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

 

1. 1/36

2. 0

3. 1/3

4. 1/6

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

1. 10!

2. 9!

3. 9\(\times\)9!

4. 10\(\times\)10!

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

1. (n–1)f

2. n(n–1)f

3. nf

4. f

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

1. 1

2. 0

3. -1

4. -xyz

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

1. 4 cos 2 \(\theta\)

2. 4

3. 2

4. 1

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

1. Annuity due

2. An immediate annuity

3. perpetual annuity

4. none of these

For the cost function C =\(\frac { 1 }{ 25 } { e }^{ 25 }\), the marginal cost is 

1. \(\frac { 1 }{ 25 } \)

2. \(\frac { 1 }{ 5 } { e }^{ 5x }\)

3. \(\frac { 1 }{ 125 } { e }^{ 5x }\)

4. 25e^5x

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

1. 4

2. 5

3. 13

4. 9

\(\left(\frac{\cos x}{cosec x}\right)-\sqrt{1-\sin^2x}\sqrt{1-\cos^2x}\) is

1. cos²x-sin²x

2. sin²x-cos²x

3. 1

4. 0

Relationship among MR, AR and ηd is

1. \({ n }_{ d }=\frac { AR }{ AR-MR } \)

2. n₄ =  AR - MR

3. MR = AR = n₄

4. \(AR=\frac { MR }{ { n }_{ 4 } } \)

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

1. Rs 60,000

2. Rs 50,000

3. Rs 10,000

4. Rs 80,000

Let a sample space of an experiment be S = { E₁,E₂,....., En} Then \(\sum _{ i=1 }^{ n }{ P({ E }_{ t }) } \) is equal to

 

1. 0

2. 1

3. \(\frac { 1 }{ 2 } \)

4. \(\frac { 1 }{ 3 } \)

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

1. 4

2. 5

3. 16

4. 25

The elasticity of demand for the demand function x = \(\frac { 1 }{ p } \) os

1. 0

2. 1

3. \(-\frac { 1 }{ p } \)

4. \(\infty \)

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

 

1. 10

2. 12

3. 14

4. 9

The equation of the circle with centre on the x axis and passing through the origin is

1. x² - 2ax + y = 0

2. y² - 2ay + x² = 0

3. x²+y²=a²

4. x² - 2ay + y = 0

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

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

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

3. \(\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\)

4. \(\begin{pmatrix} 0 & 2 \\ 2 & 0 \end{pmatrix}\)

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

1. a₁₁ A₃₁ + a₁₂ A₃₂ + a₁₃ A₃₃

2. a₁₁ A₁₁ + a₁₂ A₂₁ + a₁₃ A₃₁

3. a₂₁ A₁₁ + a₂₂ A₁₂ + a₂₃ A₁₃

4. a₁₁ A₁₁ + a₂₁ A₂₁ + a₃₁ A₃₁

The Income on 7 % stock at 80 is

1. 9%

2. 8.75%

3. 8%

4. 7%

A solution which maximizes or minimizes the given LPP is called

1. a solution

2. a feasible solution

3. an optimal solution

4. none of these

If the average revenue of a certain firm is Rs 50 and its elasticity of demand is 2, then their marginal revenue is

1. Rs 50 

2. Rs 25

3. Rs 100

4. Rs 75

Which of the following is positional measure?

 

1. Range

2. Mode

3. Mean deviation

4. Percentiles

If the centre of the circle is (-a, -b) and radius \(\sqrt{a^2-b^2}\)then the equation of circle is

1. x²+y²+2ax+2by+2b²=0

2. x²+y²+2ax +2by-2b²=0

3. x² +y² - 2ax - 2by - 2b² = 0

4. x² + y - 2ax - 2by + 2b² = 0

If A is an invertible matrix of order 2, then det (A^-1) be equal to

1. det (A)

2. \({{1}\over{det(A)}}\)

3. 1

4. 0

If A = \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)such that ad - bc \(\neq\) 0 then A^-1 is

1. \({{1}\over{ad-bc}}\begin{pmatrix} d & b \\-c & a\end{pmatrix}\)

2. \({{1}\over{ad-bc}}\begin{pmatrix} d & b \\c & a\end{pmatrix}\)

3. \({{1}\over{ad-bc}}\begin{pmatrix} d & -b \\-c & a\end{pmatrix}\)

4. \({{1}\over{ad-bc}}\begin{pmatrix} d & -b \\c & a\end{pmatrix}\)

The minimum value of the function f(x)= Ixl is

1. 0

2. -1

3. +1

4. \(-\infty \)

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

1. -5

2. -125

3. -25

4. 0

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

1. 1

2. 0

3. \(\frac{\sqrt3}{2}\)

4. \(\frac{1}{\sqrt{2}}\)

The constant term in the expansion of \({ \left( x+\frac { 2 }{ x } \right) }^{ 6 }\)

1. 156

2. 165

3. 162

4. 160

The graph of y = e^x intersect the y-axis at

1. (0,0)

2. (1,0)

3. (0,1)

4. (1,1)

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

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

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

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

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

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

1. -1

2. 8

3. 1000

4. 1000 - p₂

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

1. 2

2. 6

3. 20

4. 24

The two events A and B are mutually exclusive if

 

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

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

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

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

If A and B are non-singular matrices then, which of the following is incorrect?

1. A² = Iimplies A^-1 = A

2. I^-1 = I

3. If AX = B, then X = B^-1 A

4. If A is square matrix of order 3 then |adj A|= |A|²

If two variables moves in decreasing direction then the correlation is

1. positive

2. negative

3. perfect negative

4. no correlation

The brokerage paid by a person on this sale of 400 shares of face value Rs.100 at 1% brokerage

1. Rs 600

2. Rs 500

3. Rs 200

4. Rs 400

If y = e^2x then \(\frac{d^2y}{dx^2}\) at x = 0 is

1. 4

2. 9

3. 2

4. 0

Harmonic mean is better than other means if the data are for

 

1. Speed or rates

2. Heights or lengths.

3. Binary values like 0 and 1.

4. Ratios or proportions.

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

1. 1

2. \(\infty\)

3. \(-\infty\)

4. \(\theta\)

Find the centre and radius of the circle x²+y²-8x+6y-24 = 0

If four dice are rolled, find the number of possible outcomes in which atleast one die shows 2.

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 the values of the following trigonometric ratios. \(\sin { { 300 }^{ o } } \)

Evaluate the following expression.\(\frac { 7! }{ 6! } \)

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

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

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

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

Express the following as sum or difference
2sin4\(\theta\)sin2\(\theta\)

If \(\sin { A } =\frac { 12 }{ 13 } \), find sin 3A

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

Let f be defined by f(x) = x - 5 and g be defined by g(x) = \(\begin{cases} \frac { { x }^{ 2 }-25 }{ x+5 } if\quad x\neq -5 \\ \lambda \quad if\quad x=-5 \end{cases}\) Find \(\lambda\) such that f(x) = g(x) for all x \(\epsilon \) R

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

Find the market value of 62 shares available at Rs 132 having the par value of Rs 100.

Determine whether the following functions are odd or even?
\(f(x)=\left( \frac { { a }^{ x }-1 }{ { a }^{ x }+1 } \right) \)

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

A factory owner purchases two types of machines A and B for his factory. The requirements and the limitations for the machines are as follows:

Machine	Area occupied	Labour Force	Daily Output (in units)
A	1000 m2	12 men	60
B	1200 m2	8 men	40

He has maximum area of 9000 m² available, and 72 skilled labourers who can operate both the machines. How many machines of each type should he buy to maximize the daily output. Formulate the above as LPP.

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

Expand the following by using binomial theorem. (2a - 3b)⁴

A person borrows Rs.5000 at 5% p.a.interest compounded half yearly and agrees to pay both the principal and interest at 10 equal instalments at the end of each six months.Find the amount of these instalments.

If xy² = 1, then prove that 2 \(\frac{dy}{dx}\) + y³ = 0.

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

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

1. If x = \(a\ \theta\) and \(y=\frac{a}{\theta}\), then prove that \(\frac{dy}{dx}+\frac{y}{x}=0\)

2. Find \(\frac { dy }{ dx } \) for the following functions:
   xy=tan(xy)

Calculate the Quartile deviation for the following:

Value (x)	0	1	2	3	4	5
Frequency(f)	8	10	11	15	21	25

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

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

Compute the co-efficient of correlation batween the variates x and y from the given data:
No. of pairs of x and y series=8.x-series A.M.=74.5, x-series assumed mean =69, x-series S.D=13.07, y-series S.D=15.85, sum of products of corresponding deviations of x and y series =2176.

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.

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) \(A\cap B=\phi \)

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).

Express each of the following as the product of sine or cosine.
sin6ፀ-sin2ፀ

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

Estimated cost	30	45	80	25	50	97	47	40
Actual cost	27	48	73	29	63	87	39	45

Calculate the value of spearman’s correlation.

Find the inverse of each of the following matrices.
\(\begin{bmatrix} 3&1\\-1&3 \end{bmatrix}\)

Find the equation of the circle which touches the line x=0, y=0 and x=a.

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.

Sundar bought 4,500 of Rs 10 shares, paying 2% per annum. 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.

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

A point moves so that it is always at a distance of 4 units from the point (3, -2)

If x^m.yⁿ =(x+y)^(m+n) ,then show that \(\frac { dy }{ dx } =\frac { y }{ x } \)

How many code symbols can be formed using 5 out of 6 of the letters of 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.

if y = 2 sin x + 3 cos x, then show that y₂ + y = 0

Draw the graph of the following function f(x) =16-x²

1. 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.

2. Using the following information you are requested to (i) obtain the linear regression of Y on X (ii) Estimate the level of defective parts delivered when inspection expenditure amounts to Rs.28,000 ΣX=424, ΣY=363, ΣX² =21926, ΣY² =15123, ΣXY=12815, N=10. Here X is the expenditure on inspection, Y is the defective parts delivered.

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

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.

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

Show that the matrices \(A=\left[ \begin{matrix} 1 & 3 & 7 \\ 4 & 2 & 3 \\ 1 & 2 & 1 \end{matrix} \right] \)and \(B=\left[ \begin{matrix} \frac { -4 }{ 35 } & \frac { 11 }{ 35 } & \frac { -5 }{ 35 } \\ \frac { -1 }{ 35 } & \frac { -6 }{ 35 } & \frac { 25 }{ 35 } \\ \frac { 6 }{ 35 } & \frac { 1 }{ 35 } & \frac { -10 }{ 35 } \end{matrix} \right] \)are inverses of each other.

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

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?

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