**Q. 1 – Q. 5 carry one mark each.**

Q.1 “The dress _________ her so well that they all immediately _________ her on her appearance. ”The words that best fill the blanks in the above sentence are

(A) complemented, complemented

(B) complimented, complemented

(C) complimented, complimented

(D) complemented, complimented

Ans :- (D)

Q.2 “The judge’s standing in the legal community, though shaken by false allegations of wrongdoing, remained _________. ”The word that best fills the blank in the above sentence is

(A) undiminished

(B) damaged

(C) illegal

(D) uncertain

Ans :- (A)

Q.3 Find the missing group of letters in the following series: BC, FGH, LMNO, _____

(A) UVWXY

(B) TUVWX

(C) STUVW

(D) RSTUV

Ans :- (B)

Q.4 The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the following statements is true?

(A) The circle has the largest area.

(B) The square has the largest area.

(C) The equilateral triangle has the largest area.

(D) All the three shapes have the same area.

Ans :- (A)

Q.5 The value of the expression \frac1{1+\log_u^{vw}}+\frac1{1+\log_v^{wu}}+\frac1{1+\log_w^{uv}} is _________.

(A) -1

(B) 0

(C) 1

(D) 3

Ans :- (C)

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

Q.6 Forty students watched films A, B and C over a week. Each student watched either only one film or all three. Thirteen students watched film A, sixteen students watched film B and nineteen students watched film C. How many students watched all three films?

(A) 0

(B) 2

(C) 4

(D) 8

Ans :- (C)

Q.7 A wire would enclose an area of 1936 m^{2}, if it is bent into a square. The wire is cut into two pieces. The longer piece is thrice as long as the shorter piece. The long and the short pieces are bent into a square and a circle, respectively. Which of the following choices is closest to the sum of the areas enclosed by the two pieces in square meters?

(A) 1096

(B) 1111

(C) 1243

(D) 2486

Ans :- (C)

Q.8 A contract is to be completed in 52 days and 125 identical robots were employed, each operational for 7 hours a day. After 39 days, five-seventh of the work was completed. How many additional robots would be required to complete the work on time, if each robot is now operational for 8 hours a day?

(A) 50

(B) 89

(C) 146

(D) 175

Ans :- Marks to All

Q.9 A house has a number which needs to be identified. The following three statements are given that can help in identifying the house number.

i. If the house number is a multiple of 3, then it is a number from 50 to 59.

ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69.

iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79.

What is the house number?

(A) 54

(B) 65

(C) 66

(D) 76

Ans :- (D)

Q.10 An unbiased coin is tossed six times in a row and four different such trials are conducted. One trial implies six tosses of the coin. If H stands for head and T stands for tail, the following are the observations from the four trials:

(1) HTHTHT

(2) TTHHHT

(3) HTTHHT

(4) HHHT__ __.

Which statement describing the last two coin tosses of the fourth trial has the highest probability of being correct?

(A) Two T will occur.

(B) One H and one T will occur.

(C) Two H will occur.

(D) One H will be followed by one T.

Ans :- (B)

**Q.1 – Q.25 carry one mark each**

Q.1 The principal value of (-1)^{(-2i/π)} is

(A) e^{2 }

(B) e^{2i }

(C) e^{-2i }

(D) e^{-2}

Ans :- (A)

Q.2 Let f : C → C be an entire function with f(0) = 1, f(1) = 2 and f”(0) = 0. If there exists M > 0 such that $latex \left|f”\left(z\right)\right|$ ≤ M for all z ∈ C, then f(2) =

(A) 2

(B) 5

(C) 2 + 5i

(D) 5 + 2i

Ans :- (B)

Q.3 In the Laurent series expansion of f(z) = 1/z(z – 1) valid for $latex \left|z-1\right|$ > 1, the coefficient of 1/z – 1 is

(A) -2

(B) -1

(C) 0

(D) 1

Ans :- (C)

Q.4 Let X and Y be metric spaces, and let f : X → Y be a continuous map. For any subset S of X; which one of the following statements is true?

(A) If S is open, then f(S) is open

(B) If S is connected, then f(S) is connected

(C) If S is closed, then f(S) is closed

(D) If S is bounded, then f(S) is bounded

Ans :- (B)

Q.5 The general solution of the differential equation

xy’ = y + $latex \sqrt{x^2+y^2}$ for x > 0

is given by (with an arbitrary positive constant k)

(A) ky^{2} = x + $latex \sqrt{x^2+y^2}$

(B) kx^{2} = x + $latex \sqrt{x^2+y^2}$

(C) kx^{2} = y + $latex \sqrt{x^2+y^2}$

(D) ky^{2} = y + $latex \sqrt{x^2+y^2}$

Ans :- (C)

Q.6 Let p_{n}(x) be the polynomial solution of the differential equation

d/dx [(1 – x^{2})y’] + n(n + 1)y = 0

with p_{n}(1) = 1 for n = 1; 2; 3; ….. If d/dx[p_{n}+2(x) – p_{n}(x)] = α_{n}p_{n}+1(x), then α_{n} is

(A) 2_{n}

(B) 2_{n} + 1

(C) 2_{n} + 2

(D) 2_{n} + 3

Ans :- (D)

Q.7 In the permutation group S_{6}; the number of elements of order 8 is

(A) 0

(B) 1

(C) 2

(D) 4

Ans :- (A)

Q.8 Let R be a commutative ring with 1 (unity) which is not a field. Let I ⊂ R be a proper ideal such that every element of R not in I is invertible in R. Then the number of maximal ideals of R is

(A) 1

(B) 2

(C) 3

(D) infinite

Ans :- (A)

Q.9 Let f : R → R be a twice continuously differentiable function. The order of convergence of the secant method for finding root of the equation f(x) = 0 is

(A) $latex \frac{1+\sqrt5}2$

(B) $latex \frac2{1+\sqrt5}$

(C) $latex \frac{1+\sqrt5}3$

(D) $latex \frac3{1+\sqrt5}$

Ans :- (A)

Q.10 The Cauchy problem uu_{x} + yu_{y} = x with u(x, 1) = 2_{x}; when solved using its characteristic equations with an independent variable t, is found to admit of a solution in the form

x = 3/2 se^{t} – 1/2 se^{-t }, y = e^{t }, u = f (s; t).

Then f (s, t) =

(A) 3/2 se^{t} + 1/2 se^{-t}

(B) 1/2 se^{t} + 3/2 se^{-t}

(C) 1/2 se^{t} – 3/2 se^{-t}

(D) 3/2 se^{t} – 1/2 se^{-t}

Ans :- (A)

Q.11 An urn contains four balls, each ball having equal probability of being white or black. Three black balls are added to the urn. The probability that five balls in the urn are black is

(A) 2/7

(B) 3/8

(C) 1/2

(D) 5/7

Ans :- (B)

Q.12 For a linear programming problem, which one of the following statements is FALSE?

(A) If a constraint is an equality, then the corresponding dual variable is unrestricted in sign

(B) Both primal and its dual can be infeasible

(C) If primal is unbounded, then its dual is infeasible

(D) Even if both primal and dual are feasible, the optimal values of the primal and the dual

can differ

Ans :- (D)

Q.13 Let $latex A=\begin{bmatrix}a&2f&0\\2f&b&3f\\0&3f&c\end{bmatrix}\;,$ where a, b, c, f are real numbers and f ≠ 0: The geometric multiplicity of the largest eigenvalue of A equals ______ .

Ans :- 1 to 1

Q.14 Consider the subspaces

W_{1} = f{(x_{1,} x_{2,} x_{3}) ∈ R^{3} : x_{1} = x_{2} + 2x_{3}}

W_{2} = f{(x_{1}, x_{2}, x_{3}) ∈ R^{3} : x_{1} = 3x_{2} + 2x_{3}}

of R^{3}. Then the dimension of W_{1} + W_{2} equals ____.

Ans :- 3 to 3

Q.15 Let V be the real vector space of all polynomials of degree less than or equal to 2 with real coefficients. Let T : V → V be the linear transformation given by

T(p) = 2p + p’ for p ∈ V,

where p’ is the derivative of p. Then the number of nonzero entries in the Jordan canonical form of a matrix of T equals_______.

Ans :- 5 to 5

Q.16 Let I = [2, 3), J be the set of all rational numbers in the interval [4, 6], K be the Cantor (ternary) set, and let L = {7 + x : x ∈ K}. Then the Lebesgue measure of the set I ∪ J ∪ L equals______.

Ans :- 1 to 1

Q.17 Let u(x, y, z) = x^{2} – 2_{y} + 4z^{2} for (x, y, z) 2 R^{3}. Then the directional derivative of u in the direction $latex \frac35\widehat i-\frac45\widehat k$ at the point (5, 1, 0) is _______.

Ans :- 6 to 6

Q.18 If the Laplace transform of y(t) is given by Y (s) = L(y(t)) = 5/2(s – 1) – 2/s – 2 + 1/2(s – 3), then y(0) + y'(0) = _____.

Ans :- 1 to 1

Q.19 The number of regular singular points of the differential equation

[(x – 1)^{2} sin x]y” + [cos x sin(x – 1)]y’ + (x – 1)y = 0

in the interval [0, π/2] is equal to _____.

Ans :- 2 to 2

Q.20 Let F be a field with 7^{6} elements and let K be a subfield of F with 49 elements. Then the dimension of F as a vector space over K is _____.

Ans :- 3 to 3

Q.21 Let C([0, 1]) be the real vector space of all continuous real valued functions on [0, 1], and let T be the linear operator on C([0, 1]) given by

(Tf)(x) = $latex \int_0^1$ sin(x + y)f(y) dy, x ∈ [0, 1]. Then the dimension of the range space of T equals _______.

Ans :- 2 to 2

Q.22 Let a ∈ (-1, 1) be such that the quadrature rule

$latex \int_{-1}^1f\left(x\right)dx\simeq$ f(-a) + f(a) is exact for all polynomials of degree less than or equal to 3. Then 3a^{2} = ______.

Ans :- 1 to 1

Q.23 Let X and Y have joint probability density function given by

fX,Y (x,y) = $latex \left\{\begin{array}{l}2,\;0\leq x\leq1-y,\;0\leq y\leq1\\0,\;otherwise.\end{array}\right.$

If f_{Y} denotes the marginal probability density function of Y , then f_{Y} (1/2) = ______.

Ans :- 1 to 1

Q.24 Let the cumulative distribution function of the random variable X be given by

Then P(X = 1/2) = ______.

Ans :- 0.25 to 0.25

Q.25 Let {Xj} be a sequence of independent Bernoulli random variables with P(X_{j} = 1) = 1/4 and let $latex Yn=\frac1n{\textstyle\sum_{j=1}^n}\;X_j^2.$ Then Y_{n} converges, in probability, to _______.

Ans :- 0.25 to 0.25

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

Q.26 Let Γ be the circle given by z = 4e^{iθ}; where θ varies from 0 to 2π. Then

(A) 2πi(e^{2} – 1)

(B) πi(1 – e^{2})

(C) πi(e^{2} – 1)

(D) 2πi(1 – e^{2})

Ans :- (C)

Q.27 The image of the half plane Re(z) + Im(z) > 0 under the map w = z – 1/z + i is given by

(A) Re(w) > 0

(B) Im(w) > 0

(C) $latex \left|\omega\right|$ > 1

(D) $latex \left|\omega\right|$ < 1

Ans :- (D)

Q.28 Let D ⊂ R^{2} denote the closed disc with center at the origin and radius 2. Then

(A) π(1 – e^{-4})

(B) π/2(1 – e^{-4})

(C) π(1 – e^{-2})

(D) π/2(1 – e^{-2})

Ans :- (A)

Q.29 Consider the polynomial p(X) = X^{4} + 4 in the ring Q[X] of polynomials in the variable X with coefficients in the field Q of rational numbers. Then

(A) the set of zeros of p(X) in C forms a group under multiplication

(B) p(X) is reducible in the ring Q[X]

(C) the splitting field of p(X) has degree 3 over Q

(D) the splitting field of p(X) has degree 4 over Q

Ans :- (B)

Q.30 Which one of the following statements is true?

(A) Every group of order 12 has a non-trivial proper normal subgroup

(B) Some group of order 12 does not have a non-trivial proper normal subgroup

(C) Every group of order 12 has a subgroup of order 6

(D) Every group of order 12 has an element of order 12

Ans :- (A)

Q.31 For an odd prime p, consider the ring

Then the element 2 in $latex Z\left[\sqrt{-p}\right]$ is

(A) a unit

(B) a square

(C) a prime

(D) irreducible

Ans :- (D)

Q.32 Consider the following two statements:

P: The matrix $latex \begin{bmatrix}0&5\\0&7\end{bmatrix}$ has infinitely many LU factorizations, where L is lower triangular with each diagonal entry 1 and U is upper triangular.

Q: The matrix $latex \begin{bmatrix}0&0\\2&5\end{bmatrix}$ has no LU factorization, where L is lower triangular with each diagonal entry 1 and U is upper triangular.

Then which one of the following options is correct?

(A) P is TRUE and Q is FALSE

(B) Both P and Q are TRUE

(C) P is FALSE and Q is TRUE

(D) Both P and Q are FALSE

Ans :- (B)

Q.33 If the characteristic curves of the partial differential equation xu_{xx} + 2x^{2}u_{xy} = u_{x} – 1 are

μ(x, y) = c_{1} and ν(x, y) = c_{2}, where c_{1} and c_{2} are constants, then

(A) μ(x, y) = x^{2} – y, ν(x, y) = y

(B) μ(x, y) = x^{2} + y, ν(x, y) = y

(C) μ(x, y) = x^{2} + y, ν(x, y) = x^{2}

(D) μ(x, y) = x^{2} – y, ν(x, y) = x^{2}

Ans :- (A)

Q.34 Let f : X → Y be a continuous map from a Hausdorff topological space X to a metric space Y . Consider the following two statements:

P: f is a closed map and the inverse image f^{-1}(y) = {x ∈ X : f(x) = y} is compact for

each y ∈ Y:

Q: For every compact subset K ⊂ Y; the inverse image f^{-1}(K) is a compact subset of X:

Which one of the following is true?

(A) Q implies P but P does NOT imply Q

(B) P implies Q but Q does NOT imply P

(C) P and Q are equivalent

(D) neither P implies Q nor Q implies P

Ans :- (C)

Q.35 Let X denote R^{2} endowed with the usual topology. Let Y denote R endowed with the co-finite topology. If Z is the product topological space Y × Y, then

(A) the topology of X is the same as the topology of Z

(B) the topology of X is strictly coarser (weaker) than that of Z

(C) the topology of Z is strictly coarser (weaker) than that of X

(D) the topology of X cannot be compared with that of Z

Ans :- (C)

Q.36 Consider R^{n} with the usual topology for n = 1, 2, 3. Each of the following options gives topological spaces X and Y with respective induced topologies. In which option is X homeomorphic to Y ?

(A) X = {(x, y, z) ∈ R^{3} : x^{2} + y^{2} = 1}, Y = {(x, y, z) ∈ R^{3} : z = 0, x^{2} + y^{2} ≠ 0}

(B) X = {(x, y) ∈ R^{2} : y = sin(1/x), 0 < x ≤ 1}∪{(x, y) ∈ R^{2} : x = 0,-1 ≤ y ≤ 1}, Y = [0, 1] ⊆ R

(C) X = {(x, y) ∈ R^{2} : y = x sin(1/x), 0 < x ≤ 1}, Y = [0, 1] ⊆ R

(D) X = {(x, y, z) ∈ R^{3} : x^{2} + y^{2} = 1}, Y = {(x, y, z) ∈ R^{3} : x^{2} + y^{2} = z^{2} ≠ 0}

Ans :- (A)

Q.37 Let {X_{i}} be a sequence of independent Poisson(λ) variables and let $latex W_n=\frac1n{\textstyle\sum_{i=1}^n}\;X_i.$ Then the limiting distribution of $latex \sqrt n\left(W_n-\lambda\right)$ is the normal distribution with zero mean and variance given by

(A) 1

(B) $latex \sqrt\lambda$

(C) λ

(D) λ^{2}

Ans :- (C)

Q.38 Let X_{1}, X_{2}, …., X_{n} be independent and identically distributed random variables with probability density function given by

Also, let $latex \overline X=\frac1n{\textstyle\sum_{i=1}^n}\;X_i.$ Then the maximum likelihood estimator of θ is

(A) $latex 1/\overline X$

(B) $latex \left(1/\overline X\right)-1$

(C) $latex 1/\left(\overline X-1\right)$

(D) $latex \overline X$

Ans :- (C)

Q.39 Consider the Linear Programming Problem (LPP):

where α is a constant. If (3, 0) is the only optimal solution, then

(A) α < -2

(B) -2 < α < 1

(C) 1 < α < 2

(D) α > 2

Ans :- (D)

Q.40 Let M_{2}(R) be the vector space of all 2 × 2 real matrices over the field R. Define the linear transformation S : M_{2}(R) → M_{2}(R) by S(X) = 2X + X^{T} ,where X^{T} denotes the transpose of the matrix X. Then the trace of S equals_____.

Ans :- 10 to 10

Q.41 Consider R^{3} with the usual inner product. If d is the distance from (1, 1, 1) to the subspace span{(1, 1, 0), (0, 1, 1)} of R^{3}, then 3d^{2} = ________.

Ans :- 1 to 1

Q.42 Consider the matrix A = I_{9} – 2u^{T} u with u = 1/3[1, 1, 1, 1, 1, 1, 1, 1, 1], where I^{9} is the 9 × 9 identity matrix and u^{T} is the transpose of u. If λ and μ are two distinct eigenvalues of A, then $latex \left|\lambda-\mu\right|$ = ________.

Ans :- 2 to 2

Q.43 Let f(z) = z^{3}e^{z2} for z ∈ C and let Γ be the circle z = e^{iθ}, where θ varies from 0 to 4π. Then

Ans :- 6 to 6

Q.44 Let S be the surface of the solid

V = {(x, y, z) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, 0 ≤ z ≤ 3}.

Let $latex \widehat n$ denote the unit outward normal to S and let

Then the surface integral $latex \int\int_s\;\overrightarrow F\cdot\widehat n$ dS equals_______.

Ans :- 18 to 18

Q.45 Let A be a 3 × 3 matrix with real entries. If three solutions of the linear system of differential equations $latex \dot x$ (t) = Ax(t) are given by

then the sum of the diagonal entries of A is equal to______.

Ans :- 2 to 2

Q.46 If y_{1}(x) = e^{-x2} is a solution of the differential equation

xy” + αy’ + βx^{3}y = 0

for some real numbers α and β, then = αβ = ______.

Ans :- 4 to 4

Q.47 Let L^{2}([0, 1]) be the Hilbert space of all real valued square integrable functions on [0, 1] with the usual inner product. Let ø be the linear functional on L^{2}([0, 1]) defined by

where μ denotes the Lebesgue measure on [0, 1]. Then $latex \parallel\phi\parallel$ = _________.

Ans :- 3 to 3

Q.48 Let U be an orthonormal set in a Hilbert space H and let x ∈ H be such that $latex \parallel x\parallel$ = 2. Consider the set

Then the maximum possible number of elements in E is

Ans :- 64 to 64

Q.49 If p(x) = 2 – (x+1)+x(x+1)βx(x+1)(x – α) interpolates the points (x, y) in the table

then α + β = _______.

Ans :- 3 to 3

Q.50 If sin(πx) = a_{0} + $latex \textstyle\sum_{n=1}^\infty$ an cos(nπx) for 0 < x < 1, then (a_{0} + a_{1}) π = _______.

Ans :- 2 to 2

Q.51 For n = 1, 2, . . . , let f_{n}(x) = 2nx^{n-1}/1 + x, x ∈ [0, 1]. Then

Ans :- 1 to 1

Q.52 Let X_{1}, X_{2}, X_{3}, X_{4} be independent exponential random variables with mean 1, 1/2, 1/3, 1/4, respectively. Then Y = min(X_{1}, X_{2}, X_{3}, X_{4}) has exponential distribution with mean equal to _________.

Ans :- 0.1 to 0.1

Q.53 Let X be the number of heads in 4 tosses of a fair coin by Person 1 and let Y be the number of heads in 4 tosses of a fair coin by Person 2. Assume that all the tosses are independent. Then the value of P(X = Y ) correct up to three decimal places is______.

Ans :- 0.272 to 0.274

Q.54 Let X_{1} and X_{2} be independent geometric random variables with the same probability mass function given by P(X = k) = p(1 – p)^{k-1}, k = 1, 2, , , .. Then the value of P(X_{1} = 2 X_{1} + X2 = 4) correct up to three decimal places is_______.

Ans :- 0.332 to 0.334

Q.55 A certain commodity is produced by the manufacturing plants P_{1} and P_{2} whose capacities are 6 and 5 units, respectively. The commodity is shipped to markets M_{1}, M_{2}, M_{3} and M_{4} whose requirements are 1, 2, 3 and 5 units, respectively. The transportation cost per unit from plant P_{i} to market M_{j} is as follows:

Then the optimal cost of transportation is_______.

Ans :- 57 to 57

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