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

(a)

\({{7}\over{30}}\begin{pmatrix} \frac { 1 }{ 2 } & \frac { 5 }{ 12 } \\ \frac { 2 }{ 5 } & \frac { 4 }{ 5 } \end{pmatrix}\)

(b)

\({{7}\over{30}}\begin{pmatrix} \frac { 1 }{ 2 } & \frac { -5 }{ 12 } \\ \frac { -2 }{ 5 } & \frac { 1 }{ 5 } \end{pmatrix}\)

(c)

\({{30}\over{7}}\begin{pmatrix} \frac { 1 }{ 2 } & \frac { 5 }{ 12 } \\ \frac { 2 }{ 5 } & \frac { 4 }{ 5 } \end{pmatrix}\)

(d)

\({{30}\over{7}}\begin{pmatrix} \frac { 1 }{ 2 } & \frac { -5 }{ 12 } \\ \frac { -2 }{ 5 } & \frac { 4 }{ 5 } \end{pmatrix}\)

Which of the following matrix has no inverse.

(a)

\(\begin{pmatrix} -1 & 1 \\ 1 &-4 \end{pmatrix}\)

(b)

\(\begin{pmatrix} 2 & -1 \\ -4 &2 \end{pmatrix}\)

(c)

\(\begin{pmatrix} cos\ a & sin\ a \\ -sin\ a & cos\ a \end{pmatrix}\)

(d)

\(\begin{pmatrix} sin\ a & cos\ a \\ -cos\ a & sin\ a \end{pmatrix}\)

The term containing x³ in the expansion of (x - 2y)⁷ is _________.

(a)

3^rd

(b)

4^th

(c)

5^th

(d)

6^th

The number of permutation of n different things taken r at a time, when the repetition is allowed is ________.

(a)

rⁿ

(b)

n^r

(c)

\(\frac { n! }{ (n-r)! } \)

(d)

\(\frac { n! }{ (n+r)! } \)

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

(a)

1/2

(b)

-1/2

(c)

2

(d)

-2

Combined equation of co-ordinate axes is _______.

(a)

x²-y² = 0

(b)

x²+y² = 0

(c)

xy = c

(d)

xy = 0

The value of sec A sin(270^o + A) is ______.

(a)

-1

(b)

cos² A

(c)

sec² A

(d)

1

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

(a)

\(\frac{\pi}{4}\)

(b)

\(\frac{\pi}{2}\)

(c)

\(\frac{\pi}{3}\)

(d)

\(\frac{\pi}{6}\)

The graph of the line y = 3 is _______.

(a)

Parallel to x-axis

(b)

Parallel to y-axis

(c)

Passing through the origin

(d)

Perpendicular to x-axis

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

(a)

an identity function

(b)

modulus function

(c)

exponential function

(d)

constant function

If demand and the cost function of a firm are p = 2–x and c = 2x² + 2x + 7 then its profit function is ________.

(a)

x² + 7

(b)

x² - 7

(c)

- x² + 7

(d)

- x² - 7

The demand function is always _______.

(a)

Increasing function

(b)

Decreasing function

(c)

Non-decreasing function

(d)

Undefined function

A invested some money in 10% stock at Rs.96. If B wants to invest in an equally good 12% stock, he must purchase a stock worth of  _______.

(a)

Rs. 80

(b)

Rs. 115.20

(c)

Rs. 120

(d)

Rs. 125.40

The present value of the perpetual annuity of Rs. 2000 paid monthly at 10 % compound interest is _______.

(a)

Rs. 2,40,000

(b)

Rs. 6,00,000

(c)

Rs. 20,40,000

(d)

Rs. 2,00,400

The correct relationship among A.M.,G.M.and H.M.is: ________.

(a)

A.M.

(b)

G.M.≥A.M.≥H.M.

(c)

H.M.≥G.M.≥A.M.

(d)

A.M.≥G.M.≥H.M.

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

(a)

\(P\left( A\cap B \right) =0\)

(b)

\(P\left( A\cap B \right) =1\)

(c)

\(P\left( A\cup B \right) =0\)

(d)

\(P\left( AUB \right) =1\)

The correlation coefficient is ________.

(a)

r(X, Y) = \(\frac { { \sigma }_{ x }{ \sigma }_{ y } }{ cov(x,y) } \)

(b)

r(X, Y) = \(\frac { cov(x,y) }{ { \sigma }_{ x }{ \sigma }_{ y } } \)

(c)

r(X, Y) = \(\frac { cov(x,y) }{ { \sigma }_{ y } } \)

(d)

r(X, Y) = \(\frac { cov(x,y) }{ { \sigma }_{ x } } \)

The term regression was introduced by ________.

(a)

R.A Fisher

(b)

Sir Francis Galton

(c)

Karl Pearson

(d)

Croxton and Cowden

In the given graph the coordinates of M₁ are

(a)

x₁ = 5, x₂ = 30

(b)

x₁ = 20, x₂ = 16

(c)

x₁ = 10, x₂ = 20

(d)

x₁ = 20, x₂ = 30

Network problems have advantage in terms of project _______.

(a)

Scheduling

(b)

Planning

(c)

Controlling

(d)

All the above

Find the minors and cofactors of all the elements of the following determinants \(\begin{vmatrix}5&20\\ 0&-1 \end{vmatrix}\)

How many chords can be drawn through 21 points on a circle?

Find the center and radius of the circle (x + 2) ( x - 5) + (y - 2 ) ( y - 1) = 0

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

Differentiate the following functions with respect to x, \(\frac{1-3x}{1+3x}\)

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

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

An aeroplane flies, along the four sides of a square at speeds of 100, 200, 300 and 400 kilometres per hour respectively. What is the average speed of the plane in its flight around the square.

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

Construct a network diagram for the following situtation A < B; B < D, E; C

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

Evaluate  \(\frac { n! }{ r!(n-r)! } \) when n = 5 and r = 2

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

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

Differentiate the following with respect to x. sin² x

The total cost C in Rupees of making x units of product is C(x) = 50 + 4x + 3√x. Find the marginal cost of the product at 9 units of output.

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

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?

Calculate the correlation co-efficient from the following data:

X	12	9	8	10	11	13	7
Y	14	8	6	9	11	12	3

Develop a network based on the following information.

Activity	A	B	C	D	B	E
Immediate Predecessor	-	-	A	C	E	F

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

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

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

2. 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.

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

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

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

2. The demand function of a commodity is p = 200 - \(\frac { x }{ 100 } \) and its cost is C = 40x + 12000 where p is a unit price in rupees and x is the number of units produced and sold. Determine (i) profit function (ii) average profit at an output of 10 units (iii) marginal profit at an output of 10 units and (iv) marginal average profit at an output of 10 units.

1. Show that the maximum value of the function f(x) = x³ - 27x + 108 is 108 more than the minimum value.

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?

1. Three coins are tossed simultaneously. Consider the events A ‘three heads or three tails’, B ‘atleast two heads’ and C ‘at most two heads’ of the pairs (A, B), (A, C) and (B, C), which are independent? Which are dependent?

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

   Advertisement expenditure	40	50	38	60	65	50	35
   Sales	38	60	55	70	60	48	30

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

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

   Activity	1-2	1-3	2-4	2-5	3-4	4-5
   Duration( in days)	8	4	10	2	5	3

2. 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:

   Type	Machine I	Machine II	Machine III
   A	12	18	6
   B	6	0	9

   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.