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QUESTION: 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

Solution:

QUESTION: 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

Solution:

QUESTION: 3

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

Solution:

QUESTION: 4

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

Solution:

QUESTION: 5

The value of the expression

Solution:

QUESTION: 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?

Solution:

QUESTION: 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?

Solution:

*Multiple options can be correct

QUESTION: 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?

Solution:

QUESTION: 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?

Solution:

QUESTION: 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?

Solution:

QUESTION: 11

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

Solution:

QUESTION: 12

Let be an entire function with f (0) = 1, f (1) = 2 and f'(0) = 0. If there exists M > 0 such that

Solution:

QUESTION: 13

In the Laurent series expansion of valid for ; the coefﬁcient of is

Solution:

QUESTION: 14

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?

Solution:

QUESTION: 15

The general solution of the differential equation

is given by (with an arbitrary positive constant k)

Solution:

QUESTION: 16

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

with p_{n}(1) = 1 for n = 1; 2; 3; ....If

then n is

Solution:

QUESTION: 17

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

Solution:

QUESTION: 18

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

Solution:

QUESTION: 19

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

Solution:

QUESTION: 20

The Cauchy problem when solved using its characteristic equations with an independent variable t; is found to admit of a solution in the form

Then f (s; t) =

Solution:

QUESTION: 21

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

Solution:

QUESTION: 22

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

Solution:

QUESTION: 23

Let , where a, b, c, f are real numbers and f ≠ 0. The geometric multiplicity of the largest eigenvalue of A equals .

Solution:

QUESTION: 24

Consider the subspaces

of

Then the dimension of W_{1 }+ W_{2} equals .

Solution:

QUESTION: 25

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 ____________.

Solution:

QUESTION: 26

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 Then the Lebesgue measure of the set I ∪ J ∪ L equals .

Solution:

QUESTION: 27

Let Then the directional derivative of u in the direction at the point (5, 1, 0) is .

Solution:

QUESTION: 28

If the Laplace transform of y(t) is given by Y (s) = L(y(t)) =

then y(0) + y'(0) = .

Solution:

QUESTION: 29

The number of regular singular points of the differential equation in the interval is equal to .

Solution:

QUESTION: 30

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 .

Solution:

QUESTION: 31

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

Then the dimension of the range space of T equals _____________.

Solution:

QUESTION: 32

Let a ∈( 1, 1) be such that the quadrature rule

is exact for all polynomials of degree less than or equal to 3. Then 3a^{2} = .

Solution:

QUESTION: 33

Let X and Y have joint probability density function given by

otherwise. If fY denotes the marginal probability density function of Y , then fY (1/2) = ___________-.

Solution:

QUESTION: 34

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

Then

Solution:

QUESTION: 35

Let {X_{j }} be a sequence of independent Bernoulli random variables with and let

Then Y_{n} converges, in probability, to .

Solution:

QUESTION: 36

Let Γ be the circle given by , where θ varies from 0 to 2π. Then

Solution:

QUESTION: 37

The image of the half plane Re(z) + Im(z) > 0 under the map is given by

Solution:

QUESTION: 38

Let denote the closed disc with center at the origin and radius 2. Then

Solution:

QUESTION: 39

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

Solution:

QUESTION: 40

Which one of the following statements is true?

Solution:

QUESTION: 41

For an odd prime p, consider the ring Then the element is

Solution:

QUESTION: 42

Consider the following two statements:

P: The matrix has inﬁnitely many LU factorizations, where L is lower triangularwith each diagonal entry 1 and U is upper triangular.

Q: The matrixhas no LU factorization, where L is lower triangular with each diag-onal entry 1 and U is upper triangular.

Then which one of the following options is correct?

Solution:

QUESTION: 43

If the characteristic curves of the partial differential equation are where c_{1} and c_{2} are constants, then

Solution:

QUESTION: 44

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 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?

Solution:

QUESTION: 45

Let X denote R^{2} end owed with the usual topology. LetY denote Rendowed with the co-ﬁnite topology. If Z is the product topological space Y × Y, then

Solution:

QUESTION: 46

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?

Solution:

QUESTION: 47

Let {X_{i}} be a sequence of independent Poisson (λ) variables and let Then the limiting distribution ofis the normal distribution with zero mean andvariance given by

Solution:

QUESTION: 48

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

Also, let Then the maximum likelihood estimator of θ is

Solution:

QUESTION: 49

Consider the Linear Programming Problem (LPP):

Maximize αx_{1} + x_{2}

Subject to

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

Solution:

QUESTION: 50

Let be the vector space of all 2 × 2 real matrices over the ﬁeld R. Deﬁne the linear transformation , where X^{T} denotes the transpose of the matrix X. Then the trace of S equals .

Solution:

QUESTION: 51

Consider 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 , then 3d^{2} = .

Solution:

QUESTION: 52

Consider the matrix 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

Solution:

QUESTION: 53

Let and let Γ be the circle where θ varies from 0 to 4π. Then

Solution:

QUESTION: 54

Let S be the surface of the solid

Let denote the unit outward normal to S and let

Then the surface integral equals

Solution:

QUESTION: 55

Let A be a 3×3 matrix with real entries. If three solutions of the linear system of differential equations are given by

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

Solution:

QUESTION: 56

If is a solution of the differential equation

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

Solution:

QUESTION: 57

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 φ be the linear functional on L^{2}([0,1]) deﬁned by

where µ denotes the Lebesgue measure on

Solution:

QUESTION: 58

Let U be an orthonormal set in a Hilbert space H and let x ∈ H be such that ||x|| = 2. Consider the set

Then the maximum possible number of elements in E is .

Solution:

QUESTION: 59

If inter polates the points (x,y) in the table

then α + β = _____________.

Solution:

QUESTION: 60

If

Solution:

QUESTION: 61

For n = 1,2,..., let

Solution:

QUESTION: 62

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 ______________.

Solution:

QUESTION: 63

Let X be the number of head sin4 tosses of a fair coin by Person1 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 _______.

Solution:

QUESTION: 64

Let X_{1} and X_{2} be independent geometric random variables with the same probability mass function given by Then the value of correct up to three decimal places is

Solution:

QUESTION: 65

Acertain commodityisproduced bythemanufacturing plants P_{1} and P_{2} whosecapacitiesare 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 ______________

Solution:

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