Find the price elasticity of demand for the demand function x = 10 – p where x is the demand and p i the price. Examine whether the demand is elastic, inelastic or unit elastic at p = 6.

Find the interval in which the function f(x) = x² – 4x + 6 is strictly increasing and strictly decreasing.

Given C(x) = \(\frac { { x }^{ 3 } }{ 6 } \)5x + 200 and p(x) = 40 – x are the cost price and selling price when x units of commodity are produced. Find the level of the production that maximize the profit.

The average cost function associated with producing and marketing x units of an item is given by AC = 2x - 11+\(\frac { 50 }{ x } \). Find the range of values of the output x, for which AC is increasing.

A manufacturing company has a contract to supply 4000 units of an item per year at uniform rate. The storage cost per unit per year amounts to Rs. 50 and the set up cost per production run is Rs. 160. If the production run can be started instantaneously and shortages are not permitted, determine the number of units which should be produced per run to minimize the total cost.

For the following observations, find the regression co-efficients b_yx and b_xy and hence find the correlation co-efficient between x and y.(4,2) (2, 3)(3, 2)(4, 4)(2, 4)

Find the regression co-efficient of x on y from the following data. \(\sum\)X=20, \(\sum\)Y=40, \(\sum\)XY=300, \(\sum\)X²=150, \(\sum\)Y²=345, N=5. Find the value of x when y=5

Calculate Co-efficient of correlation for the following data:

Calculate the co-efficient of correlation between x and y on the basis of the following observations. \(\sum\)x=125, \(\sum\)x²=1650, \(\sum\)y=100, \(\sum\)y²=1460, \(\sum\)xy=50, n=25.

In a correlation study, the following values are obtained

The co-efficient of correlation is 0.5. Find the lines of regression.

Kamal sold Rs.9000 worth 7% stock at 80 and invested the proceeds in 15% stock at 120. Find the change in his income?

A person sells a 20% stock of face value Rs.5000 at a premium of 62% with the money obtained he buys a 15% stock at a discount of 22% what is the change in his income.(Brokerage 2%)

Equal amounts are invested in 12% stock at 95 (brokerage). If 12% stock brought at Rs.120 more by way of dividend income than the other, find the amount invested in each stock?

Rani sold Rs.8000 worth 7% stock at 96 and invested the amount realised in the shares of FV Rs.100 os a 10% stock by which her income increased by Rs.80. Find the purchase price of 10% stock.

A company has a total capital of Rs.5,00,000 divide into 1000 preference shares of 6% dividend with par value RS.100 each and 4000 ordinary shares of per value Rs.100 each. The company delares an annual dividend of Rs.40,000. Find the dividend received by Sundar having 100 preference shares and 200 ordinary shares.

Find the equilibrium price and equilibrium quantity for the following functions. Demand: x = 100 – 2p and supply: x = 3p – 50

Find the stationary values and stationary points for the function f(x) = 2x³ + 9x² + 12x + 1

For a particular process, the cost function is given by C = 56 - 8x+x² where C is cost per unit and x, the number of unit’s produced. Find the minimum value of the cost and the corresponding number of units to be produced.

A company has to supply 1000 item per month at a uniform rate and for each time, a production run is started with the cost of Rs. 200. Cost of holding is Rs. 20 per item per month. The number of items to be produced per run has to be ascertained. Determine the total of setup cost and average inventory cost if the run size is 500, 600, 700, 800. Find the optimal production run size using EOQ formula.

If u = x²(y–x) + y² (x–y), then show that  \(\frac { \partial u }{ \partial x } +\frac { \partial u }{ \partial y } =-2\left( x-y \right) ^{ 2 }\)

Solve the following LPP graphically. Maximize Z = 6x₁ + 5x₂ Subject to the constraints 3x₁ + 5x₂ ≤ 15, 5x₁ + 2x₂ ≤ 10 and x₁,x₂ ≥ 0

Solve the following LPP graphically. Maximize Z =−x₁ + 2x₂
Subject to the constraints −x₁ + 3x₂ ≤ 10, x₁ + x₂ ≤ 6,x₁ − x₂ ≤ 2 and x₁,x₂ ≥ 0

Solve the following LPP graphically. Minimize Z = x₁ − 5x₂ + 20
Subject to the constraints x₁ − x₂ ≥ 0,−x₁ + 2x₂ ≥ 2,x₁ ≥ 3,x₂ ≤ 4 and x₁,x₂ ≥ 0.

Solve the following LPP graphically. ∴ Maximize Z = 3x₁ + 4x₂  subject to the constraints x₁ + x₂ ≤ 4 and x₁,x₂ ≥ 0.

Solve the following LPP graphically. Minimize\(Z=-3{ x }_{ 1 }+4{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }+2{ x }_{ 2 }\le 8\quad ,{ 3x }_{ 1 }+{ 2x }_{ 2 }\le 12\quad and\quad \quad { x }_{ 1 }\ge 0,{ x }_{ 2 }\ge 2.\)

Find D₆,D₈,P₇ and P₂₀ for the data 57, 58, 61, 42, 38, 65, 72, 66. 

In a Shooting test, the probabilities of hitting the target are \(\frac { 1 }{ 2 } \) for A, \(\frac { 2 }{ 3 } \) for B and \(\frac { 3 }{ 4 } \)  for C. If all of them fire at the same target, calculate the probabilities that only one of them hit the target.

A factory has 3 machines A₁, A₂, A₃ producing 1000, 2000, 3000 bolts per day respectively. A₁ produces 1% defectives, A₂ produces 1.5% and A₃ produces 2% defectives. A bolt is chosen at random and found defective. What is the probability that it comes from machine A₁?

There are 3 boxes containing respectively 1 white, 2 red, 3 black balls; 2 white, 3 red and 1 black ball; 2 white, 1 red, 3 black balls. A box is chosen t random and from it 2 balls are drawn at random. The 2 balls are 1 red and 1 white. What is the probability that they come from the second box?

A box contains 4 red, 6 green balls. Two balls are picked out one by one at random without replacement. What is the probability that the second is green given that the first one is green?

Solve the following LPP graphically. Minimize\(Z=-3{ x }_{ 1 }+4{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }+2{ x }_{ 2 }\le 8\quad ,{ 3x }_{ 1 }+{ 2x }_{ 2 }\le 12\quad and\quad \quad { x }_{ 1 }\ge 0,{ x }_{ 2 }\ge 2.\)

Solve the following LPP graphically, Minimize \(Z=3{ x }_{ 1 }+5{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }+3{ x }_{ 2 }\ge 3,\quad { x }_{ 1 }+{ x }_{ 2 }\ge 2\quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0.\)

Solve the following LPP graphically. Maximize \(Z={ x }_{ 1 }+{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }-{ x }_{ 2 }\le -1,{ -x }_{ 1 }+{ x }_{ 2 }\le 0\quad and\quad { x }_{ 1 }+{ x }_{ 2 }\ge 0\)

Construct the network for the projects consisting of various activities and their precedence relationships are as given below:

Construct the network for the following:

A factory has 3 machines A₁, A₂, A₃ producing 1000, 2000, 3000 bolts per day respectively. A₁ produces 1% defectives, A₂ produces 1.5% and A₃ produces 2% defectives. A bolt is chosen at random and found defective. What is the probability that it comes from machine A₁?

Find the Quartile deviation.

Two identical boxes containing respectively 4 white and 3 red balls, 3 white and 7 red balls. A box is chosen at random and a ball is drawn from it. Find the probability that the ball is white. If the ball is white. If the ball is white, what is the probability that it is from first box?

Three events A, B and C have probabilities \(\frac { 2 }{ 5 } ,\frac { 1 }{ 3 } \) and \(\frac { 1 }{ 2 } \)  respectively. Given that \(P(A\cap C)=\frac { 1 }{ 5 } \) and \(P(B\cap C)=\frac { 1 }{ 4 } ,\)find P(C/B) and \(P(\bar { A } \cap \bar { C } )\)

Two dice are thrown together. Let A be the event "getting 6 on the first die" and B be the event "getting 2 on the second die". Are the events A and B independent?

How much will be required to buy 125 of Rs. 25 shares at a discount of Rs. 7

Which is better investment? 7% of Rs. 100 shares at Rs. 120 (or) 8% of Rs. 100 shares at Rs. 135.

Find the amount of an ordinary annuity of Rs 500 payable at the end of each year for 7 years at 7% per year compounded annually.

Find the amount of an ordinary annuity of Rs 600 is made at the end of every quarter for 10 years at the rate of 4% per year compounded quarterly.

A cash prize of Rs. 1,500 is given to the student standing first in examination of Business Mathematics by a person every year. Find out the sum that the person has to deposit to meet this expense. Rate of interest is 12% p.a

Determine the values of x for which the matrix A =\(\left[ \begin{matrix} x+1 & -3 & 4 \\ -5 & x+2 & 2 \\ 4 & 1 & x-6 \end{matrix} \right] \)is singular.

Without expanding show that \(\Delta =\left| \begin{matrix} { cosec }^{ 2 }\theta & { cot }^{ 2 }\theta & 1 \\ { cot }^{ 2 }\theta & { cosec }^{ 2 }\theta & -1 \\ 42 & 40 & 2 \end{matrix} \right| =0\)

If a, b, c are in A.P, find the value of \(\left| \begin{matrix} 2y+4 & \quad 5y+7 & 8y+a \\ 3y+5 & 6y+8 & 9y+b \\ 4y+6 & 7y+9 & 10y+c \end{matrix} \right| \)

Let a, b and c denote the sides BC, CA and AB repectively of \(\Delta\) ABC. If \(\left| \begin{matrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{matrix} \right| =0\), then find the value of sin² A + sin²B + sin²C.

If \(A=\left[ \begin{matrix} 1 & tan\quad x \\ -tan\quad x & \quad \quad \quad 1 \end{matrix} \right] \), then show that A^TA^-1 = \(\left[ \begin{matrix} cos\quad 2x & -sin2x \\ sin\quad 2x & cos2x \end{matrix} \right] .\)

For the data on price (in rupees) and demand (in tonnes) for a commodity, calculate the co-efficient of correlations.

Obtain the two regression lines from the following

With the help of the regression equation for the data below, calculate the value of X when Y=20

A computer while calculating the correlation co-efficient between two variables x and y from 25 pairs of observations, obtained the following results. \(\sum\)x=125, \(\sum\)x²=650, \(\sum\)y=100, \(\sum\)y²=460, xy=508. It was later found out that it had copied down two pairs as while the correct values are 

Obtain the correlation co-efficient for the correct value.

The equations of two regression lines are 4x+3y+7=0 and 3x+4y+8=0.
Find (i) the mean of x and the mean of y
(ii) the regression co-efficient b_xy and b_yx 
(iii) the correlation co-efficient between x and y.

If a, b, c are in A.P, find the value of \(\left| \begin{matrix} 2y+4 & \quad 5y+7 & 8y+a \\ 3y+5 & 6y+8 & 9y+b \\ 4y+6 & 7y+9 & 10y+c \end{matrix} \right| \)

Show that the matrix A =\(\left[ \begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix} \right] \)satisfies the equation A² - 4A - 5I₃ = 0 and hence find A^-1.

Use the product \(\left[ \begin{matrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{matrix} \right] \left[ \begin{matrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{matrix} \right] \)to solve the system of equations x - 2y + 2z = 1, 2y - 3z = 1, 3x - 2y + 4z = 2.

Two types of radio values A, B are available and two types of radios P and Q are assembled in a small factory. The factory uses 2 valves of type A and 3 valves of type B for the type B for the type of radio P, and for the radio Q it uses 3 valves of type A and 4 valves of type B. If the number of valves of type A and B used by the factory are 130 and 180 respectively, find out the number of radios assembled use matrix method.

An amount of Rs. 5000 is put into three investment at the rate of interest of 6%, 7% and 8% per annum respectively. The total annual income is Rs. 358. If the combined income from the first two investment is Rs. 70 more than the income from the third, find the amount of each investment by matrix method.

For the following data, (i) the regression equation of X on Y regression equation of Y on X (iii) the correlation co-efficient between X and Y (iv) the value of x when y=5 (v) the value of y when x=6

Ten competitors in a musical test were ranked by the judges x, y, and z in the following order.

Out the following two regression lines, find the line of regression of X on Y, 2x+3y=7 and 5x+54=9.

For the following observations, find the regression co-efficient b_yx and b_xy and hence find the correlation co-efficient (4,2)(2,3)(3,2)(4,4)(2,4).

You are given the following data:

Correlation co-efficient between x and y=0.8 (i) Find the two regression lines.
(ii) Estimate the value of y, when x=70  (iii)Estimate the value of x, when y=90

If m parallel lines in a plane are intersected by a family of n parallel lines. Find the number of parallelogram formed?

If the fourth term in the expansion of \({ \left( ax+\frac { 1 }{ x } \right) }^{ n }\) is \(\frac { 5 }{ 2 } \)  then find the values of a and n.

Using the principle of mathematical induction, prove that 1.3 + 2.3² + 3.3³ + ... + n.3ⁿ =\(\frac { (2n-1){ 3 }^{ n+1 }+3 }{ 4 } for\ all\ n\in N\)

Using binomial theorem, find the value of \({ \left( \sqrt { 2 } +1 \right) }^{ 5 }+{ \left( \sqrt { 2 } -1 \right) }^{ 5 }\)

Find the 11^th term from the end in \({ \left( 2x-\frac { 1 }{ { x }^{ 2 } } \right) }^{ 25 }\)

A manufacturer produces two types of steel trunks. He has two machine A and B. For completing, the first type of the trunk requires 3 hours on machine A and 2 hours on machine B, whereas the second type of the trunk requires 3 hours on machine A and 3 hours on machine B. Machines A and B can work at the most for 18 hours and 14 hours per day respectively. He earns a profit of Rs.30 andRs.40 per trunk of the first type and second type respectively. How many trunks of the each type must he make each day to make maximum profit?

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 1 kg of vitamin A and 5 units 1 kg of vitamin B while food Q contains 4 units 1 kg of vitamin A and 2 units 1 kg of vitamin B. Determine the minimum cost of the mixture.

One kind of the 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 other ingredients used in making the cakes?

A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (min) required for each toy on the machine is given below:

Each machine is available for a maximum of 6 hours/day. If the profit on each toy of type A is Rs.7.50 and for B is Rs.5. Show that 15 toys of type A and 30 of type B should be manufactured in a day to get maximum profit.

Every gram of wheat provides 0.1 g of proteins and 0.25 g of carbohydrates. The corresponding values of rice are 0.05 g and 0.5 g respectively. Wheat cost Rs.4 per kg and rice cost Rs.6 per kg. The minimum daily requirements of proteins and carbohydrate for an average child are 50 g and 200 g respectively. In what quantities should wheat and rice be mixed in the daily diet to provide minimum daily requirements of proteins and carbohydrate at minimum cost. Frame an LPP and solve it graphically. 

Find the annual rate of interest, to get a perpetuity of Rs. 675 for every half yearly from the present value of Rs. 30,000

Find the amount of an ordinary annuity of 12 monthly payments of Rs. 1, 500 that earns interest at 12% per annum compounded monthly. [(1.01)¹² = 1.1262 ]

Find the present value of an annuity of Rs. 900 payable at the end of 6^th months for 6 years. The money compounded at 8% per annum. [(1.04)^–12 = 0.6252 ]

A man buys 500 shares of amount Rs. 100 at Rs. 14 below par. How much money does he pay?

Which is better investment: 12% Rs. 20 shares at Rs. 16 (or) 15% Rs. 20 shares at Rs. 24.

Resolve into partial factors : \(\frac { { x }^{ 2 }+x+1 }{ { x }^{ 2 }+2x+1 } \)

In how many ways can the following prizes be given away to a class of 30 students, first and second in mathematics, first and second in physics, first in chemistry and first in English?

How many numbers are there between 100 and 1000 such that atleast one of their digits is 7?

If 22P_r+1:20P_r+2=11: 52, find r.

How many numbers greater than a million can be formed with the digits 2, 3, 0, 3, 4, 2, 3?

Solve graphically: Minimize Z = 20x₁ + 40x_2.
Subject to the constraints 36x₁ + 6x₂ ≥ 108,3x₁ + 12x₂ ≥ 36,20x₁ + 10x₂ ≥ 100 and x₁,x₂≥0

Minimize and maximize Z =x₁+ 2x₂ . Subject to the constraints x₁ + 2x₂ ≥ 100,2x₁ − x₂ ≤ 0,2x₁ + x₂ ≤ 200 and x₁,x₂ ≥ 10.

Compute the earliest start time, earliest finish time, latest start time and latest finish time of each activity of the project given below:

Calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity of the project given below and determine the critical path of the project and duration to complete the project.

The following table use the activities in a building project.

Draw the network for the project, calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity and find the critical path. Compute the project duration.

As the number of units manufactured increases from 6000 to 8000, the total cost of production increases from Rs. 33,000 to Rs. 40,000. Find the relationship between the cost (y) and the number of units made (x) if the relationship is linear.

Prove that the tangents to the circle x² + y²  = 169 at (5,12) and (12,-5) are perpendicular to each other.

Find the equation of the parabola whose vertex is (0, 0) passing through the point (2, 3) and axis is along X-axis.

If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.

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 side is 2 m from the vertex of the parabola?

Evaluate \(\begin{vmatrix} 2 &-1 &-2 \\0 & 2 & -1\\3 & -5& 0 \end{vmatrix}.\)

Show that 10P₃ = 9 P₃ + 3. 9P₂

Convert the parabola y²=4x+4y into standard form.

Evaluate \(\cot\left(\frac{-15\pi}{4}\right)\)

Evaluate : \(\cos\left[\frac{\pi}{3}-\cos^{-1}\left(\frac{1}{2}\right)\right]\)

Prove that cot x cot 2x - cot 2x cot 3x - cot 3x cot x = 1.

Prove that cos 4x = 1 - 8 sin²x cos²x.

Prove that cos 6x = 32 cos⁶x - 48 cos⁴x + 18 cos²x - 1.

If  \(sin\left( { sin }^{ -1 }\left( \frac { 1 }{ 5 } \right) +{ cos }^{ -1 }(x) \right) =1\)  then find the value of x

Prove that \({ cot }^{ -1 }\left[ \frac { \sqrt { 1+sinx } +\sqrt { 1-sinx } }{ \sqrt { 1+sinx } -\sqrt { 1-sinx } } \right] =\frac { x }{ 2 } \) where \(x\in \left( 0,\frac { \pi }{ 4 } \right) \)

Find the values of x if \(\begin{vmatrix} 2 & 4 \\5 & 1 \end{vmatrix}=\begin{vmatrix} 2x & 4\\6 & x \end{vmatrix}.\)

Show that 10P₃ = 9 P₃ + 3. 9P₂

Find the number of diagonals that can be drawn by joining the angular points of octagon ?

Find the angle between the pair of lines represented by the equation 3x²+10xy+8y²+14x+22y+15=0.

Find the value of \(\cos\left(\frac{5\pi}{12}\right)\)

From the following data compute the value of Harmonic Mean.

An automobile driver travels from plain to hill station 100km distance at an average speed of 30km per hour. He then makes the return trip at average speed of 20km per hour what is his average speed over the entire distance (200km)?

Calculate the value of quartile deviation and its coefficient from the following data

Find Q₁, Q₃, D₈ and P₆₇ of the following data :

The price of a commodity increased by 5% from 2004 to 2005, 8% from 2005 to 2006 and 77% from 2006 to 2007. Calculate the average increase from 2004 to 2007?

Prove that \(\frac { 4tan\ x(1-{ tan }^{ 2 }x) }{ 1-6{ tan }^{ 2 } x+{ tan }^{ 4 } x } =tanx\)

Prove that (sin 3x + sin x) sin x + (cos 3x - cos x) cos x = 0.

Prove that sin(n+1)x sin(n+2) x+cos(n+1)xcos(n+2)x=cosx.

If tan x = \(\frac { -4 }{ 3 } \) and x is in II quadrant,find \(sin\frac { x }{ 2 } ,cos\frac { x }{ 2 } \) and \(tan\frac { x }{ 2 } \)

Prove that cos²2x - cos²6x = sin 4x.sin 8x

Verify that A(adj A) = (adj A) A = IAI·I for the matrix A = \(\begin{bmatrix}2 & 3 \\-1 & 4\end{bmatrix}\)

It the letters of the word are arranged as in dictionary, find the rank of the word "AGAIN".

A point moves so that its distance from the point (-1, 0) is always three times its distance from the point (0, 2). Find its locus.

Prove that\(\left\{1+\cot x-\sec\left(\frac{\pi}{2}+x\right)\right\}\left\{1+\cot x+\sec\left(\frac{\pi}{2}+x\right)\right\}=2\cot x\)

Is the function defined by f(x) = x² -sin x + 5 is continuous at x =\(\pi\)?

Probability of solving specific problem independently by A and B are \(\frac{1}{2}\) and \(\frac{1}{3}\) respectively. If both try to solve the problem independently, find the probability that the problem is
(i) solved
(ii) exactly one of them solves the problem

Two urns contains the set of balls as given in the following table

One ball is drawn from each urn and find the probability that
(i) both balls are red
(ii) both balls are of the same colour.

Ten cards numbered 1 to 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 4. What is the probability that it is an even number?

Find out the GM for the following

Calculate Quartile deviation and Coefficient of Quartile deviation of the following data.

If the function \(f\left( x \right) =\begin{cases} 6ax+3b\quad if\quad x>1 \\ ax-2b\quad if\quad x<1\quad is\quad continuous\quad at\quad x=1 \\ 15\quad if\quad x=1 \end{cases}\) Find the value of a and b.

Evaluate \(\begin{matrix} \underset { x\rightarrow 1 }{ lim } & \frac { { x }^{ 7 }-2{ x }^{ 5 }+1 }{ { x }^{ 3 }-{ 3x }^{ 2 }+2 } \end{matrix}\)

Differentiate: xy + y² = tan x + y.

Differentiate sin (tan^-1 (e^-x))

Differentiate: x² (x + 1)³ (x + 2)⁴ with respect to 'x'.

Find the numbers a and b such that A² + aA + bI = 0 for the matrix A =\(\begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}\)

Find the value of tan \(\left( {{\pi}\over{8}} \right)\)

Find the locus of a point such that the sum of its distances from the points (0, 2) and (0, -2) is 6.

Show that \(\tan\left(\frac{\pi}{3}+x\right)\tan\left(\frac{\pi}{3}-x\right)=\frac{2\cos2x+1}{2\cos2x-1}\)

Find \(\frac{dy}{dx}\) if x = 15(t - sin t); y = 18(1 - cos t).

The two regression lines are 3X + 2Y = 26 and 6X + 3Y = 31. Find the correlation coefficient.

A survey was conducted to study the relationship between expenditure on accommodation (X) and expenditure on Food and Entertainment (Y) and the following results were obtained:

Write down the regression equation and estimate the expenditure on Food and Entertainment, if the expenditure on accommodation is Rs. 200.

Find the coefficient of correlation for the following data:

X and Y are a pair of correlated variables. Ten observations of their values (X, Y) have the following results. ΣX = 55, ΣXY = 350, ΣX² = 385, ΣY = 55, Predict the value of Y when the value of X is 6.

The following information is given

Coefficient of correlation between X and Y is \(\frac{8}{15}\) . Find (i) The regression Coefficient of Y on X (ii) The most likely value of Y when X = Rs.100.

If \(\begin{matrix} \underset { x\rightarrow 1 }{ lim } & \frac { { x }^{ 4 }-1 }{ x-1 } \end{matrix}=\begin{matrix} \underset { x\rightarrow k }{ lim } & \frac { { x }^{ 3 }-{ k }^{ 3 } }{ { x }^{ 2 }-{ k }^{ 2 } } \end{matrix}\),then find the value of K

Evaluate \(\underset { h\rightarrow 0 }{ lim } \frac { \sqrt { x+h } -\sqrt { x } }{ h } \)

Differentiate (sec x -1) (sec x +1)

If \(x=a\left( t+\frac { 1 }{ t } \right) ;y=a\left( t-\frac { 1 }{ t } \right) \) show that \(\frac { dy }{ dx } =\frac { x }{ y } \)

If \(y={ e }^{ a\cos ^{ -1 }{ x } }\) , show that \(\left( 1-{ x }^{ 2 } \right) \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -x\frac { dy }{ dx } -{ a }^{ 2 }y=0\)

Without expanding show that \(\Delta =\left| \begin{matrix} { cosec }^{ 2 }\theta & { cot }^{ 2 }\theta & 1 \\ { cot }^{ 2 }\theta & { cosec }^{ 2 }\theta & -1 \\ 42 & 40 & 2 \end{matrix} \right| =0\)

In how many ways can the following prizes be given away to a class of 30 students, first and second in mathematics, first and second in physics, first in chemistry and first in English?

Prove that the tangents to the circle x² + y²  = 169 at (5,12) and (12,-5) are perpendicular to each other.

Prove that (sin 3x + sin x) sin x + (cos 3x - cos x) cos x = 0.

If the function \(f\left( x \right) =\begin{cases} 6ax+3b\quad if\quad x>1 \\ ax-2b\quad if\quad x<1\quad is\quad continuous\quad at\quad x=1 \\ 15\quad if\quad x=1 \end{cases}\) Find the value of a and b.

Calculate coefficient of correlation from the following data

Find coefficient of correlation for the following:

Calculate rank correlation coefficient of the following data.

There are two series of index numbers P for price index and S for stock of the commodity. The mean and standard deviation of P are 100 and 8 and of S are 103 and 4 respectively. The correlation coefficient between the two series is 0.4. With these data obtain the regression lines of P on S and S on P.

You are given the following data:

If the Correlation coefficient between X and Y is 0.66, then find (i) the two regression coefficients, (ii) the most likely value of Y when X = 10

An amount of Rs. 5000 is put into three investment at the rate of interest of 6%, 7% and 8% per annum respectively. The total annual income is Rs. 358. If the combined income from the first two investment is Rs. 70 more than the income from the third, find the amount of each investment by matrix method.

Using binomial theorem, find the value of \({ \left( \sqrt { 2 } +1 \right) }^{ 5 }+{ \left( \sqrt { 2 } -1 \right) }^{ 5 }\)

If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.

Prove that cos 6x = 32 cos⁶x - 48 cos⁴x + 18 cos²x - 1.

If \(y={ e }^{ a\cos ^{ -1 }{ x } }\) , show that \(\left( 1-{ x }^{ 2 } \right) \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -x\frac { dy }{ dx } -{ a }^{ 2 }y=0\)

A company is producing three products P₁, P₂ and P₃, with profit contribution of Rs.20, Rs.25 and Rs.15 per unit respectively. The resource requirements per unit of each of the products and total availability are given below.

Formulate the above as a linear programming model.

A soft drink company has two bottling plants C₁ and C₂. Each plant produces three different soft drinks S₁, S₂ and S₃. The production of the two plants in number of bottles per day are:

A market survey indicates that during the month of April there will be a demand for 24000 bottles of S₁, 16000 bottles of S₂ and 48000 bottles of S₃. The operating costs, per day, of running plants C₁ and C₂ are respectively Rs.600 and Rs.400. How many days should the firm run each plant in April so that the production cost is minimized while still meeting the market demand? Formulate the above as a linear programming model.

A company produces two types of products say type A and B. Profits on the two types of product are Rs.30/- and Rs.40/- per kg respectively. The data on resources required and availability of resources are given below.

Formulate this problem as a linear programming problem to maximize the profit.

Construct a network diagram for the following situation:
A < D, E; B, D < F; C < G and B < H.

A firm manufactures pills in two sizes A and B. Size A contains 2 mgs of aspirin, 5 mgs of bicarbonate and 1 mg of codeine. Size B contains 1 mg. of aspirin, 8 mgs. of bicarbonate and 6 mgs. of codeine. It is found by users that it requires atleast 12 mgs. of aspirin, 74 mgs.of bicarbonate and 24 mgs. of codeine for providing immediate relief. It is required to determine the least number of pills a patient should take to get immediate relief. Formulate the problem as a standard LLP.

A furniture dealer deals only two items viz., tables and chairs. He has to invest Rs.10,000/- and a space to store atmost 60 pieces. A table cost him Rs.500/– and a chair Rs.200/–. He can sell all the items that he buys. He is getting a profit of Rs.50 per table and Rs.15 per chair. Formulate this problem as an LPP, so as to maximize the profit.

A dietician wishes to mix two types of food F₁ and F₂ in such a way that the vitamin contents of the mixture contains atleast 6 units of vitamin A and 9 units of vitamin B. Food F₁ costs Rs.50 per kg and F₂ costs Rs 70 per kg. Food F₁ contains 4 units per kg of vitamin A and 6 units per kg of vitamin B while food F₂ contains 5 units per kg of vitamin A and 3 units per kg of vitamin B. Formulate the above problem as a linear programming problem to minimize the cost of mixture.

A company produces two types of pens A and B. Pen A is of superior quality and pen B is of lower quality. Profits on pens A and B are Rs. 5 and Rs. 3 per pen respectively. Raw materials required for each pen A is twice as that of pen B. The supply of raw material is sufficient only for 1000 pens per day. Pen A requires a special clip and only 400 such clips are available per day. For pen B, only 700 clips are available per day. Formulate this problem as a linear programming problem.

Draw a network diagram for the project whose activities and their predecessor relationships are given below:

A firm manufactures two products A and B on which the profits earned per unit are Rs. 3 and Rs. 4 respectively. Each product is processed on two machines M₁ and M₂. Product A requires one minute of processing time on M₁ and two minutes on M₂, While B requires one minute on M₁ and one minute on M₂. Machine M₁ is available for not more than 7 hrs 30 minutes while M₂ is available for 10 hrs during any working day. Formulate this problem as a linear programming problem to maximize the profit.

Evaluate:\(\begin{vmatrix} 1&a&a^2-bc\\1&b&b^2-ca\\1&c&c^2-ab \end{vmatrix}\)

Prove that \(\begin{vmatrix} -a^{ 2 } & ab & ac \\ ab & -b^{ 2 } & bc \\ ac & bc & -c^{ 2 } \end{vmatrix}=4a^{ 2 }b^{ 2 }{ c }^{ 2 }\)

If A = \(\begin{bmatrix}1 & 1 & 1 \\ 3 & 4 & 7\\1 & -1 & 1 \end{bmatrix}\) verify that A ( adj A ) = ( adj A ) A = |A| I₃.

Suppose the inter-industry flow of the product of two industries are given as under.

Determine the technology matrix and test Hawkin's -Simon conditions for the viability of the system. If the domestic demand changes to 80 and 40 units respectively, what should be the gross output of each sector in order to meet the new demands.

The cost of 2 Kg of Wheat and 1 Kg of Sugar is Rs.70. The cost of 1 Kg of Wheat and 1 Kg of Rice is Rs.70 The cost of 3 Kg of Wheat, 2 Kg of Sugar and 1 Kg of rice is Rs.170. Find the cost of per kg each item using matrix inversion method.

If X \(=\begin{bmatrix} 8 &-1&-3 \\-5 &1&2\\10&-1&-4 \end{bmatrix}\)  and Y = \(\begin{bmatrix} 2 & 1 & -1\\0 & 2 & 1\\ 5& p & q \end{bmatrix}\) then, find p, q if Y =  X^-1

If \(A=\left[ \begin{matrix} 2 & 4 & 4 \\ 2 & 5 & 4 \\ 2 & 5 & 3 \end{matrix} \right] \) then find A ^-1.

If \(A=\left[ \begin{matrix} 1 & 2 \\ 1 & 1 \end{matrix} \right] ,B=\left[ \begin{matrix} 0 & -1 \\ 1 & 2 \end{matrix} \right] \)then, show that (AB)^-1 = B^-1A^-1

Solve by using matrix inversion method:
\(3 x-2 y+3 z=8 ; 2 x+y-z=1\)
\(4 x-3 y+2 z=4\)

An economy produces only coal and steel. These two commodities serve as intermediate inputs in each other’s production. 0.4 tonne of steel and 0.7 tonne of coal are needed to produce a tonne of steel. Similarly 0.1 tonne of steel and 0.6 tonne of coal are required to produce a tonne of coal. No capital inputs are needed. Do you think that the system is viable? 2 and 5 labour days are required to produce a tonnes of coal and steel respectively. If economy needs 100 tonnes of coal and 50 tonnes of steel, calculate the gross output of the two commodities and the total labour days required.

Resolve into partial fractions for the following : \(\frac{2 x^2-5 x-7}{(x-2)^3}\)

Resolve into partial fractions for the following : \(\frac{x^2-6 x+2}{x^2(x+2)}\)

Resolve into partial fractions for the following : \(\frac{1}{\left(x^2+4\right)(x+1)}\)

Show that the middle term in the expansion of (1 + x)²ⁿ is \(\frac { 1.3.5....(2n-1){ 2 }^{ n }.{ x }^{ n } }{ n! } \)

How many 6-digit telephone numbers can be constructed with the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if each numbers starts with 35 and no digit appear more than once?

Resolve into partial fraction :\(\frac{x+4}{\left(x^2-4\right)(x+1)}\)

Resolve into partial fraction \(\frac{x+1}{(x-2)^2(x+3)}\)

Resolve into partial fractions:\(\frac{2x+1}{(x-1)(x^2+1)}\)

A Cricket team of 11 players is to be formed from 16 players including 4 bowlers and 2 wicket-keepers. In how many different ways can a team be formed so that the team contains at least 3 bowlers and at least one wicket-keeper?

By the principle of Mathematical Induction, prove that 1 + 3 + 5 …+ (2n – 1) = n², for all n ∈ N.

Determine whether the points P(1,0), Q(2,1) and R(2,3) lie outside the circle, on the circle or inside the circle x² + y² - 4x- 6y + 9 = 0

Find the equation of the parabola which is symmetrical about x-axis and passing through (-2, -3).

A private company appointed a clerk in the year 2012, his salary was fixed as Rs.20,000. In 2017 his salary raised to Rs.25,000.
(i) Express the above information as a linear function in x and y where y represent the salary of the clerk and x-represent the year.
(ii) What will be his salary in 2020?

Find the equation of the circle passing through the points (0,0), (1, 2) and (2,0).

Find the axis, vertex, focus, equation of directrix and the length of latus rectum for the parabola x² + 6x - 4y + 21 = 0

Show that the equation 12x² - 10xy + 2y² + 14x - 5y + 2 = 0 represents a pair of straight lines and also find the separate equations of the straight lines.

Find the equation of the circle passing through the points (0, 1) , (4 ,3) and (1, -1).

As the number of units produced increases from 500 to 1000 and the total cost of production increases from. Rs 6000 to Rs 9000. Find the relationship between the cost (y) and the number of units produced (x) if the relationship is linear.

Find whether the points (-1,-2), (1,0) and (-3, -4) lie above, below or on the line 3x + 2y + 7 = 0.

Find the axis, vertex, focus, equation of directrix and the length of latus rectum of the parabola (y - 2)² = 4(x - 1).

Prove that \(\frac{\cos4x+\cos3x+\cos2x}{\sin4x+\sin3x+\sin2x}=\cot3x\)

Prove that:  \(\frac { \sin { \left( 180-\theta \right) } \cos { \left( 90+\theta \right) } \tan { \left( 270-\theta \right) \cot { \left( 360-\theta \right) } } }{ \sin { \left( 360-\theta \right) \cot { \left( 360+\theta \right) \sin { \left( 270-\theta \right) \csc { \left( -\theta \right) } } } } } =-1\)

Prove that \(\tan { \left( \pi +x \right) } \cot { \left( x-\pi \right) } -\left( \cos { \left( 2\pi -x \right) } \cos { \left( 2\pi +x \right) } \right) =\sin ^{ 2 }{ x } \)

If \(\sin A=\frac13,\sin B=\frac14\), then find the value of sin(A + B) where A and B are acute angles.

If \(\cos(\alpha+\beta)=\frac45\) and \(\sin(\alpha-\beta)=\frac{5}{13}\) where \((\alpha+\beta)\) and \((\alpha-\beta)\) are acute, then find \(\tan2\alpha\)

If sin (y + z - x), sin (z + x - y), sin (x + y - z) are in A.P., then prove that tan x, tan y, tan z are also in A. P

If sin \(\alpha +\sin { \beta } =a\) and \(\cos { \alpha + } \cos { \beta } =b\) , then prove that \(\cos { \left( a-\beta \right) } =\frac { { a }^{ 2 }+{ b }^{ 2 }-2 }{ 2 } \)

If tan \(\alpha={{1}\over{7}},\sin\beta={{1}\over{\sqrt{10}}},\) Prove that \(\alpha+2\beta={{\pi}\over{4}}\) where \(0<\alpha<{{\pi}\over{2}}\) and \(0<\beta<{{\pi}\over{2.}}\)

If cosA = \(\frac{4}{5}\)and cosB = \(\frac{12}{13}\),\(\frac{3\pi}{3}<(A, B)<2 \pi,\) find the value of cos(A+B).

Solve \({\tan}^{-1}\left( \frac { x-1 }{ x-2 } \right) +{ \tan }^{ -1 }\left( \frac { x+1 }{ x+2 } \right) =\frac { \pi }{ 4 } \)

If x\(\sqrt { 1+y } +y\sqrt { 1+x } =0\) and x ≠ y, then prove that \(\frac { dy }{ dx } =\frac { -1 }{ (x+1)^{ 2 } } \).

Differentiate the following with respect to x \(\sqrt { \frac { (x-1)(x-2) }{ (x-3)({ x }^{ 2 }+x+1) } } \)

Show that \( f(x)= \begin{cases}5 x-4, & \text { if } 0< x \leq 1 \\ 4 x^3-3 x, & \text { if } 1< x< 2\end{cases}\) is continuous at x = 1

Show that the function f(x) = |x| is not differentiable at x = 0.

Differentiate: \(\sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}}\)

Examine the following functions for continuity at indicated points
\(f(x)=\left\{\begin{array}{cl} \frac{x^2-4}{x-2}, & \text { if } x \neq 2 \\ 0, & \text { if } x=2 \end{array} \right.\) at x = 2

If y = acos mx + b sin mx, then show that y₂ + m²y = 0.

If y = sin (log x), then show that x²y₂ + xy₁ + y = 0.

Find the derivative of the following functions from first principle. log (x + 1)

Verify the continuity and differentiability of \(f(x)= \begin{cases}1-x & \text { if } x<1 \\ (1-x)(2-x) & \text { if } 1 \leq x \leq 2 \\ 3-x & \text { if } x>2\end{cases}\) at x = 1 and x = 2

A company buys in lots of 500 boxes which is a 3 month supply. The cost per box is Rs. 125 and the ordering cost in Rs. 150. The inventory carrying cost is estimated at 20% of unit value.
(i) Determine the total amount cost of existing inventory policy
(ii) Determine EOQ in units
(iii) How much money could be saved by applying the economic order quantity?

A dealer has to supply his customer with 400 units of a product per every week. The dealer gets the product from the manufacturer at a cost of Rs. 50 per unit. The cost of ordering from the manufacturers in Rs. 75 per order. The cost of holding inventory is 7.5 % per year of the product cost. Find (i) EOQ (ii) Total optimum cost.

Let u = log\(\frac { { x }^{ 4 }+{ y }^{ 4 } }{ x+y } \). By using Euler’s theorem show that \(x.\frac { \partial u }{ \partial x } +y.\frac { \partial u }{ \partial y } =3\) .

let u = x²y³ cos \(\left( \frac { x }{ y } \right) \) by using Euler’s theorem show that  \(x.\frac { \partial u }{ \partial x } +y.\frac { \partial u }{ \partial y } =5u\)

 The demand for a commodity x is q = 5-2p₁+ P₂ -\({ p }_{ 1 }^{ 2 }{ p }_{ 2 }\). Find the partial elasticities \(\frac { Eq }{ { EP }_{ 1 } } \) and \(\frac { Eq }{ { EP }_{ 2 } } \) when p₁= 3 and p₂ = 7

For the production function P = 3(L)^0.4 (K)^0.6, find the marginal productivities of labour (L) and capital (K) when L = 10 and K = 6. [use; (0.6)^0.6 = 0.736, (1.67)^0.4  = 1.2267]

The demand for a commodity A is q = 80 - \({ p }_{ 1 }^{ 2}\) + 5p₂ - p₁p₂. Find the partial elasticities \(\frac { { E }q }{ { E }p_{ 1 } } \) and \(\frac { { E }q }{ { E }p_{ 2 } } \) when p₁ = 2, p₂ = 1.

The total cost function y for x units is given by y = 4x\(\left( \frac { x+2 }{ x+1 } \right) +6\), Prove that marginal cost decreases as x increases.

A certain manufacturing concern has total cost function C = 15 + 9x - 6x² + x³ . find Find x, when the total cost is minimum 

Verify Euler’s theorem for the function \(u=\frac{1}{\sqrt{x^2+y^2}}\)

A man invests Rs. 13,500 partly in 6% of Rs. 100 shares at Rs. 140 and the remaining in 5% of Rs. 100 shares at Rs 125. If his total income is Rs. 560, how much has he invested in each?

An equipment is purchased on an installment basis such that Rs. 5000 on the signing of the contract and four yearly installments of Rs. 3000 each payable at the end of first, second, third and the fourth year. If the interest is charged at 5% p.a find the cash  down price.[(1.05)^–4 = 0.8227]

Machine A costs Rs. 15,000 and machine B costs Rs. 20,000. The annual income from A and B are Rs. 4,000 and Rs. 7,000 respectively. Machine A has a life of 4 years and B has a life of 7 years. Find which machine may be purchased. (Assume discount rate 8% p.a)

Gopal invested Rs. 8,000 in 7% of Rs. 100 shares at Rs. 80. After a year he sold these shares at Rs. 75 each and invested the proceeds (including his dividend) in 18% for Rs. 25 shares at Rs. 41. Find
(i) his dividend for the first year
(ii) his annual income in the second year
(iii) The percentage increase in his return on his original investment

A man sells 2000 ordinary shares (par value Rs. 10) of a tea company which pays a dividend of 25% at Rs. 33 per share. He invests the proceeds in cotton textiles (par value Rs. 25) ordinary shares at 44 per share which pays a dividend of 15%. Find
(i) the number of cotton textiles shares purchased and
(ii) change in his dividend income.

Sundar bought Rs 4,500,12 % of Rs. 10 shares at par. He sold them when the price rose to Rs. 23 and invested the proceeds in Rs. 25 shares paying 10% per annum at Rs. 18. Find the change in his income.

Babu sold some Rs. 100 shares at 10% discount and invested his sales proceeds in 15% of Rs. 50 shares at Rs. 33. Had he sold his shares at 10% premium instead of 10% discount, he would have earned Rs. 450 more. Find the number of shares sold by him.

Naveen deposits Rs. 250 at the beginning of each month in an account that pays an interest of 6% per annum compounded monthly, how many months will be required for the deposit to amount to atleast Rs. 6390?

Vijay wants to invest Rs. 27,000 in buying shares. The shares of the following companies are available to him. Rs. 100 shares of company A at par value ; Rs. 100 shares of company B at a premium of Rs. 25. Rs. 100 shares of company C at a discount of Rs. 10. Rs. 50 shares of company D at a premium of 20%. Find how many shares will he get if he buys shares of company (i) A (ii) B (iii) C (iv) D

The capital of a company is made up of 50,000 preferences shares with a dividend of 16% and 2,500 ordinary shares. The par value of each of preference and ordinary shares is Rs. 10. The company had a total profit of Rs. 1,60,000. If Rs. 20,000 were kept in reserve and Rs. 10,000 in depreciation, what percent of dividend is paid to the ordinary share holders

Calculate the quartile deviation and its coefficient from the following data:

Compute mean deviation about median from the following data:

Find out the coefficient of mean deviation about median in the following series

Bag I contains 3 Red and 4 Black balls while another Bag II contains 5 Red and 6 Black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from Bag I.

Three boxes B₁, B₂, B₃ contain lamp bulbs some of which are defective. The defective proportions in box B₁, box B₂ and box B₃ are respectively \(\frac { 1 }{ 2 } ,\frac { 1 }{ 8 } \ and \ \frac { 3 }{ 4 } \). A box is selected at random and a bulb drawn from it. If the selected bulb is found to be defective, what is the probability that box B₁ was selected?

A company has three machines A, B, C which produces 20%, 30% and 50% of the product respectively. Their respective defective percentages are 7, 3 and 5. From these products one is chosen and inspected. If it is defective what is the probability that it has been made by machine C?

In a shooting test the probability of hitting the target are \(\frac{3}{4}\) for A, \(\frac{1}{2}\) for B and \(\frac{2}{3}\) for C. If all of them fire at the same target, calculate the probabilities that
(i) All the three hit the target
(ii) Only one of them hits the target
(iii) At least one of them hits the target

In a screw factory machines A, B, C manufacture respectively 30%, 40% and 30% of the total output of these 2%, 4% and 6% percent are defective screws. A screws is drawn at random from the product and is found to be defective. What is the probability that it was manufactured by Machine C?

Data on readership of a magazine indicates that the proportion of male readers over 30 years old is 0.30 and the proportion of male reader under 30 is 0.20. If the proportion of readers under 30 is 0.80. What is the probability that a randomly selected male subscriber is under 30?

A can solve 90 per cent of the problems given in a book and B can solve 70 per cent. What is the probability that at least one of them will solve a problem selected at random?

For the given lines of regression 3X – 2Y = 5 and X – 4Y = 7. Find
(i) Regression coefficients
(ii) Coefficient of correlation

The heights ( in cm.) of a group of fathers and sons are given below

Find the lines of regression and estimate the height of son when the height of the father is 164 cm.

The following data relate to advertisement expenditure(in lakh of rupees) and their corresponding sales( in crores of rupees)

Estimate the sales corresponding to advertising expenditure of Rs. 30 lakh.

For 5 observations of pairs of (X, Y) of variables X and Y the following results are obtained. ΣX = 15, ΣY = 25, ΣX² = 55, ΣY² = 135, ΣXY = 83. Find the equation of the lines of regression and estimate the values of X and Y if Y = 8; X = 12.

A random sample of recent repair jobs was selected and estimated cost, actual cost were recorded.

Calculate the value of spearman’s correlation.

From the data given below

Find (a) The two regression equations, (b) The coefficient of correlation between marks in Economics and statistics, (c) The mostly likely marks in Statistics when the marks in Economics is 30.

The following data give the height in inches (X) and the weight in lb. (Y) of a random sample of 10 students from a large group of students of age 17 years:

Estimate weight of the student of a height 69 inches.

Find the equation of the regression line of Y on X, if the observations ( X_i, Y_i) are the following (1, 4) (2, 8) (3, 2) ( 4, 12) (5, 10) (6, 14) (7, 16) ( 8, 6) (9, 18).

The equations of two lines of regression obtained in a correlation analysis are the following 2X = 8 – 3Y and 2Y = 5 – X. Obtain the value of the regression coefficients and correlation coefficient.

The following data pertains to the marks in subjects A and B in a certain examination. Mean marks in A = 39.5, Mean marks in B = 47.5, standard deviation of marks in A = 10.8 and Standard deviation of marks in B = 16.8. coefficient of correlation between marks in A and marks in B is 0.42. Give the estimate of marks in B for candidate who secured 52 marks in A.

Solve the following LPP. Maximize Z = 2 x₁ + 5x₂ subject to the conditions x₁+ 4x₂ ≤ 24. 3x₁+x₂ ≤ 21, x₁+x₂ ≤ 9 and x₁, x₂ ≥ 0.

Solve the following LPP.
Maximize Z= 2 x₁ + 3x₂ subject to constraints x₁ + x₂ ≤ 30; x₂ ≤ 12; x₁ ≤ 20 and x₁, x₂ ≥ 0.

A company manufactures two models of voltage stabilizers viz., ordinary and autocut. All components of the stabilizers are purchased from outside sources, assembly and testing is carried out at company’s own works. The assembly and testing time required for the two models are 0.8 hour each for ordinary and 1.20 hours each for auto-cut. Manufacturing capacity 720 hours at present is available per week. The market for the two models has been surveyed which suggests maximum weekly sale of 600 units of ordinary and 400 units of auto-cut. Profit per unit for ordinary and auto-cut models has been estimated at Rs. 100 and Rs. 150 respectively. Formulate the linear programming problem.

Compute the earliest start time, earliest finish time, latest start time and latest finish time of each activity of the project given below:

A project schedule has the following characteristics

Construct the network and calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity and determine the Critical path of the project and duration to complete the project.

A Project has the following time schedule

Construct the network and calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity and determine the Critical path of the project and duration to complete the project.

Calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity of the project given below and determine the Critical path of the project and duration to complete the project.

Solve the following linear programming problem graphically.
Minimize Z = 200x₁+ 500x₂ subject to the constraints x₁+2x₂ ≥ 10; 3x₁+4x₂ ≤ 24 and x₁ ≥ 0, x₂ ≥ 0

Solve the following linear programming problem graphically. Maximize Z = 60x₁ + 15x₂ subject to the constraints: x₁ + x₂ ≤ 50; 3x₁ + x₂ ≤ 90 and x₁, x₂ ≥ 0.

The following table gives the characteristics of project

Draw the network for the project, calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity and find the critical path. Compute the project duration.

The value of \(\begin{vmatrix} 2x+y & x & y \\ 2y+z & y & z \\ 2z+x & z & x \end{vmatrix}\) is ________.

The value of n, when nP₂ = 20 is _______.

If kx² + 3xy - 2y² = 0 represent a pair of lines which are perpendicular then k is equal to _______.

The value of cos²45^o - sin²45^o is_______.

f(x) = - 5 , for all \(x\in R\), is a ________.

The inverse matrix of \(\begin{pmatrix} \frac { 1 }{ 5 } & \frac { 5 }{ 25 } \\ \frac { 2 }{ 5 } & \frac { 1 }{ 2 } \end{pmatrix}\) is ________.

The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is _________.

The equation of the circle with centre (3,-4) and touches the x - axis is _______.

The value of \(\frac{3 \tan 10^{\circ}-\tan ^3 10^{\circ}}{1-3 \tan ^2 10^{\circ}}\) is _______,

\(\frac{d}{dx}(\frac{1}{x})\) is equal to ________.

The technology matrix of an economic system of two industries is \(\begin{bmatrix} 0.50 & 0.25 \\ 0.40 & 0.67 \end{bmatrix}\). Test whether the system is viable as per Hawkins-Simon conditions.

Find the rank of the word 'CHAT' in dictionary.

Find the equation of the circle having (4,7) and (-2,5) as the extremities of a diameter.

Convert the following degree measure into radian measure -320^o

Determine whether the following functions are odd or even?
 \(f(x)=log({ x }^{ 2 }+\sqrt { { x }^{ 2 } } +1)\)

Show that\(\left| \overset { x }{ 2x\underset { a }{ + } 2a } \quad \overset { y }{ 2y\underset { b }{ + } 2b } \quad \overset { z }{ 2z\underset { c }{ + } 2c } \right| =0\)

Find the number of 4 letter words, with or without meaning, which can be formed out of the letters of the word “ NOTE”, where the repetition of the letters is not allowed.

A point in the plane moves so that its distance from the origin is thrice its distance from the y- axis. Find its locus.

Convert the following degree measure into radian measure -320^o

Differentiate the following with respect to x 3x⁴ - 2x³ + x + 8

Solve:\(\begin{vmatrix}7&4&11\\-3&5&x\\-x&3&1 \end{vmatrix}=0\)

Find the 5^th term in the expansion of (x - 2y)^13.

Find the angle between the lines whose slopes are \(\frac { 1 }{ 2 } \) and 3

Prove that \(\sqrt3\) cosec 20^o- sec 20^o = 4

Differentiate sin³ x with respect to cos³x.

If \(A=\left[ \begin{matrix} 2 & 4 \\ -3 & 2 \end{matrix} \right] \)then, find A ^-1.

How many distinct words can be formed using all the letters of the following words.
MATHEMATICS.

For what value of k does 2x² + 5xy + 2y² + 15x + 18y + k = 0 represent a pair of straight lines.

If tan A = m tanB, prove that \(\frac { sin(A+B) }{ sin(A-B) } =\frac { m+1 }{ m-1 } \)

Differentiate the following with respect to x. sin (x²)

Prove that \(\begin{vmatrix} {1\over a}&bc&b+c\\{1\over b}&ca&c+a\\{1\over c}&ab&a+b \end{vmatrix}=0\)

Resolve into partial fractions for the following : \(\frac{1}{x^2-1}\)

Show that the equation 12x² - 10xy + 2y² + 14x - 5y + 2 = 0 represents a pair of straight lines and also find the separate equations of the straight lines.

Prove that:  \(\sin { \theta } \cos { \theta } \left\{ \sin { \left( \frac { \pi }{ 2 } -\theta \right) } \csc { \theta } +\cos { \left( \frac { \pi }{ 2 } -\theta \right) \sec { \theta } } \right\} =1\)

Examine the following functions for continuity at indicated points
\(f(x)=\left\{\begin{array}{cl} \frac{x^2-4}{x-2}, & \text { if } x \neq 2 \\ 0, & \text { if } x=2 \end{array} \right.\) at x = 2

A sales person Ravi has the following record of sales for the month of January, February and March 2009 for three products A, B and C. He has been paid a commission at fixed rate per unit but at varying rates for products A, B and C.

Find the rate of commission payable on A, B and C per unit sold using matrix inversion method.

How many code symbols can be formed using 5 out of 6 letters A, B, C, D, E, F so that the letters
a) cannot be repeated
b) can be repeated
c) cannot be repeated but must begin with E
d) cannot be repeated but end with CAB.

Determine whether the points P(0,1), Q(5,9), R(–2, 3) and S(2, 2) lie outside the circle, on the circle or inside the circle x²+y²-4x+4y-8 = 0.

If cosA =\(\frac{4}{5}\)and cosB =\(\frac{12}{13}\),\(\frac{3 \pi}{2}<(A, B)<2 \pi\), find the value of sin(A - B)

Show that the function f(x) = |x| is not differentiable at x = 0.

The value of \(\begin{vmatrix} x & x^2 & -yz & 1 \\ y & y^2 & -zx & 1 \\ z & z^2 & -xy &1 \end{vmatrix}\) is ________.

If A = \(\begin{vmatrix}cos \theta & sin \theta \\ -sin \theta&cons\theta \end{vmatrix},\) then |2A| is equal to ________.

If \(\triangle=\begin{vmatrix} {a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & {a}_{23} \\ {a}_{31} & {a}_{32} & {a}_{33} \end{vmatrix}\) and A_ij is cofactor of a_ij, then value of \(\triangle\) is given by ________.

If \(\begin{vmatrix} 4 & 3 \\ 3 & 1 \end{vmatrix}=-5\) then value of \(\begin{vmatrix} 20 & 15 \\ 15 & 5 \end{vmatrix}\) is ________.

If any three rows or columns of a determinant are identical then the value of the determinant is ________.

The number of Hawkins-Simon conditions for the viability of an input - output analysis is ________.

The inventor of input-output analysis is ________.

Which of the following matrix has no inverse.

The inverse matrix of \(\begin{pmatrix} 3 & 1 \\ 5 & 2\end{pmatrix}\) is ________.

If A \(=\begin{pmatrix} -1 & 2 \\ 1 & -4 \end{pmatrix}\) then A (adj A) is ________.

The greatest positive integer which divide n(n + 1) (n + 2) (n + 3) for n \(\in\) N is ________.

For all n > 0, nC₁ + nC₂ + nC₃ + ... +nC_n is equal to _______

The middle term in the expansion of \({ \left( x+\frac { 1 }{ x } \right) }^{ 10 }\) is _______.

The last term in the expansion of (3 +\(\sqrt{2}\) )⁸ is ________

The number of 3 letter words that can be formed from the letters of the word number when the repetition is allowed are ________.

There are 10 true or false questions in an examination. Then these questions can be answered in ______.

The total number of 9 digit number which have all different digit is ________.

Thirteen guests has participated in a dinner. The number of handshakes happened in the dinner is __________.

Sum of Binomial co-efficient in a particular expansion is 256, then number of terms in the expansion is ________.

Sum of the binomial coefficients is ________.

The angle between the pair of straight lines x² - 7xy + 4y² = 0 _______.

The x-intercept of the straight line 3x + 2y - 1 = 0 is _______.

The locus of the point P which moves such that P is at equidistance from their coordinate axes is _______.

If kx² + 3xy - 2y² = 0 represent a pair of lines which are perpendicular then k is equal to _______.

The length of the tangent from (4,5) to the  circle x² + y² = 16 is _______.

Combined equation of co-ordinate axes is _______.

In the equation of the circle x² + y² = 16 then y intercept is (are) _______.

The equation of the circle with centre (3,-4) and touches the x - axis is _______.

The eccentricity of the parabola is _______.

The distance between directrix and focus of a parabola y² = 4ax is _______.

The value of cos²45^o - sin²45^o is_______.

The value 4cos³40^o - 3cos40^o is ________.

If \(\sin A=\frac{1}{2}\) then \(4\cos^3A-3\cos A\) is ________.

The value of \(cosec^{-1}\left(\frac{2}{\sqrt{3}}\right)\) is ________.

If \(\alpha\) and \(\beta\) be between 0 and \(\frac{\pi}{2}\) and if \(\cos(\alpha+\beta)=\frac{12}{13}\) and \(\sin(\alpha-\beta)=\frac{3}{5}\) then \(\sin2\alpha\) is  _____.

The value of \(\frac{3 \tan 10^{\circ}-\tan ^3 10^{\circ}}{1-3 \tan ^2 10^{\circ}}\) is _______,

\(\sec^{-1}\frac{2}{3}+cosec^{-1}\frac{2}{3}=\) ______.

If \(\tan A=\frac{1}{2}\) and \(\tan B=\frac{1}{3}\) then tan(2A + B) is equal to ______.

\(\sin\left(\cos^{-1}\frac{3}{5}\right)\) is _____.

If p sec 50^o = tan 50^o then p is _______.

If \(f\left( x \right) =\begin{cases} { x }^{ 2 }-4x\quad ifx\ge 2 \\ x+2\quad ifx<2 \end{cases}\), then f(5) is _______.

If f(x) = \(\frac{1-x}{1+x}\) then f(-x) is equal to _______.

The graph of y = 2x² is passing through _______.

The minimum value of the function f(x) = |x| is _______.

If f(x) = 2^x and get g(x) = \(\frac{1}{2^x}\) then (fg)(x) is ________.

The graph of f(x) = e^x is identical to that of ________.

\(\lim _{ x\rightarrow \infty }{ \frac { \tan { \theta } }{ \theta } } =\)________.

For what value of x, f(x) = \(\frac{x+2}{x-1}\) is not continuous?

\(\frac{d}{dx}(\frac{1}{x})\) is equal to ________.

If y = x and z = \(\frac{1}{x}\) then \(\frac{dy}{dz}=\)________.

If the demand function is said to be inelastic, then _______.

Relationship among MR, AR and η_d is ______.

Instantaneous rate of change of y = 2x² + 5x with respect to x at x = 2 is _________.

Profit P(x) is maximum when ________.

If f(x,y) is a homogeneous function of degree n, then \(x\frac { \partial f }{ \partial x } +y\frac { \partial f }{ \partial y } \) is equal to ________.

The maximum value of f(x) = sinx is ________.

If u = 4x² + 4xy + y² + 32 + 16 , then \(\frac { \partial ^{ 2 }u }{ \partial y\partial x } \) is equal to ________.

If u = \({ e }^{ { x }^{ 2 } }\) then \(\frac { \partial u }{ \partial x } \) is equal to _______.

A company begins to earn profit at _______.

if q = 1000 + 8p₁ - p₂ then, \(\frac { \partial q }{ \partial { p }_{ 1 } } \) is _______.

What is the amount relalised on selling 8% stock 200 shares of face value Rs. 100 at Rs. 50.

If a man received a total dividend of Rs. 25,000 at 10% dividend rate on a stock of face value Rs.100, then the number of shares purchased.

Purchasing price of one share of face value 100 available at a discount of \(9\frac{1}{2}\%\) with brokerage \(\frac{1}{2}\%\) is ________.

The % Income on 7 % stock at Rs. 80 is _______.

Rs. 5000 is paid as perpetual annuity every year and the rate of C.I 10 %. Then present value P of immediate annuity is _______.

A person brought 100 shares of 9% stock of face value Rs. 100, at a discount of 10%, then the stock purchased is _______.

The annual income on 500 shares of face value Rs.100 at 15% is _______.

If ‘a’ is the annual payment, ‘n’ is the number of periods and ‘i’ is compound interest for Rs. 1 then future amount of the annuity is  _______.

An annuity in which payments are made at the beginning of each payment period is called _______.

Example of contingent annuity is _______.

The two events A and B are mutually exclusive if _________.

If two events A and B are dependent then the conditional probability of P(B/A) is _________.

If the outcome of one event does not influence another event then the two events are _________.

The probability of obtaining an even prime number on each die, when a pair of dice is rolled is _________.

Probability that at least one of the events A, B occur is _________.

When calculating the average growth of economy, the correct mean to use is?

The best measure of central tendency is _________.

The geometric mean of two numbers 8 and 18 shall be _________.

Harmonic mean is the reciprocal of _________.

The median of 10,14,11,9,8,12,6 is _________.

If the values of two variables move in opposite direction then the correlation is said to be ______.

If r(X,Y) = 0 the variables X and Y are said to be ______.

From the following data, N = 11, ΣX = 117, ΣY = 260, ΣX² = 1313, ΣY² = 6580, ΣXY = 2827 the correlation coefficient is ________.

The variable whose value is influenced (or) is to be predicted is called ________.

The correlation coefficient ________.

The variable which influences the values or is used for prediction is called________.

The regression coefficient of Y on X ________.

The lines of regression of X on Y estimates ________.

If two variables moves in decreasing direction then the correlation is ________.

The term regression was introduced by ________.

Maximize: z = 3x₁ + 4x₂ subject to 2x₁ + x₂ ≤ 40, 2x₁+ 5x₂ ≤ 180, x₁, x₂ ≥ 0. In the LPP, which one of the following is feasible corner point?

In constructing the network which one of the following statement is false?

A solution which maximizes or minimizes the given LPP is called ______.

The maximum value of the objective function Z = 3x + 5y subject to the constraints x > 0 , y > 0 and 2x + 5y ≤ 10 is ______.

Which of the following is not correct?

The technology matrix of an economic system of two industries is\(\begin{bmatrix} 0.50 & 0.30 \\ 0.41 & 0.33 \end{bmatrix}\). Test whether the system is viable as per Hawkins Simon conditions.

The technology matrix of an economic system of two industries is \(\begin{bmatrix} 0.50 & 0.25 \\ 0.40 & 0.67 \end{bmatrix}\). Test whether the system is viable as per Hawkins-Simon conditions.

Find the adjoint of the matrix \(A=\begin{bmatrix}2&3\\1&4 \end{bmatrix}\)

Evaluate:\(\left| \begin{matrix} 2 & 4 \\ -1 & 4 \end{matrix} \right| \)

Show that\(\left| \overset { x }{ 2x\underset { a }{ + } 2a } \quad \overset { y }{ 2y\underset { b }{ + } 2b } \quad \overset { z }{ 2z\underset { c }{ + } 2c } \right| =0\)

Find the inverse of each of the following matrices.\(\left[\begin{array}{rr} 1 & -1 \\ 2 & 3 \end{array}\right]\)

Evaluate: \(\left| \begin{matrix} 1 & 2 & 4 \\ -1 & 3 & 0 \\ 4 & 1 & 0 \end{matrix} \right| \)

Evaluate:\(\left| \begin{matrix} x & x+1 \\ x-1 & x \end{matrix} \right| \)

Evaluate\(\left| \begin{matrix} 1 & 3 & 4 \\ 102 & 18 & 36 \\ 17 & 3 & 6 \end{matrix} \right| \)

Show that \(\left[ \begin{matrix} 8 & 2 \\ 4 & 3 \end{matrix} \right] \)is non – singular.

Resolve into partial fractions for the following : \(\frac{x-2}{(x+2)(x-1)^2}\)

a) In how many ways can 8 identical beads be strung on a necklace?
b) In how many ways can 8 boys form a ring?

Verify that 8C₄ + 8C₃ = 9C₄

Evaluate the following expression.\(\frac { 8! }{ 5! } \)

Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

Resolve into partial fractions : \(\frac { 7x-1 }{ { x }^{ 2 }-5x+6 } \)

Find the number of 4 letter words, with or without meaning, which can be formed out of the letters of the word “ NOTE”, where the repetition of the letters is not allowed.

Find n, if \(\frac{1}{9!}+\frac{1}{10!}=\frac{n}{11!}\)

Evaluate: 8P₃ 

In how many ways 7 pictures can be hung from 5 picture nails on a wall ?

Find the equation of the circle when the end points of the diameter are (2, 4) and (3, –2).

If the centre of the circle x² + y² + 2x - 6y + 1 = 0 lies on a straight line ax + 2y + 2 = 0, then find the value of ‘a’

If the equation of a circle x² + y² + ax + by = 0 passing through the points (1, 2) and (1, 1), find the values of a and b

Find the length of the tangent from the point (2 ,3) to the circle x² + y² + 8x + 4y + 8 = 0

The supply of a commodity is related to the price by the relation x = \(\sqrt{5p-15}\) . Show that the supply curve is a parabola.

Find the acute angle between the lines 2x - y + 3 = 0 and x + y + 2 = 0.

Find the locus of the point which is equidistant from (2, –3) and (3, –4).

Show that perpendicular distances of the line x - y + 5 = 0 from origin and from the point P(2, 2) are equal.

Find the combined equation of the given straight lines whose separate equations are 2x + y -1 = 0 and x + 2y -5 = 0.

Find the equation of the circle with centre at origin and radius is 3 units.

Prove that \(\tan^{-1}\left(\frac{4}{3}\right)-\tan^{-1}\left(\frac{1}{7}\right)=\frac{\pi}{4}\)

Convert the following degree measure into radian measure 150^o

Convert the following degree measure into radian measure -320^o

Find the degree measure corresponding to the following radian measure. \(\frac { 9\pi }{ 5 } \)

Find the degree measure corresponding to the following radian measure.  \(\frac { 11\pi }{ 18 } \)

If three angles A, B and C are in arithmetic progression, Prove that \(cotB=\frac { sinA-sinC }{ cosC-cosA } \)

Find sin105^o + cos105^o

Find the principal value of tan^-1(-\(\sqrt{3}\))

Evaluate the following tan\(\left(\cos ^{-1} \frac{8}{17}\right)\)

Prove that \({ \cos }^{ -1 }\left( \frac { 4 }{ 5 } \right) +{ \cos }^{ -1 }\left( \frac { 3 }{ 5 } \right) ={ \cos }^{ -1 }\left( \frac { 27 }{ 11 } \right) \)

Determine whether the following functions are odd or even?
 \(f(x)=log({ x }^{ 2 }+\sqrt { { x }^{ 2 } } +1)\)

Determine whether the following functions are odd or even?
f(x) = x² - |x|

Find \(\frac{dy}{dx}\) of the following functions: x = ct, y = \(\frac{c}{t}\)

Find  \(\frac{dy}{dx}\)  of the following functions: x = acos³θ, y = asin³θ

If f(x) = e^x and g(x) = log_ex ,then find
i) (f + g)(1)
ii) (fg)(1)
iii) (3f)(1)
iv) (5g)(1)

Evaluate: \(\underset { x\rightarrow \infty }{ lim } x\tan { \left( \frac { 1 }{ x } \right) } \)

Differentiate the following functions with respect to x, 7e^x

Differentiate the following functions with respect to x, x² sin x

Differentiate the following functions with respect to x, \(\sqrt{x^2+x+1}\)

Find \(\frac{dy}{dx}\) if \(x=a\cos\theta, y=a\sin\theta\)

If the demand law is given by p = 10e^{\(-\frac { x }{ 2 } \)} then find the elasticity of demand.

Show that the function f(x) = x³ − 3x² + 4x, x \(\in\) R is strictly increasing function on R.

The profit function of a firm in producing x units of a product is given by\(p(x)=\frac { { x }^{ 3 } }{ 3 } +{ x }^{ 2 }+x\). Check whether the firm is running a profitable business or not. 

A tour operator charges Rupees 136 per passenger with a discount of 40 paisa for each passenger in excess of 100. The operator requires at least 100 passengers to operate the tour. Determine the number of passenger that will maximize the amount of money the tour operator receives.

Find out the elasticity of demand for the following functions
(i) p = xe^x
(ii) p = xe^-x
(iii) p = 10\({ e }^{ -\frac { x }{ 3 } }\)

Find the elasticity of supply for the supply function x = 2p² - 5p + 1, p > 3.

For the function y = x³ +19 find the values of x when its marginal value is equal to 27.

A firm produces x tonnes of output at a total cost of C(x) = \(\frac { 1 }{ 10 } \)x³ - 4x² - 20x + 7 find the
(i) average cost function
(ii) average variable cost function
(iii) average fixed cost function
(iv) marginal cost function and
(v) marginal average cost function.

Find the elasticity of demand in terms of x for the following demand laws and also find the output (x), when the elasticity is equal to unity. 
(i) p = (a - bx)²
(ii) p = a - bx²

Find the stationary value and the stationary points f(x) = x² + 2x – 5.

Find the amount of an ordinary annuity of Rs. 3,200 per annum for 12 years at the rate of interest of 10% per year. [(1.1)¹² = 3.3184]

What is the present value of an annuity due of Rs. 1,500 for 16 years at 8% per annum? [(1.08)^-16 = 0.2919]

Find the market value of 325 shares of amount Rs. 100 at a premium of Rs. 18.

Find the number of shares which will give an annual income of Rs. 3,600 from 12% stock of face value Rs. 100.

Find the amount of annuity of Rs. 2000 payable at the end of each year for 4 years of money is worth 10% compounded annually [(1.1)⁴ = 1.4641]

If A \(=\begin{bmatrix} 1 \\ -4\\3 \end{bmatrix}\) and  B = [-1 2 1], verify that (AB)^T = B^T. A^T

If \(A=\begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}\) then show that |2A| = 4 |A|.

Evaluate \(\begin{vmatrix} 2 &-1 &-2 \\0 & 2 & -1\\3 & -5& 0 \end{vmatrix}.\)

Find the values of x if \(\begin{vmatrix} 2 & 4 \\5 & 1 \end{vmatrix}=\begin{vmatrix} 2x & 4\\6 & x \end{vmatrix}.\)

Using the property of determinant, evaluate \(\begin{vmatrix} 6 &5 &12 \\ 2 & 4 &4 \\2 & 1 & 4 \end{vmatrix}.\)

A person pays Rs 64,000 per annum for 12 years at the rate of 10% per year. Find the annuity [(1.1)¹² = 3.3184]

A limited company wants to create a fund to help their employees in critical circumstances. The estimated expenses per month is Rs. 18,000. Find the amount to be deposited by the company if the rate of compound interest is 15%.

Find the present value of Rs. 2,000 per annum for 14 years at the rate of interest of 10% per annum. If the payments are made at the end of each payment period. [ (1.1)^–14 = 0.2632]

What is the amount of perpetual annuity of Rs. 50 at 5% compound interest per year?

If the dividend received from 10% of Rs. 25 shares is Rs. 2000. Find the number of shares.

ResoIve into partial fractions :\(\frac { 12x-17 }{ (x-2)(x-1) } \)

Evaluate\(\frac { 1 }{ 5! } +\frac { 1 }{ 6! } +\frac { 1 }{ 7! } \)

Find the number of permutations of English vowels A, E, I, 0, U taking two at a time?

In how many ways can 10 beads of different colours form a necklace?

Find the number of diagonals that can be drawn by joining the angular points of octagon ?

In a railway compartment, 6 seats are vacant on a bench. In how many ways can 3 passengers sit on them?

Show that 10P₃ = 9 P₃ + 3. 9P₂

How many permutations can be made out of the letters of the word "TRIANGLE" beginning with T?

Find n if 25 C_n+5 = 25 C_2n-1.

In the expansion of \({ \left( x+\frac { 1 }{ x } \right) }^{ 6 }\), find the third term.

Find the first quartile and third quartile for the given observations
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22

A die is thrown twice and the sum of the number appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once?

An unbiased die is thrown. If A is the event ‘the number appearing is a multiple of 3’ and B be the event ‘the number appearing is even’ number then find whether A and B are independent?

A bag contains 5 white and 3 black balls. Two balls are drawn at random one after the other without replacement. Find the probability that both balls drawn are black.

A committee of two persons is formed from 3 men and 2 women. What is the probability that the committee will have
(i) No woman (ii) One man (iii) No man

Prove that \(\frac{\tan 69^o+\tan 66^o}{1-\tan 69^o\tan 66^o}=-1\)

Prove that \(\cos18^o-\sin18^o=\sqrt{2}.\sin27^o\)

If \(\tan^2x=2\tan^2\phi+1\), prove that \(\cos2x+sin^2\phi=0\)

Evaluate : \(\cos\left[\frac{\pi}{3}-\cos^{-1}\left(\frac{1}{2}\right)\right]\)

Find the principal value of \(\cos^{-1}\left(\frac{-1}{\sqrt2}\right)\)

A person purchases tomatoes from each of the 4 places at the rate of 1kg., 2kg., 3kg., and 4kg. per rupee respectively. On the average, how many kilograms has he purchased per rupee?

A family has two children. What is the probability that both the children are girls given that at least one of them is a girl?

From a pack of 52 cards, two cards are drawn at random. Find the probability that one is a king and the other is a queen.

Let P(A) = \(\frac{3}{5}\) and P(B) = \(\frac{1}{5}\) . Find P(A∩B) if A and B are independent events.

An investor buys Rs. 1,500 worth of shares in a company each month. During the first four months he bought the shares at a price of Rs. 10, 15, 20 and 30 per share. What is the average price paid for the shares bought during these four months? Verify your result.

If \(\cos x=-\frac{1}{2}\) and \(\pi , find the value of \(4\tan^2x-3cosec^2x\)

Evaluate \(\cot\left(\frac{-15\pi}{4}\right)\)

In any quadrilateral ABCD, prove that sin (A + B) + sin (C + D) = 0

Find the value of \(\cos\left(\frac{5\pi}{12}\right)\)

Prove that \(sin^2\left(\frac{\pi}{8}+\frac x2\right)-sin^2\left(\frac{\pi}{8}-\frac x2\right)=\frac{1}{\sqrt2}\sin x.\)

Calculate the correlation coefficient from the following data
N = 9, ΣX = 45, ΣY = 108, ΣX² = 285, ΣY² = 1356, ΣXY = 597

From the following data calculate the correlation coefficient Σxy = 120, Σx² = 90, Σy² = 640

The following table shows the sales and advertisement expenditure of a form

Coefficient of correlation r = 0.9. Estimate the likely sales for a proposed advertisement expenditure of Rs. 10 crores.

Given the following data, what will be the possible yield when the rainfall is 29"

Coefficient of correlation between rainfall and production is 0.8.

Calculate the correlation coefficient from the following data:
ΣX = 125, ΣY = 100, ΣX² = 650, ΣY² = 436, ΣXY = 520, N = 25

Show that the functions f(x) = 5x - \(\left| x \right| \) is continuous at x = 0

For what value of k, the following function is continous at x =0?
f(x) = \(\begin{cases} \frac { 1-cos4x }{ 8{ x }^{ 2 } } \quad ifx\neq 0 \\ k\quad \quad \quad ifx=0 \end{cases}\)

Show that the function f(x) = 5x -3 is continous at x = +3

Prove that the function given by f(x) = \(\left| x-1 \right| \), x \(\in\) R is not differentiable at x =1

Evaluate \(\underset { x\rightarrow \frac { 1 }{ 2 } }{ lim } \frac { { 4x }^{ 2 }-1 }{ 2x-1 } \)

Calculate the correlation coefficient from the following data
N = 9, ΣX = 45, ΣY = 108, ΣX² = 285, ΣY² = 1356, ΣXY = 597

Calculate the coefficient of correlation between X and Y series from the following data.

Summation of product deviations of X and Y series from their respective arithmetic means is 122.

Obtain the two regression lines from the following data N = 20, ΣX = 80, ΣY = 40, ΣX² = 1680, ΣY² = 320 and ΣXY = 480

Calculate the coefficient of correlation from the following data:
ΣX = 50, ΣY = –30, ΣX² = 290, ΣY² = 300, ΣXY = –115, N = 10

Calculate the correlation coefficient from the following data:
ΣX = 125, ΣY = 100, ΣX² = 650, ΣY² = 436, ΣXY = 520, N = 25

Differentiate: sin x.sin 2x. sin 3x with respect to 'x'.

Find \(\frac{dy}{dx}\) if x = 15(t - sin t); y = 18(1 - cos t).

If e^y (x + 1) = 1, show that \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } ={ \left( \frac { dy }{ dx } \right) }^{ 2 }\)

Evaluate \(\underset { x\rightarrow -3 }{ lim } \frac { { x }^{ 3 }+27 }{ { x }^{ 5 }+243 } \)

Evaluate \(\underset { x\rightarrow 0 }{ lim } \frac { 2sinx-sin2x }{ { x }^{ 3 } } \)

Draw the logic network for the following:
Activities C and D both follow A, activity E follows C, activity F follows D, activity E and F precedes B.

Develop a network based on the following information:

Draw the event oriented network for the following data:

Construct the network for the projects consisting of various activities and their precedence relationships are as given below:
A, B, C can start simultaneously A < F, E; B < D, C; E, D < G

Draw a network diagram for the following activities.

Using the properties of determinants, show that \(\left| \begin{matrix} 2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86 \end{matrix} \right| \) = 0

Using co-factors of elements of  second column evaluate \(\left| \begin{matrix} 6 & -1 & 5 \\ 3 & 0 & 4 \\ -2 & 7 & -3 \end{matrix} \right| \)

Find the adjoint of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ 0 & 5 & 1 \\ 3 & 6 & 8 \end{matrix} \right] \)

If A = \(\begin{bmatrix} 3 & 2 \\ 7 & 5 \end{bmatrix}\) and B = \(\begin{bmatrix} 4 & 6 \\ 3 & 2 \end{bmatrix}\), verify that (AB)^-1 = B^-1A^-1

Find the numbers a and b such that A² + aA + bI = 0 for the matrix A =\(\begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}\)

Prove that \(\left| \begin{matrix} x & sin\theta & cos\theta \\ -sin\theta & -x & 1 \\ cos\theta & 1 & x \end{matrix} \right| \) is independent of \(\theta\)

Using matrix method, solve x + 2y + z = 7, x + 3z = 11 and 2x - 3y =1.

if A =\(\left[ \begin{matrix} cos\ \alpha & sin\ \alpha \\ -sin\ \alpha & \ cos\ \alpha \ \end{matrix} \right] \) is such that A^T = A^-1, find \(\alpha\)

Write the minors and co-factors of the elements of \(\begin{vmatrix}5 & 3 \\-6 & 2\end{vmatrix}\)

Verify that A(adj A) = (adj A) A = IAI·I for the matrix A = \(\begin{bmatrix}2 & 3 \\-1 & 4\end{bmatrix}\)

Resolve into partial factors:\(\frac { x+4 }{ ({ x }^{ 2 }-4)(x+1) } \)

Solve : \(\frac { (2x+1)! }{ (x+2)! } .\frac { (x-1)! }{ (2x-1)! } =\frac { 3 }{ 5 } \)

How may different numbers between 100 and 1000 can be formed using the digits 0, 1,2,3,4, 5, 6 assuming that in any number, the digits are not repeated.

It the letters of the word are arranged as in dictionary, find the rank of the word "AGAIN".

There are 6 gentlemen and 4 ladies to line at a round table. In how many ways can they seat themselves so that no two ladies together?

Draw the network for the project whose activities with their relationships are given below:
Activities A, D, E can start simultaneously; B, C > A; G, F > D, C; H > E, F.

Draw the event oriented network for the following data:

Construct the network for each the projects consisting of various activities and their precedence relationships are as given below:

Draw a network diagram for the following activities.

Draw the network diagram for the following activities

In how many ways can n prizes be given to n boys, when a boy may receive any number of prizes?

In an examination, Yamini has to select 4 questions from each part. There are 6, 7 and 8 questions is Part I, Part II and Part III respectively. What is the number of possible combinations in which she can choose the questions?

In how many ways can 12 things be equally divided among 4 persons?

If p(n) is the statement "12n + 3" is a multiple of 5, then show that P (3) is false, whereas P(6) is true.

Let p(n) be the statement "n² + n is even". If P(k) is true, then show that P(k+1) is true.

A point moves so that its distance from the point (-1, 0) is always three times its distance from the point (0, 2). Find its locus.

For what value of \(\lambda \) are the three lines 2x-5y+3 = 0, 5x-9y+\(\lambda \)=0 and x-2y+1=0 are concurrent?

Find the value of k if the straight line 2x + 3y + 4 + k(6x - y + 12) = 0 is perpendicular to the line 7x + 5y - 4 = 0

For what value of k does 12x² + 7xy + ky² + 13x - y + 3 = 0 represents a pair of straight lines?

Find the equation of a circle of radius 5 whose centre lies on X-axis and passes through the point (2, 3).

Evaluate: \(\begin{bmatrix} 3&-2&4\\2&0&1\\1&2&3 \end{bmatrix}\)

Find |AB| if \(A=\begin{bmatrix} 3&-1\\2&1 \end{bmatrix} \) and \(B =\begin{bmatrix} 3&0\\1&-2 \end{bmatrix}\)

Solve: \(\begin{vmatrix} x & 2 & -1 \\ 2 & 5 & x \\ -1 & 2 & x \end{vmatrix}=0.\)

Without actual expansion show that the value of the determinant \(\begin{vmatrix}5 &5^2 &5^3 \\5^2 & 5^3 & 5^4\\5^4&5^5&5^6 \end{vmatrix}\)is zero.

If A \(= \begin{bmatrix} 1 & -1 \\2 & 3 \end{bmatrix}\) show that A² - 4A + 5I₂ = 0 and also find A^-1.

Find the equation of a circle whose diameters are 2x - 3y + 12 = 0 and x + 4y - 5 = 0 and area is 154 square units.

Find the equation of the parabola whose focus is (-3, 2) and the directrix is x + y = 4.

Find the locus of a point which moves in such a way that the square of its distance from the point (3, -2) is numerically equal to its distance from the line 5x - 12y = 13

A point moves such that its distance from the point (4, 0) is half that of its distance from the line x = 16, find its locus.

Find the separate equations of the pair of lines given by 3x² + 7xy + 2y² + 5x + 5y + 2 = 0.

If \(\alpha\) and \(\beta\) are acute angles such that \(\tan\alpha=\frac{m}{m+1}\) and \(\tan\beta=\frac{1}{2m+1}\), prove that \(\alpha+\beta=\frac{\pi}{4}\)

Show that \(\tan\left(\frac{\pi}{3}+x\right)\tan\left(\frac{\pi}{3}-x\right)=\frac{2\cos2x+1}{2\cos2x-1}\)

Prove that \(\frac{\sin5x+\sin3x}{\cos5x+\cos3x}=\tan4x\)

Prove that \(\frac{\sin5x-2\sin3x+sinx}{\cos5x-\cos x}=\tan x\)

Show that \(\cos^{-1}\left(\frac{3}{5}\cos x+\frac45\sin x\right)=x-\tan^{-1}\left(\frac43\right)\)

Solve by matrix inversion method: 2x + 3y - 5 = 0, x - 2y + 1 = 0.

Find the inverse of each of the following matrices \(\left[ \begin{matrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{matrix} \right] \)

Find the minor and cofactor of each element of the determinant\(\left| \begin{matrix} 1 & -2 \\ 4 & 3 \end{matrix} \right| \)

Find adj A for \(A=\left[ \begin{matrix} 2 & 3 \\ 1 & 4 \end{matrix} \right] \)

If \(A=\left[ \begin{matrix} 2 & 4 \\ -3 & 2 \end{matrix} \right] \)then, find A ^-1.

Find all other trigonometrical ratios if \(\sin x=\frac{-2\sqrt6}{5}\) and x lies in III quadrant?

Prove that \(\sin^2\frac{\pi}{6}+\cos^2\frac{\pi}{3}-\tan^2\frac{\pi}{4}=-\frac12\)

Prove that\(\left\{1+\cot x-\sec\left(\frac{\pi}{2}+x\right)\right\}\left\{1+\cot x+\sec\left(\frac{\pi}{2}+x\right)\right\}=2\cot x\)

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

Prove that \(\frac{\sin(x+y)}{\sin(x-y)}=\frac{\tan x+\tan y}{\tan x-\tan y}\)

How many distinct words can be formed using all the letters of the following words.
MISSISSIPPI

In how many ways 8 students can be arranged in a line

If nC₄ = nC₆ , find 12C_n

If 4(nC₂) = (n + 2)C₃ , find n

Find the middle term in the expansion of \((\frac{x}{3}+9y)^2\)

Differentiate: sin x.sin 2x. sin 3x with respect to 'x'.

Find \(\frac{dy}{dx}\) if x = 15(t - sin t); y = 18(1 - cos t).

If e^y (x + 1) = 1, show that \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } ={ \left( \frac { dy }{ dx } \right) }^{ 2 }\)

Evaluate \(\underset { x\rightarrow -3 }{ lim } \frac { { x }^{ 3 }+27 }{ { x }^{ 5 }+243 } \)

Evaluate \(\underset { x\rightarrow 0 }{ lim } \frac { 2sinx-sin2x }{ { x }^{ 3 } } \)

Find the rank of the word ‘RANK’ in dictionary.

Using binomial theorem, expand \({ \left( { x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } \right) }^{ 4 }\)

Find the 5^th term in the expansion of \({ \left( x-\frac { 3 }{ { x }^{ 2 } } \right) }^{ 10 }\)

Find the Coefficient of x¹⁰ in the binomial expansion of \({ \left( 2x^2-\frac { 3 }{ { x }^{ } } \right) }^{ 11 }\)

Using 9 digits from 1, 2, 3, ……,9 taking 3 digits at a time, how many 3 digits numbers can be formed when repetition is allowed?

If \(y=\sqrt { x } +\frac { 1 }{ \sqrt { x } } \) show that \(2x\frac { dy }{ dx } +y=2\sqrt { x } \).

Show that the function f(x) = [x] where [x] denotes the greatest integer function is discontinuous at all integral points

Is the function defined by f(x) = x² -sin x + 5 is continuous at x =\(\pi\)?

Differentiate: \(\sin ^{ -1 }{ \left( \sqrt { \cos { x } } \right) } \)

Differentiate: sin² x + cos² y = 1.

The profit Rs.y accumulated in thousand in x months is given by y = -x²  + 10x - 15. Find the best time to end the project.

Show that the given lines 3x - 4y - 13 = 0, 8x - 11y = 33 and 2x - 3y - 7 = 0 are concurrent and find the concurrent point.

For what value of k does 2x² + 5xy + 2y² + 15x + 18y + k = 0 represent a pair of straight lines.

The slope of one of the straight lines ax² + 2hxy + by² = 0 is twice that of the other, show that 8h² = 9ab.

For what values of a and b does the equation (a - 2)x² + by² + (b - 2)xy + 4x + 4y - 1 = 0 represents a circle? Write down the resulting equation of the circle.

Find the stationary points and stationary values of the function f(x) = x³ - 3x² - 9x + 5.

Show that the function x³ + 3x² + 3x + 7 is an increasing function for all real values of x.

Separate the intervals in which the function x³ + 8x² + 5x - 2 is increasing or decreasing.

Find the maximum and minimum values of the function x² + 16/x

For the production function P= 5(L)^0.7(K)^0.3.Find the marginal productivities of Labour (L) and Capital (K) when L = 10, K = 3 [Use (0.3)^0·3 = 0.6968; (3.33)^0·7 = 2.2322]

Find the value of p for which the straight lines 8px + (2 - 3p)y + 1 = 0 and px + py - 7 = 0 are perpendicular to each other.

Find the equation of the parabola whose focus is the point F(-1, -2) and the directrix is the line 4x - 3y + 2 = 0

If the slope of one of the straight lines ax² + 2hxy + by² = 0 is thrice that of the other, then show that 3h² = 4ab.

If (4,1) is one extremity of a diameter of the circle x² + y² - 2x + 6y -15 = 0, find the other extremity.

The average variable cost of a monthly output of x tonnes of a firm producing a valuable metal is  Rs. \(\frac { 1 }{ 5 } { x }^{ 2 }-6x+100\). Show that the average variable cost curve is a parabola. Also find the output and the  average cost at the vertex of the parabola.

If I deposit Rs.500 every year for a period of 10 years in a bank which gives C.I. 5% per year, find out the amount I will receive at the end of 10 years.

If I deposit Rs.500 every year for a period of 10 years in a bank which gives C.I 5% per year, Find out the amount I will receive at the end of 10 years

Find the present value of an annuity due of Rs.200 p.a. payable annually for 2 years at 4%p.a

A man wishes to pay back his depts of Rs.3783 due after 3 years by 3 equal yearly instalments. Find the amount of each instalments,money being worth 5% p.a. compounded annually

Find the yearly income on 120 shares of 7% stock of face value Rs.100?

Prove that cos² A + cos²(A + 120^o) + cos²(A -120^o) = \(\frac{3}{2}\)

Show that sin20^osin 40^o sin 60^o sin80^o = \(\frac{3}{16}\)

Convert the following into the product of trigonometric functions sin7\(\theta\) - sin4\(\theta\)

Convert the following into the product of trigonometric functions cos4\(\alpha\) - cos8\(\alpha\)

Convert the following into the product of trigonometric functions cos75^o + cos 45^o

A sum of Rs.1000 is deposited at the beginning of each quarter in a S.B. account that pays C.I 8% compounded quarterly. Find the account at the end of 3 years.

Find the future value of an ordinary annuity of Rs.1000 a year for 5 years for 5 years at 7% p.a compounded annually.

What is the present value of an annuity that pays 250 per month at the end of each month for 5 years assuming money to be worth 6% compounded monthly?

Find the number of shares which will give an annual income of Rs.360 from 6% stock of face value Rs.100

Find the rate of dividend which gives an annual income of Rs.1200 for 150 shares of face value Rs.100

If \(\sin { A } =\frac { 3 }{ 5 } \) , find the values of cos 3A and tan 3A.

Prove that sin600^o cos 390^o + cos 480^o sin 150^o = –1

Find the values of each of the following trigonometric ratios –sec165^o

If tanA =\(\frac{1}{7}\) and tanB =\(\frac{1}{3}\), show that cos2A = sin4B

If tan \(\alpha\) =\(\frac{1}{2}\)and tan \(\beta\) = \(\frac{1}{7}\)then prove that (2\(\alpha\)+\(\beta\)) = \(\frac{\pi}{4}\).

A fair die is rolled. A = {1, 3, 5} B = {2, 3} and C = {2, 3, 4, 5}. Find (i) P(A/B) and P(B/A) (ii) P(A/C) and p(C/A).

Mother, father and son line up at random for a family picture. Find P(A/B) if A and B are defined as follows A = Son on one end, B = Father in the middle.

A and B are two events such that P(A)\(\neq \)0. find P(B/A) if (i) A is a subset of B (ii) \(A\cap B=\phi \)

Events A and B are such that P(A)=\(\frac { 1 }{ 2 } \), P(B)=\(\frac { 7 }{ 12 }\), and P(not A or not B) = \(\frac { 1 }{ 4 }\), state whether A and B are independent?

Find the probability of drawing a one-rupee coin from a purse with two compartments one of which contains 3 fifty-paise coins and 2 one-rupee coins and other contains 2 fifty paise coins and 3 one-rupee coins.

If y=1+1/x, show that y is a strictly decreasing function for all real values of x(x\(\neq\)0).

Prove that 75-12x+6x²-x³ always decreases as x increases.

Find the maximum and minimum values of x³-6x²+7

A certain manufacturing concern has the toal cost function C = \({1\over5}x^2-6x+100\).Find when the tatal cost is minimum.

A manufacturer has to supply 12,000 units of a product per year to his customer. The demand is fixed and known and no shortages are allowed. The inventory holding cost is 20 paise per unit per month and the set up cost per run is Rs.350. Determine (i) the optimum run size q₀. (ii) Optimum scheduling period t₀ (iii) minimum total variable yearly cost.

Evaluate: \(\underset { x\rightarrow 0 }{ lim } \frac { \sin { 3x } }{ \sin { 5x } } \)

Show that \(\underset { x\rightarrow 0 }{ lim } \frac { \log { \left( 1+{ x }^{ 2 } \right) } }{ \sin ^{ 3 }{ x } } =1\)

Find \(\frac{d}{dx}(e^{3x})\) from first principle.

If y = \(\frac{1}{u^2}\) and u = x² - 9, then find \(\frac{dy}{dx}\).

If x³ + y³ = 3axy, then find \(\frac{dy}{dx}\).

Find the geometric mean of 3, 6, 24, 48

Find the harmonic mean of 6, 14, 21, 30

Find Q₂ for 37, 32, 45, 36, 39, 37, 46, 57, 27, 34, 28, 30, 21

Two integers are selected at random from integers 1 to 11. If the sum is even find the probability that both the numbers are odd.

Find the geometric mean for the following data

Find the marginal productivities for Capital (K) and Labour (L) if P = 10K-K² + KL when K = 2 and L = 6.

The demand for a quantity A is q = 16- 3P_I - 2P₂². Find the partial elasticities \({Eq\over EP_1}\) and \({Eq\over EP_2}\)

The relationship between Profit P and advertising cost x is given by \(P={4000x\over 500+x}-x\) . Find x which maximises P.

A Company uses annually 24,000 units of raw materials which costs Rs.1.25 per unit, placing each order costs Rs.22.50 and the holding costs is 5.4% per year of the average inventory. Find the EOQ, time between each order and total number of orders per year. Also, verify that at EOQ, carrying cost is equal to ordering cost

A firm has revenue function R = 8x and production cost function \(C = 150000 + 60\left(x^2\over 900\right)\) Find the total profit function and the number of units to be sold to get the maximum profit.

If ax² + 2hxy + by² + 2gx + 2fy + c = 0 then find \(\frac{dy}{dx}\).

Find \(\frac{dy}{dx}\) at (1, 1) to the curve 2x² + 3xy + 5y² = 10.

Differentiate the following with respect to x.
(i) x^x
(ii) (log x)^{cos x}

Find the second order derivative of the following functions with respect to x, 3 cos x + 4 sin x

Find the second order derivative of the following functions with respect to x, x sin x

Calculate the covariance of the following pairs of observation of two variates X and Y. (1, 5)(2, 4)(3, 3)(4, 2)(5, 1)

Explain the concept of correlation and co-efficient of correlation.

Calulate the co-efficient of correlation between x and y on the basis of the following observations. \(\sum\)\(\sum\)x=10, \(\sum\)x²=250, \(\sum\)y=70, \(\sum\)y²=300, \(\sum\)xy=75 and n =20

The co-efficient of correlation between two variables X and Y is 0.5 and their co-variance is 5.81. The variance of X is 12. Find the satandard deviation of the Y-series.

prove that the correlation co-efficient is the geometric mean of regression co-efficients.

a bank pays 8% interest compounded quarterly. Determine the equal deposits to be made at the end of each quarter for 3 years so as to receive Rs.300 at the end of 3 years.

What equal payments made at the beginning of each month for 3 years will accumulate to Rs.4,00,000 if money is worth 15% compounded monthly

A man, deposits Rs.75 at the end of 6 months in a bank which pays interest at 8% compounded semiannually. How much is to his credit at the end of 10 years?

Machine a costs Rs.25,000 and machine B costs Rs.40,000. The annual income fro machine A and B are 8000 and 10,000 respectively. Machine A has life of 5 years and machine B has life of 7 years. Which machine may be purchased, discount  rate being 10% p.a?

A bank pays interest at the rate of 8% p.a. compounded quarterly. Find how much should be deposited in the bank at the beginning of each of 3 months for 5 years in order to accumulate to Rs.10,000 at the of 5 years.

An amount of Rs. 5000 is put into three investment at the rate of interest of 6%, 7% and 8% per annum respectively. The total annual income is Rs. 358. If the combined income from the first two investment is Rs. 70 more than the income from the third, find the amount of each investment by matrix method.

Using binomial theorem, find the value of \({ \left( \sqrt { 2 } +1 \right) }^{ 5 }+{ \left( \sqrt { 2 } -1 \right) }^{ 5 }\)

If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.

Prove that cos 6x = 32 cos⁶x - 48 cos⁴x + 18 cos²x - 1.

If \(y={ e }^{ a\cos ^{ -1 }{ x } }\) , show that \(\left( 1-{ x }^{ 2 } \right) \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -x\frac { dy }{ dx } -{ a }^{ 2 }y=0\)

Without expanding show that \(\Delta =\left| \begin{matrix} { cosec }^{ 2 }\theta & { cot }^{ 2 }\theta & 1 \\ { cot }^{ 2 }\theta & { cosec }^{ 2 }\theta & -1 \\ 42 & 40 & 2 \end{matrix} \right| =0\)

In how many ways can the following prizes be given away to a class of 30 students, first and second in mathematics, first and second in physics, first in chemistry and first in English?

Prove that the tangents to the circle x² + y²  = 169 at (5,12) and (12,-5) are perpendicular to each other.

Prove that (sin 3x + sin x) sin x + (cos 3x - cos x) cos x = 0.

If the function \(f\left( x \right) =\begin{cases} 6ax+3b\quad if\quad x>1 \\ ax-2b\quad if\quad x<1\quad is\quad continuous\quad at\quad x=1 \\ 15\quad if\quad x=1 \end{cases}\) Find the value of a and b.

Find the numbers a and b such that A² + aA + bI = 0 for the matrix A =\(\begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}\)

Find the value of tan \(\left( {{\pi}\over{8}} \right)\)

Find the locus of a point such that the sum of its distances from the points (0, 2) and (0, -2) is 6.

Show that \(\tan\left(\frac{\pi}{3}+x\right)\tan\left(\frac{\pi}{3}-x\right)=\frac{2\cos2x+1}{2\cos2x-1}\)

Find \(\frac{dy}{dx}\) if x = 15(t - sin t); y = 18(1 - cos t).

Verify that A(adj A) = (adj A) A = IAI·I for the matrix A = \(\begin{bmatrix}2 & 3 \\-1 & 4\end{bmatrix}\)

It the letters of the word are arranged as in dictionary, find the rank of the word "AGAIN".

A point moves so that its distance from the point (-1, 0) is always three times its distance from the point (0, 2). Find its locus.

Prove that\(\left\{1+\cot x-\sec\left(\frac{\pi}{2}+x\right)\right\}\left\{1+\cot x+\sec\left(\frac{\pi}{2}+x\right)\right\}=2\cot x\)

Is the function defined by f(x) = x² -sin x + 5 is continuous at x =\(\pi\)?

Find the values of x if \(\begin{vmatrix} 2 & 4 \\5 & 1 \end{vmatrix}=\begin{vmatrix} 2x & 4\\6 & x \end{vmatrix}.\)

Show that 10P₃ = 9 P₃ + 3. 9P₂

Find the number of diagonals that can be drawn by joining the angular points of octagon ?

Find the angle between the pair of lines represented by the equation 3x²+10xy+8y²+14x+22y+15=0.

Find the value of \(\cos\left(\frac{5\pi}{12}\right)\)

Evaluate \(\begin{vmatrix} 2 &-1 &-2 \\0 & 2 & -1\\3 & -5& 0 \end{vmatrix}.\)

Show that 10P₃ = 9 P₃ + 3. 9P₂

Convert the parabola y²=4x+4y into standard form.

Evaluate \(\cot\left(\frac{-15\pi}{4}\right)\)

Evaluate : \(\cos\left[\frac{\pi}{3}-\cos^{-1}\left(\frac{1}{2}\right)\right]\)

Solve graphically: Minimize Z = 20x₁ + 40x_2.
Subject to the constraints 36x₁ + 6x₂ ≥ 108,3x₁ + 12x₂ ≥ 36,20x₁ + 10x₂ ≥ 100 and x₁,x₂≥0

Minimize and maximize Z =x₁+ 2x₂ . Subject to the constraints x₁ + 2x₂ ≥ 100,2x₁ − x₂ ≤ 0,2x₁ + x₂ ≤ 200 and x₁,x₂ ≥ 10.

Compute the earliest start time, earliest finish time, latest start time and latest finish time of each activity of the project given below:

Calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity of the project given below and determine the critical path of the project and duration to complete the project.

The following table use the activities in a building project.

Draw the network for the project, calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity and find the critical path. Compute the project duration.

A manufacturer produces two types of steel trunks. He has two machine A and B. For completing, the first type of the trunk requires 3 hours on machine A and 2 hours on machine B, whereas the second type of the trunk requires 3 hours on machine A and 3 hours on machine B. Machines A and B can work at the most for 18 hours and 14 hours per day respectively. He earns a profit of Rs.30 andRs.40 per trunk of the first type and second type respectively. How many trunks of the each type must he make each day to make maximum profit?

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 1 kg of vitamin A and 5 units 1 kg of vitamin B while food Q contains 4 units 1 kg of vitamin A and 2 units 1 kg of vitamin B. Determine the minimum cost of the mixture.

One kind of the 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 other ingredients used in making the cakes?

A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (min) required for each toy on the machine is given below:

Each machine is available for a maximum of 6 hours/day. If the profit on each toy of type A is Rs.7.50 and for B is Rs.5. Show that 15 toys of type A and 30 of type B should be manufactured in a day to get maximum profit.

Every gram of wheat provides 0.1 g of proteins and 0.25 g of carbohydrates. The corresponding values of rice are 0.05 g and 0.5 g respectively. Wheat cost Rs.4 per kg and rice cost Rs.6 per kg. The minimum daily requirements of proteins and carbohydrate for an average child are 50 g and 200 g respectively. In what quantities should wheat and rice be mixed in the daily diet to provide minimum daily requirements of proteins and carbohydrate at minimum cost. Frame an LPP and solve it graphically. 

For the following data, (i) the regression equation of X on Y regression equation of Y on X (iii) the correlation co-efficient between X and Y (iv) the value of x when y=5 (v) the value of y when x=6

Ten competitors in a musical test were ranked by the judges x, y, and z in the following order.

Out the following two regression lines, find the line of regression of X on Y, 2x+3y=7 and 5x+54=9.

For the following observations, find the regression co-efficient b_yx and b_xy and hence find the correlation co-efficient (4,2)(2,3)(3,2)(4,4)(2,4).

You are given the following data:

Correlation co-efficient between x and y=0.8 (i) Find the two regression lines.
(ii) Estimate the value of y, when x=70  (iii)Estimate the value of x, when y=90

For the data on price (in rupees) and demand (in tonnes) for a commodity, calculate the co-efficient of correlations.

Obtain the two regression lines from the following

With the help of the regression equation for the data below, calculate the value of X when Y=20

A computer while calculating the correlation co-efficient between two variables x and y from 25 pairs of observations, obtained the following results. \(\sum\)x=125, \(\sum\)x²=650, \(\sum\)y=100, \(\sum\)y²=460, xy=508. It was later found out that it had copied down two pairs as while the correct values are 

Obtain the correlation co-efficient for the correct value.

The equations of two regression lines are 4x+3y+7=0 and 3x+4y+8=0.
Find (i) the mean of x and the mean of y
(ii) the regression co-efficient b_xy and b_yx 
(iii) the correlation co-efficient between x and y.

A factory has 3 machines A₁, A₂, A₃ producing 1000, 2000, 3000 bolts per day respectively. A₁ produces 1% defectives, A₂ produces 1.5% and A₃ produces 2% defectives. A bolt is chosen at random and found defective. What is the probability that it comes from machine A₁?

Find the Quartile deviation.

Two identical boxes containing respectively 4 white and 3 red balls, 3 white and 7 red balls. A box is chosen at random and a ball is drawn from it. Find the probability that the ball is white. If the ball is white. If the ball is white, what is the probability that it is from first box?

Three events A, B and C have probabilities \(\frac { 2 }{ 5 } ,\frac { 1 }{ 3 } \) and \(\frac { 1 }{ 2 } \)  respectively. Given that \(P(A\cap C)=\frac { 1 }{ 5 } \) and \(P(B\cap C)=\frac { 1 }{ 4 } ,\)find P(C/B) and \(P(\bar { A } \cap \bar { C } )\)

Two dice are thrown together. Let A be the event "getting 6 on the first die" and B be the event "getting 2 on the second die". Are the events A and B independent?

Find D₆,D₈,P₇ and P₂₀ for the data 57, 58, 61, 42, 38, 65, 72, 66. 

In a Shooting test, the probabilities of hitting the target are \(\frac { 1 }{ 2 } \) for A, \(\frac { 2 }{ 3 } \) for B and \(\frac { 3 }{ 4 } \)  for C. If all of them fire at the same target, calculate the probabilities that only one of them hit the target.

A factory has 3 machines A₁, A₂, A₃ producing 1000, 2000, 3000 bolts per day respectively. A₁ produces 1% defectives, A₂ produces 1.5% and A₃ produces 2% defectives. A bolt is chosen at random and found defective. What is the probability that it comes from machine A₁?

There are 3 boxes containing respectively 1 white, 2 red, 3 black balls; 2 white, 3 red and 1 black ball; 2 white, 1 red, 3 black balls. A box is chosen t random and from it 2 balls are drawn at random. The 2 balls are 1 red and 1 white. What is the probability that they come from the second box?

A box contains 4 red, 6 green balls. Two balls are picked out one by one at random without replacement. What is the probability that the second is green given that the first one is green?

Kamal sold Rs.9000 worth 7% stock at 80 and invested the proceeds in 15% stock at 120. Find the change in his income?

A person sells a 20% stock of face value Rs.5000 at a premium of 62% with the money obtained he buys a 15% stock at a discount of 22% what is the change in his income.(Brokerage 2%)

Equal amounts are invested in 12% stock at 95 (brokerage). If 12% stock brought at Rs.120 more by way of dividend income than the other, find the amount invested in each stock?

Rani sold Rs.8000 worth 7% stock at 96 and invested the amount realised in the shares of FV Rs.100 os a 10% stock by which her income increased by Rs.80. Find the purchase price of 10% stock.

A company has a total capital of Rs.5,00,000 divide into 1000 preference shares of 6% dividend with par value RS.100 each and 4000 ordinary shares of per value Rs.100 each. The company delares an annual dividend of Rs.40,000. Find the dividend received by Sundar having 100 preference shares and 200 ordinary shares.

a bank pays 8% interest compounded quarterly. Determine the equal deposits to be made at the end of each quarter for 3 years so as to receive Rs.300 at the end of 3 years.

What equal payments made at the beginning of each month for 3 years will accumulate to Rs.4,00,000 if money is worth 15% compounded monthly

A man, deposits Rs.75 at the end of 6 months in a bank which pays interest at 8% compounded semiannually. How much is to his credit at the end of 10 years?

Machine a costs Rs.25,000 and machine B costs Rs.40,000. The annual income fro machine A and B are 8000 and 10,000 respectively. Machine A has life of 5 years and machine B has life of 7 years. Which machine may be purchased, discount  rate being 10% p.a?

A bank pays interest at the rate of 8% p.a. compounded quarterly. Find how much should be deposited in the bank at the beginning of each of 3 months for 5 years in order to accumulate to Rs.10,000 at the of 5 years.

Find the marginal productivities for Capital (K) and Labour (L) if P = 10K-K² + KL when K = 2 and L = 6.

The demand for a quantity A is q = 16- 3P_I - 2P₂². Find the partial elasticities \({Eq\over EP_1}\) and \({Eq\over EP_2}\)

The relationship between Profit P and advertising cost x is given by \(P={4000x\over 500+x}-x\) . Find x which maximises P.

A Company uses annually 24,000 units of raw materials which costs Rs.1.25 per unit, placing each order costs Rs.22.50 and the holding costs is 5.4% per year of the average inventory. Find the EOQ, time between each order and total number of orders per year. Also, verify that at EOQ, carrying cost is equal to ordering cost

A firm has revenue function R = 8x and production cost function \(C = 150000 + 60\left(x^2\over 900\right)\) Find the total profit function and the number of units to be sold to get the maximum profit.

If y=1+1/x, show that y is a strictly decreasing function for all real values of x(x\(\neq\)0).

Prove that 75-12x+6x²-x³ always decreases as x increases.

Find the maximum and minimum values of x³-6x²+7

A certain manufacturing concern has the toal cost function C = \({1\over5}x^2-6x+100\).Find when the tatal cost is minimum.

A manufacturer has to supply 12,000 units of a product per year to his customer. The demand is fixed and known and no shortages are allowed. The inventory holding cost is 20 paise per unit per month and the set up cost per run is Rs.350. Determine (i) the optimum run size q₀. (ii) Optimum scheduling period t₀ (iii) minimum total variable yearly cost.

If \(\begin{matrix} \underset { x\rightarrow 1 }{ lim } & \frac { { x }^{ 4 }-1 }{ x-1 } \end{matrix}=\begin{matrix} \underset { x\rightarrow k }{ lim } & \frac { { x }^{ 3 }-{ k }^{ 3 } }{ { x }^{ 2 }-{ k }^{ 2 } } \end{matrix}\),then find the value of K

Evaluate \(\underset { h\rightarrow 0 }{ lim } \frac { \sqrt { x+h } -\sqrt { x } }{ h } \)

Differentiate (sec x -1) (sec x +1)

If \(x=a\left( t+\frac { 1 }{ t } \right) ;y=a\left( t-\frac { 1 }{ t } \right) \) show that \(\frac { dy }{ dx } =\frac { x }{ y } \)

If \(y={ e }^{ a\cos ^{ -1 }{ x } }\) , show that \(\left( 1-{ x }^{ 2 } \right) \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -x\frac { dy }{ dx } -{ a }^{ 2 }y=0\)

If the function \(f\left( x \right) =\begin{cases} 6ax+3b\quad if\quad x>1 \\ ax-2b\quad if\quad x<1\quad is\quad continuous\quad at\quad x=1 \\ 15\quad if\quad x=1 \end{cases}\) Find the value of a and b.

Evaluate \(\begin{matrix} \underset { x\rightarrow 1 }{ lim } & \frac { { x }^{ 7 }-2{ x }^{ 5 }+1 }{ { x }^{ 3 }-{ 3x }^{ 2 }+2 } \end{matrix}\)

Differentiate: xy + y² = tan x + y.

Differentiate sin (tan^-1 (e^-x))

Differentiate: x² (x + 1)³ (x + 2)⁴ with respect to 'x'.

Prove that \(\frac { 4tan\ x(1-{ tan }^{ 2 }x) }{ 1-6{ tan }^{ 2 } x+{ tan }^{ 4 } x } =tanx\)

Prove that (sin 3x + sin x) sin x + (cos 3x - cos x) cos x = 0.

Prove that sin(n+1)x sin(n+2) x+cos(n+1)xcos(n+2)x=cosx.

If tan x = \(\frac { -4 }{ 3 } \) and x is in II quadrant,find \(sin\frac { x }{ 2 } ,cos\frac { x }{ 2 } \) and \(tan\frac { x }{ 2 } \)

Prove that cos²2x - cos²6x = sin 4x.sin 8x

Prove that cot x cot 2x - cot 2x cot 3x - cot 3x cot x = 1.

Prove that cos 4x = 1 - 8 sin²x cos²x.

Prove that cos 6x = 32 cos⁶x - 48 cos⁴x + 18 cos²x - 1.

If  \(sin\left( { sin }^{ -1 }\left( \frac { 1 }{ 5 } \right) +{ cos }^{ -1 }(x) \right) =1\)  then find the value of x

Prove that \({ cot }^{ -1 }\left[ \frac { \sqrt { 1+sinx } +\sqrt { 1-sinx } }{ \sqrt { 1+sinx } -\sqrt { 1-sinx } } \right] =\frac { x }{ 2 } \) where \(x\in \left( 0,\frac { \pi }{ 4 } \right) \)

As the number of units manufactured increases from 6000 to 8000, the total cost of production increases from Rs. 33,000 to Rs. 40,000. Find the relationship between the cost (y) and the number of units made (x) if the relationship is linear.

Prove that the tangents to the circle x² + y²  = 169 at (5,12) and (12,-5) are perpendicular to each other.

Find the equation of the parabola whose vertex is (0, 0) passing through the point (2, 3) and axis is along X-axis.

If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.

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 side is 2 m from the vertex of the parabola?

Resolve into partial factors : \(\frac { { x }^{ 2 }+x+1 }{ { x }^{ 2 }+2x+1 } \)

In how many ways can the following prizes be given away to a class of 30 students, first and second in mathematics, first and second in physics, first in chemistry and first in English?

How many numbers are there between 100 and 1000 such that atleast one of their digits is 7?

If 22P_r+1:20P_r+2=11: 52, find r.

How many numbers greater than a million can be formed with the digits 2, 3, 0, 3, 4, 2, 3?

If m parallel lines in a plane are intersected by a family of n parallel lines. Find the number of parallelogram formed?

If the fourth term in the expansion of \({ \left( ax+\frac { 1 }{ x } \right) }^{ n }\) is \(\frac { 5 }{ 2 } \)  then find the values of a and n.

Using the principle of mathematical induction, prove that 1.3 + 2.3² + 3.3³ + ... + n.3ⁿ =\(\frac { (2n-1){ 3 }^{ n+1 }+3 }{ 4 } for\ all\ n\in N\)

Using binomial theorem, find the value of \({ \left( \sqrt { 2 } +1 \right) }^{ 5 }+{ \left( \sqrt { 2 } -1 \right) }^{ 5 }\)

Find the 11^th term from the end in \({ \left( 2x-\frac { 1 }{ { x }^{ 2 } } \right) }^{ 25 }\)

If a, b, c are in A.P, find the value of \(\left| \begin{matrix} 2y+4 & \quad 5y+7 & 8y+a \\ 3y+5 & 6y+8 & 9y+b \\ 4y+6 & 7y+9 & 10y+c \end{matrix} \right| \)

Show that the matrix A =\(\left[ \begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix} \right] \)satisfies the equation A² - 4A - 5I₃ = 0 and hence find A^-1.

Use the product \(\left[ \begin{matrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{matrix} \right] \left[ \begin{matrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{matrix} \right] \)to solve the system of equations x - 2y + 2z = 1, 2y - 3z = 1, 3x - 2y + 4z = 2.

Two types of radio values A, B are available and two types of radios P and Q are assembled in a small factory. The factory uses 2 valves of type A and 3 valves of type B for the type B for the type of radio P, and for the radio Q it uses 3 valves of type A and 4 valves of type B. If the number of valves of type A and B used by the factory are 130 and 180 respectively, find out the number of radios assembled use matrix method.

An amount of Rs. 5000 is put into three investment at the rate of interest of 6%, 7% and 8% per annum respectively. The total annual income is Rs. 358. If the combined income from the first two investment is Rs. 70 more than the income from the third, find the amount of each investment by matrix method.

Determine the values of x for which the matrix A =\(\left[ \begin{matrix} x+1 & -3 & 4 \\ -5 & x+2 & 2 \\ 4 & 1 & x-6 \end{matrix} \right] \)is singular.

Without expanding show that \(\Delta =\left| \begin{matrix} { cosec }^{ 2 }\theta & { cot }^{ 2 }\theta & 1 \\ { cot }^{ 2 }\theta & { cosec }^{ 2 }\theta & -1 \\ 42 & 40 & 2 \end{matrix} \right| =0\)

If a, b, c are in A.P, find the value of \(\left| \begin{matrix} 2y+4 & \quad 5y+7 & 8y+a \\ 3y+5 & 6y+8 & 9y+b \\ 4y+6 & 7y+9 & 10y+c \end{matrix} \right| \)

Let a, b and c denote the sides BC, CA and AB repectively of \(\Delta\) ABC. If \(\left| \begin{matrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{matrix} \right| =0\), then find the value of sin² A + sin²B + sin²C.

If \(A=\left[ \begin{matrix} 1 & tan\quad x \\ -tan\quad x & \quad \quad \quad 1 \end{matrix} \right] \), then show that A^TA^-1 = \(\left[ \begin{matrix} cos\quad 2x & -sin2x \\ sin\quad 2x & cos2x \end{matrix} \right] .\)

Solve the following LPP graphically. Minimize\(Z=-3{ x }_{ 1 }+4{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }+2{ x }_{ 2 }\le 8\quad ,{ 3x }_{ 1 }+{ 2x }_{ 2 }\le 12\quad and\quad \quad { x }_{ 1 }\ge 0,{ x }_{ 2 }\ge 2.\)

Solve the following LPP graphically, Minimize \(Z=3{ x }_{ 1 }+5{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }+3{ x }_{ 2 }\ge 3,\quad { x }_{ 1 }+{ x }_{ 2 }\ge 2\quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0.\)

Solve the following LPP graphically. Maximize \(Z={ x }_{ 1 }+{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }-{ x }_{ 2 }\le -1,{ -x }_{ 1 }+{ x }_{ 2 }\le 0\quad and\quad { x }_{ 1 }+{ x }_{ 2 }\ge 0\)

Construct the network for the projects consisting of various activities and their precedence relationships are as given below:

Construct the network for the following:

Solve the following LPP graphically. Maximize Z = 6x₁ + 5x₂ Subject to the constraints 3x₁ + 5x₂ ≤ 15, 5x₁ + 2x₂ ≤ 10 and x₁,x₂ ≥ 0

Solve the following LPP graphically. Maximize Z =−x₁ + 2x₂
Subject to the constraints −x₁ + 3x₂ ≤ 10, x₁ + x₂ ≤ 6,x₁ − x₂ ≤ 2 and x₁,x₂ ≥ 0

Solve the following LPP graphically. Minimize Z = x₁ − 5x₂ + 20
Subject to the constraints x₁ − x₂ ≥ 0,−x₁ + 2x₂ ≥ 2,x₁ ≥ 3,x₂ ≤ 4 and x₁,x₂ ≥ 0.

Solve the following LPP graphically. ∴ Maximize Z = 3x₁ + 4x₂  subject to the constraints x₁ + x₂ ≤ 4 and x₁,x₂ ≥ 0.

Solve the following LPP graphically. Minimize\(Z=-3{ x }_{ 1 }+4{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }+2{ x }_{ 2 }\le 8\quad ,{ 3x }_{ 1 }+{ 2x }_{ 2 }\le 12\quad and\quad \quad { x }_{ 1 }\ge 0,{ x }_{ 2 }\ge 2.\)

For the following observations, find the regression co-efficients b_yx and b_xy and hence find the correlation co-efficient between x and y.(4,2) (2, 3)(3, 2)(4, 4)(2, 4)

Find the regression co-efficient of x on y from the following data. \(\sum\)X=20, \(\sum\)Y=40, \(\sum\)XY=300, \(\sum\)X²=150, \(\sum\)Y²=345, N=5. Find the value of x when y=5

Calculate Co-efficient of correlation for the following data:

Calculate the co-efficient of correlation between x and y on the basis of the following observations. \(\sum\)x=125, \(\sum\)x²=1650, \(\sum\)y=100, \(\sum\)y²=1460, \(\sum\)xy=50, n=25.

In a correlation study, the following values are obtained

The co-efficient of correlation is 0.5. Find the lines of regression.

Calculate the covariance of the following pairs of observation of two variates X and Y. (1, 5)(2, 4)(3, 3)(4, 2)(5, 1)

Explain the concept of correlation and co-efficient of correlation.

Calulate the co-efficient of correlation between x and y on the basis of the following observations. \(\sum\)\(\sum\)x=10, \(\sum\)x²=250, \(\sum\)y=70, \(\sum\)y²=300, \(\sum\)xy=75 and n =20

The co-efficient of correlation between two variables X and Y is 0.5 and their co-variance is 5.81. The variance of X is 12. Find the satandard deviation of the Y-series.

prove that the correlation co-efficient is the geometric mean of regression co-efficients.

Find the geometric mean of 3, 6, 24, 48

Find the harmonic mean of 6, 14, 21, 30

Find Q₂ for 37, 32, 45, 36, 39, 37, 46, 57, 27, 34, 28, 30, 21

Two integers are selected at random from integers 1 to 11. If the sum is even find the probability that both the numbers are odd.

Find the geometric mean for the following data

A fair die is rolled. A = {1, 3, 5} B = {2, 3} and C = {2, 3, 4, 5}. Find (i) P(A/B) and P(B/A) (ii) P(A/C) and p(C/A).

Mother, father and son line up at random for a family picture. Find P(A/B) if A and B are defined as follows A = Son on one end, B = Father in the middle.

A and B are two events such that P(A)\(\neq \)0. find P(B/A) if (i) A is a subset of B (ii) \(A\cap B=\phi \)

Events A and B are such that P(A)=\(\frac { 1 }{ 2 } \), P(B)=\(\frac { 7 }{ 12 }\), and P(not A or not B) = \(\frac { 1 }{ 4 }\), state whether A and B are independent?

Find the probability of drawing a one-rupee coin from a purse with two compartments one of which contains 3 fifty-paise coins and 2 one-rupee coins and other contains 2 fifty paise coins and 3 one-rupee coins.

A sum of Rs.1000 is deposited at the beginning of each quarter in a S.B. account that pays C.I 8% compounded quarterly. Find the account at the end of 3 years.

Find the future value of an ordinary annuity of Rs.1000 a year for 5 years for 5 years at 7% p.a compounded annually.

What is the present value of an annuity that pays 250 per month at the end of each month for 5 years assuming money to be worth 6% compounded monthly?

Find the number of shares which will give an annual income of Rs.360 from 6% stock of face value Rs.100

Find the rate of dividend which gives an annual income of Rs.1200 for 150 shares of face value Rs.100

If I deposit Rs.500 every year for a period of 10 years in a bank which gives C.I. 5% per year, find out the amount I will receive at the end of 10 years.

If I deposit Rs.500 every year for a period of 10 years in a bank which gives C.I 5% per year, Find out the amount I will receive at the end of 10 years

Find the present value of an annuity due of Rs.200 p.a. payable annually for 2 years at 4%p.a

A man wishes to pay back his depts of Rs.3783 due after 3 years by 3 equal yearly instalments. Find the amount of each instalments,money being worth 5% p.a. compounded annually

Find the yearly income on 120 shares of 7% stock of face value Rs.100?

Find the stationary points and stationary values of the function f(x) = x³ - 3x² - 9x + 5.

Show that the function x³ + 3x² + 3x + 7 is an increasing function for all real values of x.

Separate the intervals in which the function x³ + 8x² + 5x - 2 is increasing or decreasing.

Find the maximum and minimum values of the function x² + 16/x

For the production function P= 5(L)^0.7(K)^0.3.Find the marginal productivities of Labour (L) and Capital (K) when L = 10, K = 3 [Use (0.3)^0·3 = 0.6968; (3.33)^0·7 = 2.2322]

If \(y=\sqrt { x } +\frac { 1 }{ \sqrt { x } } \) show that \(2x\frac { dy }{ dx } +y=2\sqrt { x } \).

Show that the function f(x) = [x] where [x] denotes the greatest integer function is discontinuous at all integral points

Is the function defined by f(x) = x² -sin x + 5 is continuous at x =\(\pi\)?

Differentiate: \(\sin ^{ -1 }{ \left( \sqrt { \cos { x } } \right) } \)

Differentiate: sin² x + cos² y = 1.

Differentiate: sin x.sin 2x. sin 3x with respect to 'x'.

Find \(\frac{dy}{dx}\) if x = 15(t - sin t); y = 18(1 - cos t).

If e^y (x + 1) = 1, show that \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } ={ \left( \frac { dy }{ dx } \right) }^{ 2 }\)

Evaluate \(\underset { x\rightarrow -3 }{ lim } \frac { { x }^{ 3 }+27 }{ { x }^{ 5 }+243 } \)

Evaluate \(\underset { x\rightarrow 0 }{ lim } \frac { 2sinx-sin2x }{ { x }^{ 3 } } \)

Find all other trigonometrical ratios if \(\sin x=\frac{-2\sqrt6}{5}\) and x lies in III quadrant?

Prove that \(\sin^2\frac{\pi}{6}+\cos^2\frac{\pi}{3}-\tan^2\frac{\pi}{4}=-\frac12\)

Prove that\(\left\{1+\cot x-\sec\left(\frac{\pi}{2}+x\right)\right\}\left\{1+\cot x+\sec\left(\frac{\pi}{2}+x\right)\right\}=2\cot x\)

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

Prove that \(\frac{\sin(x+y)}{\sin(x-y)}=\frac{\tan x+\tan y}{\tan x-\tan y}\)

If \(\alpha\) and \(\beta\) are acute angles such that \(\tan\alpha=\frac{m}{m+1}\) and \(\tan\beta=\frac{1}{2m+1}\), prove that \(\alpha+\beta=\frac{\pi}{4}\)

Show that \(\tan\left(\frac{\pi}{3}+x\right)\tan\left(\frac{\pi}{3}-x\right)=\frac{2\cos2x+1}{2\cos2x-1}\)

Prove that \(\frac{\sin5x+\sin3x}{\cos5x+\cos3x}=\tan4x\)

Prove that \(\frac{\sin5x-2\sin3x+sinx}{\cos5x-\cos x}=\tan x\)

Show that \(\cos^{-1}\left(\frac{3}{5}\cos x+\frac45\sin x\right)=x-\tan^{-1}\left(\frac43\right)\)

Find the equation of a circle whose diameters are 2x - 3y + 12 = 0 and x + 4y - 5 = 0 and area is 154 square units.

Find the equation of the parabola whose focus is (-3, 2) and the directrix is x + y = 4.

Find the locus of a point which moves in such a way that the square of its distance from the point (3, -2) is numerically equal to its distance from the line 5x - 12y = 13

A point moves such that its distance from the point (4, 0) is half that of its distance from the line x = 16, find its locus.

Find the separate equations of the pair of lines given by 3x² + 7xy + 2y² + 5x + 5y + 2 = 0.

A point moves so that its distance from the point (-1, 0) is always three times its distance from the point (0, 2). Find its locus.

For what value of \(\lambda \) are the three lines 2x-5y+3 = 0, 5x-9y+\(\lambda \)=0 and x-2y+1=0 are concurrent?

Find the value of k if the straight line 2x + 3y + 4 + k(6x - y + 12) = 0 is perpendicular to the line 7x + 5y - 4 = 0

For what value of k does 12x² + 7xy + ky² + 13x - y + 3 = 0 represents a pair of straight lines?

Find the equation of a circle of radius 5 whose centre lies on X-axis and passes through the point (2, 3).

In how many ways can n prizes be given to n boys, when a boy may receive any number of prizes?

In an examination, Yamini has to select 4 questions from each part. There are 6, 7 and 8 questions is Part I, Part II and Part III respectively. What is the number of possible combinations in which she can choose the questions?

In how many ways can 12 things be equally divided among 4 persons?

If p(n) is the statement "12n + 3" is a multiple of 5, then show that P (3) is false, whereas P(6) is true.

Let p(n) be the statement "n² + n is even". If P(k) is true, then show that P(k+1) is true.

Resolve into partial factors:\(\frac { x+4 }{ ({ x }^{ 2 }-4)(x+1) } \)

Solve : \(\frac { (2x+1)! }{ (x+2)! } .\frac { (x-1)! }{ (2x-1)! } =\frac { 3 }{ 5 } \)

How may different numbers between 100 and 1000 can be formed using the digits 0, 1,2,3,4, 5, 6 assuming that in any number, the digits are not repeated.

It the letters of the word are arranged as in dictionary, find the rank of the word "AGAIN".

There are 6 gentlemen and 4 ladies to line at a round table. In how many ways can they seat themselves so that no two ladies together?

Prove that \(\left| \begin{matrix} x & sin\theta & cos\theta \\ -sin\theta & -x & 1 \\ cos\theta & 1 & x \end{matrix} \right| \) is independent of \(\theta\)

Using matrix method, solve x + 2y + z = 7, x + 3z = 11 and 2x - 3y =1.

if A =\(\left[ \begin{matrix} cos\ \alpha & sin\ \alpha \\ -sin\ \alpha & \ cos\ \alpha \ \end{matrix} \right] \) is such that A^T = A^-1, find \(\alpha\)

Write the minors and co-factors of the elements of \(\begin{vmatrix}5 & 3 \\-6 & 2\end{vmatrix}\)

Verify that A(adj A) = (adj A) A = IAI·I for the matrix A = \(\begin{bmatrix}2 & 3 \\-1 & 4\end{bmatrix}\)

Using the properties of determinants, show that \(\left| \begin{matrix} 2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86 \end{matrix} \right| \) = 0

Using co-factors of elements of  second column evaluate \(\left| \begin{matrix} 6 & -1 & 5 \\ 3 & 0 & 4 \\ -2 & 7 & -3 \end{matrix} \right| \)

Find the adjoint of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ 0 & 5 & 1 \\ 3 & 6 & 8 \end{matrix} \right] \)

If A = \(\begin{bmatrix} 3 & 2 \\ 7 & 5 \end{bmatrix}\) and B = \(\begin{bmatrix} 4 & 6 \\ 3 & 2 \end{bmatrix}\), verify that (AB)^-1 = B^-1A^-1

Find the numbers a and b such that A² + aA + bI = 0 for the matrix A =\(\begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}\)

Differentiate: sin x.sin 2x. sin 3x with respect to 'x'.

Find \(\frac{dy}{dx}\) if x = 15(t - sin t); y = 18(1 - cos t).

If e^y (x + 1) = 1, show that \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } ={ \left( \frac { dy }{ dx } \right) }^{ 2 }\)

Evaluate \(\underset { x\rightarrow -3 }{ lim } \frac { { x }^{ 3 }+27 }{ { x }^{ 5 }+243 } \)

Evaluate \(\underset { x\rightarrow 0 }{ lim } \frac { 2sinx-sin2x }{ { x }^{ 3 } } \)

Show that the functions f(x) = 5x - \(\left| x \right| \) is continuous at x = 0

For what value of k, the following function is continous at x =0?
f(x) = \(\begin{cases} \frac { 1-cos4x }{ 8{ x }^{ 2 } } \quad ifx\neq 0 \\ k\quad \quad \quad ifx=0 \end{cases}\)

Show that the function f(x) = 5x -3 is continous at x = +3

Prove that the function given by f(x) = \(\left| x-1 \right| \), x \(\in\) R is not differentiable at x =1

Evaluate \(\underset { x\rightarrow \frac { 1 }{ 2 } }{ lim } \frac { { 4x }^{ 2 }-1 }{ 2x-1 } \)

If \(\cos x=-\frac{1}{2}\) and \(\pi , find the value of \(4\tan^2x-3cosec^2x\)

Evaluate \(\cot\left(\frac{-15\pi}{4}\right)\)

In any quadrilateral ABCD, prove that sin (A + B) + sin (C + D) = 0

Find the value of \(\cos\left(\frac{5\pi}{12}\right)\)

Prove that \(sin^2\left(\frac{\pi}{8}+\frac x2\right)-sin^2\left(\frac{\pi}{8}-\frac x2\right)=\frac{1}{\sqrt2}\sin x.\)

Prove that \(\frac{\tan 69^o+\tan 66^o}{1-\tan 69^o\tan 66^o}=-1\)

Prove that \(\cos18^o-\sin18^o=\sqrt{2}.\sin27^o\)

If \(\tan^2x=2\tan^2\phi+1\), prove that \(\cos2x+sin^2\phi=0\)

Evaluate : \(\cos\left[\frac{\pi}{3}-\cos^{-1}\left(\frac{1}{2}\right)\right]\)

Find the principal value of \(\cos^{-1}\left(\frac{-1}{\sqrt2}\right)\)

In a railway compartment, 6 seats are vacant on a bench. In how many ways can 3 passengers sit on them?

Show that 10P₃ = 9 P₃ + 3. 9P₂

How many permutations can be made out of the letters of the word "TRIANGLE" beginning with T?

Find n if 25 C_n+5 = 25 C_2n-1.

In the expansion of \({ \left( x+\frac { 1 }{ x } \right) }^{ 6 }\), find the third term.

ResoIve into partial fractions :\(\frac { 12x-17 }{ (x-2)(x-1) } \)