -3\overset { \wedge }{ j } \), \(\overset { \rightarrow }{ OB } =\overset { \wedge }{ i } +\overset { \wedge }{ j } -\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ OC } =3\overset { \wedge }{ i } -\overset { \wedge }{ k } \) is _____________
3)
The angle between the vector \(3\overset { \wedge }{ i } +4\overset { \wedge }{ j } +\overset { \wedge }{ 5k } \) and the z-axis is ___________
4)
The value of \({ \left| \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right| }^{ 2 }+{ \left| \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \right| }^{ 2 }\) is _____________
5)
The straight lines \(\frac { x-3 }{ 2 } =\frac { y+5 }{ 4 } =\frac { z-1 }{ -13 } \) and \(\frac { x+1 }{ 3 } =\frac { y-4 }{ 5 } =\frac { z+2 }{ 2 } \) are _____________

12th Standard English Medium Maths Subject Applications of Vector Algebra Creative 1 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The vector, d\(\overset { \wedge }{ i } +\overset { \wedge }{ j } +2\overset { \wedge }{ k } ,\overset { \wedge }{ i } +\lambda \overset { \wedge }{ j } -\overset { \wedge }{ k } \) t and \(2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\lambda \overset { \wedge }{ k } \) are co-planar if _____________
2)
If \(\left| \overset { \rightarrow }{ a } \right| =\left| \overset { \rightarrow }{ b } \right| =1\)such that \(\overset { \rightarrow }{ a } +2\overset { \rightarrow }{ b } \) and \(5\overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \) are perpendicular to each other, then the angle between \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) is _______________
3)
The p.v, OP of a point P make angles 60^o and 45^owith X and Y axis respectively. The angle of inclination of \(\overset { \rightarrow }{ OP } \) with z-axis is ___________
4)
The value of \({ \left| \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right| }^{ 2 }+{ \left| \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \right| }^{ 2 }\) is _____________
5)
The two planes 3x + 3y - 3z - 1 = 0 and x + y - z + 5 = 0 are _____________

12th Standard English Medium Maths Subject Applications of Vector Algebra Creative 2 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
A force of magnitude 6 units acting parallel to \(\overset { \wedge }{ 2i } -\overset { \wedge }{ 2j } +\overset { \wedge }{ k } \) displaces the point of application from (1, 2, 3) to (5, 3, 7). Find the work done.
2)
Forces \(2\overset { \wedge }{ i } +7\overset { \wedge }{ j } \), \(2\overset { \wedge }{ i } -5\overset { \wedge }{ j } +6\overset { \wedge }{ k } \), \(-\overset { \wedge }{ i } +2\overset { \wedge }{ j } -\overset { \wedge }{ k } \) act at a point P whose position vector is \(4\overset { \wedge }{ i } -3\overset { \wedge }{ j } -2\overset { \wedge }{ k } \). Find the vector moment of the resultant of these forces acting at P about this point Q whose position vector is \(6\overset { \wedge }{ i } +\overset { \wedge }{ j } -3\overset { \wedge }{ k } \)
3)
Find the parametric form of vector equation of a line passing through a point (2, -1, 3) and parallel to line \({ \overset { \rightarrow }{ r } }=\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) +t\left( 2\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \right) \)
4)
If the planes \({ \overset { \rightarrow }{ r } }.\left( \overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k } \right) =7\) and \({ \overset { \rightarrow }{ r } }.\left( \lambda \overset { \wedge }{ i } +2\overset { \wedge }{ j } -7\overset { \wedge }{ k } \right) =26\) are perpendicular. Find the value of λ.
5)
Let \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) be unit vectors such \(\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } =\overset { \rightarrow }{ a } .\overset { \rightarrow }{ c } =0\) and the angle between \(\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \) is \(\frac { \pi }{ 6 } \). Prove that \(\overset { \rightarrow }{ a } =\pm 2\left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) \)

12th Standard English Medium Maths Subject Applications of Vector Algebra Creative 2 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k } \), \(\overset { \rightarrow }{ b } =-\overset { \wedge }{ i } +2\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ c } =3\overset { \wedge }{ i } +\overset { \wedge }{ j } \) find \(\frac { \lambda }{ c } \) such that \(\overset { \rightarrow }{ a } +\lambda \overset { \rightarrow }{ b } \) is perpendicular to \(\overset { \rightarrow }{ c } \)
2)
Find the area of the triangle whose vertices are A(3, -1, 2) B(1, -1, -3) and C(4, -3, 1)
3)
Find the Cartesian equation of a line passing through the points A(2, -1, 3) and B(4, 2, 1)
4)
Find the parametric form of vector equation of the plane passing through the point (1, -1, 2) having 2, 3, 3 as direction ratios of normal to the plane.
5)
Find the equation of the plane containing the line of intersection of the planes x + y + Z - 6 = 0 and 2x + 3y + 4z + 5 = 0 and passing through the point (1, 1, 1)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 1 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1}\) is
2)
If \(x = \frac{1}{5}\), the value of cos (cos^-1x+2sin^-1x) is
3)
\(\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)\) is equal to
4)
\(\sin ^{-1}\left(\tan \frac{\pi}{4}\right)-\sin ^{-1}\left(\sqrt{\frac{3}{x}}\right)=\frac{\pi}{6}\). Then x is a root of the equation
5)
sin (tan^-1x), |x| < 1 is equal to

12th Standard English Medium Maths Subject Applications of Vector Algebra Creative 3 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the Cartesian form of the equation of the plane \(\overset { \rightarrow }{ r } =\left( s-2t \right) \overset { \wedge }{ i } +\left( 3-t \right) \overset { \wedge }{ j } +\left( 2s+t \right) \overset { \wedge }{ k } \)
2)
Find the angle between the line \(\frac { x-2 }{ 3 } =\frac { y-1 }{ -1 } =\frac { z-3 }{ 2 } \) and the plane 3x + 4y + z + 5 = 0
3)
Prove that \(\left[ \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } \right] \)=\(\left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] \)
4)
If \(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } =0\) then show that \(\overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } =\overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } =\overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \)
5)
Show that the lines \(\frac { x-1 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-1 }{ 5 } \) and \(\frac { x+2 }{ 4 } =\frac { y-1 }{ 3 } =\frac { z+1 }{ -2 } \) do not intersect

12th Standard English Medium Maths Subject Applications of Vector Algebra Creative 3 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Dot product of a vector with vector \(\overset { \wedge }{ 3i } -5\overset { \wedge }{ k } \), \(2\overset { \wedge }{ i } +7\overset { \wedge }{ j } \) and \(\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) are respectively -1, 6 and 5. Find the vector.
2)
Find the equation of the plane through the intersection of the planes 2x-3y+ z-4 -0 and x - y + z + 1 = 0 and perpendicular to the plane x + 2y - 3z + 6 = 0
3)
If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } -\overset { \wedge }{ j } ,\overset { \rightarrow }{ b } =\overset { \wedge }{ j } -\overset { \wedge }{ k } ,\overset { \rightarrow }{ c } =\overset { \wedge }{ k } -\overset { \wedge }{ i } \) then find \(\left[ \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ b } -\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } -\overset { \rightarrow }{ a } \right] \)
4)
Prove by vector method, that in a right angled triangle the square of the hypotenuse is equal to the sum of the square of the other two sides.
5)
Show that the four points whose position vectors are \(6\overset { \wedge }{ i } -7\overset { \wedge }{ j } ,16\overset { \wedge }{ i } -29\overset { \wedge }{ j } -4\overset { \wedge }{ k } ,3\overset { \wedge }{ i } -6\overset { \wedge }{ j } \) are co-planar

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 1 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The value of sin^-1 (cos x), \(0\le x\le\pi\) is
2)
If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\); then cos^-1x + cos^-1 y is equal to
3)
If sin⁻¹x = 2sin⁻¹\(\alpha\) has a solution, then
4)
\(\sin ^{-1}(\cos x)=\frac{\pi}{2}-x\) is valid for
5)
If sin^-1 x+sin^-1 y+sin^-1 \(z = \frac{3\pi}{2}\), the value of x²⁰¹⁷+y²⁰¹⁸+z²⁰¹⁹\(-\frac { 9 }{ { x }^{ 101 }+{ y }^{ 101 }+{ z }^{ 101 } } \)is

12th Standard English Medium Maths Subject Applications of Vector Algebra Creative 5 Mark Questions with Solution updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Show that the points A, B, C with position vector \(2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\overset { \wedge }{ k } ,\overset { \wedge }{ i } -3\overset { \wedge }{ j } -5\overset { \wedge }{ k } \) and \(3\overset { \wedge }{ i } -4\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) respectively are the vector of a right angled, triangle. Also, find the remaining angles of the triangle.
2)
ABCD is a quadrilateral with \(\overset { \rightarrow }{ AB } =\overset { \rightarrow }{ \alpha } \) and \(\overset { \rightarrow }{ AD } =\overset { \rightarrow }{ \beta } \) and \(\overset { \rightarrow }{ AC } =2\overset { \rightarrow }{ \alpha } +3\overset { \rightarrow }{ \beta } \). If the area of the quadrilateral is λ times the area of the parallelogram with \(\overset { \rightarrow }{ AB } \) and \(\overset { \rightarrow }{ AD } \) as adjacent sides, then prove that \(\lambda =\frac { 5 }{ 2 } \)
3)
If \(\left| \overset { \rightarrow }{ A } \right| =\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \wedge }{ i } =\overset { \wedge }{ j } -\overset { \wedge }{ k } \) are two given vector, then find a vector B satisfying the equations \(\overset { \rightarrow }{ A } \times \overset { \rightarrow }{ B } \)= \(\overset { \rightarrow }{ C } \) and \(\overset { \rightarrow }{ A } \).\(\overset { \rightarrow }{ B } \) = 3
4)
Find the shortest distance between the following pairs of lines \(\frac { x-3 }{ 3 } =\frac { y-8 }{ -1 } =\frac { z-3 }{ 1 } \)and \(\frac { x+3 }{ -3 } =\frac { y+7 }{ 2 } =\frac { z-6 }{ 4 } \)
5)
Find the vector and Cartesian equation of the plane passing through the point (1,1, -1) and perpendicular to the planes x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 1 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The law of linear motion of a particle is given s = \(\frac{1}{3}\) t³-16t, the acceleration at the time when the velocity vanishes is __________
2)
Equation of the normal to the curve y = 2x²+3 sin x at x = 0 is __________
3)
The value of \(\underset { x\rightarrow \infty }{ lim } { e }^{ -x }\) is __________
4)
The critical points of the function f(x) = \((x-2)^{ \frac { 2 }{ 3 } }(2x+1)\) are __________
5)
In LMV theorem, we have f'(x₁) = \(\frac { f(b)-f(a) }{ b-a } \) then a < x₁ _________

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 1 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If a particle moves in a straight line according to s = t³-6t²-15t, the time interval during which the velocity is negative and acceleration is positive is __________
2)
The equation of the tangent to the curve x = t cost, y = t sin t at the origin is __________
3)
The function -3x+12 is ________ function on R.
4)
\(\underset { x\rightarrow 0 }{ lim } \frac { x }{ tanx } \) is _________
5)
The statement "If f has a local extremum at c and if f'(c) exists then f'(c) = 0" is ________

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 2 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the principal value of \({ tan }^{ -1 }\left( \frac { -1 }{ \sqrt { 3 } } \right) \)
2)
Find the principal value of sin^-1(-1).
3)
Find the principal value of \({ cos }^{ -1 }\left( \frac { -1 }{ 2 } \right) \)
4)
If \({ cot }^{ -1 }\left( \frac { 1 }{ 7 } \right) =\theta \) find the value of cos \(\theta \)
5)
If \({ sin }^{ -1 }\left( \frac { 1 }{ 2 } \right) ={ tan }^{ -1 }x\) then find the value of x,

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 2 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the maximum and minimum values of f(x) = |x+3| ∀ \(x\in R\).
2)
Find the point at which the curve y-e^xy+x=0 has a vertical tangent.
3)
Using Rolle’s theorem find the value of c for f(x) = sin x in[0,2π]
4)
Obtain Maclaurin’s Series expansion for e^2x.
5)
Evaluate the following limits, if necessary using L’Hopitalrule
(i) \(\underset { x\rightarrow 2 }{ lim } \cfrac { sin\pi x }{ 2-x } \)
(ii) \(\cfrac { lim }{ x\rightarrow 2 } \cfrac { { x }^{ n }-{ a }^{ n } }{ x-2 } \)
(iii) \(\underset { x\rightarrow \infty }{ lim } \cfrac { sin\frac { 2 }{ x } }{ \frac { 1 }{ x } } \)
(iv) \(\underset { x\rightarrow \infty }{ lim } \cfrac { { x }^{ 2 } }{ { e }^{ x } } \)

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 2 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the intervals of increasing and decreasing function for f(x) = x³ + 2x² - 1.
2)
Find the point on the parabola y²=18x at which the ordinate increases at twice the rate of the abscissa.
3)
Find the equation of the tangent to the curve y²=4x+5 and which is parallel to y=2x+7
4)
Verify Lagrange’s Mean Value theorem for \(f(x)=\sqrt { x-2 } \) in the interva [2,6]
5)
Expand the polynomial f(x)=x²-3x+2 in power of (x-2)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 2 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the principal value of sin^-1\(\left( -\frac { 1 }{ 2 } \right) \)(in radians and degrees).
2)
Find the principal value of sin^-1(2), if it exists.
3)
Find the principal value of
\({ Sin }^{ -1 }\left( \frac { 1 }{ \sqrt { 2 } } \right) \)
4)
Find all the values of x such that -10\(\pi\)\(\le x\le\)10\(\pi\) and sin x = 0
5)
Find the period and amplitude of y = sin 7x

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 3 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
A ball is thrown vertically upwards, moves according to the law s = 13.8 t - 4.9 t² where s
is in metres and t is in seconds.
(i) Find the acceleration at t = 1
(ii) Find velocity at t = 1
(iii) Find the maximum height reached by the ball?
2)
The side of a square is equal to the diameter of a circle. If the side and radius change at the same rate then find the ratio of the change of their areas.
3)
Find the equation of normal to the curve y⁴=ax²at(a,a)
4)
A cylindrical hole 4 mm in diameter and 12 mm deep in a metal block is reboared to increase the diameter to 4.12mm. Estimate the amount of metal removed
5)
Obtain Maclaurin’s series for \(\frac { 1 }{ 1+x } \)

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 3 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Prove that \(\frac { x }{ 1+x } \) < < log (1+ x) for x > 0.
2)
Verify LMV theorem for f(x) = x³ - 2x² - x + 3 in [0, 1].
3)
The volume of a cube is increasing at the rate of 8cm³/s.How fast is the surface area increasing when the length of an edge is 12cm?
4)
Find the angle between two curves 2x²+y²=20 and x²-4y²+8=0
5)
Write down the Taylor series expansion of the function cos x in ascending powers \(x-\frac { \pi }{ 4 } \) upto three non-zero terms.

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 3 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the domain of sin⁻¹(2−3x²)
2)
Sketch the graph of y = sin\((\frac{1}{3}x)\) for 0\(\le x <6\pi\).
3)
Find the domain of the following
\(f\left( x \right) { =sin }^{ -1 }\left( \frac { { x }^{ 2 }+1 }{ 2x } \right) \)
4)
Find the domain of cos^-1\((\frac{2+sinx}{3})\)
5)
Find the domain of f(x) = sin^-1 \((\frac{|x|-2}{3})+ \) cos^-1\((\frac{1-|x|}{4})\)

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 5 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the angle of intersection of the curves 2y²= x³ and y² = 32x.
2)
Prove that the semi-vertical angle of a cone of maximum volume and of given slant height is tan^-1(\(\sqrt { 2 } \)).
3)
Sand is pouring from a pipe at the rate of 12 cm³/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing, when the height is 4 cm?
4)
A water tank has a shape of an inverted cone with its axis vertical and vertex lower most. Its semi vertical angle is tan⁻¹(0.5). Water is poured into it at a constant rate of 5 cm3/hr. Find the rate at which the level of the water is rising at that instant when the depth of the water is 4 m.

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 5 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the intervals of concavity and points of inflexion for f(x)=x³-15x²+75x-50.
2)
Find the local maximum and local minimum values of f(x)=x⁴-3x+3x²-x.

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 3 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If cos⁻¹x + cos⁻¹y + cos⁻¹ z = \(\pi \) and 0 < x, y, z < 1, show that x² + y²+ z²+ 2xyz = 1
2)
Solve sin^-1 x > cos^-1x
3)
Solve tan^-1 2x + tan^-1 3x = \(\frac{\pi}{4}\), if 6x²< 1
4)
Find the value of the expression in terms of x, with the help of a reference triangle.
sin(cos⁻¹(1-x))
5)
Find the domain of the following
g(x) = 2sin⁻¹(2x−1)−\(\frac{\pi}{4}\)

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 1 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If the radius of the sphere is measured as 9 cm with an error of 0.03 cm, the approximate error in calculating its volume is _____________
2)
If u = log (x³ + y³+ z³ - 3xyz) then \(\frac { { \partial }u }{ \partial { x } } +\frac { { \partial }u }{ { \partial y } }+ \frac { { \partial }u }{ \partial z } \) = _____________
3)
If f(x, y) = 2x²- 3xy + 5y + 7 then f(0, 0) and f(1, 1) is _____________
4)
If x = r cos θ, y = r sin, then \(\frac { \partial r }{ \partial x } \) = ....................
5)
If is a homogeneous function of x and y of degree n, then \(x\frac { { \partial }^{ 2 }u }{ \partial { x }^{ 2 } } +y\frac { { \partial }^{ 2 }u }{ \partial x\partial y } \) = ...... \(\frac { { \partial }u }{ \partial { x } } \)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 5 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the principal value of cos⁻¹\(\left( \frac { \sqrt { 3 } }{ 2 } \right) \)
2)
Find cos^-1\((-\frac{1}{\sqrt2})\)
3)
Find the principal value of cosec⁻¹(−1)
4)
If a₁, a₂, a₃, ... an is an arithmetic progression with common difference d, prove that tan\( \left[ tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +....tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ n }{ a }_{ n-1 } } \right) \right] =\frac { { a }_{ n }-{ a }_{ 1 } }{ 1+{ a }_{ 1 }{ a }_{ n } } \)
5)
Solve \(tan^{ -1 }\left( \frac { x-1 }{ x-2 } \right) +tan^{ -1 }\left( \frac { x+1 }{ x+2 } \right) =\frac { \pi }{ 4 } \)

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 1 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If log_e4 = 1.3868, then log_e4.01 = _____________
2)
The approximate value of (627)^\(\frac14\) is ................
3)
If y = sin x and x changes from \(\frac{\pi}{2}\) to ㅠ the approximate change in y is ..............
4)
If u = sin-1 \(\left( \frac { { x }^{ 4 }+{ y }^{ 4 } }{ { x }^{ 2 }+{ y }^{ 2 } } \right) \) and f = sin u then f is a homogeneous function of degree ..................
5)
The percentage error in the 11^th root of the number 28 is approximately .......... times the percentage error in 28.

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 2 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Use differentials to find \(\sqrt{25.2}\)
2)
If of f(x, y) = x² + y³ + 2xy² find f_xx, f_yy, f_xy and f_yx.
3)
If w=e^xy,x=at²,y=2at, find \(\frac { dw }{ dt } \)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 5 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the principal value of cosec⁻¹(−1)
2)
If a₁, a₂, a₃, ... an is an arithmetic progression with common difference d, prove that tan\( \left[ tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +....tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ n }{ a }_{ n-1 } } \right) \right] =\frac { { a }_{ n }-{ a }_{ 1 } }{ 1+{ a }_{ 1 }{ a }_{ n } } \)
3)
Solve \(tan^{ -1 }\left( \frac { x-1 }{ x-2 } \right) +tan^{ -1 }\left( \frac { x+1 }{ x+2 } \right) =\frac { \pi }{ 4 } \)
4)
Solve \(cos\left( sin^{ -1 }\left( \frac { x }{ \sqrt { 1+{ x }^{ 2 } } } \right) \right) =sin\left\{ cot^{ -1 }\left( \frac { 3 }{ 4 } \right) \right\} \)
5)
Find the principal value of
sec⁻¹(−2).

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 2 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If w=log(x²+y²),x=cosθ,y=sinθ, find \(\frac { dw }{ d\theta } \)
2)
If w=xye^xy find \(\frac { { \partial }^{ 2 }u }{ \partial x\partial y } \)

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 3 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the approximate value of f (3.02) where f(x) = 3x² + 5x +3.
2)
If f = \(\frac { x }{ { x }^{ 2 }+{ y }^{ 2 } } \) then show that = \(x\frac { \partial f }{ \partial x } +y\frac { \partial f }{ \partial y } \) = -f
3)
Find the linear approximation to \(g(z)=\sqrt [ 4 ]{ zat } z=2\)
4)
If (x,y)=3x²+4y²+6xy-x²y²+5 then find
(i) fx(1,-1)
(ii) f_xy(1,1)
(iii)fxy(2,1)

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 3 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If w=xy+z and x=cot, y=sint, z=t then find \(\frac { dw }{ dt } \)
2)
Using linear approximation find \(\sqrt { 0.082 } \)
3)
If w=x²+y² and x=u²-v^2,y=2uv then find \(\frac { \partial w }{ \complement u } and\frac { \partial w }{ \partial v } \)
4)
Evaluate : \(\underset { \left( x,y,z \right) \rightarrow \left( -1,0,4 \right) }{ lim } \frac { { x }^{ 2 }-{ ze }^{ zy } }{ 6x+2y-2z } \)

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 5 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find \(\frac { \partial f }{ \partial x } ,\frac { \partial f }{ \partial y } ,\frac { { \partial }^{ 2 }f }{ \partial { x }^{ 2 } } ,\frac { { \partial }^{ 2 }f }{ { \partial y }^{ 2 } } \) at x = 2, y = 3 if f(x,y) = 2x² + 3y² - 2xy
2)
If V = log r and r² = x² +y² + z², then prove that \(\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } } +\frac { { \partial }^{ 2 } }{ \partial { z }^{ 2 } } =\frac { 1 }{ { r }^{ 2 } } \)
3)
Find \(\frac { \partial w }{ \partial u } ,\frac { \partial w }{ \partial v } \) if w=sin^-1(x,y) where x=u+v,y=u-v

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 5 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If u = tan^-1 \(\left( \frac { { x }^{ 3 }+{ y }^{ 3 } }{ x-y } \right) \) Prove that \(x\frac { \partial u }{ \partial x } +y\frac { \partial u }{ \partial y } \) sin 2u.
2)
Find the approximate value of \(\sqrt [ 3 ]{ 1.02 } +\sqrt { 1.02 } \)

12th Standard English Medium Maths Subject Applications of Integration Creative 1 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The value of \(\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ \left( \frac { { 2x }^{ 7 }-{ 3x }^{ 5 }+{ 7x }^{ 3 }-x+1 }{ { cos }^{ 2 }x } \right) dx } \) is
2)
The area between y² = 4x and its latus rectum is
3)
The value of \(\int _{ 0 }^{ \pi }{ \frac { dx }{ 1+{ 5 }^{ cos\ x } } } \) is
4)
The value of \(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ { cos }^{ 3 }3x\ dx }\ is\)
5)
The value of \(\int _{ 0 }^{ \infty }{ { e }^{ -3x }{ x }^{ 2 }dx } \) is

12th Standard English Medium Maths Subject Applications of Integration Creative 1 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If \(f(x)=\int_{0}^{x} t \cos t d t, \text { then } \frac{d f}{d x}=\)
2)
The value of \(\int _{ 0 }^{ \pi }{ { sin }^{ 4 }xdx } \) is
3)
If \(f(x)=\int_{1}^{x} \frac{e^{\sin u}}{u} d u, x>1 \text { and }\int_{1}^{3} \frac{e^{\sin x^{2}}}{x} d x=\frac{1}{2}[f(a)-f(1)]\), then one of the possible value of a is
4)
The value of \(\int _{ 0 }^{ a }{ { (\sqrt { { a }^{ 2 }-{ x }^{ 2 } } ) }^{ 3 } } dx\) is
5)
If \(\int _{ 0 }^{ x }{ f(t)dt=x+\int _{ x }^{ 1 }{ tf } (t)dt } \), then the value of f (1) is

12th Standard English Medium Maths Subject Applications of Integration Creative 2 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Evaluate \(\int _{ 0 }^{ 1 }{ \frac { { e }^{ x } }{ 1+{ e }^{ 2x } } dx } \)
2)
Find the area of the region enclosed by the curve y = \(\sqrt x\) + 1, the axis of x and the lines x = 0, x = 4.
3)
Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ log(tanx)dx } =0\)
4)
Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ 3x } } cosxdx\)
5)
Evaluate \(\int _{ 0 }^{ \infty }{ \left( { a }^{ -x }-{ b }^{ -x } \right) } dx\)

12th Standard English Medium Maths Subject Applications of Integration Creative 2 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Evaluate \(\int _{ 1 }^{ 2 }{ \frac { 3x }{ { 9x }^{ 2 }-1 } dx } \)
2)
Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ 2x }cosxdx } \)
3)
Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ -x } } \)
4)
Evaluate \(\int _{ 0 }^{ 1 }{ \left( \frac { { e }^{ 5logx }-{ e }^{ 4logx } }{ { e }^{ 3logx }-{ e }^{ 2logx } } \right) } \)
5)
Find the area of the region bounded by the curves y=2^x.y=-2x-x² and the lines x=0 and x=2

12th Standard English Medium Maths Subject Applications of Integration Creative 3 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Evaluate \(\int _{ -1 }^{ 1 }{ log\left( \frac { 2-x }{ 2+x } \right) } dx\)
2)
Evaluate \(\int _{ 0 }^{ 1 }{ \sqrt { 9-4{ x }^{ 2 } } dx } \)
3)
Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { dx }{ 1+{ tan }^{ 3 }x } } \)
4)
Evaluate \(\int _{ 0 }^{ 1 }{ x\left( 1-x \right) ^{ 10 }dx } \)
5)
Evaluate \(\int _{ 0 }^{ 50 }{ \left[ x-\left| x \right| \right] dx } \)

12th Standard English Medium Maths Subject Applications of Integration Creative 3 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sin \ x }{ 9+{ cos }^{ 2 } } dx } \)
2)
Evaluate \(\int _{ 0 }^{ 1 }{ { xe }^{ -2x } } dx\)
3)
If \(\int _{ 0 }^{ \infty }{ \frac { dx }{ \left( { x }^{ 2 }+4 \right) \left( { x }^{ 2 }+9 \right) } } =k\pi \) then find the value of k.
4)
Evaluate \(\int _{ 0 }^{ \pi }{ { cos }^{ 3 }xdx } \)
5)
Evaluate \(\int _{ -2 }^{ 3 }{ \left| 1-{ x }^{ 2 } \right| } dx\)

12th Standard English Medium Maths Subject Applications of Integration Creative 5 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the area bounded by x = at², y = at between the ordinates corresponding to t = 1 and t = 2
2)
Using integration, find the area of the triangle with sides y = 2x + 1, y = 3x + 1 and x = 4.
3)
Find the area of the curve y²=(x-5)²(x-6) between
(i) x=5 and x=6
(ii) x=6 and x=7
4)
Find the ratio of the area between the curves y=cosx and y=cos2x and x- axis from x=0 to \(x=\frac { \pi }{ 3 } \)
5)
Find the value of ‘c’ for which the area bounded by the curve y=8x²-x⁵,the lines x=1,x=c and x-axis \(\frac { 16 }{ 3 } \)

12th Standard English Medium Maths Subject Applications of Integration Creative 5 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Show that the area under the curve y = sin x and y = sin 2x between x = 0 and x = \(\frac { \pi }{ 3 } \) and x axis are as 2:3
2)
Find the volume of the solid generated by the revolution of the loop of the curve x = t² y = t - \(\frac { { t }^{ 3 } }{ 3 } \) about x-axis.
3)
Find the area bounded by the curves y=|x|-1 and y=-|x|+1
4)
Find the area bounded by the curve y2(2a-x)=x² and the line x=2a.
5)
Find the area enclosed by the parabolas 5x²-y=0 and 2x²-y+9=0.

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 1 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If cosx is an integrating factor of the differential equation \(\frac{dy}{dx}+Py= Q\), then P = ___________
2)
The I.F. of cosec x \(\frac{dy}{dx}+y\) sec²x = 0 is ___________
3)
The general solution of \(4\frac{d^2 y}{dx^2}\) + y = 0 is _________
4)
The I.F of \(\frac{dy}{dx}-y\) tan x = cos x is _________
5)
The solution of log \(\left( \frac { dy }{ dx } \right) \) = ax + by is______.

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 1 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The solution of sec²x tan y dx + sec²y tan x dy = 0 is _________
2)
The transformation y = vx reduces \(\\ \\ \\ \frac { dy }{ dx } =\frac { x+y }{ 3x } \) __________
3)
The solution of \(\frac{dy}{dx}+y\) cot x = sin 2x is ___________
4)
The I.F of y log y \(\frac{dx}{dy}+x-log\ y=0\) is __________
5)
The order and degree of y'+(y")²= (x + t")² are _________.

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 2 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
A curve passing through the origin has its slope e^x, Find the equation of the curve.
2)
Solve: x \(\frac{dy}{dx}=x+y\)
3)
Determine the order and degree of \(\frac { \left[ 1+\left( \frac { dy }{ dx } \right) ^{ 2 } \right] ^{ \frac { 3 }{ 2 } } }{ \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } } =k\)
4)
Solve :\(\frac { dy }{ dx } =\frac { 2x }{ { x }^{ 2 }+1 } \)
5)
Solve:\(\frac { dy }{ dx } +y=1\)

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 2 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Solve: \(\frac{dy}{dx}=1+e^{x-y}\)
2)
Solve: \(\frac{dy}{dx}+y=e^{-x}\)
3)
Find the order and degree of \(\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 2 }+cos\left( \frac { dy }{ dx } \right) =0\)
4)
Form the D.E corresponding to y=e^mx by eliminating 'm'.
5)
solve: x dy + y dx = xy dx

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 3 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Form the differential equation for y = e^-2x [A cos 3x-B sin 3x]
2)
Solve: x\(\frac{dy}{dx}\)+ 2y = x²
3)
Form the D.E of the family of curves c(y + c)²= x², where c is the parameter.
4)
Form the D.E to y²=a(b-x)(b+x) by eliminating a and b as its parameters.
5)
Find the D.E of all circles touching y-axis at the origin.

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 3 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Solve: \(\frac{dy}{dx}+y=cos x\)
2)
Form the D.E of family of curves represented by y=c(x-c)².where c is the parameter.
3)
Find the D.E of all circles touching x-axis at the origin.
4)
Solve : \(\left( { x }^{ 2 }-1 \right) \frac { dy }{ dx } +2xy=\frac { 1 }{ { x }^{ 2 }-1 } \)
5)
Solve : ydx+(x-y²)dy=0

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 5 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If cosx is an integrating factor of the differential equation \(\frac{dy}{dx}+Py= Q\), then P = ___________
2)
The I.F. of cosec x \(\frac{dy}{dx}+y\) sec²x = 0 is ___________
3)
The general solution of \(4\frac{d^2 y}{dx^2}\) + y = 0 is _________
4)
The I.F of \(\frac{dy}{dx}-y\) tan x = cos x is _________
5)
The differential equation corresponding to xy = c² where c is an arbitrary constant is ________.

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 5 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The surface area of a balloon being inflated changes at a constant rate. If initially, its radius 3 units and after 2 seconds it is 5 units, find the radius after t seconds.
2)
Solve: \(\frac { dy }{ dx } \) = (3x+2y+1)²
3)
Solve : \(\frac { dy }{ dx } =\left( { sin }^{ 2 }x{ cos }^{ 2 }x+{ xe }^{ x } \right) dx\)
4)
Solve : \(\left( 1+{ x }^{ 2 } \right) \frac { dy }{ dx } -x={ 2tan }^{ -1 }x\)
5)
Solve : (1+y²)(1 + log x)dx + x dy = 0, given that x = 1,y = 1.

12th Standard English Medium Maths Subject Probability Distributions Creative 1 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
IfF(x) is the probability distribution function, then \(F\left( -\infty \right) \) is _____________
2)
If \(f(x)={ Cx }^{ 2 }={ cx }^{ 2 },0 is the p.d.f, of x then c is _____________
3)
In eight throws of a die, 1 or 3 is considered a success. Then the mean number of success is _____________
4)
In a binomial distribution, if the mean is 8 and the variance is 6, then the number of trials is _____________
5)
A die is tossed 5 times. Getting an odd number is considered a success. Then the variance of distribution of number of success is _____________

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 1 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x²+ y²− 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to
2)
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is
3)
The circle x²+ y²= 4x + 8y +5 intersects the line 3x−4y = m at two distinct points if
4)
The length of the diameter of the circle which touches the x - axis at the point (1, 0) and passes through the point (2, 3).
5)
The centre of the circle inscribed in a square formed by the lines x²− 8x − 12 = 0 and y²− 14y + 45 = 0 is

12th Standard English Medium Maths Subject Probability Distributions Creative 1 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If \(f(x)=\frac { 1 }{ 2 } \) ,\(E\left( { x }^{ 2 } \right) =\frac { 1 }{ 4 } \) then var(x) is _____________
2)
In a binomial distribution,\(n=4,P(X=0)=\frac { 16 }{ 81 } \),then \(P(X=4)\) _____________
3)
Var (2x ± 5) is =________
4)
A coin is tossed 3 times. The probability of getting exactly 2 heads is________
5)
If the mean and S.D of a binomial distribution are 12 and 2 respectively, then ___________

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 1 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The eccentricity of the ellipse (x−3)² +(y−4)² \(=\frac { { y }^{ 2 } }{ 9 } \) is
2)
If the two tangents drawn from a point P to the parabola y² = 4x are at right angles then the locus of P is
3)
The locus of a point whose distance from (-2,0) is \(\frac { 2 }{ 3 } \) times its distance from the line x = \(\frac { -9 }{ 2 } \) is
4)
The values of m for which the line y = mx + \(2\sqrt { 5 } \) touches the hyperbola 16x²− 9y²= 144 are the roots of x²− (a + b)x − 4 = 0, then the value of (a+b) is
5)
If the coordinates at one end of a diameter of the circle x²+ y²− 8x − 4y + c = 0 are (11, 2), the coordinates of the other end are

12th Standard English Medium Maths Subject Probability Distributions Creative 2 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 2 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the general equation of a circle with centre (-3, -4) and radius 3 units.
2)
Determine whether x + y − 1 = 0 is the equation of a diameter of the circle x²+ y²− 6x + 4y + c = 0 for all possible values of c .
3)
Find the general equation of the circle whose diameter is the line segment joining the points (−4, −2)and (1, 1) is x²+y²+5x+3y+6=0
4)
Examine the position of the point (2, 3) with respect to the circle x²+ y²− 6x − 8y + 12 = 0.
5)
The line 3x+4y−12 = 0 meets the coordinate axes at A and B. Find the equation of the circle drawn on AB as diameter.

12th Standard English Medium Maths Subject Probability Distributions Creative 2 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 2 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Identify the type of conic section for each of the equations.
3x²+3y²−4x+3y+10 = 0
2)
Identify the type of the conic for the following equations:
3x²+2y²= 14
3)
Identify the type of the conic for the following equations :
11x²−25y²−44x+50y−256 = 0
4)
Find centre and radius of the following circles.
x²+ y²+ 6x − 4y + 4 = 0
5)
Find centre and radius of the following circles.
2x²+2y²−6x+4y+2 = 0

12th Standard English Medium Maths Subject Probability Distributions Creative 3 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 3 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the equation of the circle described on the chord 3x + y + 5 = 0 of the circle x²+ y²= 16 as diameter.
2)
A circle of radius 3 units touches both the axes. Find the equations of all possible circles formed in the general form.
3)
Find the centre and radius of the circle 3x²+ (a + 1)y²+ 6x − 9y + a + 4 = 0.
4)
Find the equation of circles that touch both the axes and pass through (-4, -2) in general form.
5)
Find the equation of the tangent and normal to the circle x²+y²−6x+6y−8 = 0 at (2, 2) .

12th Standard English Medium Maths Subject Probability Distributions Creative 3 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 3 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Identify the type of conic and find centre, foci, vertices, and directrices of each of the following:
\(\frac { { x }^{ 2 } }{ 25 } +\frac { { y }^{ 2 } }{ 9 } =1\)
2)
Prove that the length of the latus rectum of the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 is \(\frac { { 2b }^{ 2 } }{ a } \).
3)
Show that the absolute value of difference of the focal distances of any point P on the hyperbola is the length of its transverse axis.
4)
Prove that the point of intersection of the tangents at ‘t₁’ and ‘t₂’ on the parabola y² = 4ax is \(\left[ at_{ 1 }t_{ 2 },a({ t }_{ 1 }+{ t }_{ 2 }) \right] .\)
5)
A semielliptical archway over a one-way road has a height of 3m and a width of 12m. The truck has a width of 3m and a height of 2.7m. Will the truck clear the opening of the archway?

12th Standard English Medium Maths Subject Probability Distributions Creative 5 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 5 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
On lighting a rocket cracker it gets projected in a parabolic path and reaches a maximum height of 4 m when it is 6 m away from the point of projection. Finally it reaches the ground 12 m away from the starting point. Find the angle of projection.
2)
Points A and B are 10 km apart and it is determined from the sound of an explosion heard at those points at different times that the location of the explosion is 6 km closer to A than B. Show that the location of the explosion is restricted to a particular curve and find an equation of it.
3)
Find the vertex, focus, equation of directrix and length of the latus rectum of the following:
x²−2x+8y+17= 0
4)
Find the vertex, focus, equation of directrix and length of the latus rectum of the following: y²−4y−8x+12 = 0
5)
Identify the type of conic and find centre, foci, vertices, and directrices of each of the following :
18x²+12y²−144x+48y+120 = 0

12th Standard English Medium Maths Subject Probability Distributions Creative 5 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 5 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the equation of the circle passing through the points (1, 1 ), (2, -1 ) and (3, 2) .
2)
A road bridge over an irrigation canal have two semi circular vents each with a span of 20m and the supporting pillars of width 2m. Use Figure to write the equations that represents the semi-verticular vents
3)
Find the equations of tangents to the hyperbola \(\frac { { x }^{ 2 } }{ 16 } -\frac { { y }^{ 2 } }{ 64 } \) = 1 which are parallel to10x − 3y + 9 = 0.
4)
Show that the line x−y+4 = 0 is a tangent to the ellipse x²+3y²= 12 . Also find the coordinates of the point of contact.
5)
The maximum and minimum distances of the Earth from the Sun respectively are 152 × 10⁶ km and 94.5 × 10⁶ km. The Sun is at one focus of the elliptical orbit. Find the distance from the Sun to the other focus.

12th Standard English Medium Maths Subject Discrete Mathematics Creative 1 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If * is defined by a * b = a² + b² + ab + 1, then (2 * 3) * 2 is _____________
2)
The identity element of \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) \right\} \) |x \(\in \) R, x ≠ 0} under matrix multiplication is __________
3)
The identity element in the group {R - {1},x} where a * b = a + b - ab is __________
4)
A binary operation * is defined on the set of positive rational numbers Q⁺ by a*b = \(\frac { ab }{ 4 } \). Then 3 * \(\left( \frac { 1 }{ 5 } *\frac { 1 }{ 2 } \right) \) is _____________
5)
'-' is a binary operation on ___________

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 1 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x²+ y²− 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to
2)
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is
3)
If P(x, y) be any point on 16x²+ 25y²= 400 with foci F₁ (3, 0) and F₂ (-3, 0) then PF₁ + PF₂ is
4)
The radius of the circle passing through the point(6, 2) two of whose diameter are x + y = 6 and x + 2y = 4 is
5)
The area of quadrilateral formed with foci of the hyperbolas \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \text { and } \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\)

12th Standard English Medium Maths Subject Discrete Mathematics Creative 1 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The number of binary operations that can be defined on a set of 3 elements is _____________
2)
Which of the following is a contradiction?
3)
Define * on Z by a * b = a + b + 1 ∀ a,b \(\in \) Z. Then the identity element of z is ________
4)
If a * b = a²b² - ab then 3 * (1 * 1)
5)
Which of the following is a statement?

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 1 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If the two tangents drawn from a point P to the parabola y² = 4x are at right angles then the locus of P is
2)
The circle passing through (1, -2) and touching the axis of x at (3, 0) passing through the point
3)
The locus of a point whose distance from (-2,0) is \(\frac { 2 }{ 3 } \) times its distance from the line x = \(\frac { -9 }{ 2 } \) is
4)
The values of m for which the line y = mx + \(2\sqrt { 5 } \) touches the hyperbola 16x²− 9y²= 144 are the roots of x²− (a + b)x − 4 = 0, then the value of (a+b) is
5)
If the coordinates at one end of a diameter of the circle x²+ y²− 8x − 4y + c = 0 are (11, 2), the coordinates of the other end are

12th Standard English Medium Maths Subject Discrete Mathematics Creative 2 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Show that p v (q ∧ r) is a contingency.
2)
Let S be the set of positive rational numbers and is defined by a * b = \(\frac{ab}{2}\). Then find the identity element and the inverse of 2.

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 2 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If \(\vec{ a } =\hat { -3i } -\hat { j } +\hat { 5k } \), \(\vec{b}=\hat{i}-\hat{2j}+\hat{k} \), \(\vec{c}=\hat{4j}-\hat{5k} \ \) find\( \ {\vec a } .(\vec { b } \times \vec { c } )\)
2)
Find the volume of the parallelepiped whose coterminus edges are given by the vectors \(\hat { 2i } -\hat { 3j } +\hat { 4k } \), \(\hat { i } +\hat { 2j } -\hat { k } \) and \(\hat {3 i } -\hat { j } +\hat { 2k } \)
3)
Show that the vectors \(\hat { i } +\hat { 2j } -\hat { 3k } \), \(\hat { 2i } -\hat { j } +\hat { 2k } \) and \(\hat { 3i } +\hat { j } -\hat { k } \)
4)
If \(\hat { 2i } -\hat { j } +\hat { 3k } ,\hat { 3i } +\hat { 2j } +\hat { k } ,\hat { i } +\hat { mj } +\hat { 4k } \) are coplanar, find the value of m.
5)
If \(\vec { a } ,\vec { b } ,\vec { c } \) are three vectors, prove that \([\vec { a } +\vec { c } ,\vec { a } +\vec { b } ,\vec { a } +\vec { b } +\vec { c } ]\) = \([\vec { a } ,\vec { b } ,\vec { c } ]\)

12th Standard English Medium Maths Subject Discrete Mathematics Creative 2 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Show that p v (q ∧ r) is a contingency.

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 2 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the angle between the planes \(\vec { r } .(\hat { i } +\hat { j } -2\hat { k } )\) = 3 and 2x - 2y + z =2
2)
Find the length of the perpendicular from the point (1, -2, 3) to the plane x - y + z = 5.
3)
Find the acute angle between the planes \(\vec { r } .(2\hat { i } +2\hat { j } +2\hat { k } )\) and 4x-2y+2z = 15.
4)
Find the distance of a point (2, 5, −3) from the plane \(\vec { r } .(6\hat { i } -3\hat { j } +2\hat { k } )\) = 5
5)
Find the distance between the parallel planes x + 2y - 2z + 1 = 0 and 2x + 4y - 4z + 5 = 0

12th Standard English Medium Maths Subject Discrete Mathematics Creative 3 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
In (z, *) where * is defined by a * b = a^b, prove that * is not a binary operation on z.
2)
In (z, *) where * is defined as a * b = a + b + 2. Verify the commutative and associative axiom.

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 3 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If D is the midpoint of the side BC of a triangle ABC, then show by vector method that \({ \left| \vec { AB } \right| }^{ 2 }+{ \left| \vec { AC } \right| }^{ 2 }=2({ \left| \vec { AD} \right| }^{ 2 }+{ \left| \vec { BD } \right| }^{ 2 })\)
2)
Prove by vector method that the diagonals of a rhombus bisect each other at right angles.
3)
Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and BD is \(\frac { 1 }{ 2 } \left| \vec { AC } \times \vec { BD } \right| \).
4)
Forces of magnit \(5\sqrt { 2 } \) and \(10\sqrt { 2 } \) units acting in the directions \(\hat { 3i } +\hat { 4j } +\hat { 5k } \) and \(\hat { 10i } +\hat { 6j } -\hat { 8k } \) respectively, act on a particle which is displaced from the point with position vector \(\hat { 4i } -\hat { 3j } -\hat { 2k } \) to the point with position vector \(\hat { 6i } +\hat { j } -\hat { 3k } \). Find the work done by the forces.
5)
Let \(\vec { a } ,\vec { b } ,\vec { c } \) be three non-zero vectors such that \(\vec { c } \) is a unit vector perpendicular to both \(\vec { a } \) and \(\vec { b } \). If the angle between \(\vec { a } \) and \(\vec { b } \) is \(\frac { \pi }{ 6 } \), show that \({ [\vec { a } ,\vec { b } ,\vec { c } ] }^{ 2 }\) = \(\frac { 1 }{ 4 } { \left| \vec { a } \right| }^{ 2 }{ \left| \vec { b } \right| }^{ 2 }\)

12th Standard English Medium Maths Subject Discrete Mathematics Creative 3 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Let G = {1, i, -1, -i} under the binary operation multiplication. Find the inverse of all the elements.
2)
Construct the truth table for (-p) v (q ∧ r)

12th Standard English Medium Maths Subject Discrete Mathematics Creative 5 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Verify (p ∧ ~p) ∧ (~q ∧ p) is a tautlogy, contradiction or contingency.
2)
In (N, *) where * is defined by x * y = max (x, y) check the closure axion and identity anion.

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 3 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the parametric form of vector equation of the straight line passing through (−1, 2,1) and parallel to the straight line \(\vec { r } =(2\hat { i } +3\hat { j } -\hat { k } )+t(\hat { i } -2\hat { j } +\hat { k } )\) and hence find the shortest distance between the lines.
2)
Find the non-parametric form of vector equation, and Cartesian equation of the plane passing through the point (0, 1, -5) and parallel to the straight lines \(\vec { r } =(\hat { i } +2\hat { j } -4\hat { k } )+s(\hat { i } +3\hat { j } +6\hat { k } )\) and \(\hat { r } =(\hat { i } -3\hat { j } +5\hat { k } )+t(\hat { i } +\hat { j } -\hat { k } )\)
3)
Find the non-parametric form of vector equation, and Cartesian equations of the plane \(\vec { r } =(6\hat { i } -\hat { j } +\hat { k } )+s(-\hat { i } +2\hat { j } +\hat { k } )+(-5\hat { i } -4\hat { j } -5\hat { k } )\)
4)
If the straight lines \(\frac { x-1 }{ 1 } =\frac { y-2 }{ 1 } =\frac { z-3 }{ { m }^{ 2 } } \) and \(\frac { x-3 }{ 1 } =\frac { y-2 }{ { m }^{ 2 } } =\frac { z-1 }{ 2 } \) are coplanar, find the distinct real values of m.
5)
Find the equation of the plane passing through the line of intersection of the planes \(\vec { r } .(2\hat { i } -7\hat { j } +4\hat { k } )=3\) and 3x - 5y + 11 = 0, and the point (-2, 1, 3)

12th Standard English Medium Maths Subject Discrete Mathematics Creative 5 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Construct the truth table for (p ∧ q) v r.

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 5 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the vector parametric, vector non-parametric and Cartesian form of the equation of the plane passing through the points (-1, 2, 0), (2, 2, -1)and parallel to the straight line \(\frac { x-1 }{ 1 } =\frac { 2y+1 }{ 2 } =\frac { z+1 }{ -1 } \)
2)
Find the non-parametric form of vector equation, and Cartesian equation of the plane passing through the point (2, 3, 6) and parallel to the straight lines \(\frac { x-1 }{ 2 } =\frac { y+1 }{ 3 } =\frac { z-3 }{ 1 } \) and \(\frac { x+3 }{ 2 } =\frac { y-3 }{ -5 } =\frac { z+1 }{ -3 } \)
3)
Find the non-parametric form of vector equation, and Cartesian equations of the plane passing through the points (2, 2, 1), (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 9
4)
Find parametric form of vector equation and Cartesian equations of the plane passing through the points (2, 2, 1), (1, −2, 3) and parallel to the straight line passing through the points (2, 1, −3) and (−1, 5, −8)
5)
Find the non-parametric form of vector equation of the plane passing through the point (1, −2, 4) and perpendicular to the plane x + 2y −3z = 11 and parallel to the line \(\frac { x+7 }{ 3 } =\frac { y+3 }{ -1 } =\frac { z }{ 1 } \)

12th Standard English Medium Maths Subject Creative 1 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then _____________
2)
If A^T is the transpose of a square matrix A, then ___________
3)
If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| = _______
4)
If A is a square matrix of order n, then |adj A| = ______________
5)
Every homogeneous system ______

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 5 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the non-parametric form of vector equation of the plane passing through the point (1, −2, 4) and perpendicular to the plane x + 2y −3z = 11 and parallel to the line \(\frac { x+7 }{ 3 } =\frac { y+3 }{ -1 } =\frac { z }{ 1 } \)
2)
Find the parametric form of vector equation and Cartesian equations of the plane containing the line \(\vec { r } =(\hat { i } -\hat { j } +3\hat { k } )+t(2\hat { i } -\hat { j } +4\hat { k } )\) and perpendicular to plane \(\vec { r } .(\hat { i } +2\hat { j } +\hat { k } )=8\)
3)
Show that the lines \(\frac { x-2 }{ 1 } =\frac { y-3 }{ 1 } =\frac { z-4 }{ 3 } \) and \(\frac{x-1}{-3}=\frac{y-4}{2}=\frac{z-5}{1}\) coplanar. Also, find the plane containing these lines.
4)
If the straight lines \(\frac { x-1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ 2 } \) and \(\frac { x-1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ \lambda } \) are coplanar, find λ and equations of the planes containing these two lines.
5)
Find the image of the point whose position vector is \(\hat { i } +2\hat { j } +3\hat { k } \) in the plane \(\vec { r } .(\hat { i } +2\hat { j } +4\hat { k } )\) = 38

12th Standard English Medium Maths Subject Creative 1 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________
2)
The system of linear equations x + y + z = 2, 2x + y - z = 3, 3x + 2y + kz = has a unique solution if __________
3)
If the system of equations x + 2y - 3x = 2, (k + 3) z = 3, (2k + 1) y + z = 2 is inconsistent then k is ___________
4)
If \(\rho\) (A) = \(\rho\) ([A/B]) = number of unknowns, then the system is _________--
5)
If \(\rho\) (A) = r then which of the following is correct?

12th Standard English Medium Maths Subject Creative 2 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.
2)
Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).
3)
Find k if the equations x + 2y + 2z = 0, x - 3y - 3z = 0, 2x + y + kz = 0 have only the trivial solution.
4)
Solve 6x - 7y = 16, 9x - 5y = 35 using (Cramer's rule).
5)
Find Re (z) and im (z) if z = 5i¹¹ + 7i³

12th Standard English Medium Maths Subject Creative 2 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If z₁ and z₂ are two complex numbers, such that |z₁| = Iz₂|, then is it necessary that z₁ = z₂?
2)
If (cosθ + i sinθ)² = x + iy, then show that x²+y² =1
3)
If z =\(\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 107 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 107 }\), then show that Im (z) = 0
4)
If sin ∝, cos ∝ are the roots of the equation ax²+ bx + c-0 (c ≠ 0), then prove that (n + c)²- b² + c²
5)
If \({ cot }^{ -1 }\left( \frac { 1 }{ 7 } \right) =\theta \) find the value of cos \(\theta \)

12th Standard English Medium Maths Subject Creative 3 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Solve: 3x+ay = 4, 2x + ay = 2, a ≠ 0 by Cramer's rule.
2)
Under what conditions will the rank of the matrix \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & h-2 & 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} h+2 \\ 3 \end{matrix} \end{matrix} \right] \) be less than 3?
3)
Solve: x + y + 3z = 4, 2x + 2y + 6z = 7, 2x + y + z = 10.
4)
Show that the complex numbers 3 + 2i, 5i, -3 + 2i and -i form a square.
5)
Solve: 2x+2^x-1+2^x-2= 7x+7^x-1+7^x-2

12th Standard English Medium Maths Subject Creative 3 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Verify that (A^-1)^T = (A^T)^-1 for A =\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).
2)
Find the principal value of -2i.
3)
Find the locus of z if Re\(\left( \frac { z+1 }{ z-i } \right) \) = 0 where z = x+iy.
4)
Find the locus of z if Re\(\\ \left( \frac { \bar { z } +1 }{ \bar { z } -i } \right) \) = 0.
5)
Find the number of real solutions of sin (e^x) -5^x + 5^-x

12th Standard English Medium Maths Subject Creative 5 Mark Questions with Solution Part - I updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Solve: \(\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z } \) = 2
2)
For what value of λ, the system of equations x + y + z = 1, x + 2y + 4z = λ, x + 4y + 10z = λ² is consistent.
3)
Using Gaussian Jordan method, find the values of λ and μ so that the system of equations 2x - 3y + 5z = 12, 3x + y + λz =μ, x - 7y + 8z = 17 has
(i) unique solution
(ii) infinite solutions and
(iii) no solution.
4)
Show that \(\left( \frac { i+\sqrt { 3 } }{ -i+\sqrt { 3 } } \right) ^{ 2\omega }+\left( \frac { i-\sqrt { 3 } }{ i+\sqrt { 3 } } \right) ^{ 2\omega }\) = -1
5)
Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]

12th Standard English Medium Maths Subject Creative 5 Mark Questions with Solution Part - II updated Creative Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c = 0.
2)
Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]
3)
Find all the roots \((2-2i)^{ \frac { 1 }{ 3 } }\) and also find the product of its roots.
4)
If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p.
5)
Solve: (2x² - 3x + 1) (2x² + 5x + 1) = 9x².

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 1 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The volume of a sphere is increasing in volume at the rate of 3 πcm³ / sec. The rate of change of its radius when radius is \(\frac { 1 }{ 2 } \) cm
2)
The abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?
3)
The slope of the line normal to the curve f(x) = 2cos 4x at \(x=\cfrac { \pi }{ 12 } \) is
4)
The function sin⁴ x + cos⁴x is increasing in the interval
5)
The number given by the Rolle's theorem for the functlon x³- 3x², x ∈ [0, 3] is

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 1 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
The function sin⁴ x + cos⁴x is increasing in the interval
2)
The number given by the Rolle's theorem for the functlon x³- 3x², x ∈ [0, 3] is
3)
The number given by the Mean value theorem for the function \(\frac { 1 }{ x } \), x ∈ [1, 9] is
4)
One of the closest points on the curve x² - y² = 4 to the point (6, 0) is
5)
The maximum product of two positive numbers, when their sum of the squares is 200, is

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 2 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the slope of the tangent to the curves at the respective given points.
x = a cos³ t, y = b sin³ t at t = \(\frac { \pi }{ 2 } \)
2)
Find the point on the curve y = x² − 5x + 4 at which the tangent is parallel to the line 3x + y = 7.
3)
A truck travels on a toll road with a speed limit of 80 km/hr. The truck completes a 164 km journey in 2 hours. At the end of the toll road the trucker is issued with a speed violation ticket. Justify this using the Mean Value Theorem.
4)
Suppose f(x) is a differentiable function for all x with f'(x) ≤ 29 and f(2) = 17. What is the maximum value of f(7)?
5)
Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals.
\(f(x)=|\frac{1}{x}|, x\in [-1,1]\)

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 2 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find two positive numbers whose sum is 12 and their product is maximum.
2)
The volume of a cylinder is given by the formula V = πr² h. Find the greatest and least values of V if r + h = 6.
3)
Evaluate the following limit, if necessary use l ’Hôpital Rule
\(\underset { x\rightarrow \infty }{ lim } \frac { x }{ logx } \)
4)
Evaluate the following limit, if necessary use l ’Hôpital Rule
\(\underset { x\rightarrow \frac { { \pi }^{ - } }{ 2 } }{ lim } \frac { secx }{ tanx } \)
5)
Prove that the function f (x) = x² − 2x − 3 is strictly increasing in \((2, \infty)\)

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 3 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
For the function f(x) = x², x∈ [0, 2] compute the average rate of changes in the subintervals [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2] and the instantaneous rate of changes at the points x = 0.5,1, 1.5, 2
2)
A particle moves so that the distance moved is according to the law s(t) = \(s(t)=\frac{t^{3}}{3}-t^{2}+3\). At what time the velocity and acceleration are zero.
3)
A particle is fired straight up from the ground to reach a height of s feet in t seconds, where s(t) = 128t −16t².
(1) Compute the maximum height of the particle reached.
(2) What is the velocity when the particle hits the ground?
4)
The price of a product is related to the number of units available (supply) by the equation Px + 3P −16x = 234, where P is the price of the product per unit in Rupees(Rs) and x is the number of units. Find the rate at which the price is changing with respect to time when 90 units are available and the supply is increasing at a rate of 15 units/week.
5)
A point moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t² + 3t metres.
(i) Find the average velocity of the points between t = 3 and t = 6 seconds.
(ii) Find the instantaneous velocities at t = 3 and t = 6 seconds.

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 3 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find the absolute extrem of the following function on the given closed interval
f(x) = x² -12x + 10; [1, 2]
2)
Show that the value in the conclusion of the mean value theorem for
f(x) = Ax²+ Bx + c on any interval [a, b] is \(\frac{a+b}{2}\)
3)
Write down the Taylor series expansion, of the function log x about x =1 upto three nonzero terms for x > 0.
4)
Expand sin x in ascending powers x - \(\frac{\pi}{4}\) upto three non-zero terms.
5)
Evaluate : \(\underset{x\rightarrow 1^{-}}{lim}(\frac{log(1-x)}{cot(\pi x)})\).

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 5 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s =16t² in t seconds
What is the average velocity with which the camera falls during the last 2 seconds?
2)
A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s =16t² in t seconds
3)
A particle moves along a line according to the law s(t) = 2t³ − 9t² +12t − 4, where t ≥ 0.
(i) At what times the particle changes direction?
(ii) Find the total distance travelled by the particle in the first 4 seconds.
(iii) Find the particle’s acceleration each time the velocity is zero.
4)
A conical water tank with vertex down of 12 metres height has a radius of 5 metres at the top. If water flows into the tank at a rate 10 cubic m/min, how fast is the depth of the water increases when the water is 8 metres deep?
5)
A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall.
(i) How fast is the top of the ladder moving down the wall?
(ii) At what rate, the area of the triangle formed by the ladder, wall and the floor is changing?

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 5 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
A hollow cone with base radius a cm and, height b em is placed on a table. Show that the volume of the largest cylinder that can be hidden underneath is \(\frac { 4 }{ 9 } \) times volume of the cone.
2)
Write the Maclaurin series expansion of the following function
tan^-1(x); -1 ≤ x ≤ 1
3)
Find the asymptotes of the following curve \(f(x)=\frac { { x }^{ 2 } }{ { x }^{ 2 }-1 } \)
4)
Evaluate the following limit, if necessary use l ’Hôpital Rule
\(\underset { x\rightarrow { \frac { \pi }{ 2 } } }{ lim } { \left( sinx \right) }^{ tanx }\)
5)
A steel plant is capable of producing x tonnes per day of a low-grade steel and y tonnes per day of a high-grade steel, where \(y=\frac { 40-5x }{ 10-x } \). If the fixed market price of low-grade steel is half that of high-grade steel, then what should be optimal productions in low-grade steel and high-grade steel in order to have maximum receipts.

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 1 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
A circular template has a radius of 10 cm. The measurement of radius has an approximate error of 0.02 cm. Then the percentage error in calculating area of this template is
2)
The percentage error of fifth root of 31 is approximately how many times the percentage error in 31?
3)
If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in our calculation of the volume is
4)
The change in the surface area S = 6x² of a cube when the edge length varies from x_o to x_o+ dx is
5)
If \(g(x, y)=3 x^{2}-5 y+2 y^{2}, x(t)=e^{t}\) and y(t) = cos t, then \(\frac{dg}{dt}\) is equal to

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 1 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If \(f(x)=\frac{x}{x+1}\), then its differential is given by
2)
If u(x, y) = x²+ 3xy + y - 2019, then \(\left.\frac{\partial u}{\partial x}\right|_{(4,-5)}\) is equal to
3)
Linear approximation for g(x) = cos x at \(x=\frac{\pi}{2}\) is
4)
If w (x, y, z) = x² (y - z) + y² (z - x) + z²(x - y), then \(\frac { { \partial }w }{ \partial x } +\frac { \partial w }{ \partial y } +\frac { \partial w }{ \partial z } \) is
5)
If f(x,y, z) = xy +yz +zx, then f_x - f_z is equal to

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 2 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
A sphere is made of ice having radius 10 cm. Its radius decreases from 10 cm to 9.8 cm. Find approximations for the following:
(i) change in the volume
(ii) change in the surface area
2)
The time T, taken for a complete oscillation of a single pendulum with length l, is given by the equation T = 2ㅠ\(\sqrt { \frac { 1 }{ g } } \), where g is a constant. Find the approximate percentage error in the calculated value of T corresponding to an error of 2 percent in the value of l.
3)
Show that the percentage error in the n^th root of a number is approximately \(\frac1n\) times the percentage error in the number.
4)
Let f, g : (a, b)→R be differentiable functions. Show that d(fg) = fdg + gdf
5)
Find df for f(x) = x²+ 3x and evaluate it for
x = 3 and dx = 0.02

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 2 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find df for f(x) = x²+ 3x and evaluate it for
x = 2 and dx = 0.1
2)
If U(x, y, z) = \(\frac { { x }^{ 2 }+{ y }^{ 2 } }{ xy } +3{ z }^{ 2 }y\), find \(\frac { \partial U }{ \partial x } ;\frac { \partial U }{ \partial y } \) and \(\frac { \partial U }{ \partial z } \)
3)
If w(x, y, z) = x² y + y²z + z²x, x, y, z∈R, find the differential dw .
4)
In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree.
\(U(x,y,z)=xy+sin\left( \frac { { y }^{ 2 }-2{ x }^{ 2 } }{ xy } \right) \)
5)
Show that F(x,y) = \(\frac { { x }^{ 2 }+5xy-10{ y }^{ 2 } }{ 3x+7y } \) is a homogeneous function of degree 1.

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 3 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Find a linear approximation for the following functions at the indicated points.
g(x) = \(g(x)=\sqrt { { x }^{ 2 }+9 } ,{ x }_{ 0 }=-4\)
2)
Find a linear approximation for the following functions at the indicated points.
\(h(x)=\frac{x}{x+1}, x_{0}=1\)
3)
The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. find the following in calculating the area of the circular plate:
Absolute error
4)
The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm.find the following in calculating the area of the circular plate:
Percentage error
5)
Find ∆f and df for the function f for the indicated values of x, ∆x and compare
(1) f(x) = x³ - 2x² ; x = 2, ∆ x = dx = 0.5
(2) f(x) = x² + 2x + 3; x = -0.5, ∆x = dx = 0.1

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 3 Mark Questions with Solution Part - II updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
Let f (x, y) = 0 if xy ≠ 0 and f (x, y) = 1 if xy = 0.
Calculate: \(\frac { \partial f }{ \partial x } (0,0),\frac { \partial f }{ \partial y } (0,0).\)
2)
Evaluate \(\begin{matrix} lim \\ (x,y)\rightarrow (0,0) \end{matrix}cos=\left( \frac { { e }^{ x }siny }{ y } \right) \), if the limit exists.
3)
For each of the following functions find the f_x, f_y and show that f_xy= f_yx
f(x, y) = cos (x² - 3xy)
4)
For each of the following functions find the g_xy, g_xx, g_yy and g_yx.
g(x, y) = xe^y + 3x²y
5)
For each of the following functions find the g_xy, g_xx, g_yy and g_yx.
g(x, y) = log (5x + 3y)

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 5 Mark Questions with Solution Part - I updated Book back Questions - by Question Bank Software - May 13, 2022 - View & Read

1)
If w(x, y) = xy + sin (xy), then prove that \(\frac { { \partial }^{ 2 }w }{ \partial y\partial x } =\frac { { \partial }^{ 2 }w }{ \partial x\partial y } \)
2)
If z(x, y) = x tan-¹ (xy), x = t², y = se^t, s, t ∈ R. Find \(\frac { \partial z }{ \partial s } \) and \(\frac { \partial z }{ \partial t } \) at s = t = 1
3)
Let U(x, y) = e^x sin y, where x = st², y = s² t, s, t ∈ R. Find \(\frac { \partial U }{ \partial s } ,\frac { \partial U }{ \partial t } \) and evaluate them at s = t = 1.
4)
W(x, y, z) = xy + yz + zx, x = u - v, y = uv, z = u + v, u ∈ R. Find \(\frac { \partial W }{ \partial u } ,\frac { \partial W }{ \partial v } \), and evaluate them at \(\left( \frac { 1 }{ 2 } ,1 \right) \)
5)
Prove that f(x, y) = x³- 2x²y + 3xy² + y³ is homogeneous; what is the degree? Verify Euler's Theorem for f.

12th Standard English Medium Maths Subject Creative 5 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c = 0.
2)
Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]
3)
Find all the roots \((2-2i)^{ \frac { 1 }{ 3 } }\) and also find the product of its roots.
4)
If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p.
5)
Solve: (2x² - 3x + 1) (2x² + 5x + 1) = 9x².

12th Standard English Medium Maths Subject Creative 5 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Solve: \(\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z } \) = 2
2)
For what value of λ, the system of equations x + y + z = 1, x + 2y + 4z = λ, x + 4y + 10z = λ² is consistent.
3)
Using Gaussian Jordan method, find the values of λ and μ so that the system of equations 2x - 3y + 5z = 12, 3x + y + λz =μ, x - 7y + 8z = 17 has
(i) unique solution
(ii) infinite solutions and
(iii) no solution.
4)
Show that \(\left( \frac { i+\sqrt { 3 } }{ -i+\sqrt { 3 } } \right) ^{ 2\omega }+\left( \frac { i-\sqrt { 3 } }{ i+\sqrt { 3 } } \right) ^{ 2\omega }\) = -1
5)
Verify that arg(1+i) + arg(1-i) = arg[(1+i) (1-i)]

12th Standard English Medium Maths Subject Creative 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Verify that (A^-1)^T = (A^T)^-1 for A =\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).
2)
Find the principal value of -2i.
3)
Find the locus of z if Re\(\left( \frac { z+1 }{ z-i } \right) \) = 0 where z = x+iy.
4)
Find the locus of z if Re\(\\ \left( \frac { \bar { z } +1 }{ \bar { z } -i } \right) \) = 0.
5)
Find the number of real solutions of sin (e^x) -5^x + 5^-x

12th Standard English Medium Maths Subject Creative 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Solve: 3x+ay = 4, 2x + ay = 2, a ≠ 0 by Cramer's rule.
2)
Under what conditions will the rank of the matrix \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & h-2 & 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} h+2 \\ 3 \end{matrix} \end{matrix} \right] \) be less than 3?
3)
Solve: x + y + 3z = 4, 2x + 2y + 6z = 7, 2x + y + z = 10.
4)
Show that the complex numbers 3 + 2i, 5i, -3 + 2i and -i form a square.
5)
Solve: 2x+2^x-1+2^x-2= 7x+7^x-1+7^x-2

12th Standard English Medium Maths Subject Creative 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If z₁ and z₂ are two complex numbers, such that |z₁| = Iz₂|, then is it necessary that z₁ = z₂?
2)
If (cosθ + i sinθ)² = x + iy, then show that x²+y² =1
3)
If z =\(\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 107 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 107 }\), then show that Im (z) = 0
4)
If sin ∝, cos ∝ are the roots of the equation ax²+ bx + c-0 (c ≠ 0), then prove that (n + c)²- b² + c²
5)
If \({ cot }^{ -1 }\left( \frac { 1 }{ 7 } \right) =\theta \) find the value of cos \(\theta \)

12th Standard English Medium Maths Subject Creative 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.
2)
Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).
3)
Find k if the equations x + 2y + 2z = 0, x - 3y - 3z = 0, 2x + y + kz = 0 have only the trivial solution.
4)
Solve 6x - 7y = 16, 9x - 5y = 35 using (Cramer's rule).
5)
Find Re (z) and im (z) if z = 5i¹¹ + 7i³

12th Standard English Medium Maths Subject Creative 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is ____________
2)
The system of linear equations x + y + z = 2, 2x + y - z = 3, 3x + 2y + kz = has a unique solution if __________
3)
If the system of equations x + 2y - 3x = 2, (k + 3) z = 3, (2k + 1) y + z = 2 is inconsistent then k is ___________
4)
If \(\rho\) (A) = \(\rho\) ([A/B]) = number of unknowns, then the system is _________--
5)
If \(\rho\) (A) = r then which of the following is correct?

12th Standard English Medium Maths Subject Creative 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then _____________
2)
If A^T is the transpose of a square matrix A, then ___________
3)
If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| = _______
4)
If A is a square matrix of order n, then |adj A| = ______________
5)
Every homogeneous system ______

12th Standard English Medium Maths Subject Discrete Mathematics Creative 5 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Construct the truth table for (p ∧ q) v r.

12th Standard English Medium Maths Subject Discrete Mathematics Creative 5 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Verify (p ∧ ~p) ∧ (~q ∧ p) is a tautlogy, contradiction or contingency.
2)
In (N, *) where * is defined by x * y = max (x, y) check the closure axion and identity anion.

12th Standard English Medium Maths Subject Discrete Mathematics Creative 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Let G = {1, i, -1, -i} under the binary operation multiplication. Find the inverse of all the elements.
2)
Construct the truth table for (-p) v (q ∧ r)

12th Standard English Medium Maths Subject Discrete Mathematics Creative 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
In (z, *) where * is defined by a * b = a^b, prove that * is not a binary operation on z.
2)
In (z, *) where * is defined as a * b = a + b + 2. Verify the commutative and associative axiom.

12th Standard English Medium Maths Subject Discrete Mathematics Creative 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Show that p v (q ∧ r) is a contingency.

12th Standard English Medium Maths Subject Discrete Mathematics Creative 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Show that p v (q ∧ r) is a contingency.
2)
Let S be the set of positive rational numbers and is defined by a * b = \(\frac{ab}{2}\). Then find the identity element and the inverse of 2.

12th Standard English Medium Maths Subject Discrete Mathematics Creative 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
The number of binary operations that can be defined on a set of 3 elements is _____________
2)
Which of the following is a contradiction?
3)
Define * on Z by a * b = a + b + 1 ∀ a,b \(\in \) Z. Then the identity element of z is ________
4)
If a * b = a²b² - ab then 3 * (1 * 1)
5)
Which of the following is a statement?

12th Standard English Medium Maths Subject Discrete Mathematics Creative 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If * is defined by a * b = a² + b² + ab + 1, then (2 * 3) * 2 is _____________
2)
The identity element of \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) \right\} \) |x \(\in \) R, x ≠ 0} under matrix multiplication is __________
3)
The identity element in the group {R - {1},x} where a * b = a + b - ab is __________
4)
A binary operation * is defined on the set of positive rational numbers Q⁺ by a*b = \(\frac { ab }{ 4 } \). Then 3 * \(\left( \frac { 1 }{ 5 } *\frac { 1 }{ 2 } \right) \) is _____________
5)
'-' is a binary operation on ___________

12th Standard English Medium Maths Subject Probability Distributions Creative 5 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

12th Standard English Medium Maths Subject Probability Distributions Creative 5 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

12th Standard English Medium Maths Subject Probability Distributions Creative 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

12th Standard English Medium Maths Subject Probability Distributions Creative 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

12th Standard English Medium Maths Subject Probability Distributions Creative 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

12th Standard English Medium Maths Subject Probability Distributions Creative 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

12th Standard English Medium Maths Subject Probability Distributions Creative 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If \(f(x)=\frac { 1 }{ 2 } \) ,\(E\left( { x }^{ 2 } \right) =\frac { 1 }{ 4 } \) then var(x) is _____________
2)
In a binomial distribution,\(n=4,P(X=0)=\frac { 16 }{ 81 } \),then \(P(X=4)\) _____________
3)
Var (2x ± 5) is =________
4)
A coin is tossed 3 times. The probability of getting exactly 2 heads is________
5)
If the mean and S.D of a binomial distribution are 12 and 2 respectively, then ___________

12th Standard English Medium Maths Subject Probability Distributions Creative 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
IfF(x) is the probability distribution function, then \(F\left( -\infty \right) \) is _____________
2)
If \(f(x)={ Cx }^{ 2 }={ cx }^{ 2 },0 is the p.d.f, of x then c is _____________
3)
In eight throws of a die, 1 or 3 is considered a success. Then the mean number of success is _____________
4)
In a binomial distribution, if the mean is 8 and the variance is 6, then the number of trials is _____________
5)
A die is tossed 5 times. Getting an odd number is considered a success. Then the variance of distribution of number of success is _____________

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 5 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
The surface area of a balloon being inflated changes at a constant rate. If initially, its radius 3 units and after 2 seconds it is 5 units, find the radius after t seconds.
2)
Solve: \(\frac { dy }{ dx } \) = (3x+2y+1)²
3)
Solve : \(\frac { dy }{ dx } =\left( { sin }^{ 2 }x{ cos }^{ 2 }x+{ xe }^{ x } \right) dx\)
4)
Solve : \(\left( 1+{ x }^{ 2 } \right) \frac { dy }{ dx } -x={ 2tan }^{ -1 }x\)
5)
Solve : (1+y²)(1 + log x)dx + x dy = 0, given that x = 1,y = 1.

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 5 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If cosx is an integrating factor of the differential equation \(\frac{dy}{dx}+Py= Q\), then P = ___________
2)
The I.F. of cosec x \(\frac{dy}{dx}+y\) sec²x = 0 is ___________
3)
The general solution of \(4\frac{d^2 y}{dx^2}\) + y = 0 is _________
4)
The I.F of \(\frac{dy}{dx}-y\) tan x = cos x is _________
5)
The differential equation corresponding to xy = c² where c is an arbitrary constant is ________.

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Solve: \(\frac{dy}{dx}+y=cos x\)
2)
Form the D.E of family of curves represented by y=c(x-c)².where c is the parameter.
3)
Find the D.E of all circles touching x-axis at the origin.
4)
Solve : \(\left( { x }^{ 2 }-1 \right) \frac { dy }{ dx } +2xy=\frac { 1 }{ { x }^{ 2 }-1 } \)
5)
Solve : ydx+(x-y²)dy=0

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Form the differential equation for y = e^-2x [A cos 3x-B sin 3x]
2)
Solve: x\(\frac{dy}{dx}\)+ 2y = x²
3)
Form the D.E of the family of curves c(y + c)²= x², where c is the parameter.
4)
Form the D.E to y²=a(b-x)(b+x) by eliminating a and b as its parameters.
5)
Find the D.E of all circles touching y-axis at the origin.

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Solve: \(\frac{dy}{dx}=1+e^{x-y}\)
2)
Solve: \(\frac{dy}{dx}+y=e^{-x}\)
3)
Find the order and degree of \(\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 2 }+cos\left( \frac { dy }{ dx } \right) =0\)
4)
Form the D.E corresponding to y=e^mx by eliminating 'm'.
5)
solve: x dy + y dx = xy dx

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
A curve passing through the origin has its slope e^x, Find the equation of the curve.
2)
Solve: x \(\frac{dy}{dx}=x+y\)
3)
Determine the order and degree of \(\frac { \left[ 1+\left( \frac { dy }{ dx } \right) ^{ 2 } \right] ^{ \frac { 3 }{ 2 } } }{ \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } } =k\)
4)
Solve :\(\frac { dy }{ dx } =\frac { 2x }{ { x }^{ 2 }+1 } \)
5)
Solve:\(\frac { dy }{ dx } +y=1\)

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
The solution of sec²x tan y dx + sec²y tan x dy = 0 is _________
2)
The transformation y = vx reduces \(\\ \\ \\ \frac { dy }{ dx } =\frac { x+y }{ 3x } \) __________
3)
The solution of \(\frac{dy}{dx}+y\) cot x = sin 2x is ___________
4)
The I.F of y log y \(\frac{dx}{dy}+x-log\ y=0\) is __________
5)
The order and degree of y'+(y")²= (x + t")² are _________.

12th Standard English Medium Maths Subject Ordinary Differential Equations Creative 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If cosx is an integrating factor of the differential equation \(\frac{dy}{dx}+Py= Q\), then P = ___________
2)
The I.F. of cosec x \(\frac{dy}{dx}+y\) sec²x = 0 is ___________
3)
The general solution of \(4\frac{d^2 y}{dx^2}\) + y = 0 is _________
4)
The I.F of \(\frac{dy}{dx}-y\) tan x = cos x is _________
5)
The solution of log \(\left( \frac { dy }{ dx } \right) \) = ax + by is______.

12th Standard English Medium Maths Subject Applications of Integration Creative 5 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Show that the area under the curve y = sin x and y = sin 2x between x = 0 and x = \(\frac { \pi }{ 3 } \) and x axis are as 2:3
2)
Find the volume of the solid generated by the revolution of the loop of the curve x = t² y = t - \(\frac { { t }^{ 3 } }{ 3 } \) about x-axis.
3)
Find the area bounded by the curves y=|x|-1 and y=-|x|+1
4)
Find the area bounded by the curve y2(2a-x)=x² and the line x=2a.
5)
Find the area enclosed by the parabolas 5x²-y=0 and 2x²-y+9=0.

12th Standard English Medium Maths Subject Applications of Integration Creative 5 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the area bounded by x = at², y = at between the ordinates corresponding to t = 1 and t = 2
2)
Using integration, find the area of the triangle with sides y = 2x + 1, y = 3x + 1 and x = 4.
3)
Find the area of the curve y²=(x-5)²(x-6) between
(i) x=5 and x=6
(ii) x=6 and x=7
4)
Find the ratio of the area between the curves y=cosx and y=cos2x and x- axis from x=0 to \(x=\frac { \pi }{ 3 } \)
5)
Find the value of ‘c’ for which the area bounded by the curve y=8x²-x⁵,the lines x=1,x=c and x-axis \(\frac { 16 }{ 3 } \)

12th Standard English Medium Maths Subject Applications of Integration Creative 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sin \ x }{ 9+{ cos }^{ 2 } } dx } \)
2)
Evaluate \(\int _{ 0 }^{ 1 }{ { xe }^{ -2x } } dx\)
3)
If \(\int _{ 0 }^{ \infty }{ \frac { dx }{ \left( { x }^{ 2 }+4 \right) \left( { x }^{ 2 }+9 \right) } } =k\pi \) then find the value of k.
4)
Evaluate \(\int _{ 0 }^{ \pi }{ { cos }^{ 3 }xdx } \)
5)
Evaluate \(\int _{ -2 }^{ 3 }{ \left| 1-{ x }^{ 2 } \right| } dx\)

12th Standard English Medium Maths Subject Applications of Integration Creative 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Evaluate \(\int _{ -1 }^{ 1 }{ log\left( \frac { 2-x }{ 2+x } \right) } dx\)
2)
Evaluate \(\int _{ 0 }^{ 1 }{ \sqrt { 9-4{ x }^{ 2 } } dx } \)
3)
Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { dx }{ 1+{ tan }^{ 3 }x } } \)
4)
Evaluate \(\int _{ 0 }^{ 1 }{ x\left( 1-x \right) ^{ 10 }dx } \)
5)
Evaluate \(\int _{ 0 }^{ 50 }{ \left[ x-\left| x \right| \right] dx } \)

12th Standard English Medium Maths Subject Applications of Integration Creative 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Evaluate \(\int _{ 1 }^{ 2 }{ \frac { 3x }{ { 9x }^{ 2 }-1 } dx } \)
2)
Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ 2x }cosxdx } \)
3)
Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ -x } } \)
4)
Evaluate \(\int _{ 0 }^{ 1 }{ \left( \frac { { e }^{ 5logx }-{ e }^{ 4logx } }{ { e }^{ 3logx }-{ e }^{ 2logx } } \right) } \)
5)
Find the area of the region bounded by the curves y=2^x.y=-2x-x² and the lines x=0 and x=2

12th Standard English Medium Maths Subject Applications of Integration Creative 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Evaluate \(\int _{ 0 }^{ 1 }{ \frac { { e }^{ x } }{ 1+{ e }^{ 2x } } dx } \)
2)
Find the area of the region enclosed by the curve y = \(\sqrt x\) + 1, the axis of x and the lines x = 0, x = 4.
3)
Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ log(tanx)dx } =0\)
4)
Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ 3x } } cosxdx\)
5)
Evaluate \(\int _{ 0 }^{ \infty }{ \left( { a }^{ -x }-{ b }^{ -x } \right) } dx\)

12th Standard English Medium Maths Subject Applications of Integration Creative 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If \(f(x)=\int_{0}^{x} t \cos t d t, \text { then } \frac{d f}{d x}=\)
2)
The value of \(\int _{ 0 }^{ \pi }{ { sin }^{ 4 }xdx } \) is
3)
If \(f(x)=\int_{1}^{x} \frac{e^{\sin u}}{u} d u, x>1 \text { and }\int_{1}^{3} \frac{e^{\sin x^{2}}}{x} d x=\frac{1}{2}[f(a)-f(1)]\), then one of the possible value of a is
4)
The value of \(\int _{ 0 }^{ a }{ { (\sqrt { { a }^{ 2 }-{ x }^{ 2 } } ) }^{ 3 } } dx\) is
5)
If \(\int _{ 0 }^{ x }{ f(t)dt=x+\int _{ x }^{ 1 }{ tf } (t)dt } \), then the value of f (1) is

12th Standard English Medium Maths Subject Applications of Integration Creative 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
The value of \(\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ \left( \frac { { 2x }^{ 7 }-{ 3x }^{ 5 }+{ 7x }^{ 3 }-x+1 }{ { cos }^{ 2 }x } \right) dx } \) is
2)
The area between y² = 4x and its latus rectum is
3)
The value of \(\int _{ 0 }^{ \pi }{ \frac { dx }{ 1+{ 5 }^{ cos\ x } } } \) is
4)
The value of \(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ { cos }^{ 3 }3x\ dx }\ is\)
5)
The value of \(\int _{ 0 }^{ \infty }{ { e }^{ -3x }{ x }^{ 2 }dx } \) is

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 5 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If u = tan^-1 \(\left( \frac { { x }^{ 3 }+{ y }^{ 3 } }{ x-y } \right) \) Prove that \(x\frac { \partial u }{ \partial x } +y\frac { \partial u }{ \partial y } \) sin 2u.
2)
Find the approximate value of \(\sqrt [ 3 ]{ 1.02 } +\sqrt { 1.02 } \)

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 5 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find \(\frac { \partial f }{ \partial x } ,\frac { \partial f }{ \partial y } ,\frac { { \partial }^{ 2 }f }{ \partial { x }^{ 2 } } ,\frac { { \partial }^{ 2 }f }{ { \partial y }^{ 2 } } \) at x = 2, y = 3 if f(x,y) = 2x² + 3y² - 2xy
2)
If V = log r and r² = x² +y² + z², then prove that \(\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } } +\frac { { \partial }^{ 2 } }{ \partial { z }^{ 2 } } =\frac { 1 }{ { r }^{ 2 } } \)
3)
Find \(\frac { \partial w }{ \partial u } ,\frac { \partial w }{ \partial v } \) if w=sin^-1(x,y) where x=u+v,y=u-v

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If w=xy+z and x=cot, y=sint, z=t then find \(\frac { dw }{ dt } \)
2)
Using linear approximation find \(\sqrt { 0.082 } \)
3)
If w=x²+y² and x=u²-v^2,y=2uv then find \(\frac { \partial w }{ \complement u } and\frac { \partial w }{ \partial v } \)
4)
Evaluate : \(\underset { \left( x,y,z \right) \rightarrow \left( -1,0,4 \right) }{ lim } \frac { { x }^{ 2 }-{ ze }^{ zy } }{ 6x+2y-2z } \)

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the approximate value of f (3.02) where f(x) = 3x² + 5x +3.
2)
If f = \(\frac { x }{ { x }^{ 2 }+{ y }^{ 2 } } \) then show that = \(x\frac { \partial f }{ \partial x } +y\frac { \partial f }{ \partial y } \) = -f
3)
Find the linear approximation to \(g(z)=\sqrt [ 4 ]{ zat } z=2\)
4)
If (x,y)=3x²+4y²+6xy-x²y²+5 then find
(i) fx(1,-1)
(ii) f_xy(1,1)
(iii)fxy(2,1)

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If w=log(x²+y²),x=cosθ,y=sinθ, find \(\frac { dw }{ d\theta } \)
2)
If w=xye^xy find \(\frac { { \partial }^{ 2 }u }{ \partial x\partial y } \)

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Use differentials to find \(\sqrt{25.2}\)
2)
If of f(x, y) = x² + y³ + 2xy² find f_xx, f_yy, f_xy and f_yx.
3)
If w=e^xy,x=at²,y=2at, find \(\frac { dw }{ dt } \)

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If log_e4 = 1.3868, then log_e4.01 = _____________
2)
The approximate value of (627)^\(\frac14\) is ................
3)
If y = sin x and x changes from \(\frac{\pi}{2}\) to ㅠ the approximate change in y is ..............
4)
If u = sin-1 \(\left( \frac { { x }^{ 4 }+{ y }^{ 4 } }{ { x }^{ 2 }+{ y }^{ 2 } } \right) \) and f = sin u then f is a homogeneous function of degree ..................
5)
The percentage error in the 11^th root of the number 28 is approximately .......... times the percentage error in 28.

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Creative 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If the radius of the sphere is measured as 9 cm with an error of 0.03 cm, the approximate error in calculating its volume is _____________
2)
If u = log (x³ + y³+ z³ - 3xyz) then \(\frac { { \partial }u }{ \partial { x } } +\frac { { \partial }u }{ { \partial y } }+ \frac { { \partial }u }{ \partial z } \) = _____________
3)
If f(x, y) = 2x²- 3xy + 5y + 7 then f(0, 0) and f(1, 1) is _____________
4)
If x = r cos θ, y = r sin, then \(\frac { \partial r }{ \partial x } \) = ....................
5)
If is a homogeneous function of x and y of degree n, then \(x\frac { { \partial }^{ 2 }u }{ \partial { x }^{ 2 } } +y\frac { { \partial }^{ 2 }u }{ \partial x\partial y } \) = ...... \(\frac { { \partial }u }{ \partial { x } } \)

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 5 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the intervals of concavity and points of inflexion for f(x)=x³-15x²+75x-50.
2)
Find the local maximum and local minimum values of f(x)=x⁴-3x+3x²-x.

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 5 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the angle of intersection of the curves 2y²= x³ and y² = 32x.
2)
Prove that the semi-vertical angle of a cone of maximum volume and of given slant height is tan^-1(\(\sqrt { 2 } \)).
3)
Sand is pouring from a pipe at the rate of 12 cm³/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing, when the height is 4 cm?
4)
A water tank has a shape of an inverted cone with its axis vertical and vertex lower most. Its semi vertical angle is tan⁻¹(0.5). Water is poured into it at a constant rate of 5 cm3/hr. Find the rate at which the level of the water is rising at that instant when the depth of the water is 4 m.

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Prove that \(\frac { x }{ 1+x } \) < < log (1+ x) for x > 0.
2)
Verify LMV theorem for f(x) = x³ - 2x² - x + 3 in [0, 1].
3)
The volume of a cube is increasing at the rate of 8cm³/s.How fast is the surface area increasing when the length of an edge is 12cm?
4)
Find the angle between two curves 2x²+y²=20 and x²-4y²+8=0
5)
Write down the Taylor series expansion of the function cos x in ascending powers \(x-\frac { \pi }{ 4 } \) upto three non-zero terms.

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
A ball is thrown vertically upwards, moves according to the law s = 13.8 t - 4.9 t² where s
is in metres and t is in seconds.
(i) Find the acceleration at t = 1
(ii) Find velocity at t = 1
(iii) Find the maximum height reached by the ball?
2)
The side of a square is equal to the diameter of a circle. If the side and radius change at the same rate then find the ratio of the change of their areas.
3)
Find the equation of normal to the curve y⁴=ax²at(a,a)
4)
A cylindrical hole 4 mm in diameter and 12 mm deep in a metal block is reboared to increase the diameter to 4.12mm. Estimate the amount of metal removed
5)
Obtain Maclaurin’s series for \(\frac { 1 }{ 1+x } \)

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the intervals of increasing and decreasing function for f(x) = x³ + 2x² - 1.
2)
Find the point on the parabola y²=18x at which the ordinate increases at twice the rate of the abscissa.
3)
Find the equation of the tangent to the curve y²=4x+5 and which is parallel to y=2x+7
4)
Verify Lagrange’s Mean Value theorem for \(f(x)=\sqrt { x-2 } \) in the interva [2,6]
5)
Expand the polynomial f(x)=x²-3x+2 in power of (x-2)

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the maximum and minimum values of f(x) = |x+3| ∀ \(x\in R\).
2)
Find the point at which the curve y-e^xy+x=0 has a vertical tangent.
3)
Using Rolle’s theorem find the value of c for f(x) = sin x in[0,2π]
4)
Obtain Maclaurin’s Series expansion for e^2x.
5)
Evaluate the following limits, if necessary using L’Hopitalrule
(i) \(\underset { x\rightarrow 2 }{ lim } \cfrac { sin\pi x }{ 2-x } \)
(ii) \(\cfrac { lim }{ x\rightarrow 2 } \cfrac { { x }^{ n }-{ a }^{ n } }{ x-2 } \)
(iii) \(\underset { x\rightarrow \infty }{ lim } \cfrac { sin\frac { 2 }{ x } }{ \frac { 1 }{ x } } \)
(iv) \(\underset { x\rightarrow \infty }{ lim } \cfrac { { x }^{ 2 } }{ { e }^{ x } } \)

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If a particle moves in a straight line according to s = t³-6t²-15t, the time interval during which the velocity is negative and acceleration is positive is __________
2)
The equation of the tangent to the curve x = t cost, y = t sin t at the origin is __________
3)
The function -3x+12 is ________ function on R.
4)
\(\underset { x\rightarrow 0 }{ lim } \frac { x }{ tanx } \) is _________
5)
The statement "If f has a local extremum at c and if f'(c) exists then f'(c) = 0" is ________

12th Standard English Medium Maths Subject Application of Differential Calculus Creative 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
The law of linear motion of a particle is given s = \(\frac{1}{3}\) t³-16t, the acceleration at the time when the velocity vanishes is __________
2)
Equation of the normal to the curve y = 2x²+3 sin x at x = 0 is __________
3)
The value of \(\underset { x\rightarrow \infty }{ lim } { e }^{ -x }\) is __________
4)
The critical points of the function f(x) = \((x-2)^{ \frac { 2 }{ 3 } }(2x+1)\) are __________
5)
In LMV theorem, we have f'(x₁) = \(\frac { f(b)-f(a) }{ b-a } \) then a < x₁ _________

12th Standard English Medium Maths Subject Applications of Vector Algebra Creative 5 Mark Questions with Solution - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Show that the points A, B, C with position vector \(2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\overset { \wedge }{ k } ,\overset { \wedge }{ i } -3\overset { \wedge }{ j } -5\overset { \wedge }{ k } \) and \(3\overset { \wedge }{ i } -4\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) respectively are the vector of a right angled, triangle. Also, find the remaining angles of the triangle.
2)
ABCD is a quadrilateral with \(\overset { \rightarrow }{ AB } =\overset { \rightarrow }{ \alpha } \) and \(\overset { \rightarrow }{ AD } =\overset { \rightarrow }{ \beta } \) and \(\overset { \rightarrow }{ AC } =2\overset { \rightarrow }{ \alpha } +3\overset { \rightarrow }{ \beta } \). If the area of the quadrilateral is λ times the area of the parallelogram with \(\overset { \rightarrow }{ AB } \) and \(\overset { \rightarrow }{ AD } \) as adjacent sides, then prove that \(\lambda =\frac { 5 }{ 2 } \)
3)
If \(\left| \overset { \rightarrow }{ A } \right| =\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \wedge }{ i } =\overset { \wedge }{ j } -\overset { \wedge }{ k } \) are two given vector, then find a vector B satisfying the equations \(\overset { \rightarrow }{ A } \times \overset { \rightarrow }{ B } \)= \(\overset { \rightarrow }{ C } \) and \(\overset { \rightarrow }{ A } \).\(\overset { \rightarrow }{ B } \) = 3
4)
Find the shortest distance between the following pairs of lines \(\frac { x-3 }{ 3 } =\frac { y-8 }{ -1 } =\frac { z-3 }{ 1 } \)and \(\frac { x+3 }{ -3 } =\frac { y+7 }{ 2 } =\frac { z-6 }{ 4 } \)
5)
Find the vector and Cartesian equation of the plane passing through the point (1,1, -1) and perpendicular to the planes x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0

12th Standard English Medium Maths Subject Applications of Vector Algebra Creative 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Dot product of a vector with vector \(\overset { \wedge }{ 3i } -5\overset { \wedge }{ k } \), \(2\overset { \wedge }{ i } +7\overset { \wedge }{ j } \) and \(\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) are respectively -1, 6 and 5. Find the vector.
2)
Find the equation of the plane through the intersection of the planes 2x-3y+ z-4 -0 and x - y + z + 1 = 0 and perpendicular to the plane x + 2y - 3z + 6 = 0
3)
If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } -\overset { \wedge }{ j } ,\overset { \rightarrow }{ b } =\overset { \wedge }{ j } -\overset { \wedge }{ k } ,\overset { \rightarrow }{ c } =\overset { \wedge }{ k } -\overset { \wedge }{ i } \) then find \(\left[ \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ b } -\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } -\overset { \rightarrow }{ a } \right] \)
4)
Prove by vector method, that in a right angled triangle the square of the hypotenuse is equal to the sum of the square of the other two sides.
5)
Show that the four points whose position vectors are \(6\overset { \wedge }{ i } -7\overset { \wedge }{ j } ,16\overset { \wedge }{ i } -29\overset { \wedge }{ j } -4\overset { \wedge }{ k } ,3\overset { \wedge }{ i } -6\overset { \wedge }{ j } \) are co-planar

12th Standard English Medium Maths Subject Applications of Vector Algebra Creative 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the Cartesian form of the equation of the plane \(\overset { \rightarrow }{ r } =\left( s-2t \right) \overset { \wedge }{ i } +\left( 3-t \right) \overset { \wedge }{ j } +\left( 2s+t \right) \overset { \wedge }{ k } \)
2)
Find the angle between the line \(\frac { x-2 }{ 3 } =\frac { y-1 }{ -1 } =\frac { z-3 }{ 2 } \) and the plane 3x + 4y + z + 5 = 0
3)
Prove that \(\left[ \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } \right] \)=\(\left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] \)
4)
If \(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } =0\) then show that \(\overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } =\overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } =\overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \)
5)
Show that the lines \(\frac { x-1 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-1 }{ 5 } \) and \(\frac { x+2 }{ 4 } =\frac { y-1 }{ 3 } =\frac { z+1 }{ -2 } \) do not intersect

12th Standard English Medium Maths Subject Applications of Vector Algebra Creative 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k } \), \(\overset { \rightarrow }{ b } =-\overset { \wedge }{ i } +2\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ c } =3\overset { \wedge }{ i } +\overset { \wedge }{ j } \) find \(\frac { \lambda }{ c } \) such that \(\overset { \rightarrow }{ a } +\lambda \overset { \rightarrow }{ b } \) is perpendicular to \(\overset { \rightarrow }{ c } \)
2)
Find the area of the triangle whose vertices are A(3, -1, 2) B(1, -1, -3) and C(4, -3, 1)
3)
Find the Cartesian equation of a line passing through the points A(2, -1, 3) and B(4, 2, 1)
4)
Find the parametric form of vector equation of the plane passing through the point (1, -1, 2) having 2, 3, 3 as direction ratios of normal to the plane.
5)
Find the equation of the plane containing the line of intersection of the planes x + y + Z - 6 = 0 and 2x + 3y + 4z + 5 = 0 and passing through the point (1, 1, 1)

12th Standard English Medium Maths Subject Applications of Vector Algebra Creative 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
A force of magnitude 6 units acting parallel to \(\overset { \wedge }{ 2i } -\overset { \wedge }{ 2j } +\overset { \wedge }{ k } \) displaces the point of application from (1, 2, 3) to (5, 3, 7). Find the work done.
2)
Forces \(2\overset { \wedge }{ i } +7\overset { \wedge }{ j } \), \(2\overset { \wedge }{ i } -5\overset { \wedge }{ j } +6\overset { \wedge }{ k } \), \(-\overset { \wedge }{ i } +2\overset { \wedge }{ j } -\overset { \wedge }{ k } \) act at a point P whose position vector is \(4\overset { \wedge }{ i } -3\overset { \wedge }{ j } -2\overset { \wedge }{ k } \). Find the vector moment of the resultant of these forces acting at P about this point Q whose position vector is \(6\overset { \wedge }{ i } +\overset { \wedge }{ j } -3\overset { \wedge }{ k } \)
3)
Find the parametric form of vector equation of a line passing through a point (2, -1, 3) and parallel to line \({ \overset { \rightarrow }{ r } }=\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) +t\left( 2\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \right) \)
4)
If the planes \({ \overset { \rightarrow }{ r } }.\left( \overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k } \right) =7\) and \({ \overset { \rightarrow }{ r } }.\left( \lambda \overset { \wedge }{ i } +2\overset { \wedge }{ j } -7\overset { \wedge }{ k } \right) =26\) are perpendicular. Find the value of λ.
5)
Let \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) be unit vectors such \(\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } =\overset { \rightarrow }{ a } .\overset { \rightarrow }{ c } =0\) and the angle between \(\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \) is \(\frac { \pi }{ 6 } \). Prove that \(\overset { \rightarrow }{ a } =\pm 2\left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) \)

12th Standard English Medium Maths Subject Applications of Vector Algebra Creative 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
The vector, d\(\overset { \wedge }{ i } +\overset { \wedge }{ j } +2\overset { \wedge }{ k } ,\overset { \wedge }{ i } +\lambda \overset { \wedge }{ j } -\overset { \wedge }{ k } \) t and \(2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\lambda \overset { \wedge }{ k } \) are co-planar if _____________
2)
If \(\left| \overset { \rightarrow }{ a } \right| =\left| \overset { \rightarrow }{ b } \right| =1\)such that \(\overset { \rightarrow }{ a } +2\overset { \rightarrow }{ b } \) and \(5\overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \) are perpendicular to each other, then the angle between \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) is _______________
3)
The p.v, OP of a point P make angles 60^o and 45^owith X and Y axis respectively. The angle of inclination of \(\overset { \rightarrow }{ OP } \) with z-axis is ___________
4)
The value of \({ \left| \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right| }^{ 2 }+{ \left| \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \right| }^{ 2 }\) is _____________
5)
The two planes 3x + 3y - 3z - 1 = 0 and x + y - z + 5 = 0 are _____________

12th Standard English Medium Maths Subject Applications of Vector Algebra Creative 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Let \(\overset { \rightarrow }{ a } \), \(\overset { \rightarrow }{ b } \)and \(\overset { \rightarrow }{ c } \)be three non- coplanar vectors and let \(\overset { \rightarrow }{ p } ,\overset { \rightarrow }{ q } ,\overset { \rightarrow }{ r } \) be the vectors defined by the relations \(\overset { \rightarrow }{ P } =\frac { \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ q } =\frac { \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ r } =\frac { \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } \) Then the value of \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) .\overset { \rightarrow }{ p } +\left( \overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) .\overset { \rightarrow }{ q } +\left( \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \right) .\overset { \rightarrow }{ r } \)= ____________
2)
The volume of the parallelepiped whose sides are given by \(\overset { \rightarrow }{ OA } =2\overset { \wedge }{ i } -3\overset { \wedge }{ j } \), \(\overset { \rightarrow }{ OB } =\overset { \wedge }{ i } +\overset { \wedge }{ j } -\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ OC } =3\overset { \wedge }{ i } -\overset { \wedge }{ k } \) is _____________
3)
The angle between the vector \(3\overset { \wedge }{ i } +4\overset { \wedge }{ j } +\overset { \wedge }{ 5k } \) and the z-axis is ___________
4)
The value of \({ \left| \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right| }^{ 2 }+{ \left| \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \right| }^{ 2 }\) is _____________
5)
The straight lines \(\frac { x-3 }{ 2 } =\frac { y+5 }{ 4 } =\frac { z-1 }{ -13 } \) and \(\frac { x+1 }{ 3 } =\frac { y-4 }{ 5 } =\frac { z+2 }{ 2 } \) are _____________

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Creative 5 Mark Questions with Solution - by Question Bank Software - Jun 19, 2021 - View & Read

1)
An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?
2)
An equilateral triangle is inscribed in the parabola y² = 4ax whose vertex is at the vertex of the parabola. Find the length of its side.
3)
The girder of a railway bridge is a parabola with its vertex at the highest point 15 m above the ends. If the span is 120 m, find the height of the bridge at 24 m from the middle point.
4)
The foci of a hyperbola coincides with the foci of the ellipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { y^{ 2 } }{ 9 } =1\). Find the equation of the hyperbola if its eccentricity is 2.
5)
A kho-kho player In a practice session while running realises that the sum of tne distances from the two kho-kho poles from him is always 8m. Find the equation of the path traced by him of the distance between the poles is 6m.

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Creative 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the circumference and area of the circle x² +y² - 2x + 5y + 7 = 0
2)
Find the condition for the line lx + my + n = 0 is tangent to the circle x² + y² = a²
3)
Find the equation of the ellipse whose e = \(\frac34\), foci ony-axis, centre at origin and passing through (6, 4).
4)
Find the equation of the hyperbola whose conjugate axis is 5 and the distance between the foci is 13.
5)
Find the value of c if y = x + c is a tangent to the hyperbola 9x² - 16y² = 144.

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Creative 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the value of p so that 3x + 4y - p = 0 is a tangent to the circle x² +y²- 64 = 0.
2)
Find the area of th triangle found by the lines Joining the vertex of the parabola x² = -36y to the ends of the latus rectum.
3)
Find the equation of the ellipse whose latus rectum is 5 and e = \(\frac { 2 }{ 3 } \)
4)
For the hyperbola 3x² - 6y² = -18, find the length of transverse and conjugate axes and eccentricity.
5)
Show that the line x + y + 1 = 0 touches the hyperbola \(\frac { { x }^{ 2 } }{ 16 } -\frac { { y }^{ 2 } }{ 15 } \) = 1 and find the co-ordinates of the point of contact

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Creative 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the equation of tangent to the circle x² +y² + 2x - 3y - 8 = 0 at (2, 3).
2)
Find the equation of the parabola with vertex at the origin, passing through (2, -3) and symmetric about x-axis
3)
If the line y = 3x + 1, touches the parabola y² = 4ax, find the length of the latus rectum?
4)
For the ellipse x² + 3y² = a², find the length of major and minor axis.
5)
Find the eccentricity of the hyperbola with foci on the x-axis if the length of its conjugate axis is \({ \left( \frac { 3 }{ 4 } \right) }^{ th }\) of the length of its tranverse axis.

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Creative 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the length of the tangent from (2, -3) to the circle x² + y²- 8x - 9y + 12 = 0.
2)
If a parabolic reflector is 24 cm in diameter and 6 cm deep, find its locus.
3)
Find the locus of a point which divides so that the sum of its distances from (-4, 0) and (4, 0) is 10 units.
4)
Find the eccentricity of the ellipse with foci on x-axis if its latus rectum be equal to one half of its major axis.
5)
Find the equation of the hyperbola whose vertices are (0, ±7) and e = \(\frac { 4 }{ 3 } \)

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Creative 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If (0, 4) and (0, 2) are the vertex and focus of a parabola then its equation is ___________
2)
Equation of tangent at (-4, -4) on x² = -4y is _____________
3)
The length of the latus rectum of the ellipse \(\frac { { x }^{ 2 } }{ 36 } +\frac { { y }^{ 2 } }{ 49 } \) = 1 is __________
4)
In an ellipse, the distance between its foci is 6 and its minor axis is 8, then e is ________
5)
The auxiliary circle of the ellipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { { y }^{ 2 } }{ 16 } \) = 1 is __________

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Creative 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
The equation of the directrix of the parabola y²+ 4y + 4x + 2 = 0 is ___________
2)
y²- 2x - 2y + 5 = 0 is a _________
3)
The eccentricity of the ellipse 9x²+ 5y² - 30y = 0 is __________
4)
If the distance between the foci is 2 and the distance between the direction is 5, then the equation of the ellipse is __________
5)
The equation 7x²- 6\(\sqrt { 3 } \) xy + 13y² - 4\(\sqrt { 3 } \) x - 4y - 12 = 0 represents ____________

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Creative 5 Mark Questions with Solution - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the domain of the following functions
(i) f(x) = sin^-1(2x - 3)
(ii) f(x) = sin^-1x + cos x
2)
Write the function \(f(x)=\tan ^{-1} \sqrt{\frac{a-x}{a+x}}-a<x<a \) in the simplest form
3)
Simplify \({ sin }^{ -1 }\left( \frac { sinx+cosx }{ \sqrt { 2 } } \right) ,\frac { \pi }{ 4 }\)
4)
If \({ tan }^{ -1 }\left( \frac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right) =a\) than prove that x²= sin 2a
5)
Prove that \({ tan }^{ -1 }\left( \frac { 1-x }{ 1+x } \right) -{ tan }^{ -1 }\left( \frac { 1-y }{ 1+y } \right) ={ sin }^{ -1 }\left( \frac { y-x }{ \sqrt { 1+{ x }^{ 2 } } .\sqrt { 1+{ y }^{ 2 } } } \right)\)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Creative 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Prove that \({ cos }^{ -1 }\left( \frac { 4 }{ 5 } \right) +{ tan }^{ -1 }\left( \frac { 3 }{ 5 } \right) ={ tan }^{ -1 }\left( \frac { 27 }{ 11 } \right) \)
2)
Prove that \({ tan }^{ -1 }\left( \frac { m }{ n } \right) -{ tan }^{ -1 }\left( \frac { m-n }{ m+n } \right) =\frac { \pi }{ 4 } \)
3)
Solve \({ tan }^{ -1 }\left( \frac { 2x }{ 1-{ x }^{ 2 } } \right) +{ cot }^{ -1 }\left( \frac { 1-{ x }^{ 2 } }{ 2x } \right) =\frac { \pi }{ 3 } ,x>0\)
4)
Prove that \({ tan }^{ -1 }\sqrt { x } =\frac { 1 }{ 2 } { cos }^{ -1 }={ \frac { 1 }{ 2 } { cos }^{ -1 }\left( \frac { 1-x }{ 1+x } \right) ,x\in \left| 0,1 \right| }\)
5)
Find the real solutions of the equation
\({ tan }^{ -1 }\sqrt { x(x+1) } +{ sin }^{ -1 }\sqrt { { x }^{ 2 }+x+1 } =\frac { \pi }{ 2 } \)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Creative 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Evaluate \(cos\left[ { sin }^{ -1 }\frac { 3 }{ 5 } +{ sin }^{ -1 }\frac { 5 }{ 13 } \right] \)
2)
Solve: \({ tan }^{ -1 }\left( \cfrac { x-1 }{ x-2 } \right) +{ tan }^{ -1 }\left( \cfrac { x+1 }{ x+2 } \right) =\cfrac { \pi }{ 4 } \)
3)
If \(sin\left( { sin }^{ -1 }\frac { 1 }{ 5 } +{ cos }^{ -1 }x \right) =1\) then find the value ofx.
4)
Evaluate \(cos\left[ { cos }^{ -1 }\left( \frac { -\sqrt { 3 } }{ 2 } +\frac { \pi }{ 6 } \right) \right] \)
5)
Solve: cos(tan^-1x) = \(sin\left( { cot }^{ -1 }\frac { 3 }{ 4 } \right) \)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Creative 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the principal value of \({ tan }^{ -1 }\left( \frac { -1 }{ \sqrt { 3 } } \right) \)
2)
Find the principal value of \({ cos }^{ -1 }\left( \frac { -1 }{ 2 } \right) \)
3)
If \({ sin }^{ -1 }\left( \frac { 1 }{ 2 } \right) ={ tan }^{ -1 }x\) then find the value of x,
4)
Evaluate \(sin\left( { cos }^{ -1 }\left( \frac { 3 }{ 5 } \right) \right) \)
5)
Prove that \(2{ tan }^{ -1 }\left( \frac { 2 }{ 3 } \right) ={ tan }^{ -1 }\left( \frac { 12 }{ 5 } \right) \)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Creative 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the principal value of sin^-1(-1).
2)
If \({ cot }^{ -1 }\left( \frac { 1 }{ 7 } \right) =\theta \) find the value of cos \(\theta \)
3)
Prove that \({ tan }^{ -1 }\left( \frac { 1 }{ 7 } \right) +{ tan }^{ -1 }\left( \frac { 1 }{ 13 } \right) ={ tan }^{ -1 }\left( \frac { 2 }{ 9 } \right) \)
4)
Evaluate \(sin\left( \frac { 1 }{ 2 } { cos }^{ -1 }\frac { 4 }{ 5 } \right) \)
5)
Evaluate \(sin\left( { cos }^{ -1 }\left( \frac { 1 }{ 2 } \right) \right) \)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Creative 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If \({ tan }^{ -1 }\left\{ \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right\} =\alpha \) then x² = _____________
2)
\({ tan }^{ -1 }\left( \frac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \frac { 2 }{ 11 } \right) \) = ____________
3)
The value of \({ cos }^{ -1 }\left( \cos\cfrac { 5\pi }{ 3 } \right) +sin^{ -1 }\left( \sin\cfrac{5\pi }{ 3 } \right) \) is ______________
4)
If \({ cos }^{ -1 }x>x>{ sin }^{ -1 }x\) then _________
5)
The value of sin 2(tan^-1 0.75) is ___________

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Creative 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If \({ sin }^{ -1 }x-cos^{ -1 }x=\frac { \pi }{ 6 } \) then ___________
2)
If \(\alpha ={ tan }^{ -1 }\left( tan\frac { 5\pi }{ 4 } \right) \) and \(\beta ={ tan }^{ -1 }\left( -tan\frac { 2\pi }{ 3 } \right) \) then ___________
3)
If \(\alpha ={ tan }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2y-x } \right) ,\beta ={ tan }^{ -1 }\left( \frac { 2x-y }{ \sqrt { 3y } } \right) \) then \(\alpha -\beta \) __________
4)
If tan^-1(3) + tan^-1(x) = tan^-1(8) then x = ____________
5)
\(sin\left\{ 2{ cos }^{ -1 }\left( \frac { -3 }{ 5 } \right) \right\} =\) __________

12th Standard English Medium Maths Subject Theory of Equations Creative 5 Mark Questions with Solution - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If the sum of the roots of the quadratic equation ax²+ bx + c = 0 (abc ≠ 0) is equal to the sum of the squares of their reciprocals, then \(\frac { a }{ c } ,\frac { b }{ a } ,\frac { c }{ b } \) are H.P.
2)
If a, b, c, d and p are distinct non-zero real numbers such that (a²+b²+c²) p²-2 (ab+bc+cd) p+(b²+c²+d²)≤ 0 then prove that a, b, c, d are in G.P and ad = bc
3)
If c ≠ 0 and \(\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c } \) has two equal roots, then find p.
4)
If the equation x² + bx + ca = 0 and x² + cx + ab = 0 have a comnion root and b≠c, then prove that their roots will satisfy the equation x² + ax + bc = 0.
5)
Solve: (2x² - 3x + 1) (2x² + 5x + 1) = 9x².

12th Standard English Medium Maths Subject Theory of Equations Creative 3 Mark Questions with Solution - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the number of real solutions of sin (e^x) -5^x + 5^-x
2)
Find the number of positive integral solutions of (pairs of positive integers satisfying) x² - y² = 353702.
3)
Solve: 2x+2^x-1+2^x-2= 7x+7^x-1+7^x-2
4)
Solve: (x-1)⁴+(x-5)⁴= 82
5)
Solve: \({ (5+2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }+{ (5-2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }=10\)

12th Standard English Medium Maths Subject Theory of Equations Creative 2 Mark Questions with Solution - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If sin ∝, cos ∝ are the roots of the equation ax²+ bx + c-0 (c ≠ 0), then prove that (n + c)²- b² + c²
2)
Find value of a for which the sum of the squares of the equation x² - (a- 2) x - a -1 = 0 assumes the least value.
3)
Find the Interval for a for which 3x²+2(a²+1) x+(a²-3a+2) possesses roots of opposite sign.
4)
Find x If \(x=\sqrt { 2+\sqrt { 2+\sqrt { 2+....+upto\infty } } } \)
5)
Find the number of positive and negative roots of the equation x⁷ - 6x⁶ + 7x⁵ + 5x²+2x+2

12th Standard English Medium Maths Subject Theory of Equations Creative 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If a, b, c ∈ Q and p +√q (p, q ∈ Q) is an irrational root of ax²+bx+c = 0 then the other root is ___________
2)
If f(x) = 0 has n roots, then f'(x) = 0 has __________ roots
3)
If ∝, β, ૪ are the roots of the equation x³-3x+11 = 0, then ∝+β+૪ is __________.
4)
If x² - hx - 21 = 0 and x² - 3hx + 35 = 0 (h > 0) have a common root, then h = ___________
5)
If p(x) = ax² + bx + c and Q(x) = -ax² + dx + c where ac ≠ 0 then p(x). Q(x) = 0 has at least _______ real roots.

12th Standard English Medium Maths Subject Theory of Equations Creative 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
The quadratic equation whose roots are ∝ and β is ___________
2)
The equation \(\sqrt { x+1 } -\sqrt { x-1 } =\sqrt { 4x-1 } \) has ____________
3)
For real x, the equation \(\left| \frac { x }{ x-1 } \right| +|x|=\frac { { x }^{ 2 } }{ |x-1| } \) has ________
4)
If \((2+\sqrt{3})^{x^{2}-2 x+1}+(2-\sqrt{3})^{x^{2}-2 x-1}=\frac{2}{2-\sqrt{3}}\) then x = _________
5)
If ∝, β, ૪ are the roots of 9x³-7x+6 = 0, then ∝ β ૪ is __________

12th Standard English Medium Maths Subject Complex Numbers Creative 5 Mark Questions with Solution - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If 1, ω, ω² are the cube roots of unity then show that (1+5ω²+ω⁴) (1+5ω+ω²) (5+ω+ω⁵) = 64
2)
Show that \(\left( \frac { i+\sqrt { 3 } }{ -i+\sqrt { 3 } } \right) ^{ 2\omega }+\left( \frac { i-\sqrt { 3 } }{ i+\sqrt { 3 } } \right) ^{ 2\omega }\) = -1
3)
Verify that 2 arg(-1) ≠ arg(-1)²
4)
Find all the roots \((2-2i)^{ \frac { 1 }{ 3 } }\) and also find the product of its roots.
5)
Find the radius and centre of the circle \(z\bar { z } \)-(2+3i)z-(2-3i)\(\bar { z } \)+9 = 0 where z is a complex number.

12th Standard English Medium Maths Subject Complex Numbers Creative 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Explain the falacy:
2)
Find the principal value of -2i.
3)
Show that the complex numbers 3 + 2i, 5i, -3 + 2i and -i form a square.
4)
Find the locus of z if Re\(\left( \frac { z+1 }{ z-i } \right) \) = 0 where z = x+iy.
5)
Find the locus of z if Re\(\\ \left( \frac { \bar { z } +1 }{ \bar { z } -i } \right) \) = 0.

12th Standard English Medium Maths Subject Complex Numbers Creative 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find the circle roots of -27.
2)
Show that \(\left| \frac { z-3 }{ z+3 } \right| \) = 2 represent a circle.
3)
If the complex number 2 + i and 1-2i are equidistant from x + iy then show that x+3y = 0.
4)
Find the locus of z if |3z - 5| = 3 |z + 1| where z = x + iy.
5)
If \(\frac { (a+i)^{ 2 } }{ 2a-i } \) = p + iq, show that p²+q² = \(\frac { ({ a }^{ 2 }+i)^{ 2 } }{ 4a^{ 2 }+1 } \).

12th Standard English Medium Maths Subject Complex Numbers Creative 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If z₁ and z₂ are two complex numbers, such that |z₁| = Iz₂|, then is it necessary that z₁ = z₂?
2)
If (cosθ + i sinθ)² = x + iy, then show that x²+y² =1
3)
If z =\(\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 107 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 107 }\), then show that Im (z) = 0
4)
If 1, ω, ω² are the cube roots of unity show that (1+ω²)³ - (1+ω)³ = 0
5)
Find the modules of (1+ 3i)³

12th Standard English Medium Maths Subject Complex Numbers Creative 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Find Re (z) and im (z) if z = 5i¹¹ + 7i³
2)
If z₁ and z₂ are 1-i, -2+4i then find Im\(\left( \frac { { z }_{ 1 }{ z }_{ 2 } }{ \bar { { z }_{ 1 } } } \right) \).
3)
Find the value of the complex number (i²⁵)³.
4)
Find the argument of -2
5)
Find the values of the real number x and y if 3x + (2x - 3y) i = 6 + 3i⁹.

12th Standard English Medium Maths Subject Complex Numbers Creative 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
The value of (1+i) (1+i²) (1+i³) (1+i⁴) is ____________
2)
The amplitude of \(\frac{1}{i}\) is equal to _______
3)
The complex number z which satisfies the condition \(\left| \frac { 1+z }{ 1-z } \right| \) = 1 lies on _________
4)
\(\frac { 1+e^{ -i\theta } }{ 1+{ e }^{ i\theta } } \) =__________
5)
If ω is the cube root of unity, then the value of (1-ω) (1-ω²) (1-ω⁴) (1-ω⁸) is _________

12th Standard English Medium Maths Subject Complex Numbers Creative 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If \(\sqrt { a+ib } \) = x + iy, then possible value of \(\sqrt { a-ib }\) is ___________
2)
If z = cos\(\frac { \pi }{ 4 } \) + i sin\(\frac { \pi }{ 6 } \), then ______
3)
.If a = 3+i and z = 2-3i, then the points on the Argand diagram representing az, 3az and - az are _________
4)
If a = 3 + i and z = 2 - 3i, then the points on the Argand diagram representing az, 3az and - az are ___________
5)
If z = 1-cos θ + i sin θ, then |z| = _____________

12th Standard English Medium Maths Subject Application of Matrices and Determinants Creative 5 Mark Questions with Solution - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Using determinants; find the quadratic defined by f(x) = ax² + bx + c, if f(1) = 0, f(2) = -2 and f(3) = -6.
2)
Solve: \(\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z } \) = 2
3)
The sum of three numbers is 20. If we multiply the third number by 2 and add the first number to the result we get 23. By adding second and third numbers to 3 times the first number we get 46. Find the numbers using Cramer's rule.
4)
Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c = 0.
5)
Using Gaussian Jordan method, find the values of λ and μ so that the system of equations 2x - 3y + 5z = 12, 3x + y + λz =μ, x - 7y + 8z = 17 has
(i) unique solution
(ii) infinite solutions and
(iii) no solution.

12th Standard English Medium Maths Subject Application of Matrices and Determinants Creative 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
Solve: 2x + 3y = 10, x + 6y = 4 using Cramer's rule.
2)
Solve: 3x+ay = 4, 2x + ay = 2, a ≠ 0 by Cramer's rule.
3)
Under what conditions will the rank of the matrix \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & h-2 & 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} h+2 \\ 3 \end{matrix} \end{matrix} \right] \) be less than 3?
4)
Verify that (A^-1)^T = (A^T)^-1 for A =\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).
5)
Solve: x + y + 3z = 4, 2x + 2y + 6z = 7, 2x + y + z = 10.

12th Standard English Medium Maths Subject Application of Matrices and Determinants Creative 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
For what value of t will the system tx +3y - z = 1, x + 2y + z = 2, -tx + y + 2z = -1 fail to have unique solution?
2)
Verify (AB)^-1 = B^-1 A^-1 for A =\(\left[ \begin{matrix} 2 & 1 \\ 5 & 3 \end{matrix} \right] \) and B =\(\left[ \begin{matrix} 4 & 5 \\ 3 & 4 \end{matrix} \right] \).
3)
Find the rank of the matrix math \(\left[ \begin{matrix} 4 \\ -2 \\ 1 \end{matrix}\begin{matrix} 4 \\ 3 \\ 4 \end{matrix}\begin{matrix} 0 \\ -1 \\ 8 \end{matrix}\begin{matrix} 3 \\ 5 \\ 7 \end{matrix} \right] \).
4)
Solve 2x - 3y = 7, 4x - 6y = 14 by Gaussian Jordan method.
5)
If the rank of the matrix \(\left[ \begin{matrix} \lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda \end{matrix} \right] \) is 2, then find ⋋.

12th Standard English Medium Maths Subject Application of Matrices and Determinants Creative 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
For any 2 \(\times\) 2 matrix, if A (adj A) =\(\left[ \begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix} \right] \) then find |A|.
2)
If A is a square matrix such that A³ = I, then prove that A is non-singular.
3)
Find the rank of the matrix \(\left[ \begin{matrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{matrix} \right] \).
4)
Show that the equations 3x + y + 9z = 0, 3x + 2y + 12z = 0 and 2x + y + 7z = 0 have nontrivial solutions also.
5)
Solve : 2x - y = 3, 5x + y = 4 using matrices.

12th Standard English Medium Maths Subject Application of Matrices and Determinants Creative 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
For the matrix A, if A³ = I, then find A^-1.
2)
Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.
3)
Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).
4)
Find k if the equations x + 2y + 2z = 0, x - 3y - 3z = 0, 2x + y + kz = 0 have only the trivial solution.
5)
Solve 6x - 7y = 16, 9x - 5y = 35 using (Cramer's rule).

12th Standard English Medium Maths Subject Application of Matrices and Determinants Creative 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 19, 2021 - View & Read

1)
The system of linear equations x + y + z = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if _________
2)
Let A be a 3 \(\times\) 3 matrix and B its adjoint matrix If |B| = 64, then |A| = ___________
3)
The system of linear equations x + y + z = 2, 2x + y - z = 3, 3x + 2y + kz = has a unique solution if __________
4)
If A is a matrix of order m \(\times\) n, then \(\rho\) (A) is _________
5)
If \(\rho\) (A) = r then which of the following is correct?

12th Standard English Medium Maths Subject Application of Matrices and Determinants Creative 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 19, 2021 - View & Read

1)
If the system of equations x = cy + bz, y = az + cx and z = bx + ay has a non - trivial solution then _____________
2)
If A^T is the transpose of a square matrix A, then ___________
3)
If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| = _______
4)
If A is a square matrix of order n, then |adj A| = ______________
5)
If A =\(\left( \begin{matrix} cosx & sinx \\ -sinx & cosx \end{matrix} \right) \) and A(adj A) =\(\lambda \) \(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \) then \(\lambda \) is ________

12th Standard English Medium Maths Subject Book Back 5 Mark Questions with Solution Part -II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8 \end{matrix} \right] \)
2)
Determine k and solve the equation 2x³-6x²+3x+k = 0 if one of its roots is twice the sum of the other two roots.
3)
For the ellipse 4x²+ y²+ 24x − 2y + 21 = 0, find the centre, vertices and the foci. Also prove that the length of latus rectum is 2
4)
Parabolic cable of a 60m portion of the roadbed of a suspension bridge are positioned as shown below. Vertical Cables are to be spaced every 6m along this portion of the roadbed. Calculate the lengths of first two of these vertical cables from the vertex.
5)
Prove by vector method that the perpendiculars (attitudes) from the vertices to the opposite sides of a triangle are concurrent.

12th Standard English Medium Maths Subject Book Back 5 Mark Questions with Solution Part -I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
If A = \(\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right] \), find x and y such that A² + xA + yI₂ = O₂. Hence, find A^-1.
2)
Solve the following system of equations, using matrix inversion method:
2x₁ + 3x₂ + 3x₃ = 5, x₁ - 2x₂ + x₃ = -4, 3x₁ - x₂ - 2x₃ = 3.
3)
(a) If A = \(\left[ \begin{matrix} -5 & 1 & 3 \\ 7 & 1 & -5 \\ 1 & -1 & 1 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 1 & 2 \\ 3 & 2 & 1 \\ 2 & 1 & 3 \end{matrix} \right] \), find the products AB and BA and hence solve the system of equations x + y + 2z = 1, 3x + 2y + z = 7, 2x + y + 3z = 2.
4)
Investigate the values of λ and μ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y - 5z = 8, 2x + 3y + λz = μ, have
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions.
5)
Determine the values of λ for which the following system of equations (3λ − 8)x + 3y + 3z = 0, 3x + (3λ − 8)y + 3z = 0, 3x + 3y + (3λ − 8)z = 0. has a non-trivial solution.

12th Standard English Medium Maths Subject Book Back 3 Mark Questions with Solution Part -II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Solve the equations
x⁴+ 3x³- 3x - 1 = 0
2)
Evaluate :\(\int _{ 0 }^{ 1 }{ \frac { 2x+7 }{ { 5x }^{ 2 }+9 } } dx\)
3)
Evaluate: \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { cos\theta }{ (1+sin\theta )(2+sin\theta ) } } d\theta \)
4)
Show that \(\int _{ 0 }^{ \pi }{ g(sinx)dx=2 } \int _{ 0 }^{ \frac { \pi }{ 2 } }{ g(sinx)dx, } \) where g(sin x) is a function of sin x.
5)
Evaluate \(\int _{ 0 }^{ x }{ { x }^{ 2 } } \)cos nx dx, where n is a positive integer.

12th Standard English Medium Maths Subject Book Back 2 Mark Questions with Solution Part -II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
If A = \(\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \) is non-singular, find A⁻¹.
2)
Find the rank of the matrix \(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{matrix} \right] \) by reducing it to a row-echelon form.
3)
Find the following \(\left| \frac { 2+i }{ -1+2i } \right| \)
4)
If \(\omega \neq 1\) is a cube root of unity, then the show that \(\cfrac { a+b\omega +c{ \omega }^{ 2 } }{ b+c\omega +{ a\omega }^{ 2 } } +\cfrac { a+b\omega +{ c\omega }^{ 2 } }{ c+a\omega +b{ \omega }^{ 2 } } =-1\)
5)
Find the square roots of −6+8i

12th Standard English Medium Maths Subject Book Back 3 Mark Questions with Solution Part -I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find the inverse of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \end{matrix} \right] \).
2)
If A = \(\left[ \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} \right] \), show that A^-1 = \(\frac {1}{2}\) (A² - 3I).
3)
Find the adjoint of the following:
\(\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right] \)
4)
Prove the following properties z is real if and only if z = \(\bar { z } \)
5)
Show that \(\left( 2+i\sqrt { 3 } \right) ^{ 10 }-\left( 2-i\sqrt { 3 } \right) ^{ 10 }\) is purely imaginary

12th Standard English Medium Maths Subject Book Back 2 Mark Questions with Solution Part -I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
If A is a non-singular matrix of odd order, prove that |adj A| is positive
2)
Find the rank of each of the following matrices:
\(\left[ \begin{matrix} 3 & 2 & 5 \\ 1 & 1 & 2 \\ 3 & 3 & 6 \end{matrix} \right] \)
3)
If z₁= 1 - 3i, z₂= - 4i, and z₃ = 5 , show that (z₁+ z₂) + z₃= z₁+ (z₂+ z₃)
4)
Simplify \(\left( \frac { 1+i }{ 1-i } \right) ^{ 3 }-\left( \frac { 1-i }{ 1+i } \right) ^{ 3 }\) into rectangular form
5)
Construct a cubic equation with roots 1, 2 and 3

12th Standard English Medium Maths Subject Book Back 1 Mark Questions with Solution Part -II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
If x < 0, y < 0 such that xy = 1, then tan^-1(x) + tan^-1(y) =_____
2)
The distance between the foci of a hyperbola is 16 and e = \(\sqrt { 2 } \). Its equation is ____________
3)
If B, B¹ are the ends of minor axis, F₁, F₂ are foci of the ellipse \(\frac { { x }^{ 2 } }{ 8 } +\frac { { y }^{ 2 } }{ 4 } \) = 1 then area of F₁BF₂B¹ is __________
4)
Let \(\overset { \rightarrow }{ a } \), \(\overset { \rightarrow }{ b } \)and \(\overset { \rightarrow }{ c } \)be three non- coplanar vectors and let \(\overset { \rightarrow }{ p } ,\overset { \rightarrow }{ q } ,\overset { \rightarrow }{ r } \) be the vectors defined by the relations \(\overset { \rightarrow }{ P } =\frac { \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ q } =\frac { \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ r } =\frac { \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } \) Then the value of \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) .\overset { \rightarrow }{ p } +\left( \overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) .\overset { \rightarrow }{ q } +\left( \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \right) .\overset { \rightarrow }{ r } \)= ____________
5)
The number of vectors of unit length perpendicular to the vectors \(\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) \) and \(\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right) \)is __________

12th Standard English Medium Maths Subject Book Back 1 Mark Questions with Solution Part -I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
If |adj(adj A)| = |A|⁹, then the order of the square matrix A is
2)
If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is
3)
If A = \(\left[ \begin{matrix} \cos { \theta } & \sin { \theta } \\ -\sin { \theta } & \cos { \theta } \end{matrix} \right] \) and A(adj A) = \(\left[ \begin{matrix} k & 0 \\ 0 & k \end{matrix} \right] \), then k =
4)
If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| = _______
5)
If z = cos\(\frac { \pi }{ 4 } \) + i sin\(\frac { \pi }{ 6 } \), then ______

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 5 Mark Questions with Solution Part -II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Using the equivalence property, show that p ↔️ q ≡ ( p ∧ q) v (ㄱp ∧ ㄱq)
2)
Let M = \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\} \) and let * be the matrix multiplication. Determine whether M is closed under ∗. If so, examine the commutative and associative properties satisfied by ∗ on M.
3)
Let M = \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\} \) and let ∗ be the matrix multiplication. Determine whether M is closed under ∗ . If so, examine the existence of identity, existence of inverse properties for the operation ∗ on M.
4)
Using truth table check whether the statements ¬(p V q) V (¬p ∧ q) and ¬p are logically equivalent.
5)
Prove p⟶(q⟶r) ☰ (p ∧ q)⟶r without using truth table.

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 5 Mark Questions with Solution Part -I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Using the equivalence property, show that p ↔️ q ≡ ( p ∧ q) v (ㄱp ∧ ㄱq)
2)
Verify whether the following compound propositions are tautologies or contradictions or contingency
((p⟶ q) ∧ (q ⟶ r)) ⟶ (p ⟶ r)
3)
Let M = \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\} \) and let * be the matrix multiplication. Determine whether M is closed under ∗. If so, examine the commutative and associative properties satisfied by ∗ on M.
4)
Let A be Q\{1}. Define ∗ on A by x*y = x + y − xy. Is ∗ binary on A? If so, examine the commutative and associative properties satisfied by ∗ on A.
5)
Check whether the statement p➝(q➝p) is a tautology or a contradiction without using the truth table.

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 3 Mark Questions with Solution Part -II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Establish the equivalence property p ➝ q ≡ ㄱp ν q
2)
Let \(A=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{matrix}\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix} \right) ,B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{matrix} \right) ,C=\left( \begin{matrix} 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix} \right) \)be any three boolean matrices of the same type.
Find AΛB
3)
Verify whether the following compound propositions are tautologies or contradictions or contingency
(p ∧ q) ∧ ¬ (p ∨ q)
4)
Let \(A=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{matrix}\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix} \right) ,B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{matrix} \right) ,C=\left( \begin{matrix} 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix} \right) \)be any three boolean matrices of the same type.
Find (A∧B)∨C
5)
Verify whether the following compound propositions are tautologies or contradictions or contingency
(( p V q)∧ ¬ p) ➝ q

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 3 Mark Questions with Solution Part -I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation + on Z.
2)
Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation - on Z.
3)
Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation + on Z_e = the set of all even integers
4)
Verify
(i) closure property
(ii) commutative property, and
(iii) associative property of the following operation on the given set. (a*b) = a^b;∀a, b∈N (exponentiation property)
5)
How many rows are needed for following statement formulae?
\(p \vee \neg t \wedge(p \vee \neg s)\)

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Let p: Jupiter is a planet and q: India is an island be any two simple statements. Give verbal sentence describing each of the following statements.
(i) ¬p
(ii) p ∧ ¬q
(iii) ¬p ∨ q
(iv) p➝ ¬q
(v) p↔q
2)
Write each of the following sentences in symbolic form using statement variables p and q.
(i) 19 is not a prime number and all the angles of a triangle are equal.
(ii) 19 is a prime number or all the angles of a triangle are not equal
(iii) 19 is a prime number and all the angles of a triangle are equal
(iv) 19 is not a prime number
3)
Fill in the following table so that the binary operation ∗ on A = {a, b, c} is commutative.

* a b c

a b

b c b a

c a c
4)
Write the converse, inverse, and contrapositive of each of the following implication.
If x and y are numbers such that x = y, then x² = y²
5)
Construct the truth table for the following statements.
(¬p ⟶ r) ∧ ( p ↔️ q)

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 2 Mark Questions with Solution Part -I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Determine whether ∗ is a binary operation on the sets given below.
(a*b) = a√b is binary on R
2)
On Z, define \(⊗ \mathrm{by}(m * n)\) = mn + nm: ∀m, n∈Z. Is ⊗ binary on Z?
3)
Determine the truth value of each of the following statements
(i) If 6 + 2 = 5 , then the milk is white.
(ii) China is in Europe or \(\sqrt3\) is an integer
(iii) It is not true that 5 + 5 = 9 or Earth is a planet
(iv) 11 is a prime number and all the sides of a rectangle are equal
4)
Which one of the following sentences is a proposition?
(i) 4 + 7 =12
(ii) What are you doing?
(iii) 3ⁿ ≤ 81, n ∈ N
(iv) Peacock is our national bird
(v) How tall this mountain is!
5)
Consider the binary operation ∗ defined on the set A = {a, b, c, d} by the following table:

* a b c d

a a c b d

b d a b c

c c d a a

d d b a c

Is it commutative and associative?

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 1 Mark Questions with Solution Part -II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
In the set Q define a⊙b = a+b+ab. For what value of y, 3⊙(y⊙5) = 7?
2)
Which one is the contrapositive of the statement (pVq)⟶r?
3)
Which one of the following is incorrect? For any two propositions p and q, we have
4)
The dual of ᄀ(p V q) V [p V (p ∧ ᄀr)] is
5)
Which one of the following is not true?

12th Standard English Medium Maths Subject Discrete Mathematics Book Back 1 Mark Questions with Solution Part -I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
If a compound statement involves 3 simple statements, then the number of rows in the truth table is
2)
Which one is the contrapositive of the statement (pVq)⟶r?
3)
Which one of the following is incorrect? For any two propositions p and q, we have
4)
The dual of ᄀ(p V q) V [p V (p ∧ ᄀr)] is
5)
Which one of the following is not true?

12th Standard English Medium Maths Subject Probability Distributions Book Back 5 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 Iitres and a maximum of 600 litres with probability density function
\(\begin{cases} \begin{matrix} k & 200\le x\le 600 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\)
Find
(i) the value of k
(ii) the distribution function
(iii) the probability that daily sales will fall between 300 litres and 500 litres?
2)
If X is the random variable with probability density function f(x) given by,
\(f(x)=\begin{cases} \begin{matrix} x+1 & -1\le x<0 \end{matrix} \\ \begin{matrix} -x+1 & 0\le x<1 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\)
then find
(i) the distribution function F(x)
(ii) P( -0.5 ≤X ≤ 0.5)
3)
If X is the random variable with distribution function F(x) given by,

then find (i) the probability density function f(x)
(ii) P(0.3 ≤ X ≤ 0.6)
4)
A retailer purchases a certain kind of electronic device from a manufacturer. The manufacturer, indicates that the defective rate of the device is 5%. The inspector of the retailer randomly picks 10 items from a shipment. What is the probability that there will be
(i) at least one defective item
(ii) exactly two defective items.
5)
The mean and standard deviation of a binomial variate X are respectively 6 and 2.
Find
(i) the probability mass function
(ii) P(X = 3)
(iii) P(X\(\ge \)2).

12th Standard English Medium Maths Subject Probability Distributions Book Back 5 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Let X be a random variable denoting the life time of an electrical equipment having probability density function
\(f(x)=\begin{cases} \begin{matrix} { ke }^{ -2x } & forx>0 \end{matrix} \\ \begin{matrix} 0 & forx\le 0 \end{matrix} \end{cases}\)
Find
(i) the value of k
(ii) Distribution function
(iii) P(X < 2)
(iv) calculate the probability that X is at least for four unit of time
(v) P(X = 3)
2)
Suppose that f (x) given below represents a probability mass function

x 1 2 3 4 5 6

f(x) c² 2c² 3c² 4c² c 2c

Find
(i) the value of c
(ii) Mean and variance.
3)
Two balls are chosen randomly from an urn containing 8 white and 4 black balls. Suppose that we win Rs. 20 for each black ball selected and we lose Rs. 10 for each white ball selected. Find the expected winning amount and variance
4)
The mean and variance of a binomial variate X are respectively 2 and 1.5. Find
(i) P(X = 0)
(ii) P(X =1)
(iii) P(X ≥1)
5)
On the average, 20% of the products manufactured by ABC Company are found to be defective. If we select 6 of these products at random and X denote the number of defective products find the probability that
(i) two products are defective
(ii) at most one product is defective
(iii) at least two products are defective.

12th Standard English Medium Maths Subject Probability Distributions Book Back 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
For the random variable X with the given probability mass function as below, find the mean and variance.
\(f(x)=\begin{cases} \begin{matrix} \cfrac { 1 }{ 2 } e^{ -\frac { x }{ 2 } } & for\quad x>0 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\)
2)
If X~ B(n, p) such that 4P(X = 4) = P(X = 2) and n = 6. Find the distribution, mean and standard deviation of X.
3)
An urn contains 2 white balls and 3 red balls. A sample of 3 balls are chosen at random from the urn. If X denotes the number of red balls chosen, find the values taken by the random variable X and its number of inverse images
4)
Two balls are chosen randomly from an urn containing 6 white and 4 black balls. Suppose that we win Rs. 30 for each black ball selected and we lose Rs. 20 for each white ball selected. If X denotes the winning amount, then find the values of X and number of points in its inverse images.
5)
Two fair coins are tossed simultaneously (equivalent to a fair coin is tossed twice). Find the probability mass function for number of heads occurred.

12th Standard English Medium Maths Subject Probability Distributions Book Back 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
In a pack of 52 playing cards, two cards are drawn at random simultaneously. If the number of black cards drawn is a random variable, find the values of the random variable and number of points in its inverse images.
2)
Two balls are chosen randomly from an urn containing 6 red and 8 black balls. Suppose that we win Rs. 15 for each red ball selected and we lose Rs. 10 for each black ball selected. X denotes the winning amount, then find the values of X and number of points in its inverse images.
3)
A six sided die is marked '2' on one face, '3' on two ofits faces, and '4' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find the values of the random variable and number of points in its inverse images.
4)
The probability density function of X is
\(f(x)=\left\{\begin{array}{cc} x & 0
find P(0.2 ≤ X< 0.6)
5)
The probability density function of X is
\(f(x)=\left\{\begin{array}{cc} x & 0
find P(0.5≤X<1.5)

12th Standard English Medium Maths Subject Probability Distributions Book Back 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Determine the order and degree (if exists) of the following differential equations:
\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3{ \left( \frac { dy }{ dx } \right) }^{ 2 }={ x }^{ 2 }log\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \)
2)
Determine the order and degree (if exists) of the following differential equations:
\(3\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ={ \left[ 4+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 3 }{ 2 } }\)
3)
Determine the order and degree (if exists) of the following differential equations:
dy + (xy − cos x)dx = 0
4)
Solve:\(\frac { dy }{ dx } \) = (3x+y+4)².
5)
Solve the following differential equations or show that the solution of
\(\\ \\ \\ \frac { dy }{ dx } =\sqrt { \frac { 1-{ y }^{ 2 } }{ 1-{ x }^{ 2 } } } \)

12th Standard English Medium Maths Subject Probability Distributions Book Back 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
For each of the following differential equations, determine its order, degree (if exists)
\(\frac { dy }{ dx } +xy=cotx\)
2)
For each of the following differential equations, determine its order, degree (if exists)
\({ \left( \frac { { d }^{ 3 }y }{ d{ x }^{ 3 } } \right) }^{ \frac { 2 }{ 3 } }-3\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +5\frac { dy }{ dx } +4=0\)
3)
For each of the following differential equations, determine its order, degree (if exists)
\(\sqrt { \frac { dy }{ dx } } -4\frac { dy }{ dx } -7x=0\)
4)
For each of the following differential equations, determine its order, degree (if exists)
\(y\left( \frac { dy }{ dx } \right) =\frac { x }{ \left( \frac { dy }{ dx } \right) +{ \left( \frac { dy }{ dx } \right) }^{ 3 } } \)
5)
Express each of the following physical statements in the form of differential equation.
(i) Radium decays at a rate proportional to the amount Q present.
(ii) The population P of a city increases at a rate proportional to the product of population and to the difference between 5,00,000 and the population.
(iii) For a certain substance, the rate of change of vapor pressure P with respect to temperature T is proportional to the vapor pressure and inversely proportional to the square of the temperature.
(iv) A saving amount pays 8% interest per year, compounded continuously. In addition, the income from another investment is credited to the amount continuously at the rate of Rs. 400 per year.

12th Standard English Medium Maths Subject Probability Distributions Book Back 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Suppose that X takes on one of the values 0, 1, and 2. If for some constant k, \(P(X=i)=k P(X=i-1) \text { for } i=1,2 \text { and } P(X=0)=\frac{1}{7}\) , then the value of k is
2)
The probability mass function of a random variable is defined as:

x -2 -1 0 1 2

f(x) k 2k 3k 4k 5k

Then E(X ) is equal to:
3)
Let X have a Bernoulli distribution with mean 0.4, then the variance of (2X - 3) is
4)
If in 6 trials, X is a binomial variable which follows the relation 9P(X = 4) = P(X = 2), then the probability of success is
5)
A computer salesperson knows from his past experience that he sells computers to one in every twenty customers who enter the showroom. What is the probability that he will sell a computer to exactly two of the next three customers?

12th Standard English Medium Maths Subject Probability Distributions Book Back 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is
2)
A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is
3)
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. Then the possible values of X are
4)
If the function \(f(x)=\frac { 1 }{ 12 } \) for a < x < b, represents a probability density function of a continuous random variable X, then which of the following cannot be the value of a and b?
5)
If X is a binomial random variable with expected value 6 and variance 2.4, then P(X = 5) is

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 5 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Solve the Linear differential equation:
\(\frac { dy }{ dx } +\frac { y }{ x } =sinx\)
2)
Solve the Linear differential equation:
\(({ x }^{ 2 }+1)\frac { d }{ y } dx+2xy=\sqrt { { x }^{ 2 }+4 } \)
3)
The growth of a population is proportional to the number present. If the population of a colony doubles in 50 years, in how many years will the population become triple?
4)
In a murder investigation, a corpse was found by a detective at exactly 8 p.m. Being alert, the detective also measured the body temperature and found it to be 70^oF. Two hours later, the detective measured the body temperature again and found it to be 60^oF. If the room temperature is 50^oF, and assuming that the body temperature of the person before death was 98.6^oF, at what time did the murder occur? [log(2.43) = 0.88789; log(0.5)=-0.69315]
5)
A tank initially contains 50 litres of pure water. Starting at time t = 0 a brine containing with 2 grams of dissolved salt per litre flows into the tank at the rate of 3 litres per minute. The mixture is kept uniform by stirring and the well-stirred mixture simultaneously flows out of the tank at the same rate. Find the amount of salt present in the tank at any time t > 0.

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 5 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Solve \((1+{ 2e }^{ x/y })dx+2{ e }^{ x/y }\left( 1-\frac { x }{ y } \right) dy=0\)
2)
Solve the following differential equations
(x³+ y³) dy-x²ydx = 0
3)
Solve the Linear differential equation:
\(\frac { dy }{ dx } +\frac { y }{ x } =sinx\)
4)
The growth of a population is proportional to the number present. If the population of a colony doubles in 50 years, in how many years will the population become triple?
5)
A pot of boiling water at 100^oC is removed from a stove at time t = 0 and left to cool in the kitchen. After 5 minutes, the water temperature has decreased to 80^oC , and another 5 minutes later it has dropped to 65^oC. Determine the temperature of the kitchen.

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find the differential equation of the family of circles passing through the origin and having their centres on the x -axis.
2)
Find the differential equation corresponding to the family of curves represented by the equation y = Ae^8x + Be^-8x, where A and B are arbitrary constants.
3)
The slope of the tangent to the curve at any point is the reciprocal of four times the ordinate at that point. The curve passes through (2, 5). Find the equation of the curve.
4)
Form the differential equation by eliminating the arbitrary constants A and B from y = A cos x + B sin x.
5)
Solve : \(\frac { dy }{ dx } =\sqrt { 4x+2y-1 } \)

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find the differential equation of the family of parabolas with vertex at (0, −1) and having axis along the y-axis.
2)
Find the differential equation corresponding to the family of curves represented by the equation y = Ae^8x + Be^-8x, where A and B are arbitrary constants.
3)
Find the differential equation of the curve represented by xy = ae^x + be^−x + x².
4)
Show that y = ae^-3x + b, where a and b are arbitary constants, is a solution of the differential equation\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3\frac { dy }{ dx } =0\)
5)
Find the differential equation of the family of all ellipses having foci on the x -axis and centre at the origin.

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find value of m so that the function y = e^mx is a solution of the given differential equation, y''− 5y' + 6y = 0
2)
Show that y = a cos bx is a solution of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ b }^{ 2 }y=0\).
3)
Determine the order and degree (if exists) of the following differential equations:
\(3\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ={ \left[ 4+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 3 }{ 2 } }\)
4)
Determine the order and degree (if exists) of the following differential equations:
dy + (xy − cos x)dx = 0
5)
Find the differential equation of the family of parabolas y² = 4ax, where a is an arbitrary constant.

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
For each of the following differential equations, determine its order, degree (if exists)
\({ \left( \frac { { d }^{ 3 }y }{ d{ x }^{ 3 } } \right) }^{ \frac { 2 }{ 3 } }-3\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +5\frac { dy }{ dx } +4=0\)
2)
For each of the following differential equations, determine its order, degree (if exists)
\({ x }^{ 2 }\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +{ \left[ 1+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 1 }{ 2 } }=0\)
3)
For each of the following differential equations, determine its order, degree (if exists)
\({ \left( \frac { d^2y }{ dx^2 } \right) }^{ 3 }=\sqrt { 1+\left( \frac { dy }{ dx } \right) } \)
4)
For each of the following differential equations, determine its order, degree (if exists)
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +5\frac { dy }{ dx } +\int { ydx } ={ x }^{ 3 }\)
5)
For each of the following differential equations, determine its order, degree (if exists)
\(x={ e }^{ xy\left( \frac { dy }{ dx } \right) }\)

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
The number of arbitrary constants in the particular solution of a differential equation of third order is
2)
Integrating factor of the differential equation \(\frac{d y}{d x}=\frac{x+y+1}{x+1}\) is
3)
The population P in any year t is such that the rate of increase in the population is proportional to the population. Then
4)
If the solution of the differential equation \(\frac{d y}{d x}=\frac{a x+3}{2 y+f}\)represents a circle, then the value of a is
5)
The slope at any point of a curve y = f (x) is given by \(\frac{dy}{dx}=3x^2\) and it passes through (-1, 1). Then the equation of the curve is

12th Standard English Medium Maths Subject Ordinary Differential Equations Book Back 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
The order and degree of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ \left( \frac { dy }{ dx } \right) }^{ 1/3 }+{ x }^{ 1/4 }=0\) are respectively
2)
The differential equation representing the family of curves y = Acos(x + B), where A and B are parameters,is
3)
The solution of \(\frac{d y}{d x}+p(x) y=0\) is
4)
The integrating factor of the differential equation \(\frac { dy }{ dx } +y=\frac { 1+y }{ \lambda } \) is
5)
The integrating factor of the differential equation \(\frac{d y}{d x}+P(x) y=Q(x)\) is x, then P(x)

12th Standard English Medium Maths Subject Applications of Integration Book Back 5 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Evaluate the following:
\(\int _{ 0 }^{ 1 }{ { x }^{ 3 }{ e }^{ -2x }dx } \)
2)
The region enclosed by the circle x² + y² = a² is divided into two segments by the line x = h. Find the area of the smaller segment.
3)
Find, by integration, the area of the region bounded by the lines 5x − 2y = 15, x + y + 4 = 0 and the x-axis
4)
Father of a family wishes to divide his square field bounded by x = 0, x = 4, y = 4 and y = 0 along the curve y² = 4x and x²= 4y into three equal parts for his wife, daughter and son. Is it possible to divide? If so, find the area to be divided among them.
5)
Find, by integration, the volume of the container which is in the shape of a right circular conical frustum.

12th Standard English Medium Maths Subject Applications of Integration Book Back 5 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Estimate the value of \(\int _{ 0 }^{ 0.5 }{ { x }^{ 2 } } dx\) using the Riemann sums corresponding to 5 subintervals of equal width and applying
(i) left-end rule
(ii) right-end rule
(iii) the mid-point rule.
2)
Evaluate\(\int _{ 1 }^{ 4 }{ ({ 2x }^{ 2 }+3) } \) dx, as the limit of a sum
3)
Show that \(\int ^{1}_{0} (tan ^{-1} x + tan ^{-1}(1-x))\) dx = \(\frac {\pi}{2}\) - log_e2
4)
Find the area of the region bounded by y = cos x, y = sin x, the lines x = \(\frac{\pi}{4}\) and x = \(\frac{5\pi}{4}\).
5)
The curve y = (x − 2)²+1 has a minimum point at P. A point Q on the curve is such that the slope of PQ is 2. Find the area bounded by the curve and the chord PQ.

12th Standard English Medium Maths Subject Applications of Integration Book Back 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Prove that \(\int _{ 0 }^{ \infty }{ { e }^{ -x }{ x }^{ n }dx=n! } \) where n is a positive integer.
2)
Find the area of the region bounded by the line 6x + 5y = 30, x − axis and the lines x = −1 and x = 3.
3)
Find the area of the region bounded by the line 7x − 5y = 35, x−axis and the lines x = −2 and x = 3.
4)
Find the area of the region bounded between the parabola y² = 4ax and its latus rectum.
5)
Find the area of the region bounded by the y-axis and the parabola x = 5 − 4y − y².

12th Standard English Medium Maths Subject Applications of Integration Book Back 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find an approximate value of \(\int _{ 1 }^{ 1.5 }{ xdx } \) by applying the left-end rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}.
2)
Evaluate :\(\int _{ 0 }^{ \frac { \pi }{ 3 } }{ \frac { sec\ x\ tan\ x }{ 1+{ sec }^{ 2 }x } dx } \)
3)
Evaluate :\(\int _{ 0 }^{ 9 }{ \frac { 1 }{ x+\sqrt { x } } dx } \)
4)
Evaluate \(\int _{ 1 }^{ 2 }{ \frac { x }{ (x+1)(x+2) } dx } \)
5)
Evaluate: \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { cos\theta }{ (1+sin\theta )(2+sin\theta ) } } d\theta \)

12th Standard English Medium Maths Subject Applications of Integration Book Back 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Evaluate \(\int _{ b }^{ \infty }{ \frac { 1 }{ { a }^{ 2 }+{ x }^{ 2 } } dx,a>0,b\in R } \)
2)
Evaluate the following
\(\int _{ 0 }^{ 1 }{ { x }^{ 2 }{ (1-x) }^{ 3 }dx } \)
3)
Find the area of the region bounded by the curve y = sin x and the ordinate x=0 \(x=\frac { \pi }{ 3 } \)
4)
Find the area enclosed between the parabola y²=4ax and the line x=a, x=9a.
5)
Find the area bounded by the curve y = cosax in one arc of the curve.

12th Standard English Medium Maths Subject Applications of Integration Book Back 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Evaluate: \(\int _{ 0 }^{ 3 }{ (3{ x }^{ 2 }-4x+5) } dx\)
2)
Evaluate :\(\int _{ 0 }^{ 1 }{ [2x] } dx\) where [⋅] is the greatest integer function
3)
Evaluate the following \(\int _{ 0 }^{ \pi /2 }{ { cos}^{ 7}x\quad dx } \)
4)
Evaluate the following
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }x{ cos }^{ 4 }xdx } \)
5)
Find the area bounded by y=x²+2,x-x-axis, x=1 and x=2

12th Standard English Medium Maths Subject Applications of Integration Book Back 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
The value of \(\int _{ 0 }^{ a }{ { (\sqrt { { a }^{ 2 }-{ x }^{ 2 } } ) }^{ 3 } } dx\) is
2)
If \(\int _{ 0 }^{ x }{ f(t)dt=x+\int _{ x }^{ 1 }{ tf } (t)dt } \), then the value of f (1) is
3)
\(\text { The value of } \int_{0}^{\frac{2}{3}} \frac{d x}{\sqrt{4-9 x^{2}}} \text { is }\)
4)
The value of \(\int _{ -1 }^{ 2 }{ |x|dx } \) is
5)
For any value of \(n \in \mathbb{Z}, \int_{0}^{\pi} e^{\cos ^{2} x} \cos ^{3}[(2 n+1) x] d x\) is

12th Standard English Medium Maths Subject Applications of Integration Book Back 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
The area between y² = 4x and its latus rectum is
2)
The value of \(\int _{ 0 }^{ 1 }{ x{ (1-x) }^{ 99 }dx } \) is
3)
The value of \(\int _{ 0 }^{ \pi }{ \frac { dx }{ 1+{ 5 }^{ cos\ x } } } \) is
4)
If \(\frac{\Gamma(n+2)}{\Gamma(n)}=90\) then n is
5)
The value of \(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ { cos }^{ 3 }3x\ dx }\ is\)

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 5 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Let f(x, y) = sin(xy²) + \(e^{{x^3}+5y}\)for all ∈ R². Calculate \(\frac { \partial f }{ \partial x } ,\frac { \partial f }{ \partial y } ,\frac { { \partial }^{ 2 }f }{ { \partial y\partial x } } \)and \(\frac { { \partial }^{ 2 }f }{ { \partial x\partial y } } \)
2)
Let w(x, y) = xy+\(\frac { { e }^{ y } }{ { y }^{ 2 }+1 } \) for all (x, y) ∈ R². Calculate \(\frac { { \partial }^{ 2 }w }{ { \partial y\partial x } } \) and \(\frac { { \partial }^{ 2 }w }{ { \partial x\partial y } } \)
3)
For each of the following functions find the f_x, f_y, and show that f_xy= f_yx
f(x, y) = \(\frac { 3x }{ y+sinx \ } \)
4)
If U(x, y, z) = log (x³ + y³ + z³), find \(\frac { \partial U }{ \partial x } +\frac { \partial U }{ \partial y } +\frac { \partial U }{ \partial z } \)
5)
If V(x,y) = e^x(x cos y - y siny), then prove that \(\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } =\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } } \) = 0

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 5 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
If w(x, y) = xy + sin (xy), then prove that \(\frac { { \partial }^{ 2 }w }{ \partial y\partial x } =\frac { { \partial }^{ 2 }w }{ \partial x\partial y } \)
2)
If z(x, y) = x tan-¹ (xy), x = t², y = se^t, s, t ∈ R. Find \(\frac { \partial z }{ \partial s } \) and \(\frac { \partial z }{ \partial t } \) at s = t = 1
3)
Let U(x, y) = e^x sin y, where x = st², y = s² t, s, t ∈ R. Find \(\frac { \partial U }{ \partial s } ,\frac { \partial U }{ \partial t } \) and evaluate them at s = t = 1.
4)
W(x, y, z) = xy + yz + zx, x = u - v, y = uv, z = u + v, u ∈ R. Find \(\frac { \partial W }{ \partial u } ,\frac { \partial W }{ \partial v } \), and evaluate them at \(\left( \frac { 1 }{ 2 } ,1 \right) \)
5)
Prove that f(x, y) = x³- 2x²y + 3xy² + y³ is homogeneous; what is the degree? Verify Euler's Theorem for f.

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Let f (x, y) = 0 if xy ≠ 0 and f (x, y) = 1 if xy = 0.
Calculate: \(\frac { \partial f }{ \partial x } (0,0),\frac { \partial f }{ \partial y } (0,0).\)
2)
Evaluate \(\begin{matrix} lim \\ (x,y)\rightarrow (0,0) \end{matrix}cos=\left( \frac { { e }^{ x }siny }{ y } \right) \), if the limit exists.
3)
For each of the following functions find the f_x, f_y and show that f_xy= f_yx
f(x, y) = cos (x² - 3xy)
4)
For each of the following functions find the g_xy, g_xx, g_yy and g_yx.
g(x, y) = xe^y + 3x²y
5)
For each of the following functions find the g_xy, g_xx, g_yy and g_yx.
g(x, y) = log (5x + 3y)

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find a linear approximation for the following functions at the indicated points.
g(x) = \(g(x)=\sqrt { { x }^{ 2 }+9 } ,{ x }_{ 0 }=-4\)
2)
Find a linear approximation for the following functions at the indicated points.
\(h(x)=\frac{x}{x+1}, x_{0}=1\)
3)
The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. find the following in calculating the area of the circular plate:
Absolute error
4)
The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm.find the following in calculating the area of the circular plate:
Percentage error
5)
Find ∆f and df for the function f for the indicated values of x, ∆x and compare
(1) f(x) = x³ - 2x² ; x = 2, ∆ x = dx = 0.5
(2) f(x) = x² + 2x + 3; x = -0.5, ∆x = dx = 0.1

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find df for f(x) = x²+ 3x and evaluate it for
x = 2 and dx = 0.1
2)
If U(x, y, z) = \(\frac { { x }^{ 2 }+{ y }^{ 2 } }{ xy } +3{ z }^{ 2 }y\), find \(\frac { \partial U }{ \partial x } ;\frac { \partial U }{ \partial y } \) and \(\frac { \partial U }{ \partial z } \)
3)
If w(x, y, z) = x² y + y²z + z²x, x, y, z∈R, find the differential dw .
4)
In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree.
\(U(x,y,z)=xy+sin\left( \frac { { y }^{ 2 }-2{ x }^{ 2 } }{ xy } \right) \)
5)
Show that F(x,y) = \(\frac { { x }^{ 2 }+5xy-10{ y }^{ 2 } }{ 3x+7y } \) is a homogeneous function of degree 1.

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
A sphere is made of ice having radius 10 cm. Its radius decreases from 10 cm to 9.8 cm. Find approximations for the following:
(i) change in the volume
(ii) change in the surface area
2)
The time T, taken for a complete oscillation of a single pendulum with length l, is given by the equation T = 2ㅠ\(\sqrt { \frac { 1 }{ g } } \), where g is a constant. Find the approximate percentage error in the calculated value of T corresponding to an error of 2 percent in the value of l.
3)
Show that the percentage error in the n^th root of a number is approximately \(\frac1n\) times the percentage error in the number.
4)
Let f, g : (a, b)→R be differentiable functions. Show that d(fg) = fdg + gdf
5)
Find df for f(x) = x²+ 3x and evaluate it for
x = 3 and dx = 0.02

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
If \(f(x)=\frac{x}{x+1}\), then its differential is given by
2)
If u(x, y) = x²+ 3xy + y - 2019, then \(\left.\frac{\partial u}{\partial x}\right|_{(4,-5)}\) is equal to
3)
Linear approximation for g(x) = cos x at \(x=\frac{\pi}{2}\) is
4)
If w (x, y, z) = x² (y - z) + y² (z - x) + z²(x - y), then \(\frac { { \partial }w }{ \partial x } +\frac { \partial w }{ \partial y } +\frac { \partial w }{ \partial z } \) is
5)
If f(x,y, z) = xy +yz +zx, then f_x - f_z is equal to

12th Standard English Medium Maths Subject Differentials and Partial Derivatives Book Back 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
A circular template has a radius of 10 cm. The measurement of radius has an approximate error of 0.02 cm. Then the percentage error in calculating area of this template is
2)
The percentage error of fifth root of 31 is approximately how many times the percentage error in 31?
3)
If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in our calculation of the volume is
4)
The change in the surface area S = 6x² of a cube when the edge length varies from x_o to x_o+ dx is
5)
If \(g(x, y)=3 x^{2}-5 y+2 y^{2}, x(t)=e^{t}\) and y(t) = cos t, then \(\frac{dg}{dt}\) is equal to

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 5 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
A hollow cone with base radius a cm and, height b em is placed on a table. Show that the volume of the largest cylinder that can be hidden underneath is \(\frac { 4 }{ 9 } \) times volume of the cone.
2)
Write the Maclaurin series expansion of the following function
tan^-1(x); -1 ≤ x ≤ 1
3)
Find the asymptotes of the following curve \(f(x)=\frac { { x }^{ 2 } }{ { x }^{ 2 }-1 } \)
4)
Evaluate the following limit, if necessary use l ’Hôpital Rule
\(\underset { x\rightarrow { \frac { \pi }{ 2 } } }{ lim } { \left( sinx \right) }^{ tanx }\)
5)
A steel plant is capable of producing x tonnes per day of a low-grade steel and y tonnes per day of a high-grade steel, where \(y=\frac { 40-5x }{ 10-x } \). If the fixed market price of low-grade steel is half that of high-grade steel, then what should be optimal productions in low-grade steel and high-grade steel in order to have maximum receipts.

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 5 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s =16t² in t seconds
What is the average velocity with which the camera falls during the last 2 seconds?
2)
A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s =16t² in t seconds
3)
A particle moves along a line according to the law s(t) = 2t³ − 9t² +12t − 4, where t ≥ 0.
(i) At what times the particle changes direction?
(ii) Find the total distance travelled by the particle in the first 4 seconds.
(iii) Find the particle’s acceleration each time the velocity is zero.
4)
A conical water tank with vertex down of 12 metres height has a radius of 5 metres at the top. If water flows into the tank at a rate 10 cubic m/min, how fast is the depth of the water increases when the water is 8 metres deep?
5)
A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall.
(i) How fast is the top of the ladder moving down the wall?
(ii) At what rate, the area of the triangle formed by the ladder, wall and the floor is changing?

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find the absolute extrem of the following function on the given closed interval
f(x) = x² -12x + 10; [1, 2]
2)
Show that the value in the conclusion of the mean value theorem for
f(x) = Ax²+ Bx + c on any interval [a, b] is \(\frac{a+b}{2}\)
3)
Write down the Taylor series expansion, of the function log x about x =1 upto three nonzero terms for x > 0.
4)
Expand sin x in ascending powers x - \(\frac{\pi}{4}\) upto three non-zero terms.
5)
Evaluate : \(\underset{x\rightarrow 1^{-}}{lim}(\frac{log(1-x)}{cot(\pi x)})\).

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
For the function f(x) = x², x∈ [0, 2] compute the average rate of changes in the subintervals [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2] and the instantaneous rate of changes at the points x = 0.5,1, 1.5, 2
2)
A particle moves so that the distance moved is according to the law s(t) = \(s(t)=\frac{t^{3}}{3}-t^{2}+3\). At what time the velocity and acceleration are zero.
3)
A particle is fired straight up from the ground to reach a height of s feet in t seconds, where s(t) = 128t −16t².
(1) Compute the maximum height of the particle reached.
(2) What is the velocity when the particle hits the ground?
4)
The price of a product is related to the number of units available (supply) by the equation Px + 3P −16x = 234, where P is the price of the product per unit in Rupees(Rs) and x is the number of units. Find the rate at which the price is changing with respect to time when 90 units are available and the supply is increasing at a rate of 15 units/week.
5)
A point moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t² + 3t metres.
(i) Find the average velocity of the points between t = 3 and t = 6 seconds.
(ii) Find the instantaneous velocities at t = 3 and t = 6 seconds.

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find two positive numbers whose sum is 12 and their product is maximum.
2)
The volume of a cylinder is given by the formula V = πr² h. Find the greatest and least values of V if r + h = 6.
3)
Evaluate the following limit, if necessary use l ’Hôpital Rule
\(\underset { x\rightarrow \infty }{ lim } \frac { x }{ logx } \)
4)
Evaluate the following limit, if necessary use l ’Hôpital Rule
\(\underset { x\rightarrow \frac { { \pi }^{ - } }{ 2 } }{ lim } \frac { secx }{ tanx } \)
5)
Prove that the function f (x) = x² − 2x − 3 is strictly increasing in \((2, \infty)\)

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find the slope of the tangent to the curves at the respective given points.
x = a cos³ t, y = b sin³ t at t = \(\frac { \pi }{ 2 } \)
2)
Find the point on the curve y = x² − 5x + 4 at which the tangent is parallel to the line 3x + y = 7.
3)
A truck travels on a toll road with a speed limit of 80 km/hr. The truck completes a 164 km journey in 2 hours. At the end of the toll road the trucker is issued with a speed violation ticket. Justify this using the Mean Value Theorem.
4)
Suppose f(x) is a differentiable function for all x with f'(x) ≤ 29 and f(2) = 17. What is the maximum value of f(7)?
5)
Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals.
\(f(x)=|\frac{1}{x}|, x\in [-1,1]\)

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
The function sin⁴ x + cos⁴x is increasing in the interval
2)
The number given by the Rolle's theorem for the functlon x³- 3x², x ∈ [0, 3] is
3)
The number given by the Mean value theorem for the function \(\frac { 1 }{ x } \), x ∈ [1, 9] is
4)
One of the closest points on the curve x² - y² = 4 to the point (6, 0) is
5)
The maximum product of two positive numbers, when their sum of the squares is 200, is

12th Standard English Medium Maths Subject Application of Differential Calculus Book Back 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
The volume of a sphere is increasing in volume at the rate of 3 πcm³ / sec. The rate of change of its radius when radius is \(\frac { 1 }{ 2 } \) cm
2)
The abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?
3)
The slope of the line normal to the curve f(x) = 2cos 4x at \(x=\cfrac { \pi }{ 12 } \) is
4)
The function sin⁴ x + cos⁴x is increasing in the interval
5)
The number given by the Rolle's theorem for the functlon x³- 3x², x ∈ [0, 3] is

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 5 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find the non-parametric form of vector equation of the plane passing through the point (1, −2, 4) and perpendicular to the plane x + 2y −3z = 11 and parallel to the line \(\frac { x+7 }{ 3 } =\frac { y+3 }{ -1 } =\frac { z }{ 1 } \)
2)
Find the parametric form of vector equation and Cartesian equations of the plane containing the line \(\vec { r } =(\hat { i } -\hat { j } +3\hat { k } )+t(2\hat { i } -\hat { j } +4\hat { k } )\) and perpendicular to plane \(\vec { r } .(\hat { i } +2\hat { j } +\hat { k } )=8\)
3)
Show that the lines \(\frac { x-2 }{ 1 } =\frac { y-3 }{ 1 } =\frac { z-4 }{ 3 } \) and \(\frac{x-1}{-3}=\frac{y-4}{2}=\frac{z-5}{1}\) coplanar. Also, find the plane containing these lines.
4)
If the straight lines \(\frac { x-1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ 2 } \) and \(\frac { x-1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ \lambda } \) are coplanar, find λ and equations of the planes containing these two lines.
5)
Find the image of the point whose position vector is \(\hat { i } +2\hat { j } +3\hat { k } \) in the plane \(\vec { r } .(\hat { i } +2\hat { j } +4\hat { k } )\) = 38

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 5 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find the vector parametric, vector non-parametric and Cartesian form of the equation of the plane passing through the points (-1, 2, 0), (2, 2, -1)and parallel to the straight line \(\frac { x-1 }{ 1 } =\frac { 2y+1 }{ 2 } =\frac { z+1 }{ -1 } \)
2)
Find the non-parametric form of vector equation, and Cartesian equation of the plane passing through the point (2, 3, 6) and parallel to the straight lines \(\frac { x-1 }{ 2 } =\frac { y+1 }{ 3 } =\frac { z-3 }{ 1 } \) and \(\frac { x+3 }{ 2 } =\frac { y-3 }{ -5 } =\frac { z+1 }{ -3 } \)
3)
Find the non-parametric form of vector equation, and Cartesian equations of the plane passing through the points (2, 2, 1), (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 9
4)
Find parametric form of vector equation and Cartesian equations of the plane passing through the points (2, 2, 1), (1, −2, 3) and parallel to the straight line passing through the points (2, 1, −3) and (−1, 5, −8)
5)
Find the non-parametric form of vector equation of the plane passing through the point (1, −2, 4) and perpendicular to the plane x + 2y −3z = 11 and parallel to the line \(\frac { x+7 }{ 3 } =\frac { y+3 }{ -1 } =\frac { z }{ 1 } \)

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find the parametric form of vector equation of the straight line passing through (−1, 2,1) and parallel to the straight line \(\vec { r } =(2\hat { i } +3\hat { j } -\hat { k } )+t(\hat { i } -2\hat { j } +\hat { k } )\) and hence find the shortest distance between the lines.
2)
Find the non-parametric form of vector equation, and Cartesian equation of the plane passing through the point (0, 1, -5) and parallel to the straight lines \(\vec { r } =(\hat { i } +2\hat { j } -4\hat { k } )+s(\hat { i } +3\hat { j } +6\hat { k } )\) and \(\hat { r } =(\hat { i } -3\hat { j } +5\hat { k } )+t(\hat { i } +\hat { j } -\hat { k } )\)
3)
Find the non-parametric form of vector equation, and Cartesian equations of the plane \(\vec { r } =(6\hat { i } -\hat { j } +\hat { k } )+s(-\hat { i } +2\hat { j } +\hat { k } )+(-5\hat { i } -4\hat { j } -5\hat { k } )\)
4)
If the straight lines \(\frac { x-1 }{ 1 } =\frac { y-2 }{ 1 } =\frac { z-3 }{ { m }^{ 2 } } \) and \(\frac { x-3 }{ 1 } =\frac { y-2 }{ { m }^{ 2 } } =\frac { z-1 }{ 2 } \) are coplanar, find the distinct real values of m.
5)
Find the equation of the plane passing through the line of intersection of the planes \(\vec { r } .(2\hat { i } -7\hat { j } +4\hat { k } )=3\) and 3x - 5y + 11 = 0, and the point (-2, 1, 3)

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
If D is the midpoint of the side BC of a triangle ABC, then show by vector method that \({ \left| \vec { AB } \right| }^{ 2 }+{ \left| \vec { AC } \right| }^{ 2 }=2({ \left| \vec { AD} \right| }^{ 2 }+{ \left| \vec { BD } \right| }^{ 2 })\)
2)
Prove by vector method that the diagonals of a rhombus bisect each other at right angles.
3)
Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and BD is \(\frac { 1 }{ 2 } \left| \vec { AC } \times \vec { BD } \right| \).
4)
Forces of magnit \(5\sqrt { 2 } \) and \(10\sqrt { 2 } \) units acting in the directions \(\hat { 3i } +\hat { 4j } +\hat { 5k } \) and \(\hat { 10i } +\hat { 6j } -\hat { 8k } \) respectively, act on a particle which is displaced from the point with position vector \(\hat { 4i } -\hat { 3j } -\hat { 2k } \) to the point with position vector \(\hat { 6i } +\hat { j } -\hat { 3k } \). Find the work done by the forces.
5)
Let \(\vec { a } ,\vec { b } ,\vec { c } \) be three non-zero vectors such that \(\vec { c } \) is a unit vector perpendicular to both \(\vec { a } \) and \(\vec { b } \). If the angle between \(\vec { a } \) and \(\vec { b } \) is \(\frac { \pi }{ 6 } \), show that \({ [\vec { a } ,\vec { b } ,\vec { c } ] }^{ 2 }\) = \(\frac { 1 }{ 4 } { \left| \vec { a } \right| }^{ 2 }{ \left| \vec { b } \right| }^{ 2 }\)

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find the angle between the planes \(\vec { r } .(\hat { i } +\hat { j } -2\hat { k } )\) = 3 and 2x - 2y + z =2
2)
Find the length of the perpendicular from the point (1, -2, 3) to the plane x - y + z = 5.
3)
Find the acute angle between the planes \(\vec { r } .(2\hat { i } +2\hat { j } +2\hat { k } )\) and 4x-2y+2z = 15.
4)
Find the distance of a point (2, 5, −3) from the plane \(\vec { r } .(6\hat { i } -3\hat { j } +2\hat { k } )\) = 5
5)
Find the distance between the parallel planes x + 2y - 2z + 1 = 0 and 2x + 4y - 4z + 5 = 0

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
If \(\vec{ a } =\hat { -3i } -\hat { j } +\hat { 5k } \), \(\vec{b}=\hat{i}-\hat{2j}+\hat{k} \), \(\vec{c}=\hat{4j}-\hat{5k} \ \) find\( \ {\vec a } .(\vec { b } \times \vec { c } )\)
2)
Find the volume of the parallelepiped whose coterminus edges are given by the vectors \(\hat { 2i } -\hat { 3j } +\hat { 4k } \), \(\hat { i } +\hat { 2j } -\hat { k } \) and \(\hat {3 i } -\hat { j } +\hat { 2k } \)
3)
Show that the vectors \(\hat { i } +\hat { 2j } -\hat { 3k } \), \(\hat { 2i } -\hat { j } +\hat { 2k } \) and \(\hat { 3i } +\hat { j } -\hat { k } \)
4)
If \(\hat { 2i } -\hat { j } +\hat { 3k } ,\hat { 3i } +\hat { 2j } +\hat { k } ,\hat { i } +\hat { mj } +\hat { 4k } \) are coplanar, find the value of m.
5)
If \(\vec { a } ,\vec { b } ,\vec { c } \) are three vectors, prove that \([\vec { a } +\vec { c } ,\vec { a } +\vec { b } ,\vec { a } +\vec { b } +\vec { c } ]\) = \([\vec { a } ,\vec { b } ,\vec { c } ]\)

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
If the two tangents drawn from a point P to the parabola y² = 4x are at right angles then the locus of P is
2)
The circle passing through (1, -2) and touching the axis of x at (3, 0) passing through the point
3)
The locus of a point whose distance from (-2,0) is \(\frac { 2 }{ 3 } \) times its distance from the line x = \(\frac { -9 }{ 2 } \) is
4)
The values of m for which the line y = mx + \(2\sqrt { 5 } \) touches the hyperbola 16x²− 9y²= 144 are the roots of x²− (a + b)x − 4 = 0, then the value of (a+b) is
5)
If the coordinates at one end of a diameter of the circle x²+ y²− 8x − 4y + c = 0 are (11, 2), the coordinates of the other end are

12th Standard English Medium Maths Subject Applications of Vector Algebra Book Back 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x²+ y²− 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to
2)
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is
3)
If P(x, y) be any point on 16x²+ 25y²= 400 with foci F₁ (3, 0) and F₂ (-3, 0) then PF₁ + PF₂ is
4)
The radius of the circle passing through the point(6, 2) two of whose diameter are x + y = 6 and x + 2y = 4 is
5)
The area of quadrilateral formed with foci of the hyperbolas \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \text { and } \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\)

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 5 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find the equation of the circle passing through the points (1, 1 ), (2, -1 ) and (3, 2) .
2)
A road bridge over an irrigation canal have two semi circular vents each with a span of 20m and the supporting pillars of width 2m. Use Figure to write the equations that represents the semi-verticular vents
3)
Find the equations of tangents to the hyperbola \(\frac { { x }^{ 2 } }{ 16 } -\frac { { y }^{ 2 } }{ 64 } \) = 1 which are parallel to10x − 3y + 9 = 0.
4)
Show that the line x−y+4 = 0 is a tangent to the ellipse x²+3y²= 12 . Also find the coordinates of the point of contact.
5)
The maximum and minimum distances of the Earth from the Sun respectively are 152 × 10⁶ km and 94.5 × 10⁶ km. The Sun is at one focus of the elliptical orbit. Find the distance from the Sun to the other focus.

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 5 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
On lighting a rocket cracker it gets projected in a parabolic path and reaches a maximum height of 4 m when it is 6 m away from the point of projection. Finally it reaches the ground 12 m away from the starting point. Find the angle of projection.
2)
Points A and B are 10 km apart and it is determined from the sound of an explosion heard at those points at different times that the location of the explosion is 6 km closer to A than B. Show that the location of the explosion is restricted to a particular curve and find an equation of it.
3)
Find the vertex, focus, equation of directrix and length of the latus rectum of the following:
x²−2x+8y+17= 0
4)
Find the vertex, focus, equation of directrix and length of the latus rectum of the following: y²−4y−8x+12 = 0
5)
Identify the type of conic and find centre, foci, vertices, and directrices of each of the following :
18x²+12y²−144x+48y+120 = 0

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 3 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Identify the type of conic and find centre, foci, vertices, and directrices of each of the following:
\(\frac { { x }^{ 2 } }{ 25 } +\frac { { y }^{ 2 } }{ 9 } =1\)
2)
Prove that the length of the latus rectum of the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 is \(\frac { { 2b }^{ 2 } }{ a } \).
3)
Show that the absolute value of difference of the focal distances of any point P on the hyperbola is the length of its transverse axis.
4)
Prove that the point of intersection of the tangents at ‘t₁’ and ‘t₂’ on the parabola y² = 4ax is \(\left[ at_{ 1 }t_{ 2 },a({ t }_{ 1 }+{ t }_{ 2 }) \right] .\)
5)
A semielliptical archway over a one-way road has a height of 3m and a width of 12m. The truck has a width of 3m and a height of 2.7m. Will the truck clear the opening of the archway?

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 3 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find the equation of the circle described on the chord 3x + y + 5 = 0 of the circle x²+ y²= 16 as diameter.
2)
A circle of radius 3 units touches both the axes. Find the equations of all possible circles formed in the general form.
3)
Find the centre and radius of the circle 3x²+ (a + 1)y²+ 6x − 9y + a + 4 = 0.
4)
Find the equation of circles that touch both the axes and pass through (-4, -2) in general form.
5)
Find the equation of the tangent and normal to the circle x²+y²−6x+6y−8 = 0 at (2, 2) .

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 2 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Identify the type of conic section for each of the equations.
3x²+3y²−4x+3y+10 = 0
2)
Identify the type of the conic for the following equations:
3x²+2y²= 14
3)
Identify the type of the conic for the following equations :
11x²−25y²−44x+50y−256 = 0
4)
Find centre and radius of the following circles.
x²+ y²+ 6x − 4y + 4 = 0
5)
Find centre and radius of the following circles.
2x²+2y²−6x+4y+2 = 0

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 2 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
Find the general equation of a circle with centre (-3, -4) and radius 3 units.
2)
Determine whether x + y − 1 = 0 is the equation of a diameter of the circle x²+ y²− 6x + 4y + c = 0 for all possible values of c .
3)
Find the general equation of the circle whose diameter is the line segment joining the points (−4, −2)and (1, 1) is x²+y²+5x+3y+6=0
4)
Examine the position of the point (2, 3) with respect to the circle x²+ y²− 6x − 8y + 12 = 0.
5)
The line 3x+4y−12 = 0 meets the coordinate axes at A and B. Find the equation of the circle drawn on AB as diameter.

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 1 Mark Questions with Solution Part - II - by Question Bank Software - Jun 01, 2021 - View & Read

1)
The eccentricity of the ellipse (x−3)² +(y−4)² \(=\frac { { y }^{ 2 } }{ 9 } \) is
2)
If the two tangents drawn from a point P to the parabola y² = 4x are at right angles then the locus of P is
3)
The locus of a point whose distance from (-2,0) is \(\frac { 2 }{ 3 } \) times its distance from the line x = \(\frac { -9 }{ 2 } \) is
4)
The values of m for which the line y = mx + \(2\sqrt { 5 } \) touches the hyperbola 16x²− 9y²= 144 are the roots of x²− (a + b)x − 4 = 0, then the value of (a+b) is
5)
If the coordinates at one end of a diameter of the circle x²+ y²− 8x − 4y + c = 0 are (11, 2), the coordinates of the other end are

12th Standard English Medium Maths Subject Two Dimensional Analytical Geometry-II Book Back 1 Mark Questions with Solution Part - I - by Question Bank Software - Jun 01, 2021 - View & Read

1)
The equation of the circle passing through (1, 5) and (4, 1) and touching y-axis is x²+ y²− 5x − 6y + 9 + \(\lambda\)(4x + 3y − 19) = 0 where λ is equal to
2)
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is
3)
The circle x²+ y²= 4x + 8y +5 intersects the line 3x−4y = m at two distinct points if
4)
The length of the diameter of the circle which touches the x - axis at the point (1, 0) and passes through the point (2, 3).
5)
The centre of the circle inscribed in a square formed by the lines x²− 8x − 12 = 0 and y²− 14y + 45 = 0 is

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 5 Mark Questions with Solution Part - II - by Question Bank Software - May 31, 2021 - View & Read

1)
Find the principal value of cosec⁻¹(−1)
2)
If a₁, a₂, a₃, ... an is an arithmetic progression with common difference d, prove that tan\( \left[ tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +....tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ n }{ a }_{ n-1 } } \right) \right] =\frac { { a }_{ n }-{ a }_{ 1 } }{ 1+{ a }_{ 1 }{ a }_{ n } } \)
3)
Solve \(tan^{ -1 }\left( \frac { x-1 }{ x-2 } \right) +tan^{ -1 }\left( \frac { x+1 }{ x+2 } \right) =\frac { \pi }{ 4 } \)
4)
Solve \(cos\left( sin^{ -1 }\left( \frac { x }{ \sqrt { 1+{ x }^{ 2 } } } \right) \right) =sin\left\{ cot^{ -1 }\left( \frac { 3 }{ 4 } \right) \right\} \)
5)
Find the principal value of
sec⁻¹(−2).

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 5 Mark Questions with Solution Part - I - by Question Bank Software - May 31, 2021 - View & Read

1)
Find the principal value of cos⁻¹\(\left( \frac { \sqrt { 3 } }{ 2 } \right) \)
2)
Find cos^-1\((-\frac{1}{\sqrt2})\)
3)
Find the principal value of cosec⁻¹(−1)
4)
If a₁, a₂, a₃, ... an is an arithmetic progression with common difference d, prove that tan\( \left[ tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +....tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ n }{ a }_{ n-1 } } \right) \right] =\frac { { a }_{ n }-{ a }_{ 1 } }{ 1+{ a }_{ 1 }{ a }_{ n } } \)
5)
Solve \(tan^{ -1 }\left( \frac { x-1 }{ x-2 } \right) +tan^{ -1 }\left( \frac { x+1 }{ x+2 } \right) =\frac { \pi }{ 4 } \)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 3 Mark Questions with Solution Part - II - by Question Bank Software - May 31, 2021 - View & Read

1)
If cos⁻¹x + cos⁻¹y + cos⁻¹ z = \(\pi \) and 0 < x, y, z < 1, show that x² + y²+ z²+ 2xyz = 1
2)
Solve sin^-1 x > cos^-1x
3)
Solve tan^-1 2x + tan^-1 3x = \(\frac{\pi}{4}\), if 6x²< 1
4)
Find the value of the expression in terms of x, with the help of a reference triangle.
sin(cos⁻¹(1-x))
5)
Find the domain of the following
g(x) = 2sin⁻¹(2x−1)−\(\frac{\pi}{4}\)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 3 Mark Questions with Solution Part - I - by Question Bank Software - May 31, 2021 - View & Read

1)
Find the domain of sin⁻¹(2−3x²)
2)
Sketch the graph of y = sin\((\frac{1}{3}x)\) for 0\(\le x <6\pi\).
3)
Find the domain of the following
\(f\left( x \right) { =sin }^{ -1 }\left( \frac { { x }^{ 2 }+1 }{ 2x } \right) \)
4)
Find the domain of cos^-1\((\frac{2+sinx}{3})\)
5)
Find the domain of f(x) = sin^-1 \((\frac{|x|-2}{3})+ \) cos^-1\((\frac{1-|x|}{4})\)

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 2 Mark Questions with Solution Part - II - by Question Bank Software - May 31, 2021 - View & Read

1)
Find the principal value of sin^-1\(\left( -\frac { 1 }{ 2 } \right) \)(in radians and degrees).
2)
Find the principal value of sin^-1(2), if it exists.
3)
Find the principal value of
\({ Sin }^{ -1 }\left( \frac { 1 }{ \sqrt { 2 } } \right) \)
4)
Find all the values of x such that -10\(\pi\)\(\le x\le\)10\(\pi\) and sin x = 0
5)
Find the period and amplitude of y = sin 7x

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 2 Mark Questions with Solution Part - I - by Question Bank Software - May 31, 2021 - View & Read

1)
Find the principal value of \({ tan }^{ -1 }\left( \frac { -1 }{ \sqrt { 3 } } \right) \)
2)
Find the principal value of sin^-1(-1).
3)
Find the principal value of \({ cos }^{ -1 }\left( \frac { -1 }{ 2 } \right) \)
4)
If \({ cot }^{ -1 }\left( \frac { 1 }{ 7 } \right) =\theta \) find the value of cos \(\theta \)
5)
If \({ sin }^{ -1 }\left( \frac { 1 }{ 2 } \right) ={ tan }^{ -1 }x\) then find the value of x,

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 1 Mark Questions with Solution Part - II - by Question Bank Software - May 31, 2021 - View & Read

1)
The value of sin^-1 (cos x), \(0\le x\le\pi\) is
2)
If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\); then cos^-1x + cos^-1 y is equal to
3)
If sin⁻¹x = 2sin⁻¹\(\alpha\) has a solution, then
4)
\(\sin ^{-1}(\cos x)=\frac{\pi}{2}-x\) is valid for
5)
If sin^-1 x+sin^-1 y+sin^-1 \(z = \frac{3\pi}{2}\), the value of x²⁰¹⁷+y²⁰¹⁸+z²⁰¹⁹\(-\frac { 9 }{ { x }^{ 101 }+{ y }^{ 101 }+{ z }^{ 101 } } \)is

12th Standard English Medium Maths Subject Inverse Trigonometric Functions Book Back 1 Mark Questions with Solution Part - I - by Question Bank Software - May 31, 2021 - View & Read

1)
The domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1}\) is
2)
If \(x = \frac{1}{5}\), the value of cos (cos^-1x+2sin^-1x) is
3)
\(\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)\) is equal to
4)
\(\sin ^{-1}\left(\tan \frac{\pi}{4}\right)-\sin ^{-1}\left(\sqrt{\frac{3}{x}}\right)=\frac{\pi}{6}\). Then x is a root of the equation
5)
sin (tan^-1x), |x| < 1 is equal to

12th Standard English Medium Maths Subject Theory of Equations Book Back 5 Mark Questions with Solution Part - II - by Question Bank Software - May 31, 2021 - View & Read

1)
Solve the equation 3x³- 16x²+ 23x - 6 = 0 if the product of two roots is 1.
2)
Form the equation whose roots are the squares of the roots of the cubic equation x³+ ax²+ bx + c = 0.
3)
Solve the equation x³− 9x²+14x + 24 = 0 if it is given that two of its roots are in the ratio 3:2.
4)
Find a polynomial equation of minimum degree with rational coefficients, having \(\sqrt{5}\)−\(\sqrt{3}\) as a root.
5)
If 2+i and 3-\(\sqrt{2}\) are roots of the equation x⁶-13x⁵+ 62x⁴-126x³+ 65x²+127x-140 = 0, find all roots.

12th Standard English Medium Maths Subject Theory of Equations Book Back 5 Mark Questions with Solution Part - I - by Question Bank Software - May 31, 2021 - View & Read

1)
Solve : (x - 5) (x - 7) (x + 6) (x + 4) = 504
2)
Find all zeros of the polynomial x⁶- 3x⁵- 5x⁴+ 22x³- 39x²- 39x + 135, if it is known that 1+2i and \(\sqrt{3}\) are two of its zeros.
3)
Solve the following equation: x⁴-10x³+ 26x²-10x + 1 = 0
4)
Solve the equation 6x⁴- 5x³- 38x²- 5x + 6 = 0 if it is known that \(\frac{1}{3}\) is a solution.
5)
Discuss the maximum possible number of positive and negative roots of the polynomial equations x²−5x+6 and x²−5x+16 . Also draw rough sketch of the graphs

12th Standard English Medium Maths Subject Theory of Equations Book Back 3 Mark Questions with Solution Part - II - by Question Bank Software - May 31, 2021 - View & Read

1)
If α and β are the roots of the quadratic equation 17x²+43x−73 = 0 , construct a quadratic equation whose roots are α + 2 and β + 2.
2)
If α and β are the roots of the quadratic equation 2x²−7x+13 = 0 , construct a quadratic equation whose roots are α² and β².
3)
If the sides of a cubic box are increased by 1, 2, 3 units respectively to form a cuboid, then the volume is increased by 52 cubic units. Find the volume of the cuboid.
4)
If α, β and γ are the roots of the cubic equation x³+2x²+3x+4 = 0, form a cubic equation whose roots are, 2α, 2β, 2γ
5)
Find the sum of squares of roots of the equation 2x⁴- 8x³+ 6x²-3 = 0.

12th Standard English Medium Maths Subject Theory of Equations Book Back 3 Mark Questions with Solution Part - I - by Question Bank Software - May 31, 2021 - View & Read

1)
Solve the equation 2x³+11x²−9x−18 = 0.
2)
Find the condition that the roots of ax³+ bx²+ cx + d = 0 are in geometric progression. Assume a, b, c, d ≠ 0.
3)
Solve the equation 3x³-26x²+52x - 24 = 0 if its roots form a geometric progression.
4)
Solve the equation
2x³- 9x²+ 10x = 3
5)
Find all real numbers satisfying 4^x- 3(2^x+2) + 2⁵= 0

12th Standard English Medium Maths Subject Theory of Equations Book Back 2 Mark Questions with Solution Part - II - by Question Bank Software - May 31, 2021 - View & Read

1)
Find the exact number of real zeros and imaginary of the polynomial x⁹+9x⁷+7x⁵+5x³+3x.
2)
Construct a cubic equation with roots 1, 1 and −2
3)
Construct a cubic equation with roots 2, −2, and 4.
4)
Examine for the rational roots of x⁸- 3x + 1 = 0
5)
Discuss the nature of the roots of the following polynomials:
x⁵-19x⁴+ 2x³+ 5x²+11

12th Standard English Medium Maths Subject Theory of Equations Book Back 2 Mark Questions with Solution Part - I - by Question Bank Software - May 31, 2021 - View & Read

1)
If α, β, and γ are the roots of the equation x³+ px²+ qx + r = 0, find the value of \(\Sigma \frac { 1 }{ \beta \gamma } \) in terms of the coefficients.
2)
Construct a cubic equation with roots 1, 2 and 3
3)
If α, β, γ and \(\delta\) are the roots of the polynomial equation 2x⁴+ 5x³− 7x²+ 8 = 0, find a quadratic equation with integer coefficients whose roots are α + β + γ + \(\delta\) and αβ૪\(\delta\).
4)
Formulate into a mathematical problem to find a number such that when its cube root is added to it, the result is 6.
5)
A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was left standing.

12th Standard English Medium Maths Subject Theory of Equations Book Back 1 Mark Questions with Solution Part - II - by Question Bank Software - May 31, 2021 - View & Read

1)
The polynomial x³- kx²+ 9x has three real zeros if and only if, k satisfies
2)
The number of real numbers in [0, 2π] satisfying sin⁴x - 2sin²x + 1 is
3)
If x³+12x²+10ax+1999 definitely has a positive zero, if and only if
4)
The polynomial x³+ 2x + 3 has
5)
The number of positive zeros of the polynomial \(\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }\)(-1)^rx^r is

12th Standard English Medium Maths Subject Theory of Equations Book Back 1 Mark Questions with Solution Part - I - by Question Bank Software - May 31, 2021 - View & Read

1)
A zero of x³ + 64 is
2)
If f and g are polynomials of degrees m and n respectively, and if h(x) = (f ^_og)(x), then the degree of h is
3)
A polynomial equation in x of degree n always has
4)
If α, β and γ are the zeros of x³+ px²+ qx + r, then \(\Sigma \frac { 1 }{ \alpha } \) is
5)
According to the rational root theorem, which number is not possible rational zero of 4x⁷+ 2x⁴- 10x³- 5?

12th Standard English Medium Maths Subject Complex Numbers Book Back 5 Mark Questions with Solution Part - II - by Question Bank Software - May 31, 2021 - View & Read

1)
Show that \(\left( \frac { 19+9i }{ 5-3i } \right) ^{ 15 }-\left( \frac { 8+i }{ 1+2i } \right) ^{ 15 }\) is purely imaginary.
2)
Let z₁, z₂ and z₃ be complex numbers such that \(\left| { z }_{ 1 } \right\| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =r>0\) and z₁+ z₂+ z₃ \(\neq \) 0 prove that \(\left| \frac { { z }_{ 1 }{ z }_{ 2 }+{ z }_{ 2 }{ z }_{ 3 }+{ z }_{ 3 }{ z }_{ 1 } }{ { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 } } \right| \) = r
3)
If z₁, z_2, and z₃ are three complex numbers such that |z₁| = 1, |z₂| = 2|z₃| = 3 and |z₁+ z₂+ z₃| = 1, show that |9z₁z₂+ 4z₁z₂+ z₂z₃| = 6
4)
If z = x + iy is a complex number such that Im \(\left( \frac { 2z+1 }{ iz+1 } \right) =0\) show that the locus of z is 2x²+ 2y²+ x - 2y = 0
5)
If z = x + iy and arg \(\left( \frac { z-i }{ z+2 } \right) =\frac { \pi }{ 4 } \), then show that x²+ y²+ 3x - 3y + 2 = 0

12th Standard English Medium Maths Subject Complex Numbers Book Back 5 Mark Questions with Solution Part - I - by Question Bank Software - May 31, 2021 - View & Read

1)
Suppose z₁, z₂and z₃are the vertices of an equilateral triangle inscribed in the circle |z| = 2. If z₁ = 1 + i\(\sqrt { 3 } \) then find z₂ and z₃.
2)
Find all cube roots of \(\sqrt { 3 } +i\)
3)
Solve the equation z³+ 8i = 0, where \(z \in \mathbb{C}\)
4)
Simplify: (1+i)¹⁸
5)
Simplify: \(\left( -\sqrt { 3 } +3i \right) ^{ 31 }\)

12th Standard English Medium Maths Subject Complex Numbers Book Back 3 Mark Questions with Solution Part - II - by Question Bank Software - May 31, 2021 - View & Read

1)
Find the value of the real numbers x and y, if the complex number (2+i)x+(1−i)y+2i −3 and x+(−1+2i)y+1+i are equal
2)
If z₁, z₂ and z₃ are complex numbers such that |z₁| = |z₂| = |z₃| = |z₁+z₂+z₃| = 1 find the value of \(\left| \frac { 1 }{ { z }_{ 1 } } +\frac { 1 }{ z_{ 2 } } +\frac { 1 }{ { z }_{ 3 } } \right| \)
3)
If |z| = 2 show that \(3\le \left| z+3+4i \right| \le 7\)
4)
Find the rectangular form of the complex numbers
\(\left( cos\frac { \pi }{ 6 } +isin\frac { \pi }{ 6 } \right) \left( cos\frac { \pi }{ 12 } +isin\frac { \pi }{ 12 } \right) \)
5)
If (x₁+ iy₁)(x₂+ iy₂)(x3 + iy₃)...(x_n+ iy_n) = a + ib, show that
(x₁²+ y₁²)(x₂²+ y₂²)(x₃²+ y₃²)...(x_n²+ y_n²) = a²+ b²

12th Standard English Medium Maths Subject Complex Numbers Book Back 3 Mark Questions with Solution Part - I - by Question Bank Software - May 31, 2021 - View & Read

1)
Given the complex number z = 2 + 3i, represent the complex numbers in Argand diagram z, iz , and z+iz
2)
Find the rectangular form of the complex numbers
\(\left( cos\frac { \pi }{ 6 } +isin\frac { \pi }{ 6 } \right) \left( cos\frac { \pi }{ 12 } +isin\frac { \pi }{ 12 } \right) \)
3)
If \(cos\alpha +cos\beta +cos\gamma =sin\alpha +sin\beta +sin\gamma =0\) then show that
(i) \(cos3\alpha +cos3\beta +cos3\gamma =3cos(\alpha +\beta +\gamma )\)
(ii) \(sin3\alpha +sin3\beta +sin3\gamma +sin3\gamma =3sin\left( \alpha +\beta +\gamma \right) \)
4)
Find the quotient \(\frac { 2\left( cos\frac { 9\pi }{ 4 } +isin\frac { 9\pi }{ 4 } \right) }{ 4\left( cos\left( \frac { -3\pi }{ 2 } + \right) isin\left( \frac { -3\pi }{ 2 } \right) \right) } \) in rectangular form
5)
Evaluate the following if z = 5−2i and w = −1+3i
(z + w)²

12th Standard English Medium Maths Subject Complex Numbers Book Back 2 Mark Questions with Solution Part - II - by Question Bank Software - May 31, 2021 - View & Read

1)
Simplify the following
i i ²i³...i²⁰⁰⁰
2)
Evaluate the following if z = 5−2i and w = −1+3i
2z + 3w
3)
Find the square roots of −6+8i
4)
Find the following \(\left| \overline { (1+i) } (2+3i)(4i-3) \right| \)
5)
Find the modulus and principal argument of the following complex numbers.
\(-\sqrt { 3 } +i\)

12th Standard English Medium Maths Subject Complex Numbers Book Back 2 Mark Questions with Solution Part - I - by Question Bank Software - May 31, 2021 - View & Read

1)
Show that the following equations represent a circle, and find its centre and radius \(\left| z-2-i \right| =3\)
2)
Write in polar form of the following complex numbers
\(2+i2\sqrt { 3 } \)
3)
Simplify the following
\({ i }^{ 59 }+\frac { 1 }{ { i }^{ 59 } } \)
4)
If z = x + iy , find the following in rectangular form.
Im(3z + 4\(\bar { z } \) − 4i)
5)
Simplify the following:
i i²i³...i⁴⁰

12th Standard English Medium Maths Subject Complex Numbers Book Back 1 Mark Questions with Solution Part - II - by Question Bank Software - May 31, 2021 - View & Read

1)
If \(\frac { z-1 }{ z+1 } \) is purely imaginary, then |z| is
2)
If z = x + iy is a complex number such that |z+2| = |z−2|, then the locus of z is
3)
If \(\alpha \) and \(\beta \) are the roots of x²+x+1 = 0, then \({ \alpha }^{ 2020 }+{ \beta }^{ 2020 }\) is
4)
The product of all four values of \(\left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) ^{ \frac { 3 }{ 4 } }\) is
5)
If \(\omega =cis\cfrac { 2\pi }{ 3 } \), then the number of distinct roots of \(\left| \begin{matrix} z+1 & \omega & { \omega }^{ 2 } \\ \omega & z+{ \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & z+\omega \end{matrix} \right| \)=0

12th Standard English Medium Maths Subject Complex Numbers Book Back 1 Mark Questions with Solution Part - I - by Question Bank Software - May 31, 2021 - View & Read

1)
iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is
2)
The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is
3)
The area of the triangle formed by the complex numbers z, iz and z+iz in the Argand’s diagram is
4)
The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is
5)
If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to

12th Standard English Medium Maths Subject Application of Matrices and Determinants Book Back 5 Mark Questions with Solution Part - II - by Question Bank Software - May 31, 2021 - View & Read

1)
The upward speed v(t)of a rocket at time t is approximated by v(t) = at² + bt + c, 0 ≤ t ≤ 100 where a, b and c are constants. It has been found that the speed at times t = 3, t = 6, and t = 9 seconds are respectively, 64, 133, and 208 miles per second respectively. Find the speed at time t = 15 seconds. (Use Gaussian elimination method.)
2)
Investigate for what values of λ and μ the system of linear equations x + 2y + z = 7 , x + y + λz = μ , x + 3y − 5z = 5 has
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions
3)
Test for consistency and if possible, solve the following systems of equations by rank method.
2x - y + z = 2, 6x - 3y + 3z = 6, 4x - 2y + 2z = 4
4)
Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 1 & -1 & 0 \\ 1 & 0 & -1 \\ 6 & -2 & -3 \end{matrix} \right] \)
5)
Solve the following systems of linear equations by Cramer’s rule:
\(\frac { 3 }{ x } -\frac { 4 }{ y } -\frac { 2 }{ z } \) -1 = 0, \(\frac { 1 }{ x } +\frac { 2 }{ y } +\frac { 1 }{ z } \) - 2 = 0, \(\frac { 2 }{ x } -\frac { 5 }{ y } -\frac { 4 }{ z } \) + 1 = 0

12th Standard English Medium Maths Subject Application of Matrices and Determinants Book Back 5 Mark Questions with Solution Part - I - by Question Bank Software - May 31, 2021 - View & Read

1)
If A = \(\left[ \begin{matrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{matrix} \right] \), verify thatA(adj A) = (adj A)A = |A| I₃.
2)
If A = \(\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right] \), find x and y such that A² + xA + yI₂ = O₂. Hence, find A^-1.
3)
If A = \(\frac { 1 }{ 7 } \left[ \begin{matrix} 6 & -3 & a \\ b & -2 & 6 \\ 2 & c & 3 \end{matrix} \right] \) is orthogonal, find a, b and c , and hence A⁻¹.
4)
If F(\(\alpha\)) = \(\left[ \begin{matrix} \cos { \alpha } & 0 & \sin { \alpha } \\ 0 & 1 & 0 \\ -\sin { \alpha } & 0 & \cos { \alpha } \end{matrix} \right] \), show that [F(\(\alpha\))]^-1 = F(-\(\alpha\)).
5)
If A = \(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] \), show that A² - 3A - 7I₂ = O₂. Hence find A⁻¹.

12th Standard English Medium Maths Subject Application of Matrices and Determinants Book Back 3 Mark Questions with Solution Part -II - by Question Bank Software - May 31, 2021 - View & Read

1)
Reduce the matrix \(\left[ \begin{matrix} 0 \\ -1 \\ 4 \end{matrix}\begin{matrix} 3 \\ 0 \\ 2 \end{matrix}\begin{matrix} 1 \\ 2 \\ 0 \end{matrix}\begin{matrix} 6 \\ 5 \\ 0 \end{matrix} \right] \) to row-echelon form.
2)
Find the rank of the matrix \(\left[ \begin{matrix} 2 \\ \begin{matrix} -3 \\ 6 \end{matrix} \end{matrix}\begin{matrix} -2 \\ \begin{matrix} 4 \\ 2 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} -2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} -1 \\ 7 \end{matrix} \end{matrix} \right] \) by reducing it to an echelon form.
3)
Solve the following system of linear equations, using matrix inversion method:
5x + 2y = 3, 3x + 2y = 5.
4)
Solve the following system of linear equations by matrix inversion method:
2x + 5y = −2, x + 2y = −3
5)
Find the adjoint of the following:
\(\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right] \)

12th Standard English Medium Maths Subject Application of Matrices and Determinants Book Back 3 Mark Questions with Solution Part - I - by Question Bank Software - May 31, 2021 - View & Read

1)
Find the inverse of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \end{matrix} \right] \).
2)
Verify the property (A^T)^-1 = (A^-1)^T with A = \(\left[ \begin{matrix} 2 & 9 \\ 1 & 7 \end{matrix} \right] \).
3)
Verify (AB)^-1 = B^-1A^-1 with A = \(\left[ \begin{matrix} 0 & -3 \\ 1 & 4 \end{matrix} \right] \), B = \(\left[ \begin{matrix} -2 & -3 \\ 0 & -1 \end{matrix} \right] \).
4)
If A = \(\frac { 1 }{ 9 } \left[ \begin{matrix} -8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & -8 & 4 \end{matrix} \right] \), prove that A⁻¹ = A^T.
5)
If A = \(\left[ \begin{matrix} 8 & -4 \\ -5 & 3 \end{matrix} \right] \), verify that A(adj A) = (adj A)A = |A|I₂.

12th Standard English Medium Maths Subject Application of Matrices and Determinants Book Back 2 Mark Questions with Solution Part -II - by Question Bank Software - May 31, 2021 - View & Read

1)
If A = \(\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \) is non-singular, find A⁻¹.
2)
If A is a non-singular matrix of odd order, prove that |adj A| is positive
3)
If A is symmetric, prove that then adj A is also symmetric.
4)
Prove that \(\left[ \begin{matrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{matrix} \right] \) is orthogonal.
5)
Find the adjoint of the following:
\(\left[ \begin{matrix} -3 & 4 \\ 6 & 2 \end{matrix} \right] \)

12th Standard English Medium Maths Subject Application of Matrices and Determinants Book Back 2 Mark Questions with Solution Part - I - by Question Bank Software - May 31, 2021 - View & Read

1)
Find the rank of the matrix \(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{matrix} \right] \) by reducing it to a row-echelon form.
2)
Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 2 & -4 \\ -1 & 2 \end{matrix} \right] \)
3)
Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 1 \\ 3 \end{matrix}\begin{matrix} -2 \\ -6 \end{matrix}\begin{matrix} -1 \\ -3 \end{matrix}\begin{matrix} 0 \\ 1 \end{matrix} \right] \)
4)
Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 1 & -2 & 3 \\ 2 & 4 & -6 \\ 5 & 1 & -1 \end{matrix} \right] \)
5)
Find the rank of the following matrices by minor method or show that the rank of matrix is 3
\(\left[ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 8 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 2 \\ 1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ 0 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 3 \\ 2 \end{matrix} \end{matrix} \right] \)

12th Standard English Medium Maths Subject Application of Matrices and Determinants Book Back 1 Mark Questions with Solution Part - II - by Question Bank Software - May 31, 2021 - View & Read

1)
If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =
2)
If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is
3)
If A = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right] \) then the value of a₂₃ is
4)
If A, B and C are invertible matrices of some order, then which one of the following is not true?
5)
If (AB)^-1 = \(\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right] \), then B^-1 =

Stateboard 12th Standard Maths Subject English Medium Public Answer Key- March 2020 - by QB Admin - May 19, 2021 - View & Read

Stateboard 12th Standard Maths Subject English Medium Public Answer Key- March 2019 - by QB Admin - May 19, 2021 - View & Read

Stateboard 12th Standard Maths Subject Public Question Paper - March 2019 - by QB Admin - May 19, 2021 - View & Read

12th Standard Maths Reduced Syllabus 2020-21 - by QB Admin - Apr 23, 2021 - View & Read

12th Standard Maths English Medium Application of Matrices and Determinants Important Questions - by Question Bank Software - Apr 23, 2021 - View & Read

1)
If \(\rho\) (A) ≠ \(\rho\) ([AIB]), then the system is _____________
2)
If A = \(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \), then adj(adj A) is
3)
If A is a matrix of order m \(\times\) n, then \(\rho\) (A) is _________
4)
If A = [2 0 1] then the rank of AA^T is ______
5)
The two lines are Parallel (non-coincident) then the solution is ___________

x	-2	-1	0	1	2
f(x)	k	2k	3k	4k	5k

x	1	2	3	4	5	6
f(x)	c²	2c²	3c²	4c²	c	2c

x	1	2	3	4	5	6
f(x)	c²	2c²	3c²	4c²	c	2c

x	-2	-1	0	1	2
f(x)	k	2k	3k	4k	5k

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