**Q. 1 – Q. 5 carry one mark each.**

Q.1 An apple costs Rs. 10. An onion costs Rs. 8. Select the most suitable sentence with respect to grammar and usage.

(A) The price of an apple is greater than an onion.

(B) The price of an apple is more than onion.

(C) The price of an apple is greater than that of an onion.

(D) Apples are more costlier than onions.

Ans. (C) The price of an apple is greater than that of an onion.

Q.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.

(A) burning

(B) igniting

(C) clutching

(D) flinging

Ans. (C) clutching

Q.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?

(A) P is the son-in-law of M.

(B) P is the grandchild of M.

(C) P is the daughter-in law of M.

(D) P is the grandfather of M.

Ans. (B) P is the grandchild of M.

Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.

(A) 324

(B) 441

(C) 97

(D) 64

Ans. C ; D

Q.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 ____________.

(A) 2.0

(B) 10.0

(C) 12.0

(D) 22.0

Ans. (A) 2.0

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

Q.6 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).?

(A) 0

(B) 3

(C) 4

(D) 5

Ans. (D) 5

Q.7. The overwhelming number of people infected with rabies in India has been flagged by World Health Organization as a source of concern. It is estimated that inoculating 70% of pets and stray dogs against rabies can lead to significant reduction in the number of people infected with rabies. Which of the following can be logically inferred from the above statement?

(A) The number of people infected with rabies in India is high.

(B) The number of people in other parts of the world who are infected with rabies is low.

(C) Rabies can be eradicated in India by vaccinating 70% of stray dogs.

(D) Stray dogs are the main sources of rabies worldwide.

Ans. (A) The number of people infected with rabies in India is high.

Q.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 left. Manu is two months older than shravan, who is three months younger than Trideep. Pavan is older than Sravan. Who should occupy extra space in the flat?

(A) Manu

(B) Sravan

(C) Trideep

(D) Pavan

Ans. (C) Trideep

Q.9 Find the area bounded by the lines 3x+2y=14, 2x-3y=5 in the first quadrant.

(A) 14.95

(B) 15.25

(C) 15.70

(D) 20.35

Ans. (B) 15.25

Q.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?

(A) −0.030

(B) −0.014

(C) 0.014

(D) 0.030

Ans. (A) −0.030

**END OF THE QUESTION PAPER **

**List of Symbols, Notations and Data**

i.i.d. : independent and identically distributed

N(µ,σ^{2 } ) : Normal distribution with mean μ and variance σ^{2}, µ ∈ (-∞, ∞), σ> 0

E(X) ∶ Expected value (mean) of the random variable X

Φ(t)= \frac1{\sqrt{2\mathrm\pi}}\int_{-\infty}^te^{-\frac{x^2}2}dx

[x] : the greatest integer less than or equal to x

\mathbb{Z} ∶ Set of integers

{\mathbb{Z}}_n ∶ Set of integers modulo n

\mathbb{R} ∶ Set of real numbers

\mathbb{C} ∶ Set of complex numbers

\mathbb{R}^n ∶ n‐ dimensional Euclidean space

Usual metric d on \mathbb{R}^n is given by d(x_{1},x_{2}, …x_{n}) , (y_{1},y_{2},….y_{n}), … , = {\textstyle\sum_{i=1}^n}\left(x_i-y_i\right)^2)^{1/2}

ℓ_{2} ∶ Normed linear space of all square‐summable real sequences

C[0,1]∶ Set of all real valued continuous functions on the interval [0,1]

\overline{B(0,1)}\;:\;\{(x,y)\;\in\mathbb{R}^2\;:\;x^2+y^2\leq1\}

M^{*} ∶ Conjugate transpose of the matrix M

M^{T} ∶ Transpose of the matrix M

Id : Identity matrix of appropriate order*R*(M) ∶ Range space of M*N*(M)∶ Null space of M

W^{⊥}: Orthogonal complement of the subspace W

**Q. 1 – Q. 25 carry one mark each.**

Q.1 Let {X,Y,Z} be a basis of \mathbb{R}^3. Consider the following statements P and Q:

(P) : {X+Y,Y+Z,X-Z} is a basis of \mathbb{R}^3.

(Q) : {X+Y+Z,X+2Y-Z,X-3Z} is a basis of \mathbb{R}^3 .

Which of the above statements hold TRUE?

Which of the above statements hold TRUE?

(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Ans. (C) only Q

Q.2 Consider the following statements P and Q:

[P]: If M= \begin{bmatrix}1&1&1\\1&2&4\\1&3&9\end{bmatrix}

[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 statement hold TRUE?

(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Ans. (C) only Q

Q.3 Consider the following statements P and Q:

(P) : If M is an n×n complex matrix, then *R*(M)=(N(M^{*}))^{⊥}

(Q) : There exists a unitary matrix with an eigenvalue λ such that |λ| <1. Which of the above statements hold TRUE?

(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Ans. (B) only P

Q.4 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}=λt, for some λ ∈ \mathbb{R} \{0}, then which of the following statements is TRUE?

(A) determinant (T)=|λ|^{n}

(B) There exists a non‐trivial subspace V1 of V such that T(X)=0 for all X ∈V_{1}

(C) T is invertible

(D) λ is the only eigenvalue of T

Ans. (B) There exists a non‐trivial subspace V1 of V such that T(X)=0 for all X ∈V_{1}

Q.5 Let S=[0, 1) ∪[2, 3] and f ∶ S → \mathbb{R} be a strictly increasing function such that *f(S)* is connected. Which of the following statements is TRUE?

(A) *f* has exactly one discontinuity

(B) *f* has exactly two discontinuities

(C)* f* has infinitely many discontinuities

(D)* f* is continuous

Ans. (D)* f* is continuous

Q.6 Let a_{1}= 1 and a_{n}=a_{n-1}+4, n≥2. Then \lim_{n\rightarrow\infty}\left[\frac1{a_1a_2}+\frac1{a_2a_3}+....+\frac1{a_{n-1}a_n}\right] is equal to ____________________.

Ans. 0.24 : 0.26

Q.7 Maximum {x+y ∶ (x,y) ∈ \overline{B\left(0,1\right)}} is equal to _________.

Ans. 1.39 : 1.43

Q.8 Let a,b,c,d ∈ \mathbb{R} such that c^{2}+d^{2}≠0. Then, the Cauchy problem *au _{x}+bu_{y}=e^{x+y}, x,y∈ \mathbb{R} *u(x,y)= 0 on cx+dy= 0 has a unique solution if

(A) ac+bd ≠0

(B) ad-bc ≠0

(C) ac-bd ≠0

(D) ad+bc ≠0

Ans. (A) ac+bd ≠0

Q.9 Let u(x,t) be the d’Alembert’s solution of the initial value problem for the wave equation

u_{tt}-c^{2}u_{xx}= 0

u(x,0)=f(x),u_{t}(x,0)

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

Ans. -0.1 : 0.1

Q.10 Let the probability density function of a random variable X be

f(x)\;=\;\left\{\begin{array}{l}x\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0\leq x\leq\frac12\\c{(2x-1)}^2\;\;\;\;\;\;\frac12\leq x\leq1\\0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{array}\right.Then, the value of c is equal to ________________________

Ans. 5.2 : 5.3

Q.11 Let V be the set of all solutions of the equation y′′ +ay′+by= 0 satisfying y(0)=y(1) , where a, b are positive real numbers. Then, dimension(V ) is equal to _____________________

Ans. 0.9 : 1.1

Q.12 Let y′′+*p(x)*y′+q(x)y = 0, x ∈ (-∞ , ∞), where p(x) and q(x) are continuous functions. If y_{1}(x) =sinx-2cosx and y_{2}(x)= 2 sin(x)+cos(x) are two linearly independent solutions of the above equation, then

| 4 *p*(0)+ 2 *q*(1) | is equal to ____________________

Ans. 1.95 : 2.05

Q.13 Let P_{n}(x) be the Legendre polynomial of degree n and I = \int_{-1}^1x^k\;P_n(x)dx , 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?

(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Ans. (A) both P and Q

Q.14 Consider the following statements P and Q:

(P) : x^{2}y′′ + xy′ + \left(x^2-\frac14\right)y=0

has two linearly independent Frobenius series solutions near x=0.

(Q) : x^{2}y′′ + 3 sin(x)y′+y= 0 has two linearly independent Frobenius series solutions near x=0.

Which of the above statements hold TRUE?

(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Ans. (B) only P

Q.15 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 ______________________

Ans. 0.22 : 0.28

Q.16 Let (z_{n}) be a sequence of distinct points in D(0,1) {z ∈ \mathbb{C} ∶ |z| < 1 with

\lim_{n\rightarrow\infty}\;z_n=0 . 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 (z_{n}) = 0 if n is even and f (z_{n}) = 1 if n is odd.

Which of the above statements hold TRUE?

(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Ans. (B) only P

Q.17 Let (\left(\mathbb{R},\tau\right)) be a topological space with the cofinite topology. Every infinite subset of \mathbb{R} is

(A) Compact but NOT connected

(B) Both compact and connected

(C) NOT compact but connected

(D) Neither compact nor connected

Ans. (B) Both compact and connected

Q.18 Let c_{o}={(x_{n}) ∶ x_{n} ∈ \mathbb{R}, x_{n} → 0 } and M={(x_{n} ∈ c_{o} ∶ x_{1}+x_{2}+ ⋯ +x_{10=}0} . Then, dimension(c_{o}/M) is equal to ______________________.

Ans. 0.9 : 1.1

Q.19 Consider ( \mathbb{R}^2,\vert\vert.\vert\vert_\infty) where ‖(x,y)‖_{∞} = maximum{|x|, |y|}. Let *f* ∶ \mathbb{R}^2 → \mathbb{R} be defined by f(x,y)= \frac{x+y}2\;and\;\widetilde f the norm preserving linear extension of *f* to

( \mathbb{R}^3,\vert\vert.\vert\vert_\infty). Then, \widetilde f(1,1,1) is equal to ___________.

Ans. 0.9 : 1.1

Q.20 *f* ∶ [0,1]→ [0,1] is called a shrinking map if *|f(x)-f(y)*| <M |*x-y*| for all *x, y* ∈ [0,1] and a contraction if there exists an α< 1 such that |*f(x)-f(y)*| ≤ α|*x-y*| for all *x, y* ∈ [0,1].

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

(A) *f* is both a shrinking map and a contraction

(B) *f* is a shrinking map but NOT a contraction

(C) *f* is NOT a shrinking map but a contraction

(D) *f* is Neither a shrinking map nor a contraction

Ans. B ; D

Q.21 Let M be the set of all n×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 M is connected.

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

Which of the above statements hold TRUE?

(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Ans. (B) only P

Q.22 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,

f(x;\;\mu,\alpha)=\left\{\begin{array}{l}\frac1{\Gamma(\alpha)}{(x-\mu)}^{\alpha-1}\\0\;\;otherwise\end{array}e^{-(x-\mu)}\right.;\;x>\mu

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

(A) Maximum likelihood estimator of only μ exists

(B) Maximum likelihood estimator of only α exists

(C) Maximum likelihood estimators of both μ and α exist

(D) Maximum likelihood estimator of Neither μ nor α exists

Ans. (D) Maximum likelihood estimator of Neither μ nor α exists

Q.23 Suppose X and Y are two random variables such that aX+bY is a normal random variable for all a, b ∈ \mathbb{R} . 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?

(A) both P and Q

(B) both Q and R

(C) both Q and S

(D) both P and S

Ans. (B) both Q and R

Q.24 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.

(Q) : If Q={±1,±i,±j,±k} is the quaternion group, then Q/{1,1} is abelian.

Which of the above statements hold TRUE?

(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Ans. (C) only Q

Q.25 Let F be a field of order 32. Then the number of non‐zero solutions (a, b) ∈ F×F of the equation x^{2}+xy+y^{2}= 0 is equal to ________________________.

Ans. -0.1 : 0.1

**Q. 26 – Q. 55 carry two marks each.**

Q.26 Let ϒ={z ∈ \mathbb{C} ∶ |z| = 2} be oriented in the counter‐clockwise direction. Let Then, the value of \frac1{2\pi i}\int_\gamma z^7\cos\left(\frac1{z^2}\right)dz is equal to ________________________.

Ans. 0.039 : 0.043

Q.27 Let *u(x,y)=x ^{3}+ax^{2}y+2y^{3}* be a harmonic function and v(x,y) its harmonic conjugate. If v(0,0)= 1, then |a+b+v(1,1)| is equal to _______________.

Ans. 9.9 : 10.1

Q.28 Let ϒ be the triangular path connecting the points (0,0), (2,2) and (0,2) in the counterclockwise direction in . Then I=\oint\limits_\gamma\sin\left(x^3\right)dx+6xydy is equal to _____________________

Ans. 15.9 : 16.1

Q.29 Let y be the solution of y′ +y =| x|, x ∈ \mathbb{R} y(-1)=0 Then y(1) is equal to

(A) \frac2e-\frac2{e^2}

(B) \frac2e-2e^2

(C) 2- \frac2e

(D) 2-2e

Ans. (A) \frac2e-\frac2{e^2}

Q.30 Let X be a random variable with the following cumulative distribution function:

F(x)=\left\{\begin{array}{l}0\;\;\;\;\;\;\;\;\;\;x<0\\x^2\;\;\;\;\;\;\;\;0\leq x\leq\frac12\\\frac34\;\;\;\;\;\;\;\;\frac12\leq x\leq1\\1\;\;\;\;\;\;\;\;\;\;\;x\geq1\end{array}\right.Then P\left(\frac14<X<1\right) is equal to ___________________

Ans. 0.65 : 0.71

Q.31 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

(A) \left(\sqrt2,0\right)

(B) \left(0,\sqrt2\right)

(C) (1,1)

(D) (-1,1)

Ans. (A) \left(\sqrt2,0\right)

Q.32 Let S(x)=a_{0}+ \sum_{n=1}^\infty\left(a_n\cos\left(nx\right)+b_n\sin\left(nx\right)\right) be the Fourier series of the 2π periodic function defined by f(x)=x^{2}+4sin(x)cos(x), -π≤x≤π. Then \left|\overset\infty{\underset{n=0}{\sum a_n}}-\overset\infty{\underset{n=1}{\sum b_n}}\right|

is equal to _____________.

Ans. 1.9 : 2.1

Q.33 Let y(t) be a continuous function on [0, ∞). If y\left(t\right)=t\left(14\int_0^ty(x)dx\right)+4\int_0^txy(x)dx,\;then\;\int_0^\frac{\mathrm\pi}2y(t)dt is equal to_______.

Ans.0.45 : 0.55

Q.34 Let S_n={\textstyle\sum_{k=1}^n}\frac1k\;and\;I_n=\int_1^n\frac{x-\lbrack x\rbrack}{x^2}dx . Then S_{10}+I_{10} is equal to

(A) ln 10+1

(B) ln 10-1

(C) \ln\left(10\right)-\frac1{10}

(D) ln 10+ \frac1{10}

Ans. (A) ln 10+1

Q.35 For any (x, y) ∈ \mathbb{R}^2\ \overline{B\left(0,1\right)}, letf(x,y)=distance ((x,y, \overline{B\left(0,1\right)})

= infimum { \sqrt{\left(x-x_1\right)^2+(y-y_1})^2: (x_{1},y_{1}) ∈ \overline{B\left(0,1\right)}} Then, ||∇f(3,4)|| is equal to _________.

Ans. 0.9 : 1.1

Q.36 Let f(x)=\left(\int_0^xe^{-t^2}dt\right)^2\;and\;g(x)=\;\int_0^1\frac{e^{-x^2(1+t^2)}}{1+t^2}dt. Then f'(√π)+g'(√π) is equal to __________.

Ans. -0.1 : 0.1

Q.37 Let \begin{bmatrix}a&b&c\\b&d&e\\c&e&f\end{bmatrix} 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 ______________.

Ans. 6.9 : 7.1

Q.38 Let

M=\begin{bmatrix}1&1&0\\0&1&1\\0&0&1\end{bmatrix}[latex] and e<sup>M</sup>=[latex] Id+M+\frac1{2!}M^2+\frac1{3!}M^3+..... If e^{M}=[b_{ij}] then, \frac1e\sum_{i=1}^3\sum_{j=1}^3b_{ij} is equal to ___________________.

Ans. 5.4 : 5.6

Q.39 Let the integral I=\int_0^4f(x)dx where f(x)=\left\{\begin{array}{lc}x&0\leq x\leq2\\4-x&2\leq x\leq4\end{array}\right.

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?

(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Ans. (B) only P

Q.40 The difference between the least two eigenvalues of the boundary value problem

y′′ +λy= 0, 0 <x<π,

y(0)=0, y′(π) 0,

is equal to _________.

Ans. 1.9 : 2.1

Q.41 The number of roots of the equation x^{2}-cos(x)= 0 in the interval \left[-\frac{\mathrm\pi}2,\frac{\mathrm\pi}2\right] is equal to __________.

Ans. 1.9 : 2.1

Q.42 For the fixed point iteration x_{k+1}=g(x_{k}), k=0,1, 2, … …, consider the following statements P and Q:

(P) : If g(x)=1+ \frac2x then the fixed point iteration converges to 2 for all x_{0} ∈ [1, 100].

(Q) : If g(x)= \frac2x then the fixed point iteration converges to 2 for all x_{0} ∈ [0, 100].

Which of the above statements hold TRUE?

(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Ans. (A) both P and Q

Q.43. Let T:l_{2}→l_{2} be defined by T((x_{1}+x_{2},...,x_{n},...))=(x_{2}-x_{1},x_{3}-x_{2},...,x_{n+1}-x_{n},... ).

Then

(A) ‖T‖ = 1

(B) ‖T‖ > 2 but bounded

(C) 1 < ‖T‖ ≤ 2

(D) ‖T‖ is unbounded

Ans. (C) 1 < ‖T‖ ≤ 2

Q.44 Minimize ω=x+ 2y subject to

2x+y ≥ 3

x+y ≥ 2

x ≥ 0, y ≥ 0.

Then, the minimum value of ω is equal to ____________________.

Ans. 1.9 : 2.1

Q.45 Maximize ω=11x-z subject to

10x+y-z≤ 1

2x- 2y +z ≤ 2

x, y, z ≥ 0.

Then, the maximum value of ω is equal to ______________________.

Ans. 1.2 : 1.3

Q.46 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)= \frac1{2^k} ; k= 1,2,3, … and it is independent of the X_{i}'s, then E(X_{1}+X_{2}+...+ X_{N}) is equal to __________.

Ans. 1.9 : 2.1

Q.47 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 ___________________.

Ans. 0.7 : 0.8

Q.48 Let X_{1}, X_{2}, X_{3}, … be a sequence of i.i.d. uniform (0,1) random variables. Then, the value of \lim_{n\rightarrow\infty}P(-\ln\left(1-X_1\right)-...-\ln\left(1-X_{\;n}\right)\geq n) is equal to ______.

Ans. 0.4 : 0.6

Q.49 Let X be a standard normal random variable. Then, P(X<0)||[X]| = 1) is equal to

(A) \frac{\phi(1)-{\displaystyle\frac12}}{\phi(2)-{\displaystyle\frac12}}

(B) \frac{\phi(1)+{\displaystyle\frac12}}{\phi(2)+{\displaystyle\frac12}}

(C) \frac{\phi(1)-{\displaystyle\frac12}}{\phi(2)+{\displaystyle\frac12}}

(D) \frac{\phi(1)+1}{\phi(2)+1}

Ans. (A) \frac{\phi(1)-{\displaystyle\frac12}}{\phi(2)-{\displaystyle\frac12}}

Q.50 Let X_{1}, X_{2}, X_{3}, …,X_{n} be a be a random sample from the probability density function f(x)=\;\left\{\begin{array}{l}\theta\;\alpha\;e^{-ax}+(1-\theta)2\alpha\;e^{-2ax};\;x\geq0\\0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;;\;otherwise,\end{array}\right.

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

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

Which of the following statements is TRUE?

(A) Uniformly Most Powerful test does NOT exist

(B) Uniformly Most Powerful test is of the form, {\textstyle\sum_{i=1}^n}X_i>c for some 0<c< ∞

(C) Uniformly Most Powerful test is of the form {\textstyle\sum_{i=1}^n}X_i<c for some 0<c< ∞

(D) Uniformly Most Powerful test is of the form c_{1}< {\textstyle\sum_{i=1}^n}X_i<c_2, for some 0<c_{1}<c_{2}< ∞

Ans. (C) Uniformly Most Powerful test is of the form {\textstyle\sum_{i=1}^n}X_i<c for some 0<c< ∞

Q.51 Let X_{1}, X_{2}, X_{3}, …, be a sequence of i.i.d. N(μ,1) random variables. Then, \lim_{n\rightarrow\infty}\frac{\sqrt{\mathrm\pi}}{2n}\sum_{i=1}^nE\left(\left|X_i-\mu\right|\right) is equal to ______________________.

Ans. 0.69 : 0.73

Q.52 Let X_{1}, X_{2}, X_{3}, …,X_{n} be a random sample from uniform [1,θ], for some θ> 1. If X_{(n)}= Maximum (X_{1}, X_{2}, X_{3}, …,X_{n} ), then the UMVUE of θ is

(A) \frac{n+1}nX_{(n)}+\frac1n

(B) \frac{n+1}nX_{(n)}-\frac1n

(C) \frac{n}nX_{(n+1)}+\frac1n

(D) \frac n{n+1}X_{(n)}+\frac{n+1}n

Ans. (B) \frac{n+1}nX_{(n)}-\frac1n

Q.53 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 θ ∈ {1, 2}. Then, the maximum likelihood estimator of θ is equal to _____________.

Ans. 1.9 : 2.1

Q.54 The remainder when 98! is divided by 101 is equal to _________________.

Ans. 49.9 : 50.1

Q.55 Let G be a group whose presentation is G=\{x,y\vert x^5=y^2=e,\;\;\;\;x^2y=yx\} Then G is isomorphic to

(A) {\mathbb{Z}}_5

(B) {\mathbb{Z}}_10

(C) {\mathbb{Z}}_2

(D) {\mathbb{Z}}_30

Ans. (C) {\mathbb{Z}}_2

**END OF THE QUESTION PAPER**

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