### 12th Standard CBSE Maths Study material & Free Online Practice Tests - View Model Question Papers with Solutions for Class 12 Session 2019 - 2020 CBSE [ Chapter , Marks , Book Back, Creative & Term Based Questions Papers - Syllabus, Study Materials, MCQ's Practice Tests etc..]

#### 12th CBSE Mathematics - Linear Programming Six Marks Model Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

A manufacture produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours for day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine, Each unit of product A is sold at Rs 7 profit and  that of B at a profit of Rs 4. Find the production level per day for maximum profit graphically.

• 2)

A manufacturer produces nuts and bolts. It takes 2 hours work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 2 hours on machine B to produce a package of bolts. He earns a profit of Rs 24 per package on nuts and Rs 18 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines both for at the most 10 hours a days. Make an LPP from above and solve it graphically?

• 3)

Find graphically, the maximum value of Z = 2x + 5y, subject to constraints given below: 2x + 4y $\le$ 8 $\Rightarrow$ x + 2y $\le$ 4
3x + y $\le$ 6
x + y $\le$ 4
x $\\ \ge$ 0, y  $\\ \ge$ 0

• 4)

A company manufactures two types of sweaters, type A and B. It costs Rs 360 to make one unit of type A and Rs 120 to make a unit of type B. The company can make at most 300 sweaters and can spend Rs 72,000 a day. The number of sweaters of type A cannot exceed the number of type B by more than 100. The company makes a profit of Rs 200 on each unit of type A. The company charging a nominal profit of Rs 20 on a unit of type B. Using LPP, solve for max. profit.

• 5)

A decorative item dealer deals in two items A and B. He has Rs 15,000 to invest and a space to store at the most 80 pieces. Item A cost him Rs 300 and item B costs him Rs 150. He can sell items A and B at respective profits of Rs 50 and Rs 28. Assuming he can sell all he buys, formulate the linear programming problem in order to maximize his profit and solve it graphically.

#### 12th Standard CBSE Mathematics - Three Dimensional Geometry Six Marks Model Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

If lines $\frac { x-1 }{ 2 } =\frac { y+1 }{ 3 } =\frac { z-1 }{ 4 }$ and $\frac { x-3 }{ 1 } =\frac { y-k }{ 32 } =\frac { z }{ 1 }$intersect, then find the value of k and hence find the equation of plane containing these lines.

• 2)

Find the distance of the point$3\hat { i } -2\hat { j } +\hat { k }$from the plane 3x+y-z+2=0 measured parallel to the line$\frac { x-1 }{ 2 } =\frac { y+2 }{ -3 } =\frac { z-1 }{ 1 }$ . Also, find the foot of the . Also, find the foot of the perpendicular from the given point upon the given plane.

• 3)

Find the equation of plane passing through the line of intersection of the planes$\vec { r } .\left( 2\hat { i } +3\hat { j } -\hat { k } \right) =-1$ and $\vec { r } .\left( \hat { i } +\hat { j } -2\hat { k } \right) =0$and passing through the point (3,- 2, -1). Also, find the angle between the two given planes

• 4)

Find the distance between the point (7,2,4) and the plane determined by the points A(2, 5, - 3) B(- 2, - 3, 5) and C(5, 3, - 3).

• 5)

Find the distance of the point (2, 12, 5) from the point of intersection of the lines $\vec { r } =\left( 2\hat { i } -4\hat { j } +2\hat { k } \right) +\lambda \left( 3\hat { i } +4\hat { j } +2\hat { k } \right)$ and the
plane .

#### CBSE 12th Mathematics - Probability Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of success.

• 2)

12 cards, numbered 1 to 12, are placed in box mixed up thoroughly and then a card is drawn at random from the box. If it is known that the number on the drawn card is more than 3, find the probability that it is an even number.

• 3)

In a bulb factory machines A,B and C manufacture 60%,30% and 10% bulbs respectively. 1%,2% and 3% of the bulbs produced respectively by A,B and C are found to be defective. Find the probability that this bulb was produced by the machine A.

• 4)

The probabilities of A,B,C solving a problem are $\frac { 1 }{ 3 } ,\frac { 2 }{ 7 } and\frac { 3 }{ 8 }$ respectively. If all the three try to solve the problem simultaneously, find the probability that exactly one of them can solve it.

• 5)

A die is throewn again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.

#### 12th Standard CBSE Mathematics - Three Dimensional Geometry Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Find the vector equation of the plane through the intersection of the planes $\vec { r } .(\hat { i } +\hat { j } +\hat { k } )=6$ and $\vec { r } .(2\hat { i } +3\hat { j } +4\hat { k } )=-5$ and the point (1,1,1)

• 2)

Find the value of $\lambda$ so that the lines $\frac { 1-x }{ 3 } =\frac { 7y-14 }{ 2\lambda } =\frac { 5z-10 }{ 11 }$ and $\frac { 7-7x }{ 3\lambda } =\frac { y-5 }{ 1 } =\frac { 6-z }{ 5 }$ are perpendicular to each other.

• 3)

Find the equation of the plane passing through the point (-1,3,2) and perpendicular to each of the planes x+2y+3z=5 and 3x+3y+z=0

• 4)

Find the shortest distance between the following two lines:

$\overrightarrow { r } =(1+\lambda )\acute { i } +(2-\lambda )\acute { j } +(\lambda +1)\acute { k } \\ \vec { r } =(2\acute { i } -\acute { j } -\acute { k } )+\mu (2\acute { i } +\acute { j } +2\acute { k } )$

• 5)

Find the equation of the plane determined by the points A(3,-1,2), B(5,2,4) and C(-1,-1,6). Also find the distance of the point P(6,5,9) from the plane.

#### 12th Standard CBSE Mathematics - Vector Algebra Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Find a unit vector perpendicular to each of the vectors $\overrightarrow { a } +\overrightarrow { b } \quad and\quad \overrightarrow { a } -\overrightarrow { b }$  where $\overrightarrow { a } =3\hat { i } +2\hat { j } +2\hat { k } \quad and\quad \overrightarrow { b } =\hat { i } +2\hat { j } -2\hat { k }$

• 2)

If two vectors $\overrightarrow { a } \quad and\quad \overrightarrow { b }$ are such that $\left| \overrightarrow { a } \right| =2,\left| \overrightarrow { b } \right| =1\quad and\quad \overrightarrow { a } .\overrightarrow { b } =1$ then find the value of ($3\overrightarrow { a } -5\overrightarrow { b }$).($2\overrightarrow { a } +7\overrightarrow { b }$).

• 3)

Using vectors find the area of the triangle with vertices A(1,1,2),B(2,3,5) and C(1,5,5).

• 4)

If $\overrightarrow { \alpha } =3\hat { i } +4\hat { j } +5\hat { k } and\quad \overrightarrow { \beta } =2\hat { i } +\hat { j } -4\hat { k }$ the express $\overrightarrow { \beta }$ in the form $\overrightarrow { \beta } ={ \overrightarrow { \beta } }_{ 1 }+{ \overrightarrow { \beta } }_{ 2 }$  where ${ \overrightarrow { \beta } }_{ 1 }$ is parallel to $\overrightarrow { \alpha }$  and ${ \overrightarrow { \beta } }_{ 2 }$ is perpendicular to $\overrightarrow { \alpha }$

• 5)

If $\overrightarrow { a } \quad and\quad \overrightarrow { b }$ are two vectors such that $\left| \overrightarrow { a } +\overrightarrow { b } \right| =\left| \overrightarrow { a } \right|$ then prove that vector $2\overrightarrow { a } +\overrightarrow { b }$ is perpendicular to vector $\overrightarrow { b }$

#### CBSE 12th Mathematics - Linear Programming Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

One kind of cake requires 200 g of flour and 25 g of fat, and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes. Formulate the above as a linear programming problem and solve graphically.

• 2)

(Manufacturing Problem) A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is atmost 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs.300 and that on a chain is Rs.190, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an LPP and solve it graphically.

• 3)

A merchant plans to sell two types of personal computers-a desktop model and a portable model that will cost Rs.25,000 and Rs.40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchantshould stock to get maximum profit if he does not want to invest more than Rs.70 lakhs and his profit on the desktop model is Rs.4,500 and on the portable model is Rs.5,000. Make an LPP and solve it graphically.

• 4)

A diet for a sick person must contain at least 4,000 units of vitamins, 50 units of minerals and 1,400 calories. Two foods X and Y are available at a cost of Rs.4 and Rs.3 per unit respectively. One unit of the food X contains 200 units of vitamins, 1 unit of minerals and 40 calories, whereas one unit of food Y contains 100 units of vitamins, 2 units of minerals and 40 calories. Find what combination of X and Y should be used to have least cost, satisfying the requirements?

• 5)

A firm deals with two kinds of fruit juices- pineapple and orange juice. These are mixed and two mixtures are sold as soft drinks A and B. One tin of A requires 4 litres of pineapple and 1 litre of orange juice. One tin of B requires 2 litres of pineapple and 3 litres of orange juice. The firm has only 46 litres of pineapple juice and 24 litres of orange juice. Each tin of A and B are sold at a profit of Rs.4 and Rs.3 respectively. How many tins of each type should the firm produce to maximise the profit? Solve the problem graphically.

#### CBSE 12th Mathematics - Differential Equations Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Solve the differential equation:$\frac { dy }{ dx } +1={ e }^{ x+y }$

• 2)

Solve the differential equation: y - x $\frac { dy }{ dx } =a\left( { y }^{ 2 }+{ x }^{ 2 }\frac { dy }{ dx } \right) ,$ where x = a, y = a.

• 3)

Solve the differential equation: sec2 y (1+x2) dy 2x tan y dx = 0, given that $y=\frac {\pi}{4}$, when x = 1.

• 4)

Solve the differential equation:​​​​​​​ $\frac {dy}{dx}$  + y cot x = 2 cos x .

• 5)

Solve the differential equation:​​​​​​​ $\frac {dy}{dx} = tan (x + y)$

#### 12th Standard CBSE Mathematics - Application of Integrals Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Draw the rough sketch of y2 = x + 1 and y2 = x + 1 and determine the area enclosed by the two curves.

• 2)

Using integration, find the area of the quadrant of the circle x2 + y2 = 4

• 3)

Find the area bounded by y = x, the x - axis and the lines x = -1 and x = 2.

• 4)

Calculate the area under the curve:
$y=\sqrt { 2 } x$ between the ordinates x =0 and x = 1.

• 5)

Find the area lying above the x - axis and included between the circle x2 + y2 = 8x and the parabola y2 = 4x.

#### 12th Standard CBSE Mathematics - Integrals Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Evaluate the integral $\int {x^2cot^{-1}x}\ dx$

• 2)

Evaluate the integral $\int {1\over a^2 sin^2\ x+b^2 cos^2 x}dx$

• 3)

Evaluate the integral: $\int {x^2+4\over x^4+16}dx.$

• 4)

Evaluate the integral: $\int {(x-4)e^x\over(x-2)^3}dx$

• 5)

Evaluate the integral: $\int{dx\over\sqrt{5-4x-2x^2}}$

#### 12th Standard CBSE Mathematics - Vector Algebra Four Mark Model Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Find the projection of $\overrightarrow { b } +\overrightarrow { c } \quad on\quad \overrightarrow { a }$ where $\overrightarrow { a } =2\hat { i } -2\hat { j } +\hat { k } ,\overrightarrow { b } =\hat { i } +2\hat { j } -2\hat { k }$ and $\overrightarrow { c } =2\hat { i } -\hat { j } +4\hat { k }$

• 2)

If $\overrightarrow { a } +\overrightarrow { b } +\overrightarrow { c } =0$ and $\left| \overrightarrow { a } \right| =3,\left| \overrightarrow { b } \right| =5\quad and\quad \left| \overrightarrow { c } \right| =7$ show that the angle between $\overrightarrow { a } \quad and\quad \overrightarrow { b }$ is 60o.

• 3)

If $\overrightarrow { a } =\hat { i } +\hat { j } +\hat { k } ,\overrightarrow { b } =4\hat { i } -2\hat { j } +3\hat { k }$  and $\overrightarrow { c } =\hat { i } -2\hat { j } +\hat { k }$ find a vector of a magnitude 6 units which is parallel to the vector $2\overrightarrow { a } -\overrightarrow { b } +3\overrightarrow { c }$

• 4)

Using vectors find the area of the triangle with vertices A(1,1,2),B(2,3,5) and C(1,5,5).

• 5)

If $\overrightarrow { a } \times \overrightarrow { b } =\overrightarrow { a } \times \overrightarrow { c } \quad and\quad \overrightarrow { a } \times \overrightarrow { c } =\overrightarrow { b } \times \overrightarrow { d }$ prove that $\overrightarrow { a } -\overrightarrow { d }$ is parallel to $\overrightarrow { b } -\overrightarrow { c }$ provided $\overrightarrow { a } \neq \overrightarrow { d } \quad and\quad \overrightarrow { b } \neq \overrightarrow { c }$

#### 12th CBSE Mathematics - Differential Equations Six Mark Model Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Obtain the differential equation of all the circles of radius r.

• 2)

Find the particular solution of the differential equation (tan-1 y - x) dy = (1 + y2) dx, given that when x = 0, y = 0.

• 3)

Find the particular solution of the differential equation $\frac { dy }{ dx } =\frac { xy }{ { x }^{ 2 }+{ y }^{ 2 } }$ given that y = 1, when x = 0.

• 4)

Solve the differential equation ${ x }^{ 2 }dy+\left( xy+{ y }^{ 2 } \right) dx=0$given y = 1, when x = 1.

• 5)

Find the particular solution of the following differential equation given that : y = 0, when x = 1 (x2 + xy) dy = (x2 + y2) dx.

#### CBSE 12th Mathematics - Application of Derivatives Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Radius of a variable circle is changing at the rate of 5cm/sec.What is the radius of the circle at the times when area is changing at the rate of 100cm2 /sec?

• 2)

The bottom of a rectangular swimming tank is 50cm $\times$20cm.Water is pumped into the tank at the rate of 500c.c/min.Find the rate at which the level of water in the tank rising

• 3)

A water tank has a shape of an inverted right circular cone with its axis vertical and vertex lowermost.It semi-vertical angle is tan-1(0.5).Water is poured into it at a constant rate of 5cubic meter per minute. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 10m.

• 4)

A woman is moving away from a tower 41.5m high at the rate of 2m/sec.Find the rate at which the angle of elevation of the top of the tower is changing,when she is at a distance of 30m from the foot of the tower.Assume that eye level is 1.5m from the ground.

• 5)

Find the intervals in which the function f given by f(x) =sin x+cos x,0$\le$x$\le$2$\pi$,is strightlly increasing or strightly decreasing.

#### CBSE 12th Mathematics - Continuity and Differentiability Four Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

If y=x2, prove that $\frac { d^{ 2 }y }{ dx^{ 2 } } \frac { 1 }{ y } \left( \frac { dy }{ dx } \right) ^{ 2 }-\frac { y }{ x } =0$

• 2)

If xmyn=(x+y)m+n,prove that $\frac { dy }{ dx } =\frac { y }{ x }$

• 3)

Differentiate log $\left( x+\sqrt { 1+{ x }^{ 2 } } \right)$

• 4)

Differentiate $\frac { \sqrt { a+x } +\sqrt { a-x } }{ \sqrt { a+x } -\sqrt { a-x } }$

• 5)

Find $\frac { dy }{ dx }$ for sin (xy)+$\frac { x }{ y }$ = x2 - y

#### 12th Standard CBSE Mathematics - Determinants Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Using properties of determinants solve for $x\begin{vmatrix} a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x \end{vmatrix}=0$

• 2)

Using properties of determinants solve the following for X:

$\begin{vmatrix} x-a & x & x \\ x & x+a & x \\ x & x & x+a \end{vmatrix}=0,a\neq 0$

• 3)

Use product $\begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}\begin{bmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{bmatrix}$ to solve the system of equations.

x-y+2z=1; 2y-3z=1; 3x-2y+4z=2

• 4)

solve for x,y,z

${2\over x}+{3\over y}+{10\over z}=4$${4\over x}-{6\over y}+{5\over z}=1$;${6\over x}+{9\over y}-{20\over z}=2$

• 5)

A school wants to award its student or the values of Honesty, Regularity and Hard Work with a total cash award of Rs.6,000. Three times the award money for Hard work added to that given for Honesty amounts to Rs.11,000. The award money given for Honesty and Hard work together is double the one given for regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hard work, suggest one more value which the school must include for awards.

#### 12th CBSE Maths - Application of Integrals Six Mark Model Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Using integration, find the area of the triangle formed by positive x-axis and tangent and normal to the circle x+y = 4 at (1, $\sqrt3$).

• 2)

Using integration, find the area of the $\triangle$PQR co-ordinates whose vertices are P(2, 0), Q(4, 5) and R(6, 3).

• 3)

Using integration, find the area of the region enclosed between the two circles x2 + y2 = 9 and (x32)2 + y2 = 9.

• 4)

Find the area of the region bounded by the two parabolas y2 = 4ax and x2 = 4ay, when a > 0.

• 5)

Using integration find the area of the region given by {(x,y):(x2$\le$y$\le$ |x|)}.

#### 12th CBSE Maths - Integrals Six Mark Model Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Find : $\int { \frac { sinx }{ sin^{ 3 }x+cos^{ 3 }x } } dx$

• 2)

Evaluate : $\int { \frac { 8 }{ (x+2){ (x }^{ 2 }+4) } } dx$

• 3)

Find : $\int { \frac { sin^{ -1 }\sqrt { x } -cos^{ -1 }\sqrt { x } }{ sin^{ -1 }\sqrt { x } +cos^{ -1 }\sqrt { x } } } dx,x\epsilon \left[ 0,1 \right]$

• 4)

Evaluate : $\int { \frac { { x }^{ 2 }+x+1 }{ (x+2)({ x }^{ 2 }+1) } } dx$

• 5)

Evaluate: $\int _{ 2 }^{ 5 }{ \left( { x }^{ 2 }+3 \right) } dx$

#### CBSE 12th Mathematics - Matrices Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Using elementary transformations, find the inverse of the matrix

$\left[ \begin{matrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{matrix} \right]$.

• 2)

If $A=\left[ \begin{matrix} \cos { \alpha } & \sin { \alpha } \\ -\sin { \alpha } & \cos { \alpha } \end{matrix} \right] ,$ then show that ${ A }^{ 2 }=\begin{bmatrix} \cos { 2\alpha } & \sin { 2\alpha } \\ -\sin { 2\alpha } & \cos { 2\alpha } \end{bmatrix}$

• 3)

If $A=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ and $l=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, prove that $(al+bA)^{ 3 }={ a }^{ 3 }l+{ 3a }^{ 2 }bA.$

• 4)

Let f(x) = x2 - 5x+6 find f(A) If, A =$\left[ \begin{matrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{matrix} \right]$

• 5)

If, A = $\left[ \begin{matrix} a & 0 \\ 1 & 1 \end{matrix} \right]$and B = $\left[ \begin{matrix} 1 & 0 \\ 5 & 1 \end{matrix} \right]$ find all those values of a for which A = B

#### CBSE 12th Mathematics - Inverse Trigonometric Functions Four Marks and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Find the value of ${ tan }^{ -1 }\left( \frac { X }{ Y } \right) -{ tan }^{ -1 }\left( \frac { X-{ Y } }{ X-Y } \right)$

• 2)

Solve for X, 2tan-1(sinX)=tan-1(2secX),$X\neq \frac { \pi }{ 2 }$

• 3)

Find the value of cos $cos\left( { 2cos }^{ -1 }x+{ sin }^{ -1 }x \right)$ at x = $\frac { 1 }{ 5 }$

• 4)

What is the +ive integral solution of ${ tan }^{ -1 }x+{ cos }^{ -1 }\frac { y }{ \sqrt { 1+{ y }^{ 2 } } } ={ sin }^{ -1 }\frac { 3 }{ \sqrt { 10 } }$

• 5)

Find value of cos-1 [cos {2cot-1($\sqrt { 2 }$-1)}]

#### 12th Standard CBSE Mathematics - Relations and Functions Four and Six Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Consider $f:R_+\rightarrow[-5,\infty)$ given by f(x)=9x2+6x-5. Show that f is invertible with $f^{-1}(y)={(\sqrt{(y+6)}-1)\over3}$

• 2)

Consider the binary operation * on the set {1,2,3,4,5} defined by a*b=min{a,b}. Write the operation table of the operation *.

• 3)

Find $fof^{ -1 }$ and $f^{ -1 }$ of for the function:
$f(x)=\frac { 1 }{ x } ,x\neq 0$. Also prove that $fof^{ -1 }$$f^{ -1 }$ of .

• 4)

Let R be a relation defined on the set of natural numbers Nas follow:
R = {(x, y) : x $\in$ N, y  $\in$N and 2x + y = 24}
Find the domain and range of the relation R. Also, find if R is an equivalence relation or not.

• 5)

LetXbe a non-empty set, Let * be a binary operation on the power set P(X) defined by A * B = A n B. What is the identify element for the operation * ? Given X is a set of people in a locality, A is a set of children and B is a set of citizens aged above 75 years in the same locality. Is * an invertible binary Feration for these sets as defined above?
What qualities would you suggest that elements of A should have towards elements of B?

#### CBSE 12th Standard Mathematics - Application of Derivatives Six Mark Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Find the equation of the tangent line to the curve $y=x^2-2x+7$ which is
(i) parallel to the line $2x-y+9=0$
(ii) perpendicular to the line $5y-15x=13$

• 2)

Find the intervals in which $f(x)-\sin3x-\cos3x,0<x<\pi$ is strictly increasing or strictly decreasing.

• 3)

A tank with rectangular base and rectangular sides open at the top is to be constructed so that its depth is 3 mand volume is 75cm3.Ifbuildingof tank costs Rs 100 per square metre for the base and Rs 50 per square metre for the sides, find the cost of least expensive tank.

• 4)

A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs 5 per cm2 and the material for the sides cost Rs 2.50/cm2. Find the least cost of the box.

• 5)

Show that semi-vertical angle of a cone of maximum volume and given slant height is $\cos^{-1}(\frac{1}{\sqrt3})$

#### 12th CBSE Mathematics - Continuity and Differentiability Six Mark Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Find the value of k, so that the function

$f(x)=\begin{cases} { kx }^{ 2 },\quad if\quad x\ge 1 \\ 4\quad ,\quad if\quad x<1 \end{cases}$ is continuous at x=1.

• 2)

For what value of is the function defined by

$f(x)=\begin{cases} \lambda ({ x }^{ 2 }-2x)\quad ,\quad if\quad x\le 0 \\ 4x+1\quad \quad \quad ,\quad if\quad x>0 \end{cases}continuious\quad at\quad x=0?$

• 3)

For what value of k is the function $f(x)=\begin{cases} \frac { { e }^{ x }+{ e }^{ -x }-2 }{ x^{ 2 } }\ \ ,if\quad x\neq 0 \\ \quad 4k \ \ \ \ \ \ \ \ \ , if\quad x=0 \end{cases}$is continuous at x=0?

• 4)

For what value of k, is the following function continuous at x=0?

$f(x)=\begin{cases} \frac { 1-cos\quad 4x }{ 8x^{ 2 } } \ , \ \ if\quad x\neq 0 \\ \quad \quad \quad k \ \ \ , \ \ if\quad x=0 \end{cases}$

• 5)

Find the derivative of each of the following function w.r.t. x, or find $\frac { dy }{ dx }$:

$y=\sqrt { \frac { sec \ \ x-1 }{ sec \ \ \ x+1 } }$

#### 12th CBSE Mathematics - Determinants Six Mark Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Using properties of determinants, show that triangle ABC is isosceles if: $\left| \begin{matrix} 1 & 1 & 1 \\ 1+cosA & 1+cosB & 1+cosC \\ \cos ^{ 2 }{ A } +cosA & \cos ^{ 2 }{ B } +cosB & \cos ^{ 2 }{ B } +cosC \end{matrix} \right| =0$

• 2)

Using properties of determinants, prove that:
$\left| \begin{matrix} { (y+z) }^{ 2 } & xy & zx \\ xy & { (x+z) }^{ 2 } & yz \\ xz & yz & { (x+y) }^{ 2 } \end{matrix} \right| =2xyz{ (x+y+z) }^{ 3 }.$

• 3)

If $A=\left[ \begin{matrix} 1 & 2 & -3 \\ 2 & 3 & 2 \\ 3 & -3 & -4 \end{matrix} \right]$find A-1 . Hence solve the following system of equations:
x+2y-3z=-4, 2x+3y+2z=2,3x-3y-4z=11.

• 4)

Determine the product $\left[ \begin{matrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{matrix} \right] \left[ \begin{matrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{matrix} \right] ,$  and use it to solve the system of equations: x-y+z=4, x-2y-2z=9, 2x+y+3z=1.

• 5)

If $A=\left[ \begin{matrix} 1 & 2 & 5 \\ 1 & -1 & -1 \\ 2 & 3 & -1 \end{matrix} \right] ,$ find A-1 . Hence solve the following system of equations:
x+2y+5z=10, x-y-z=-2, 2x+3y-z=-11.

#### CBSE 12th Standard Mathematics - Matrices Six Mark Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

If $A=\left( \begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix} \right)$, prove that A3 - 6A2 + 7A + 2I = 0

• 2)

If $A=\left( \begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix} \right)$ and A3 - 6A2 + 7A + kI3 = 0, find k.

• 3)

Using elementary column operations, find the inverse of the following matrix :
$\left[ \begin{matrix} -1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{matrix} \right]$

• 4)

If A = $\left[ \begin{matrix} 3 & 1 \\ 7 & 5 \end{matrix} \right]$ find x, y such that A2 +xI = yA Hence find A-1

• 5)

If A(x1,y1), B(x2,y2) and C(x3,y3) are the vertices of an equilateral triangle with each side equal to ‘a’ units, prove that, $\left[ \begin{matrix} { x }_{ 1 } & { y }_{ 1 } & { z }_{ 1 } \\ { x }_{ 2 } & { y }_{ 2 } & { z }_{ 2 } \\ { x }_{ 3 } & { y }_{ 3 } & { z }_{ 3 } \end{matrix} \right]$ = $\sqrt { 3 }$a2

#### 12th Standard CBSE Mathematics - Probability Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

An urn contains 10 black and 5 white balls.Two balls are drawn from the urn one after the other without replacement.What is the probability that both drawn balls are black?

• 2)

Three cards are drawn successively, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are king and the third card drawn is an ace?

• 3)

Six balls are drawn successively from an urn containing 7 red and 9 black balls.Tell whether or not the trials of drawing black balls are Bernoulli trials when after each draws the ball drawn is:
(i) replaced (ii) not replaced in the urn.

• 4)

If a machine is correctly set up, it produces 90%  acceptable item.If it is incorrectly set up,it produces only 40% acceptable items.Past experience shows that 80% of the setups are correctly done.If after a certain setup, the machine produces 2 acceptable items, find the probability that the machine is correctly set up.

• 5)

An instructor has a question bank consiting of 300 easy True/False questions,200 difficult True/False questions,500 easy multiple choice question and 400 difficult multiple choice questions.If a question is selected at random from the question bank,waht is the probability that it will be an easy question,given that it is a multiple choice question?

#### 12th Standard CBSE Mathematics - Three Dimensional Geometry Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

If a line has direction -ratios<2,-1,-2>,determine its direction cosines.

• 2)

Find the direction cosines of the line passing through the two points(-2,4,-5) and (1,2,3)

• 3)

Find the direction-cosines of x,y, and z-axis.

• 4)

Find the direction cosines of the unit  vector perpendicular to the plane $\vec { r } .\left( 6\hat { i } -3\hat { j } -2\hat { k } \right) +1=0$
through the origin.

• 5)

Find the distance of a point (2, 5, - 3) from the plane:
$\hat { r } .(6\hat { i } -3\hat { j } +2\hat { k }) =4$

#### 12th Standard CBSE Mathematics - Vector Algebra Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Write two different having same magnitude.

• 2)

Find the values of x and y so that the vector  $\overset { \wedge }{ 2i } +\overset { \wedge }{ 3j } \quad and\quad \overset { \wedge }{ xi } +\overset { \wedge }{ yj }$   are equal.

• 3)

Find out the vector in the direction of vector $\overset { \rightarrow }{ PQ }$  where P and Q are the points(1, 2, 3) and (4, 5, 6) respectively.

• 4)

Find a vecto in a direction of vector $\overset { \wedge }{ 5i } -\overset { \wedge }{ j } -\overset { \wedge }{ 2k }$,which has magnitude 8 units.

• 5)

Find the direction-consines of the vector $\overset { \wedge }{ i } +\overset { \wedge }{ 2j } +\overset { \wedge }{ 3k }$ .

#### 12th Standard CBSE Mathematics - Differential Equations Three Marks - by Shalini Sharma - Udaipur - View & Read

• 1)

Find the general solution of differential equation $\log { \left( \frac { dy }{ dx } \right) } =x+1$

• 2)

Find the general solution of differential equation
$\frac { dy }{ dx } +y={ e }^{ -x }$

• 3)

Find the general solution of the differential equation: $(x-y){dy\over dx}={x+2y}$

• 4)

Form the differential equation of the family of parabolas having vertex at the orgin and axis along positive y-axis.

• 5)

Show that the differential equation 2y ex/y dx + (y-2x ex/y) dy = 0 is homogeneous. Find the particular solution of this differential equation, given that x =0 when y=1.

#### 12th Standard CBSE Mathematics - Application of Integrals Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x - axis in the first quadrant.

• 2)

Find the area of the region in the first quadrant enclosed by x - axis and $x=\sqrt { 3 } y$ by the circle ${ x }^{ 2 }+{ y }^{ 2 }=4$.

• 3)

Find the area of the region bounded by the parabola y = x2 and $y=\left| x \right|$

• 4)

Find the area bounded by the curve x2 = 4y and the line x = 4y - 2.

• 5)

Find the area under the given curves and given lines :
y = x2, x = 1, x = 2 and x - axis

#### CBSE 12th Mathematics - Linear Programming Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

The objective function is maximum or minimum, which lies on the boundary of the feasible region.

• 2)

Solve the following linear programming problem graphically:
Maximise Z=4x+y subject to the constraints:
$\\ x+y\le 50,3x+y\le 90,x\ge 0,y\ge 0.$

• 3)

(Manufacturing Problem) A manufacturer has Three machines I,II and III installed in his factory. Machines I and II are capable of being operated for at most 12 hours whereas machine III must be operated for at least 5 hours a day. She produces only two items M and N each on the three machines are given in the following table:

 Items Number of hours required on machines I II III M 1 2 1 N 2 1 1.25

She makes a profit of RS.600 and Rs 400 on items M and N respectively. How many of each item should she produce so as to maximise her profit assuming that she can sell all the items that she produced? What will be the maximum profit?

• 4)

Maximise Z=3x+4y subject to the constraints:
$x+y\le 4,x\ge 0,y\ge 0.$

• 5)

Minimise Z=-3x+4y
Subject to $x+2y\le 8,\quad 3x+2y\le 12,\quad x\ge 0,y\ge 0.$

#### CBSE 12th Mathematics - Integrals Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Write an antiderivative for each of the followings functions, using method of inspection :

(i) $\cos { 2x }$      (ii)${ 3x }^{ 2 }+{ 4x }^{ 3 }$      (iii)  $\frac { 1 }{ x } ,x\neq 0$

• 2)

Find the following integrals:

$(i)\quad \int { \frac { { x }^{ 3 }-1 }{ { x }^{ 2 } } } dx\quad \quad \quad \quad (ii)\quad \int { { (x }^{ 2/3 }+1) } dx\quad (iii)\quad \int { { (x }^{ 3/2 }+{ 2e }^{ x }-\frac { 1 }{ x } ) } dx$

• 3)

Find: $\int { \frac { { x }^{ 2 }+1 }{ { x }^{ 2 }-5x+6 } dx }$

• 4)

Find: $\int { \frac { { x }^{ 2 } }{ ({ x }^{ 2 }+1)({ x }^{ 2 }+4) } } dx$

• 5)

Find:$\int { \sqrt { 3-2x-{ x }^{ 2 } } } dx.$

#### CBSE 12th Mathematics - Application of Derivatives Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

The volume of a cube is increasing at the rate of 9 cubic centimetres per second.How fast is the surface area increasing when the length of an edge is 10 centimetres?

• 2)

Show that the function  given by:
$f(x)=7x-3$ is strictly increasing on R.

• 3)

The total cost C(x) associated with the production of 'x' units of an item is given by:
$C(x)=0.007x^{ 3 }-0.003x^{ 2 }+15x+4000$
Find the marginal cost when 17 units are produced.

• 4)

The total revenue received from the sale of 'x' units  of a product ,is given by:
$R(x)=13x^{ 2 }+26x+15$

Find the marginal revenue when x=7

• 5)

Prove that the function given by
$f(x)=x^{ 3 }-3x^{ 2 }+3x-100\quad$

#### 12th Standard CBSE Mathematics - Relations and Functions Five Mark Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Are-:$R\times R\rightarrow$ and $\div :R\times R$ commutative binary operations?

• 2)

Let f:N$\rightarrow$N be defined by$f(n)=\begin{cases} \frac { n+1 }{ 2 } ,if\quad n\quad is\quad odd. \\ \frac { n }{ 2 } ,\quad if\quad n\quad is\quad even \end{cases}\quad for\quad all\quad n\in N.$ State whether the functions f is onto, one-one or bijective.Justify your answer

• 3)

Let f:$R\rightarrow R$ be defined as f(x)=10x+7. Find the function g:$R\rightarrow R$ such that gof=og=IR

• 4)

Let A=R- (3) ,B=R-[1] Let $f:A\rightarrow B$ be defined by $f(x)=\left( \frac { x-2 }{ x-3 } \right) \forall x\in A$ Then show that f is bijective Hence find  $f^{ -1 }(x)$

• 5)

L~tXbe a non-empty set and * be a binary operation on P(X) (the power set of set X) defined by
$A*B=A\bigcup B\quad for\quad all\quad A,B\in P(x)$
Prove that '*' is both commutative and associative on P(X). Find the identity element with respect to on P(X). Also, show that <1>E P(X) is the only invertible element of P(X).

#### 12th CBSE Mathematics - Inverse Trigonometric Functions Five Mark Question Paper - by Asha Mady - Secunderabad - View & Read

• 1)

Show that:
${ sin }^{ -1 }(2x\sqrt { 1-{ x }^{ 2 } } )={ 2sin }^{ -1 }x,\frac { 1 }{ \sqrt { 2 } } \le x\le \frac { 1 }{ \sqrt { 2 } }$

• 2)

Find the value of cos $({ sec }^{ -1 }x+{ cosec }^{ -1 }x),\left| x \right| \ge 1$

• 3)

Find te principal values of the following: ${ sec }^{ -1 }\left( \frac { 2 }{ \sqrt { 3 } } \right)$

• 4)

Solve for X, $\quad { tan }^{ -1 }\left( \frac { X-1 }{ X-2 } \right) +{ tan }^{ -1 }\left( \frac { X+1 }{ X+2 } \right) =\frac { \pi }{ 4 }$

• 5)

Solve the following equation: $cos\left( { tan }^{ -1 }X \right) =sin\left( { cot }^{ -1 }\frac { 3 }{ 4 } \right)$

#### 12th Standard CBSE Mathematics - Continuity and Differentiability Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Prove that the identity function on real numbers given by:
f(x)=x is continuous at every real number.

• 2)

Discuss the continuity of the function f defined by:
$f(x)={ x }^{ 3 }+{ x }^{ 2 }-1$

• 3)

Find the derivative of tan(2x+3).

• 4)

Differentiate $sin(cos({ x }^{ 2 }))\quad with\quad respect\quad to\quad x.$

• 5)

$Find\quad \frac { dy }{ dx } \quad if\quad x-y=\pi$

#### 12th Standard CBSE Mathematics - Determinants Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Prove that $\left| \begin{matrix} 1! & 2! & 3! \\ 2! & 3! & 4! \\ 3! & 4! & 5! \end{matrix} \right| =4!$

• 2)

If $f(x)=\left| \begin{matrix} a & -1 & 0 \\ ax & a & -1 \\ { a }x^{ 2 } & ax & a \end{matrix} \right|$, using properties of determinants, find the value of f(2x)-f(x).

• 3)

Prove that: $\left| \begin{matrix} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2ab \end{matrix} \right| =2(a+b+c{ ) }^{ 3 }$

• 4)

Using the properties of determinants, solve the following for 'x' $\left| \begin{matrix} x+2 & x+6 & x-1 \\ x+6 & x-1 & x+2 \\ x-1 & x+2 & x+6 \end{matrix} \right| =0$

• 5)

Prove that $\left| \begin{matrix} a & b & c \\ { a }^{ 2 } & { b }^{ 2 } & c^{ 2 } \\ b+c & c+a & a+b \end{matrix} \right| =(a+b+c)(a-b)(b-c)(c-a)$

#### CBSE 12th Mathematics - Matrices Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

• 2)

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

• 3)

If $x\left[ \begin{matrix} 2 \\ 3 \end{matrix} \right] +y\left[ \begin{matrix} -1 \\ 1 \end{matrix} \right] =\left[ \begin{matrix} 10 \\ 5 \end{matrix} \right]$ , find the values of x and y.

• 4)

Given:$3\begin{bmatrix} x & y \\ z & w \end{bmatrix}=\begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix}+\begin{bmatrix} 4 & x+y \\ z+w & 3 \end{bmatrix},$ find the values of x, y, z and w.

• 5)

Given: $3\begin{bmatrix} x & y \\ z & w \end{bmatrix}=\begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix}+\begin{bmatrix} 4 & x+y \\ z+w & 3 \end{bmatrix},$find the values of x,y,z and w.

#### CBSE 12th Standard Mathematics - Inverse Trigonometric Functions Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Find the principal value of ${ cot }^{ -1 }\left( -\frac { 1 }{ \sqrt { 3 } } \right)$

• 2)

Show that:
${ sin }^{ -1 }(2x\sqrt { 1-{ x }^{ 2 } } )={ 2sin }^{ -1 }x,\frac { 1 }{ \sqrt { 2 } } \le x\le \frac { 1 }{ \sqrt { 2 } }$

• 3)

Express $({ tan }^{ -1 }\left( \frac { cosx }{ 1-sinx } \right) ,-\frac { 3\pi }{ 2 }$   in the simplest form :

• 4)

Write ${ cot }^{ -1 }\left( \frac { 1 }{ \sqrt { { x }^{ 2 }-1 } } \right) ,x>1$ in the simplest form.

• 5)

Prove that :
${ tan }^{ -1 }x+{ tan }^{ -1 }\frac { 2x }{ { 1-x }^{ 2 } } ={ tan }^{ -1 }\left( \frac { { 3x-x }^{ 3 } }{ { 1-3x }^{ 2 } } \right) ,\left| x \right| <\frac { 1 }{ \sqrt { 3 } }$

#### 12th Standard CBSE Mathematics Unit 1 Relations and Functions Three Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Show that the function $f\quad :\quad N\rightarrow N$ given by $f(x)=2x$ is one-one but not onto.

• 2)

Show that the function:
$f:N\rightarrow N$ given by $f(2)=1$ and $f(x)=x-1$ , for every $x>2$ is onto but not one-one.

• 3)

Show that if $f$ :$A\rightarrow B$ and $B\rightarrow C$ are one-one, then $gof:A\rightarrow C$ is also one-one.

• 4)

Show that the relation R in the set {1,2,3} given by:
R={(1,1), (2,2), (3,3), (1,2), (2,3)} is reflexive but neither symmetric nor transitive.

• 5)

Show that $\vee :R\times R\rightarrow R$ given by $(a,b)\rightarrow$ max. {a,b} and $\wedge :R\times R\rightarrow R$ given by $(a,b)\rightarrow$ min. (a,b} are binary operations.

#### CBSE 12th Mathematics - Probability Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

One card is drawn is drawn from a pack of 52 cards. Find the probability of getting :
(a) a red card
(b) a jack of hearts
(c) a black face card
(d) a king.

• 2)

A bag contain 2 red, 6 black and 8 green balls. A ball is drawn at random from the bag. Find the probabilty:
(a) a red ball
(b) a black ball
(c) a green ball
(d) a non-red ball

• 3)

If E and F are two events such that $P(E)=\frac { 1 }{ 4 } ,$ $P(E)=\frac { 1 }{ 2 }$ and $P(E\cap F)=\frac { 1 }{ 8 }$,
find
(a) P(E or F)
(b) P(not E and not F).

• 4)

If P(E) =$\frac { 6 }{ 11 }$, P(F) =$\frac { 5 }{ 11 }$ and P(E$\cup$F)=$\frac { 7 }{ 11 }$ then find (a) P(E/F), (b) P(F/E)

• 5)

If P(E) =$\frac { 7 }{ 13 }$,P(F)=$\frac { 9 }{ 13 }$ and P(E$\cap$F)=$\frac { 4 }{ 13 }$,then evaluate :
(a) $P(\overline { E } /F)$  (b) $P(\overline { E } /F)$

#### 12th Standard CBSE Mathematics - Three Dimensional Geometry Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

If a lines makes angle 60° and 45° with the positive directions of x-axis and z-axis
respectively, then find the angle that it makes with the y-axis.

• 2)

If a line makes angle $\alpha ,\beta ,\gamma$ with the coordinates axis ,then find the value of cos $cos2\alpha +cos2\beta +cos2\gamma$

• 3)

Let $I_{ e }m_{ i }n_{ i }i=1,2,3$ be the direction cosines of three mutually perpendicular vector ion space$\quad \left[ \begin{matrix} { l }_{ 4 } & { m }_{ 1 } & { n }_{ 1 } \\ { l }_{ 2 } & { m }_{ 2 } & { n }_{ 2 } \\ { l }_{ 3 } & { m }_{ 3 } & { n }_{ 3 } \end{matrix} \right]$
Show that AA'=l3 ,where A =

• 4)

If the equation of a line $\frac { x-2 }{ 2 } =\frac { 2y-5 }{ -3 } ,z=-1$ then find the ratio of the line and a point on the line.

• 5)

If the equation of a line is x=ay+b z=cy+d, then find direction ratios of the line
and a point on the line.

#### 12th Standard CBSE Mathematics - Vector Algebra Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Classify the following on scalar and vector quantities:
(i) Work
(ii) Force
(iii) Velocity
(iv) Displacement.

• 2)

In a triangle ABC, Show that $\overset\rightarrow {AB}+\overset\rightarrow {BC}+\overset\rightarrow {CA}=0$

• 3)

Find the unit vector in the direction of $\overset\rightarrow a+\overset\rightarrow b$if $\overset\rightarrow a= 2\overset\wedge i+\overset\wedge j+3\overset\wedge k$, and $\overset\rightarrow b= \overset\wedge i+2\overset\wedge j-\overset\wedge k$

• 4)

Find the position vector of c which divides the line segment joining A & B whose position vectors are $3\overset\rightarrow a+\overset\rightarrow b$and $\overset\rightarrow a-3\overset\rightarrow b$ internally in the ratio 2:3.

• 5)

Find the direction cosines of the vector joining the points A(I, 2, - 3) and B(- 1, - 2, 1) directed from B to A.

#### 12th Standard CBSE Mathematics - Differential Equations Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Show that the solution of differential equation :
$y=\left( { 2x }^{ 2 }-1 \right) +c{ e }^{ { -x }^{ 2 } }$ is $\frac { dy }{ dx } +2xy-4{ x }^{ 3 }=0$

• 2)

From the differential equation of equation y = a cos2x + b sin2x, where a and b are constant.

• 3)

Find the sum of the order and degree of the following differential equations :
$\frac { { d }^{ 2 }y }{ dx^{ 2 } } +\sqrt [ 3 ]{ \frac { dy }{ dx } } +\left( 1+x \right) =0$

• 4)

Obtain the differential equation of the family of circles passing through the points (a,0) and (- a, 0).

• 5)

Solve the differential equation $\frac { dy }{ dx } ={ e }^{ x-y }+x{ e }^{ -y }$.

#### 12th Standard CBSE Mathematics - Application of Integrals Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

$\int { sin2xcos3xdx }$

• 2)

$\int { \frac { dx }{ 1+sinx } }$

• 3)

$\int { tan^{ -1 } } \sqrt { \frac { 1-cos2x }{ 1+cos2x } } dx$

• 4)

$\int { { cos }^{ 3 } } xdx$

• 5)

$\int { \frac { dx }{ 1+{ e }^{ x } } }$

#### 12th Standard CBSE Mathematics Integrals Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

$\int { sin2xcos3xdx }$

• 2)

$\int { \frac { dx }{ 1+sinx } }$

• 3)

$\int { sin^{ -1 }(cosx)dx }$

• 4)

$\int { tan^{ -1 } } \sqrt { \frac { 1-cos2x }{ 1+cos2x } } dx$

• 5)

$\int { \frac { a }{ b+ce^{ x } } } dx$

#### 12th Standard CBSE Mathematics Unit 6 Application of Derivatives Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

The sides of an equilateral triangle are increasing at the rate of 2 cm/see, Find the rate at which its area increases, when side is 10 cm long.

• 2)

If x changes from 4 to 4.01, then find the approximate change in log, x.

• 3)

The side of an equilateral triangle is increasing at the rate of 5 cm/sec. At what rate its area increasing when the side of the triangle is 10 cm.

• 4)

The length x of a rectangle in decreasing at the rate of 5 cm/min and the width y increasing at the rate of 4 cm/min. find the rate of change its area when x = 5 cm and y = 8 cm.

• 5)

The volume of a cube increasing at the rate of 9 cm3/sec. How fast is the surface area increasing when the length of an edge is 10 cm.

#### 12th CBSE Mathematics Continuity and Differentiability Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

if y = $f({ e }^{ { { sin }^{ -1 } } }2x)$, find dy/dx.

• 2)

If y = log(sin x), find $\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } }$

• 3)

If x = $\theta$sin$\theta$, y = $\theta$cos$\theta$ find dy/dx at $\theta$ = $\pi/4$
​​​​​​​

• 4)

If y= tan-1$\sqrt { \frac { 1-x }{ 1+x } } find\frac { dy }{ dx }$

• 5)

If y = tan-1$\sqrt { \frac { sinx }{ 1+cosx } , } find\frac { dy }{ dx }$

#### CBSE 12th Mathematics - Matrices Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Find the value of x, y, z if
$\left[ \begin{matrix} 2x+y & x-y \\ x-z & x+y+z \end{matrix} \right] =\left[ \begin{matrix} 10 & -1 \\ 2 & 8 \end{matrix} \right]$

• 2)

If $A=\left[ \begin{matrix} 1 & 4 \\ 3 & 2 \\ 2 & 1 \end{matrix} \right] B=\left[ \begin{matrix} 5 & 2 \\ -1 & 0 \\ 1 & 1 \end{matrix} \right]$, then find the matrix X for which A + B - X = 0.

• 3)

Solve the matrix equation $\left[ \begin{matrix} { x }^{ 2 } \\ { y }^{ 2 } \end{matrix} \right] -3\left[ \begin{matrix} x \\ 2y \end{matrix} \right] =\left[ \begin{matrix} -2 \\ -9 \end{matrix} \right]$

• 4)

If $A=\left[ \begin{matrix} 1 & 2 & 3 \end{matrix} \right]$ and $B=\left[ \begin{matrix} -2 \\ 3 \\ 1 \end{matrix} \right]$, find AB and BA.

• 5)

Find the value of x and y in each if AB exist
(i) ${ A }_{ 3\times x },{ B }_{ 4\times y }$ and ${ AB }_{ 3\times 3 }$
(ii) ${ A }_{ x\times 2 },{ B }_{ y\times 4 }$ and ${ AB }_{ 3\times 4 }$

#### CBSE 12th Mathematics Unit 2 Inverse Trigonometric Functions Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Write in the simplest form : ${ tan }^{ -1 }\left[ \frac { cos\quad x }{ 1+sin\quad x } \right] ,x\left[ -\frac { \pi }{ 2 } ,\frac { \pi }{ 2 } \right]$

• 2)

Show that ${ tan }^{ -1 }\frac { x }{ y } -{ tan }^{ -1 }\frac { x-y }{ x+y } =\frac { \pi }{ 4 }$

• 3)

Show that : ${ tan }^{ -1 }\frac { 2 }{ 3 } =\frac { 1 }{ 2 } { tan }^{ -1 }\frac { 12 }{ 5 }$

• 4)

Show that :${ tan }^{ -1 }\left( \frac { 3a^{ 2 }x-{ x }^{ 3 } }{ { a }^{ 3 }-3a{ x }^{ 2 } } \right) =3tan^{ -1 }\left( \frac { x }{ a } \right)$

• 5)

show that : ${ tan }^{ -1 }\frac { 1 }{ 4 } +{ tan }^{ -1 }\frac { 2 }{ 9 } =\frac { 1 }{ 2 } { tan }^{ -1 }\frac { 4 }{ 3 }$

#### CBSE 12th Mathematics Unit 1 Relations and Functions Two Marks Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Define Reflexive.Give one example.

• 2)

Define symmetric Relation.Give one example

• 3)

Define Transitive Relation. Give one example.

• 4)

Consider the relation perpendicular on a set  of lines in a plane. Show that this relation is symmetric and neither reflexive and nor transitive.

• 5)

What is meant by one-one function?

#### CBSE Class 12th Mathematics Unit 13 Probability Book Back Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Given P(A)=$1\over2$,P(B)=$1\over3$ and $P(A\cap B)={1\over6}$  Are the events A and B independent?

• 2)

If P(A)=0.4, P(B)=p and $P(A\cup B)=0.7$ find the value of p, if A and B are independent events.

• 3)

Given P(A)=0.2, P(B)=0.3 and $P(A\cap B)=0.3$ Find P(A/B)

• 4)

Given P(A)=0.4, P(B)=0.7 and P(B/A)=0.6, Find $P(A\cup B)$

• 5)

Events E and F are given to be independent. Find P(F) if it is given that P(E)=0.60 and P(E$\cap$F)=0.35

#### CBSE Class 12th Mathematics Unit 12 Linear Programming Book Back Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

The objective function is maximum or minimum, which lies on the boundary of the feasible region.

• 2)

Solve the following linear programming problem graphically:
Minimise Z=200x+500y subject to the constraints:
$x+2y\ge 10,3x+4y\le 24,x\ge 0,y\ge 0.$

• 3)

Maximise Z=3x+4y subject to the constraints:
$x+y\le 4,x\ge 0,y\ge 0.$

• 4)

Minimise Z=3x+5y
subject to $x+3y\ge 3,x+3y\ge 2,x,y\ge 0.$

• 5)

A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts while It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs.17.50 per package on nuts and RS. 7per package of bolts. How many packages of each should be produced each day so as to maximise his profits if he operates his machines for at the most 12 hours a day? Formulate this mathematically and then solve it.

#### CBSE Class 12th Mathematics Unit 11 Three Dimensional Geometry Book Back Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

The Cartesian equation of a line AB is $\frac { 2x-1 }{ \sqrt { 3 } } =\frac { y+2 }{ 2 } \frac { z-3 }{ 3 }$. Find the direction cosines of a line parallel to AB.

• 2)

Find the direction cosines of the line

$\frac { 4-x }{ 2 } =\frac { y }{ 6 } =\frac { 1-z }{ 3 }$

• 3)

Write the Cartesian equation of the plane $\vec { r } .(3\hat { i } +2\hat { j } +5\hat { k } )=7.$

• 4)

Using direction ratios, show that the points (2,3,4),(-1,-2,1) and (5,8,7) are collinear.

• 5)

Write the direction ratio's of the vector $3\overset { \rightarrow }{ a } +2\overset { \rightarrow }{ b }$ where $\overset { \rightarrow }{ a } +\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k }$ and $\overset { \rightarrow }{ b } +\overset { \wedge }{ 2i } +\overset { \wedge }{ 4j } +5\overset { \wedge }{ k } \\$

#### CBSE Class 12th Mathematics Unit 10 Vector Algebra Book Back Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Find a unit vector in the direction of $\overrightarrow { a } =3\overrightarrow { i } -2\overrightarrow { j } +6\overrightarrow { k }$

• 2)

Find a vector in the direction of $\overrightarrow { a } =\overrightarrow { i } -2\overrightarrow { j }$ whose magnitude is 7.

• 3)

Write the value of p for which $\overrightarrow { a } =3\hat { i } +2\hat { j } +9\hat { k } \quad and\quad \overrightarrow { b } =\hat { i } +p\hat { j } +3\hat { k }$  are parallel vectors.

• 4)

Vectors $\overrightarrow { a } \quad and\quad \overrightarrow { b }$ are such that $\left| \overrightarrow { a } \right| =\sqrt { 3 } ,\left| \overrightarrow { b } \right| =\frac { 2 }{ 3 } and\quad (\overrightarrow { a } \times \overrightarrow { b } )$ is a unit vector. write the angle between $\overrightarrow { a } \quad and\quad \overrightarrow { b }$?

• 5)

If $\overrightarrow { a } =x\hat { i } +2\hat { j } -z\hat { k } \quad and\quad \overrightarrow { b } =3\hat { i } -y\hat { j } +\hat { k }$ are two equal vectors then write the value of x+y+z.

#### 12th Standard CBSE Mathematics Unit 9 Differential Equations Book Back Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Write the degree of the differential equation $\left( \frac { { d }^{ 2 }s }{ { dt }^{ 2 } } \right) ^{ 2 }+\left( \frac { ds }{ dt } \right) ^{ 3 }+4=0$

• 2)

Write the differential equation formed from the equation y = mx + c, where m and c are arbitrary constants.

• 3)

Write the differential equation obtained by eliminating the arbitrary constant C in the equation representing the family of curves xy = C cos x.

• 4)

Find the differential equation of the family of lines passing through the origin.

• 5)

Find the integrating factor of the differential equation $\left( \frac { { e }^{ -2\sqrt { x } } }{ \sqrt { x } } -\frac { y }{ \sqrt { x } } \right) \frac { dx }{ dy } =1$ .

#### 12th Standard CBSE Mathematics Unit 7 Integrals Book Back Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Evaluate the integral:  $\int { {2cos\ x\over 3sin^2}dx. }$

• 2)

Evaluate the integral: $\int { x\ +\ cos\ 6x \over 3x^2\ + sin\ 6x} dx.$

• 3)

Evaluate the integral: $\int {sec^2x\over3+tan\ x}dx.$

• 4)

Evaluate the integral: $\int {sec\ x\ (sec\ x\ +\ tan\ x)}dx$

• 5)

Given $\int {e^x (tan\ x\ +\ 1)\ sec\ x\ dx\ =\ e^x}\ f(x)\ +\ c.$ Write the value of f(x).

#### 12th Standard CBSE Mathematics Unit 8 Application of Integrals Book Back Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Find the area of the region by the curve $y=\frac { 1 }{ x }$ , X-axis and between X = 1, X = 4.

• 2)

On sketching the graph of  $y=\left| x-2 \right|$  and evaluating $\int _{ -1 }^{ 3 }{ \left| x-2 \right| } dx$ , what does $\int _{ -1 }^{ 3 }{ \left| x-2 \right| } dx$ represent on the graph ?

• 3)

Find the area of the region bounded by the curve y2 = x and the lines x = 1 , x = 4 and the x -  axis.

• 4)

Find the area of the region in the first quadrant enclosed by x - axis and $x=\sqrt { 3 } y$ by the circle ${ x }^{ 2 }+{ y }^{ 2 }=4$.

• 5)

The area between x = y2 and x = 4 is divided into two equal parts by the line x = a, find the value of a.

#### 12th Standard CBSE Mathematics Unit 6 Application of Derivatives Book Back Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

The amount of pollution content added in air in a city due to X diesel vehicles is given by P(x)=0.005x3+0.02x2 +30x.Find the marginal increase   in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question?

• 2)

A balloon which always remains spherical has a variable diameter ${3\over2}(2x+1)$.Find the rate of change of its volume with respect to x.

• 3)

Find the point on the curve y=x2-4x+5,where tangent to the curve is parallel to the x-axis.

• 4)

f(x)=-(x-1)2+10

• 5)

f(x)=9x2+12x+2

#### 12th Standard CBSE Mathematics Unit 5 Continuity and Differentiability Book Back Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Examine the continuity of the function f (x) = x2+5 at x=-1

• 2)

Give an example of a function which is continuous at x=1, but not differentiable at x=1.

• 3)

Show that the function f (x)= $\begin{cases} { x }^{ 3 }+3\quad ,\quad if\quad x\neq 0 \\ 1\quad \quad \quad ,\quad if\quad x=0 \end{cases}$ is not continuous at x=0.

• 4)

If y= sec-1 $\left( \frac { \sqrt { x } +1 }{ \sqrt { x } -1 } \right) +\quad sin^{ -1 }\left( \frac { \sqrt { x } -1 }{ \sqrt { x } +1 } \right) ,\quad find\frac { dy }{ dx } .$

• 5)

If y=500 e7x + 600 e-7x , show that $\frac { d^{ 2 }y }{ dx^{ 2 } } =49y$.

#### 12th Standard CBSE Mathematics - Determinants Book Back Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Evaluate $\begin{vmatrix} a+ib & c+id \\ c-id & a-ib \end{vmatrix}$

• 2)

Find the cofactor of a12 in the following.$\begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{vmatrix}$

• 3)

Find the minor of the element of second row and third column (a23)in the following determinant:

$\begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & 7 \end{vmatrix}$

• 4)

What positive value of x makes the following pair of determinants equal?

$\begin{vmatrix} 2x & 3 \\ 5 & x \end{vmatrix},\begin{vmatrix} 16 & 3 \\ 5 & 2 \end{vmatrix}$

• 5)

Find x, if $\begin{vmatrix} -1 & 2 \\ 4 & 8 \end{vmatrix}=\begin{vmatrix} 2 & x \\ x & -4 \end{vmatrix}$

#### 12th Standard CBSE Mathematics Unit 3 Matrices Book Back Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

If $\left[ \begin{matrix} x & +3y & y \\ 7 & -x & 4 \end{matrix} \right]$=$\begin{bmatrix} 4 & -1 \\ 0 & 4 \end{bmatrix}$, find the values of x and y.

• 2)

If $\begin{bmatrix} 3 & 4 \\ 2 & x \end{bmatrix}\left[ \begin{matrix} x \\ 1 \end{matrix} \right] =\left[ \begin{matrix} 19 \\ 15 \end{matrix} \right]$, find the value of x

• 3)

Write the value of x-y+z from the following equation :

$\left[ \begin{matrix} x+y+z \\ x+z \\ y+z \end{matrix} \right] =\left[ \begin{matrix} 9 \\ 5 \\ 7 \end{matrix} \right]$

• 4)

Find the value of x+y from the following equation:

$2\begin{bmatrix} x & 5 \\ 7 & y-3 \end{bmatrix}+\begin{bmatrix} 3 & -4 \\ 1 & 2 \end{bmatrix}=\begin{bmatrix} 7 & 6 \\ 15 & 14 \end{bmatrix}$

• 5)

If A is a $3\times 3$ matrix, whose elements are given by ${ a }_{ ij }=\frac { 1 }{ 3 } \left| -3i+j \right|$, then write the value ${ a }_{ 23 }$.

#### 12th Standard CBSE Mathematics Unit 2 Inverse Trigonometric Functions Book Back Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

Evaluate  $sin\left[ \frac { \pi }{ 3 } -{ sin }^{ -1 }\left( -\frac { 1 }{ 2 } \right) \right]$ .

• 2)

Solve for X, $\quad { tan }^{ -1 }\frac { 1-X }{ 1+X } =\frac { 1 }{ 2 } { tan }^{ -1 }X,\quad X>0.$

• 3)

Write the principal values of the following: sec-1(-2).

• 4)

Write the principal value of ${ tan }^{ -1 }\left( 1 \right) +{ cos }^{ -1 }\left( -\frac { 1 }{ 2 } \right) \quad$

• 5)

Write the value of  ${ tan }\left( 2{ tan }^{ -1 }\frac { 1 }{ 5 } \right)$ .

#### 12th Standard CBSE Mathematics Unit 1 Relations and Functions Book Back Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

If f(X)=X+7 and g(X)=X-7, $X\in R,$ find fog(7). ?

• 2)

If the binary operation * on the set of integers Z is defined by a*b=a+3bthen find the value of 2*4.

• 3)

Let * be a binary operation on N given by a*b=HCF(a,b), $a,b\in N$. Write the value of 22*4.

• 4)

If the binary operation * defined on Q is defined as a*b=2a+b-ab, for all $a,b\in Q,$ find the value of 3*4.

• 5)

If f:$R\to R$ be defined by $f(X)=(3-X^3)^{1\over3}$ then find fof(X).

#### 12th Standard CBSE Mathematics - Probability One Mark Question with Answer Key - by Shalini Sharma - Udaipur - View & Read

• 1)

Given P(A)=$1\over2$,P(B)=$1\over3$ and $P(A\cap B)={1\over6}$  Are the events A and B independent?

• 2)

If P(A)=0.4, P(B)=p and $P(A\cup B)=0.7$ find the value of p, if A and B are independent events.

• 3)

Given P(A)=0.2, P(B)=0.3 and $P(A\cap B)=0.3$ Find P(A/B)

• 4)

Given P(A)=0.4, P(B)=0.7 and P(B/A)=0.6, Find $P(A\cup B)$

• 5)

Events E and F are given to be independent. Find P(F) if it is given that P(E)=0.60 and P(E$\cap$F)=0.35

#### 12th Standard CBSE Mathematics - Three Dimensional Geometry One Mark Question and Answer - by Shalini Sharma - Udaipur - View & Read

• 1)

The Cartesian equation of a line AB is $\frac { 2x-1 }{ \sqrt { 3 } } =\frac { y+2 }{ 2 } \frac { z-3 }{ 3 }$. Find the direction cosines of a line parallel to AB.

• 2)

Find the distance of the point (2,3,4) from the plane $\overrightarrow { r } .(3\acute { i } -6\acute { j } +2\acute { k } )=-11$.

• 3)

Find the direction cosines of the line passing through the following points:

(-2,4,-5),(1,2,3).

• 4)

Write the vector equation of the following line:$\frac { x-5 }{ 3 } =\frac { y+4 }{ 7 } =\frac { 6-z }{ 2 }$

• 5)

Write the Cartesian equation of the following line given in vector form:

$\overrightarrow { r } =2\hat { i } +\hat { j } +4\hat { k } +\lambda (\hat { i } +\hat { j } -\hat { k } )$

#### 12th Standard CBSE Mathematics - Vector Algebra One Mark Question with Answer Key - by Shalini Sharma - Udaipur - View & Read

• 1)

Find a unit vector in the direction of $\overrightarrow { a } =3\overrightarrow { i } -2\overrightarrow { j } +6\overrightarrow { k }$

• 2)

For what value of $\lambda$ are the vectors $\overrightarrow { a } =2\overrightarrow { i } +\lambda \overrightarrow { j } +\overrightarrow { k }$ and $\overrightarrow { b } =\overrightarrow { i } -2\overrightarrow { j } +3\overrightarrow { k }$  perpendicular to each other?

• 3)

If $\overrightarrow { a } =\overrightarrow { i } +2\overrightarrow { j } -\overrightarrow { k } \quad and\quad \overrightarrow { b } =3\overrightarrow { i } +\overrightarrow { j } -5\overrightarrow { k }$ find a unit vector in the direction of $\overrightarrow { a } -\overrightarrow { b }$

• 4)

If $\left| \overrightarrow { a } \right| =2,\left| \overrightarrow { b } \right| =\sqrt { 3 }$  and $\overrightarrow { a } .\overrightarrow { b } =\sqrt { 3 }$ find the angle between $\overrightarrow { a } \quad and\quad \overrightarrow { b }$

• 5)

If $\overrightarrow { a } =\hat { i } +2\hat { j } -3\hat { k }$ and $\overrightarrow { b } =2\hat { i } +4\hat { j } +9\hat { k }$ find a unit vector parallel to $\overrightarrow { a } +\overrightarrow { b }$

#### 12th Standard CBSE Mathematics - Differential Equations One Mark Question and Answer - by Shalini Sharma - Udaipur - View & Read

• 1)

What is the degree of the following differential equation?

${ { 5x\left( \frac { dy }{ dx } \right) } }^{ 2 }-\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -6y=logx$

• 2)

Write the degree of the differential equation: $5x{ \left( \frac { dy }{ dx } \right) }^{ 2 }-\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =0.$

• 3)

Write the degree of the differential equation: $x{ \left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) }^{ 2 }+y{ \left( \frac { dy }{ dx } \right) }^{ 2 }+{ x }^{ 3 }=0.$

• 4)

Write the degree of the differential equation:${ \left( \frac { dy }{ dx } \right) }^{ 4 }+3x\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =0$​​​​​​​

• 5)

Write the order and degree of the differential equation $(\frac{{d}^{2}y}{{dx}^{2}})^{3}$ -5$\frac{dy}{dx}$+6=0.

#### 12th Standard CBSE Mathematics Unit 7 Integrals One Mark Question with Answer Key - by Shalini Sharma - Udaipur - View & Read

• 1)

Evaluate the integral: $\int { {x^2\over1+x^3}dx. }$

• 2)

Evaluate the integral: $\int { x\ +\ cos\ 6x \over 3x^2\ + sin\ 6x} dx.$

• 3)

Evaluate the integral: $\int {sec^2\ x \over cosec^2 x}dx.$

• 4)

Evaluate the integral: $\int {2-3\ sin\ x\over cos^2\ x}dx$

• 5)

$\int {1+tan\ x\over 1-tan\ x}dx$.

#### 12th Standard CBSE Mathematics Unit 6 Application of Derivatives One Mark Question with Answer - by Shalini Sharma - Udaipur - View & Read

• 1)

The amount of pollution content added in air in a city due to X diesel vehicles is given by P(x)=0.005x3+0.02x2 +30x.Find the marginal increase   in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question?

• 2)

Prove that,the function f(x)=x3-3x2+3x+107 is increasing on R.

• 3)

Show that the function y=${3\over x}+7$ is strictly decreasing for $\epsilon$R (x$\neq$0).

• 4)

Show that,the function f(x)=log(cos x) is decreasing,in $[0,{\pi\over2}]$.

• 5)

Show that f(x)=(x-1) ex+1 is an increasing function for x > 0.

#### 12th Standard CBSE Mathematics Unit 5 Continuity and Differentiability One Mark Question with Answer Key - by Shalini Sharma - Udaipur - View & Read

• 1)

Examine the continuity of the function f (x) = x2+5 at x=-1

• 2)

Examine the continuity of the function f (x)=$\frac { 1 }{ x+3 } ,\quad x\quad \varepsilon \quad R$.

• 3)

Give an example of a function which is continuous at x=1, but not differentiable at x=1.

• 4)

State the points of discountinuity for the function $f(x)= [x]$  in $-3 < x < 3.$

• 5)

Discuss the continuity of the function defined by f(x) =$\frac { 1 }{ x } ,\quad x\neq 0$

#### 12th CBSE Mathematics Unit 4 Determinants One Mark Question Paper with Answer Key - by Shalini Sharma - Udaipur - View & Read

• 1)

Evaluate $\begin{vmatrix} a+ib & c+id \\ c-id & a-ib \end{vmatrix}$

• 2)

Write the value of determinant $\begin{vmatrix} 2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x \end{vmatrix}$.

• 3)

What positive value of x makes the following pair of determinants equal?

$\begin{vmatrix} 2x & 3 \\ 5 & x \end{vmatrix},\begin{vmatrix} 16 & 3 \\ 5 & 2 \end{vmatrix}$

• 4)

If A=$\begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}$ write A-1 in terms of .

• 5)

Given $\begin{vmatrix} 4 & -1 & 0 \\ 2 & 1 & 4 \\ 1 & 0 & 3 \end{vmatrix}$find (i)minor of an element a23 (ii)cofactor of an element a32

#### 12th Standard CBSE Mathematics - Matrices One Mark Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

If $\left[ \begin{matrix} x & +3y & y \\ 7 & -x & 4 \end{matrix} \right]$=$\begin{bmatrix} 4 & -1 \\ 0 & 4 \end{bmatrix}$, find the values of x and y.

• 2)

If matrix A=$[\begin{matrix} 1 & 2 & 3 \end{matrix}]$ write AA' , where A' is the transpose of matrix A.

• 3)

If $\left[ \begin{matrix} y & +2x & 5 \\ & -x & 3 \end{matrix} \right] =\begin{bmatrix} 7 & 5 \\ -2 & 3 \end{bmatrix}$, find the value of y.

• 4)

If $A=\left[ { a }_{ ij } \right] =\left[ \begin{matrix} 2 & 3 & -5 \\ 1 & 4 & 9 \\ 0 & 7 & -2 \end{matrix} \right] andb=\left[ { b }_{ ij } \right] =\left[ \begin{matrix} 2 & 1 & -1 \\ -3 & 4 & 4 \\ 1 & 5 & 2 \end{matrix} \right] ,then\quad find\quad { a }_{ 22 }+{ b }_{ 21 }.$

• 5)

If $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} 3 & 1 \\ 2 & 5 \end{bmatrix}=\begin{bmatrix} 7 & 11 \\ k & 23 \end{bmatrix}$, find the value of k.

#### 12th CBSE Mathematics Unit 2 Inverse Trigonometric Functions One Mark Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

Write the principal values of the following: ${ sin }^{ -1 }\left( sin\frac { 4\pi }{ 5 } \right)$

• 2)

Write the value of  ${ tan }^{ -1 }\left[ 2sin\left( 2{ cos }^{ -1 }\frac { \sqrt { 3 } }{ 2 } \right) \right]$.

• 3)

Write the value of ${ sin }\left( 2{ sin }^{ -1 }\frac { 3 }{ 5 } \right)$

• 4)

Write ${ cot }^{ -1 }\left( \frac { 1 }{ \sqrt { { X }^{ 2 }-1 } } \right) ,\quad \left| X \right| >1$ in the simplest form.

• 5)

Evaluate $cot\left( { sec }^{ -1 }X+{ sin }^{ -1 }\frac { 1 }{ X } \right)$

#### 12th CBSE Mathematics Relations and Functions One Mark Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

If f(X)=X+7 and g(X)=X-7, $X\in R,$ find fog(7). ?

• 2)

If the binary operation * on the set of integers Z is defined by a*b=a+3bthen find the value of 2*4.

• 3)

Write fog, if f:$R\to R$ and g:$R\to R$ are given by f(x)=8x2 and g(x)=x1/3

• 4)

Let f and g be two real functions defined as f(x)=2x-3; g(x)=$3+x\over2$. Find fog and gof. Can you say one is inverse of the other?

• 5)

Let f:$R\rightarrow R$ is defined by f(x)=x2. Is f one-one?

#### 12th CBSE Mathematics Three Dimensional Geometry Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

Write the Cartesian equation of the following line given in vector form:

$\overrightarrow { r } =2\hat { i } +\hat { j } +4\hat { k } +\lambda (\hat { i } +\hat { j } -\hat { k } )$

• 2)

Write the intercept cut off by the plane 2x+y-z=5 on the x-axis.

• 3)

What are the direction cosines of a line, which makes equal angles with the coordinate axes?

• 4)

Find the distance of the plane 3x-4y+12z=3 from the origin.

• 5)

Write the direction cosines of a line parallel to the z-axis.

#### 12th Standard Mathematics Vector Algebra Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

Find a unit vector in the direction of $\overrightarrow { a } =3\overrightarrow { i } -2\overrightarrow { j } +6\overrightarrow { k }$

• 2)

Write the value of p for which $\overrightarrow { a } =3\hat { i } +2\hat { j } +9\hat { k } \quad and\quad \overrightarrow { b } =\hat { i } +p\hat { j } +3\hat { k }$  are parallel vectors.

• 3)

If $\overrightarrow { a } =x\hat { i } +2\hat { j } -z\hat { k } \quad and\quad \overrightarrow { b } =3\hat { i } -y\hat { j } +\hat { k }$ are two equal vectors then write the value of x+y+z.

• 4)

Find the projection of vector $\overset\wedge i+3\overset\wedge j+7\overset\wedge k$ on the vector $2\overset\wedge i-3\overset\wedge j+6\overset\wedge k$.

• 5)

Find $\overrightarrow a.(\overrightarrow b\times\overrightarrow c)$, if $\overrightarrow a=2\overset\wedge i+\overset\wedge j+3\overset\wedge k,\overrightarrow b=-\overset\wedge i+2\overset\wedge j+\overset\wedge k and \overrightarrow c=3\overset\wedge i+\overset\wedge j+2\overset\wedge k$

#### 12th CBSE Mathematics Differential Equations Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

Write the degree of the differential equation: $5x{ \left( \frac { dy }{ dx } \right) }^{ 2 }-\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =0.$

• 2)

Write the sum of the order and degree of the differential equation $1+\left( \frac { dy }{ dx } \right) ^{ 4 }=7\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 3 }$

• 3)

Write the degree of the differential equation ${ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 2 }+x\left( \frac { dy }{ dx } \right) ^{ 4 }=0$

• 4)

Find the differential equation representing the family of curves $V=\frac { A }{ r } +B$ , where A and B arbitrary constants.

• 5)

Find the integrating factor of the differential equation $\left( \frac { { e }^{ -2\sqrt { x } } }{ \sqrt { x } } -\frac { y }{ \sqrt { x } } \right) \frac { dx }{ dy } =1$ .

#### 12th Standard CBSE Mathematics Unit 8 Application of Integrals Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

Find the area of the region by the curve $y=\frac { 1 }{ x }$ , X-axis and between X = 1, X = 4.

• 2)

Find the area of the region bounded by the curve y2 = x and the lines x = 1 , x = 4 and the x -  axis.

• 3)

The area between x = y2 and x = 4 is divided into two equal parts by the line x = a, find the value of a.

• 4)

Using integration, find the area of the triangular region whose sides have the equations:
y = 2x + 1, y = 3x + 1and x = 4.

• 5)

Smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2 is :
(A) $2(\pi -2)$
(B) $\pi -2$
(C) $2\pi -1$
(D) $2(\pi +2)$

#### 12th Standard CBSE Mathematics Unit 7 Integrals Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

Evaluate the integral: $\int { {x^2\over1+x^3}dx. }$

• 2)

Evaluate the integral: $\int {(ax\ +\ b)^3}dx$

• 3)

$\int cos^2x\ cosec^2x\ dx.$

• 4)

$\int {x\over \sqrt {x+2}}dx.$

• 5)

Given $\int { { e }^{ x }(tanx+1)secxdx={ e }^{ x }f(x)+c }$. Write f(x) satisfying the above.

#### 12th CBSE Mathematics Unit 6 Application of Derivatives Model Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

The amount of pollution content added in air in a city due to X diesel vehicles is given by P(x)=0.005x3+0.02x2 +30x.Find the marginal increase   in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question?

• 2)

For what value of m,The function f(x)=mx+c,is decreasing for x $\epsilon$ R.

• 3)

Show that f(x)=(x-1) ex+1 is an increasing function for x > 0.

• 4)

For the function y=x3, if x=5 and $\Delta$x=0.01,find $\Delta$y

• 5)

f(x)=9x2+12x+2

#### 12th CBSE Mathematics Unit 5 Continuity and Differentiability Important Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

Examine the continuity of the function f (x)=$\frac { 1 }{ x+3 } ,\quad x\quad \varepsilon \quad R$.

• 2)

Give an example of a function which is continuous at x=1, but not differentiable at x=1.

• 3)

State the points of discountinuity for the function $f(x)= [x]$  in $-3 < x < 3.$

• 4)

If y= sec-1 $\left( \frac { \sqrt { x } +1 }{ \sqrt { x } -1 } \right) +\quad sin^{ -1 }\left( \frac { \sqrt { x } -1 }{ \sqrt { x } +1 } \right) ,\quad find\frac { dy }{ dx } .$

• 5)

Differentiate the following w.r.t. x, or find $\frac { dy }{ dx }$.

$y={ e }^{ x }+{ e }^{ { x }^{ 2 } }+{ e }^{ { x }^{ 3 } }+{ e }^{ { x }^{ 4 } }+{ e }^{ { x }^{ 5 } }.$

#### 12th CBSE Mathematics Unit 4 Determinants Important Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

If A is a square matrix of order 3 and |3A|=k|A|, then write the value of k.

• 2)

What positive value of x makes the following pair of determinants equal?

$\begin{vmatrix} 2x & 3 \\ 5 & x \end{vmatrix},\begin{vmatrix} 16 & 3 \\ 5 & 2 \end{vmatrix}$

• 3)

Write the adjoint of the following matrix $\begin{bmatrix} 2 & -1 \\ 4 & 3 \end{bmatrix}$

• 4)

Evaluate $\begin{vmatrix} cos15^{ o } & sin15^{ o } \\ sin75^{ o } & cos75^{ o } \end{vmatrix}$

• 5)

Given determinant $\begin{vmatrix} a_{ 11 } & a_{ 12 } & a_{ 13 } \\ a_{ 21 } & a_{ 22 } & a_{ 23 } \\ a_{ 31 } & a_{ 32 } & a_{ 33 } \end{vmatrix}$.Find the value of a11A21+a12A22+a13A23, where Aij is cofactor of element aij

#### 12th Standard CBSE Mathematics Unit 3 Matrices Important Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

If $\left[ \begin{matrix} x & +3y & y \\ 7 & -x & 4 \end{matrix} \right]$=$\begin{bmatrix} 4 & -1 \\ 0 & 4 \end{bmatrix}$, find the values of x and y.

• 2)

If matrix A=$[\begin{matrix} 1 & 2 & 3 \end{matrix}]$ write AA' , where A' is the transpose of matrix A.

• 3)

If $\left[ \begin{matrix} y & +2x & 5 \\ & -x & 3 \end{matrix} \right] =\begin{bmatrix} 7 & 5 \\ -2 & 3 \end{bmatrix}$, find the value of y.

• 4)

If $A=\left[ { a }_{ ij } \right] =\left[ \begin{matrix} 2 & 3 & -5 \\ 1 & 4 & 9 \\ 0 & 7 & -2 \end{matrix} \right] andb=\left[ { b }_{ ij } \right] =\left[ \begin{matrix} 2 & 1 & -1 \\ -3 & 4 & 4 \\ 1 & 5 & 2 \end{matrix} \right] ,then\quad find\quad { a }_{ 22 }+{ b }_{ 21 }.$

• 5)

If $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} 3 & 1 \\ 2 & 5 \end{bmatrix}=\begin{bmatrix} 7 & 11 \\ k & 23 \end{bmatrix}$, find the value of k.

#### 12th CBSE Maths Unit 2 Inverse Trigonometric Functions Important Question Paper - by Shalini Sharma - Udaipur - View & Read

• 1)

Evaluate  $sin\left[ \frac { \pi }{ 3 } -{ sin }^{ -1 }\left( -\frac { 1 }{ 2 } \right) \right]$ .

• 2)

Using principal value, evaluate the following:  ${ cos }^{ -1 }\left( cos\frac { 2\pi }{ 3 } \right) +{ sin }^{ -1 }\left( sin\frac { 2\pi }{ 3 } \right)$

• 3)

Show that ${ sin }^{ -1 }\left( 2X\sqrt { 1-{ X }^{ 2 } } \right) =2{ sin }^{ -1 }X$

• 4)

Solve for X, $\quad { tan }^{ -1 }\frac { 1-X }{ 1+X } =\frac { 1 }{ 2 } { tan }^{ -1 }X,\quad X>0.$

• 5)

Write the principal values of the following: ${ cos }^{ -1 }\left( cos\frac { 7\pi }{ 6 } \right)$

#### CBSE 12th Standard Mathematics Unit 1 Important Questions - by Shalini Sharma - Udaipur - View & Read

• 1)

If f(X)=X+7 and g(X)=X-7, $X\in R,$ find fog(7). ?

• 2)

If the binary operation * on the set of integers Z is defined by a*b=a+3bthen find the value of 2*4.

• 3)

Let * be a binary operation on N given by a*b=HCF(a,b), $a,b\in N$. Write the value of 22*4.

• 4)

If the binary operation * defined on Q is defined as a*b=2a+b-ab, for all $a,b\in Q,$ find the value of 3*4.

• 5)

If f:$R\to R$ be defined by $f(X)=(3-X^3)^{1\over3}$ then find fof(X).

#### Maths CBSE 12 Class Revision Test paper - by Bala - View & Read

• 1)

For the set A={1,2,3} define a relation R in the set A is follows:

R={(1,1).(2,2),(3,3),(1,3)}. Write the ordered pairs to be added to R to make it the smallest equivalence relation.

• 2)

If matrix A=$[\begin{matrix} 1 & 2 & 3 \end{matrix}]$ write AA' , where A' is the transpose of matrix A.

• 3)

Evaluate : $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ x }Isinx+cosx) } dx$

• 4)

Find $\lambda$, if the vectors
$\overrightarrow a=\overset\wedge i+3\overset\wedge j+\overset\wedge k,\overrightarrow b=2\overset\wedge i-\overset\wedge j-\overset\wedge k$ and $\overrightarrow c=\lambda \overset\wedge j+3\overset\wedge k$ are coplanar.

• 5)

How many equivalence relations on the set {1,2, 3} containing (I, 2) and (2, 1) are there in all ? Justify your answer.

#### CBSE Maths First Full Test Class 12 - by Bala - View & Read

• 1)

Let * be a binary operation on N given by a * b = LCM(a,b) for all a,b $\in$ N. Find 5*7

• 2)

Use elementary column operation ${ C }_{ 2 }\rightarrow { C }_{ 2 }+2{ C }_{ 1 }$ in the following matrix equation :
$\left( \begin{matrix} 4 & 2 \\ 3 & 3 \end{matrix} \right) =\left( \begin{matrix} 1 & 2 \\ 0 & 3 \end{matrix} \right) \left( \begin{matrix} 2 & 0 \\ 1 & 1 \end{matrix} \right)$

• 3)

Evaluate the integral: $\int {(1-x)\sqrt x\ dx}$

• 4)

Find $\lambda$ when the projection of $\overrightarrow { a } =\lambda \hat { i } +\hat { j } +4\hat { k } \quad on\quad \overrightarrow { b } =2\hat { i } +6\hat { j } +3\hat { k }$ is 4 units.

• 5)

Define Reflexive.Give one example.

#### CBSE Class 12 Mathematics Important Question Paper - by Bala - View & Read

• 1)

Show that the relation R:{1,2,3}$\rightarrow${1,2,3} given by R={(1,1),(2,2),(3,3),(1,2),(2,3)} is reflexive but neither symmetric nor transitive.

• 2)

If $A=\begin{bmatrix} cos\alpha - & sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}$, then for what value of $\alpha$, A is an identify matrix?

• 3)

Evaluate the integral: $\int {(1-x)\sqrt x\ dx}$

• 4)

Write the value of the area of the parallelogram determined by the vectors $2\hat { i } \quad and\quad 3\hat { j }$

• 5)

Let f and g be real function be $f(x)=\sqrt { x+4 } ,x\ge 4$ find the function fg, $\frac { f }{ g }$

#### Vector Algebra Important Questions CBSE 12th Mathematics - by Bala - View & Read

• 1)

Find a unit vector in the direction of $\overrightarrow { a } =3\overrightarrow { i } -2\overrightarrow { j } +6\overrightarrow { k }$

• 2)

If p(1,5,4) and Q(4,1,-2) find the direction ratios of $\overrightarrow { PQ }$

• 3)

Write the direction cosines of the vector $-2\hat { i } +\hat { j } -5\hat { k }$

• 4)

If $\overrightarrow { a }$ is a unit vector and $(\overrightarrow { x } -\overrightarrow { a } ).(\overrightarrow { x } +\overrightarrow { a } )=8\quad find\quad \left| \overrightarrow { x } \right|$

• 5)

Find the sum of the vectors $\overset\rightarrow a=\overset\wedge i-2\overset\wedge j+\overset\wedge k,\overset\rightarrow b=-2\overset\wedge i+4\overset\wedge j+5\overset\wedge k,\overset\rightarrow c=\overset\wedge i-6\overset\wedge j-7\overset\wedge k.$

#### 12th CBSE Mathematics Three Dimensional Geometry Important Question Paper - by Bala - View & Read

• 1)

Find the direction ratios of the line $\frac { x+2 }{ 1 } =\frac { 2y-1 }{ 3 } =\frac { 3-z }{ 5 } .$

• 2)

Find the angle between the line $\vec { r } =(2\hat { i } -\hat { j } +3\hat { k } )+\lambda (3\hat { i } -\hat { j } +2\hat { k } )$  and the plane $\vec { r } .(\hat { i } +\hat { j } +\hat { k } )=3.$

• 3)

What is the distance of the point (p,q,r) from the x-axis?

• 4)

Find the distance of a point (2, 5, - 3) from the plane $\overset { \rightarrow }{ r } (6\hat { i } +3\hat { j+6\hat { 2k } )=4 }$

• 5)

Find the length of the perpendicular drawrt from the origin to the plane 2x - 3y + 6z + 21 = 0

#### 12th CBSE Mathematics Linear Programming Model Question Paper - by Bala - View & Read

• 1)

The objective function is maximum or minimum, which lies on the boundary of the feasible region.

• 2)

Solve the following problem graphically:
Minimise and Maximise Z=3x+9y subject to the constraints:
$x+3y\le 60,x+y\ge 10,x\le y,x\ge 0,y\ge 0.$

• 3)

Maximise Z=3x+2y
subject to $x+2y\le 10,3x+y\le 15,x,y\ge 0.$

• 4)

Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs RS. 60/kg and Food Q costs RS. 80/kg. Food P contains 3 units/kg of Vitamin A AND 4 units/kg of vitamin B. Food Q contains 5 units/kg Vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.

• 5)

A farmer mixes two brands P and Q of cattle feed. Brand P, costing RS.250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of vitamin C. Brand Q costing RS.200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A,B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

#### 12th CBSE Mathematics Probability Important Question Paper - by Bala - View & Read

• 1)

If P(A)=0.4, P(B)=p and $P(A\cup B)=0.7$ find the value of p, if A and B are independent events.

• 2)

Given P(A)=0.4, P(B)=0.7 and P(B/A)=0.6, Find $P(A\cup B)$

• 3)

A bag contain 2 red, 6 black and 8 green balls. A ball is drawn at random from the bag. Find the probabilty:
(a) a red ball
(b) a black ball
(c) a green ball
(d) a non-red ball

• 4)

If P(E) =$\frac { 6 }{ 11 }$, P(F) =$\frac { 5 }{ 11 }$ and P(E$\cup$F)=$\frac { 7 }{ 11 }$ then find (a) P(E/F), (b) P(F/E)

• 5)

If P(F) = 0.35 and P(E$\cup$F) = 0.85 and E and F are independent events. Find P(E).

#### 12th Standard Mathematics Revision Model Question Paper 1 - by Bala - View & Read

• 1)

State the reason for the relation R in the set {1,2,3} given by R={(1,2),(2,1),} not to be transitive.

• 2)

Find the principal value of ${ sin }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2 } \right)$

• 3)

If ${ X }_{ m\times 3 }{ Y }_{ p\times 4 }={ Z }_{ 2\times b }$, for three matrices X,Y and Z, find the values of m,p and b.

• 4)

$\int {x^2-1\over x^2+1}dx.$

• 5)

Let f:$X\rightarrow Y$ be a function Define a relation R on X given be R=[(a,b) ; (f(b)] Show that R is an equivalence relation ?

#### 12th Standard CBSE Mathematics Term Test Question Paper - by Bala - View & Read

• 1)

If the binary operation * on the set of integers Z is defined by a*b=a+3bthen find the value of 2*4.

• 2)

Let * be a binary operation on N given by a*b=HCF(a,b), $a,b\in N$. Write the value of 22*4.

• 3)

If the binary operation * defined on Q is defined as a*b=2a+b-ab, for all $a,b\in Q,$ find the value of 3*4.

• 4)

Let f:$R\rightarrow R$ is defined by f(x)=|x|. Is function f onto? Give reasons.

• 5)

Let R be a relation in the set of natural numbers N defined by R={(a,b)$\in$NXN;a

#### Probability - Important Questions Model Paper In 12th Maths - by Bala - View & Read

• 1)

Given P(A)=$1\over2$,P(B)=$1\over3$ and $P(A\cap B)={1\over6}$  Are the events A and B independent?

• 2)

Given P(A)=0.2, P(B)=0.3 and $P(A\cap B)=0.3$ Find P(A/B)

• 3)

Given P(A)=0.4, P(B)=0.7 and P(B/A)=0.6, Find $P(A\cup B)$

• 4)

Events E and F are given to be independent. Find P(F) if it is given that P(E)=0.60 and P(E$\cap$F)=0.35

• 5)

Does the following represent a probability distribution? Give reasons.

 X 0 1 2 P(x) 1/3 1/3 1/6

#### Linear Programming - Important Questions Model Question Paper 1 In 12th Maths - by Bala - View & Read

• 1)

Solve the following linear programming problem graphically:
Maximise Z=4x+y subject to the constraints:
$\\ x+y\le 50,3x+y\le 90,x\ge 0,y\ge 0.$

• 2)

Solve the following problem graphically:
Minimise and Maximise Z=3x+9y subject to the constraints:
$x+3y\le 60,x+y\ge 10,x\le y,x\ge 0,y\ge 0.$

• 3)

Determine the minimum value of Z=3x+2y (if any), if the feasible region for an LLP is shown in the figure:

• 4)

Solve the following LLP graphically:
Maximise Z=2x+3y, subject to $x+y\le 4,x\ge 0,y\ge 0.$

• 5)

Minimize and Maximize Z=5x+2y, subject to the following constraints:
$x-2y\le 2,3x+2y\le 12,-3x+2y\le 3,x\ge 0,y\ge 0.$

#### Three Dimensional Geometry - Important Questions Model Question Paper 1 In 12th Maths - by Bala - View & Read

• 1)

Find the distance of the point (2,3,4) from the plane $\overrightarrow { r } .(3\acute { i } -6\acute { j } +2\acute { k } )=-11$.

• 2)

Write the Cartesian equation of the following line given in vector form:

$\overrightarrow { r } =2\hat { i } +\hat { j } +4\hat { k } +\lambda (\hat { i } +\hat { j } -\hat { k } )$

• 3)

Write the vector equation of a line whose Cartesian equation is $\frac { x+3 }{ 2 } =\frac { y-1 }{ 4 } =\frac { z+1 }{ 5 }$.

• 4)

Find the vector normal to the plane $\vec { r } .(3\hat { i } -7\hat { k } )+5=0$.

• 5)

Find the coordinates of a point, where the line $\frac { x+2 }{ 1 } =\frac { y-5 }{ 3 } =\frac { z+1 }{ 5 }$ cuts yz-plane.

#### Vector Algebra - Important Questions Model Question Paper 1 In 12th Maths - by Bala - View & Read

• 1)

Find the angle between the vectors $\overrightarrow { a } =\overrightarrow { i } -\overrightarrow { j } +\overrightarrow { k } and\quad \overrightarrow { b } =\overrightarrow { i } +\overrightarrow { j } -\overrightarrow { k }$

• 2)

Find a vector in the direction of $\overrightarrow { a } =\overrightarrow { i } -2\overrightarrow { j }$ whose magnitude is 7.

• 3)

Vectors $\overrightarrow { a } \quad and\quad \overrightarrow { b }$ are such that $\left| \overrightarrow { a } \right| =\sqrt { 3 } ,\left| \overrightarrow { b } \right| =\frac { 2 }{ 3 } and\quad (\overrightarrow { a } \times \overrightarrow { b } )$ is a unit vector. write the angle between $\overrightarrow { a } \quad and\quad \overrightarrow { b }$?

• 4)

If $\overrightarrow { a } =x\hat { i } +2\hat { j } -z\hat { k } \quad and\quad \overrightarrow { b } =3\hat { i } -y\hat { j } +\hat { k }$ are two equal vectors then write the value of x+y+z.

• 5)

If $\left| \overrightarrow { a } \right| =3$ interpret the following:

(i)2$\overrightarrow { a }$   (ii)-5$\overrightarrow { a }$

#### Differential Equations - Important Questions Model Question Paper 1 In 12th Maths - by Bala - View & Read

• 1)

What is the degree of the following differential equation?

${ { 5x\left( \frac { dy }{ dx } \right) } }^{ 2 }-\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -6y=logx$

• 2)

Write the order and degree of the differential equation $(\frac{{d}^{2}y}{{dx}^{2}})^{3}$ -5$\frac{dy}{dx}$+6=0.

• 3)

Find the differential equation of the family of lines passing through the origin.

• 4)

Write the differential equation representing the curve y2 = 4ax, where a is an arbitrary constant.

• 5)

Find the integrating factor of the differential equation $\left( \frac { { e }^{ -2\sqrt { x } } }{ \sqrt { x } } -\frac { y }{ \sqrt { x } } \right) \frac { dx }{ dy } =1$ .

#### Application Of Integrals - Important Questions Model Question Paper 1 In 12th Maths - by Bala - View & Read

• 1)

Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x - axis in the first quadrant.

• 2)

Find the area of the region in the first quadrant enclosed by x - axis and $x=\sqrt { 3 } y$ by the circle ${ x }^{ 2 }+{ y }^{ 2 }=4$.

• 3)

The area between x = y2 and x = 4 is divided into two equal parts by the line x = a, find the value of a.

• 4)

Find the area bounded by the curve x2 = 4y and the line x = 4y - 2.

• 5)

Area lying between the curves y2 = 4x and y = 2x is :
(A) $\frac { 2 }{ 3 }$
(B) $\frac { 1 }{ 3 }$
(C) $\frac { 1 }{ 4 }$
(D) $\frac { 3 }{ 4 }$

#### Integrals - Important Questions Model Question Paper 1 In 12th Maths - by Bala - View & Read

• 1)

Evaluate the integral: $\int { {x^2\over1+x^3}dx. }$

• 2)

$\int {1+tan\ x\over 1-tan\ x}dx$.

• 3)

$\int {dx\over x\ cos^2 (1+log\ x)}$.

• 4)

$\int {x^2-1\over x^2+1}dx.$

• 5)

$\int {e^x\over 1+e^x}dx.$

### CBSE Education Study Materials

#### 12th CBSE Mathematics 2019 - 2020 Academic Syllabus - by Shalini Sharma - Udaipur Aug 21, 2019 Aug 21, 2019

Mathematics 2019 - 2020 Academic Syllabus

#### Revision Plan to score full Marks in CBSE Class 12 Mathematics - by Bala Jan 23, 2019 Jan 23, 2019

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#### Tips to crack CBSE Class 12 Mathematics with 100/100 - by Bala Jan 23, 2019 Jan 23, 2019

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#### Tips to crack CBSE Class 12 Mathematics with 100/100 - by Bala Jan 12, 2019 Jan 12, 2019

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#### CBSEStudy Material - Sample Question Papers with Solutions for Class 12 Session 2019 - 2020

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