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QUESTION: 1

**Q. 1 – Q. 5 carry one mark each.**

**Q. **An apple costs Rs. 10. An onion costs Rs. 8.

Select the most suitable sentence with respect to grammar and usage.

Solution:

QUESTION: 2

The Buddha said, “Holding on to anger is like "__grasping"__ a hot coal with the intent of throwing it at someone else; you are the one who gets burnt.”

Select the word below which is closest in meaning to the word underlined above.

Solution:

QUESTION: 3

M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-inlaw

of M. How is P related to M?

Solution:

*Multiple options can be correct

QUESTION: 4

The number that least fits this set: (324, 441, 97 and 64) is ________.

**Note: Question may have more than one correct answer)**

Solution:

QUESTION: 5

It takes 10 s and 15 s, respectively, for two trains travelling at different constant speeds to

completely pass a telegraph post. The length of the first train is 120 m and that of the second train is

150 m. The magnitude of the difference in the speeds of the two trains (in m/s) is ____________.

Solution:

QUESTION: 6

**Q. 6 – Q. 10 carry two marks each.**

**Q.**

The velocity V of a vehicle along a straight line is measured in m/s and plotted as shown with respect to time in seconds. At the end of the 7 seconds, how much will the odometer reading increase by (in m)?

Solution:

QUESTION: 7

The overwhelming number of people infected with rabies in In dia has been flagged by the World Health Organization as a source of concern. It is estimated that inoculating 70% of pets and stray dogs against rabies can lead to a significant reduction in the number of people infected w ith rabies.

Which of the fo llowing can be logically inferred from the above sentences?

Solution:

QUESTION: 8

A flat is shared by four first year undergraduate students. They agreed to allow the oldest of them to enjoy some extra space in the flat. Manu is two months older than Sravan, who is three months younger than Trideep. Pavan is one month older than Sravan. Who should occupy the extra space in the flat?

Solution:

QUESTION: 9

Find the area bounded by the lines 3x+2 y=14, 2x-3y =5 in the first quadrant

Solution:

QUESTION: 10

A straight line is fit to a data set (ln *x*, *y*) . This line intercepts the abscissa at ln *x* = 0.1 and has a slope of −0.02. What is the value of *y* at *x* = 5 from the fit?

Solution:

QUESTION: 11

**Q. 11 – Q. 35 carry one mark each.**

**Q.**

**Let {X,Y,Z} be a basis of ****. Consider the following statements P and Q:
(P) : {X+Y, Y+Z,X-Z} is a basis of **

(Q) : {X+Y+Z, X+2Y-Z, X-3Z} is a basis of

Which of the above statements hold TRUE?

Solution:

QUESTION: 12

Consider the following statements P and Q:

(P) : If M= then M is singular

(Q) : Let S be a diagonalizable matrix. If T is a matrix such that S + 5 T = Id, then T is

diagonalizable

Which of the above statements hold TRUE?

Solution:

QUESTION: 13

Consider the following statements P and Q:

(P) : If M is an n x n complex matrix, then

(Q) : There exists a unitary matrix with an eigenvalue λ such that |λ| <1.

Which of the above statements hold TRUE?

Solution:

QUESTION: 14

Consider a real vector space V of dimension n and a non-zero linear transformation

T : V → V. If dimension(T(V)) < n and T^{2} = λ T, for some then which of the following statements is TRUE?

Solution:

QUESTION: 15

Let and and be a strictly increasing function such that f(S) is connected. Which of the following statements is TRUE?

Solution:

*Answer can only contain numeric values

QUESTION: 16

Let a_{1}=1 and then

is equal to _____________________

Solution:

*Answer can only contain numeric values

QUESTION: 17

Maximum is equal to _________________

Solution:

QUESTION: 18

Let a,b,c such that c^{2}+d^{2} 0. Then, the Cauchy problem

has a unique solution if

Solution:

*Answer can only contain numeric values

QUESTION: 19

Let u(x,t) be the d'Alembert's solution of the initial value problem for the wave

equation

where c is a positive real number and f, g are smooth odd functions. Then, u(0,1) is

equal to ___________

Solution:

*Answer can only contain numeric values

QUESTION: 20

Let the probability density function of a random variable X be

Then, the value of c is equal to ________________________

Solution:

*Answer can only contain numeric values

QUESTION: 21

Let V be the set of all solutions of the equation y" +a y'+by=0 satisfying y(0) =y(1), where a, b are positive real numbers. Then, dimension(V ) is equal to

_____________________

Solution:

*Answer can only contain numeric values

QUESTION: 22

Let where p(x) and q(x)are continuous functions. If cos(x) are two linearly independent solutions of the above equation, then |4p(0)+2q(1)| is equal to ____________________

Solution:

QUESTION: 23

Let P_{n}(x) be the Legendre polynomial of degree n and where k

is a non-negative integer. Consider the following statements P and Q:

(P) : I = 0 if k < n.

(Q) : I = 0 if n - k is an odd integer.

Which of the above statements hold TRUE?

Solution:

QUESTION: 24

Consider the following statements P and Q:

(P) : has two linearly independent Frobenius series

solutions near x=0.

(Q) : has two linearly independent Frobenius series

solutions near x=0.

Which of the above statements hold TRUE?

Solution:

*Answer can only contain numeric values

QUESTION: 25

Let the polynomial x^{4} be approximated by a polynomial of degree 2, which

interpolates x^{4} at x=-1, 0 and 1. Then, the maximum absolute interpolation error

over the interval [-1, 1] is equal to ______________________

Solution:

QUESTION: 26

Let (z_{n})be a sequence of distinct points in with

Consider the following statements P and Q:

(P) : There exists a unique analytic function f on D(0,1) such that f(z_{n}) = sin(z_{n}) for

all n.

(Q) : There exists an analytic function f on D(0,1) such that f(zn) = 0 if n is even

and f(z_{n}) = 1 if n is odd.

Which of the above statements hold TRUE?

Solution:

QUESTION: 27

Let be a topological space with the cofinite topology. Every infinite subset of

is

Solution:

*Answer can only contain numeric values

QUESTION: 28

Let

Then, dimension(C_{0}/M) is equal to _______________________

Solution:

*Answer can only contain numeric values

QUESTION: 29

Solution:

*Multiple options can be correct

QUESTION: 30

Which of the following statements is TRUE for the function f(x) =x-x^{2}/2 ?

**Note: Question may have more than one correct answer)**

Solution:

QUESTION: 31

Let be the set of all n x n real matrices with the usual norm topology. Consider the

following statements P and Q:

(P) : The set of all symmetric positive definite matrices in is connected.

(Q) : The set of all invertible matrices in is compact.

Which of the above statements hold TRUE?

Solution:

QUESTION: 32

Let X_{1}, X_{2}, X_{3}, … , X_{n} be a random sample from the following probability density

function for 0 < μ < ∞, 0 < α < 1,

Here α and μ are unknown parameters. Which of the following statements is TRUE?

Solution:

QUESTION: 33

Suppose X and Y are two random variables such that aX+bY is a normal random a,b

variable for all. Consider the following statements P, Q, R and S:

(P) : X is a standard normal random variable.

(Q) : The conditional distribution of X given Y is normal.

(R) : The conditional distribution of X given X+Y is normal.

(S) : X-Y has mean 0.

Which of the above statements ALWAYS hold TRUE?

Solution:

QUESTION: 34

Consider the following statements P and Q:

(P) : If H is a normal subgroup of order 4 of the symmetric group S_{4}, then S_{4}/H is

abelian.

Which of the above statements hold TRUE?

Solution:

*Answer can only contain numeric values

QUESTION: 35

Let F be a field of order 32. Then the number of non-zero solutions (a,b) ∈ FxF of

the equation x^{2}+xy+y^{2 }=0 is equal to __________________

Solution:

*Answer can only contain numeric values

QUESTION: 36

**Q. 36 – Q. 65 carry two marks each.**

**Q.**

Let be oriented in the counter-clockwise direction. Let

Then, the value of I is equal to __________________________

Solution:

*Answer can only contain numeric values

QUESTION: 37

Let be a harmonic function and v(x,y) its harmonic conjugate. If is equal to _______________

Solution:

*Answer can only contain numeric values

QUESTION: 38

Let λ be the triangular path connecting the points (0,0), (2,2) and (0,2) in the counterclockwise

direction in R^{2}. Then

is equal to _____________________

Solution:

QUESTION: 39

Let y be the solution of

Then y(1) is equal to

Solution:

*Answer can only contain numeric values

QUESTION: 40

Let X be a random variable with the following cumulative distribution function:

Then P(1/4 <X<1) is equal to ___________________

Solution:

QUESTION: 41

Let γ be the curve which passes through (0,1) and intersects each curve of the family y=cx^{2} orthogonally. Then γ also passes through the point

Solution:

*Answer can only contain numeric values

QUESTION: 42

Let be the Fourier series of the

2 π periodic function defined by then

is equal to ________________

Solution:

*Answer can only contain numeric values

QUESTION: 43

Let y(t) be a continuous function on [0, ∞). If

then is equal to _______________.

Solution:

QUESTION: 44

Let then, S_{10 }+ I_{10 } is equal to

Solution:

*Answer can only contain numeric values

QUESTION: 45

For any

Then, is equal to ____________________

Solution:

*Answer can only contain numeric values

QUESTION: 46

Let

equal to ______________

Solution:

*Answer can only contain numeric values

QUESTION: 47

Let M= be a real matrix with eigenvalues 1, 0 and 3. If the eigenvectors

corresponding to 1 and 0 are (1,1,1)^{T} and (1, -1,0)^{T} respectively, then the value of

3*f* is equal to _________________

Solution:

*Answer can only contain numeric values

QUESTION: 48

Let M= and then

is equal to ________________________

Solution:

QUESTION: 49

Let the integral

Consider the following statements P and Q:

(P) : If I_{2} is the value of the integral obtained by the composite trapezoidal rule with

two equal sub-intervals, then I_{2} is exact.

(Q) : If I_{3} is the value of the integral obtained by the composite trapezoidal rule with

three equal sub-intervals, then I_{3} is exact.

Which of the above statements hold TRUE?

Solution:

*Answer can only contain numeric values

QUESTION: 50

The difference between the least two eigenvalues of the boundary value problem

is equal to ______________________________

Solution:

*Answer can only contain numeric values

QUESTION: 51

The number of roots of the equation x^{2}- cos(x)= 0 in the interval

is equal to ____________

Solution:

QUESTION: 52

For the fixed point iteration consider the following statements P and Q:

(P) : if g(x) =1+2/x then the fixed point iteration converges to 2 for all

(Q) : if g(x) = then the fixed point iteration converges to 2 for all

Which of the above statements hold TRUE?

Solution:

QUESTION: 53

Let T: l_{1}-l_{2 }be defined by

Then

Solution:

*Answer can only contain numeric values

QUESTION: 54

Minimize w =x+2y subject to

2x+y 3

x+y2

x0, y0

Then, the minimum value of w is equal to _________________________

Solution:

*Answer can only contain numeric values

QUESTION: 55

Maximize w=11 x-z subject to

10x+y-z1

2x-2y+z2

x,y,z0

Then, the maximum value of w is equal to _________________________

Solution:

*Answer can only contain numeric values

QUESTION: 56

Let X_{1}, X_{2}, X_{3}, … be a sequence of i.i.d. random variables with mean 1. If N is a geometric random variable with the probability mass function P(N=K) =1/2^{K}. K=1,2,3, .... and it is independent of the X_{I}'s then E(X_{1}+X_{2}+X_{3}) is equal to ____________

Solution:

*Answer can only contain numeric values

QUESTION: 57

Let x_{1} be an exponential random variable with mean 1 and x_{2} a gamma random

variable with mean 2 and variance 2. If x_{1} and x_{2} are independently distributed, then

P(x_{1} < x_{2)} is equal to _________________________

Solution:

*Answer can only contain numeric values

QUESTION: 58

Let x_{1}, x_{2}, x_{3}, … be a sequence of i.i.d. uniform (0,1) random variables. Then, the value

of

is equal to ____________________

Solution:

QUESTION: 59

Let X be a standard normal random variable. Then is equal to

Solution:

QUESTION: 60

Let X_{1},X_{2},X_{3}... Xn be a random sample from the probability density function

where α>0, 0 θ 1 are parameters. Consider the following testing problem:

H_{o}: θ = 1, α = 1 versus H_{1}: θ = 0, α = 2.

Which of the following statements is TRUE?

Solution:

*Answer can only contain numeric values

QUESTION: 61

Let X_{1},X_{2},X_{3}... be a sequence of i.i.d. N(μ,1) random variables. Then,

is equal to _____________________________

Solution:

QUESTION: 62

Let X_{1},X_{2},X_{3}... Xn be a random sample from uniform [1,θ] for some θ >1. if X_{n }= Maximum (X_{1},X_{2},X_{3}... Xn) then the UMVUE of θ is

Solution:

*Answer can only contain numeric values

QUESTION: 63

Let x_{1}=x_{2}=x_{3}=1, x_{4}=x_{5}=x_{6 =2 }be a random sample from a Poisson random

variable with mean θ, where Then, the maximum likelihood estimator of θ

is equal to ____________________

Solution:

*Answer can only contain numeric values

QUESTION: 64

The remainder when 98! is divided by 101 is equal to ____________________________

Solution:

QUESTION: 65

Let G be a group whose presentation is

Then G is isomorphic to

Solution:

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