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# Mathematics - (MA) 2018 GATE Paper (Practice Test)

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Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 3

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 4

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 5

The value of the expression

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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?

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 7

A wire would enclose an area of 1936 m2, 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?

*Multiple options can be correct
Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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?

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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?

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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?

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 11

The principal value of (-1)(-2i/π) is

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 13

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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?

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 15

The general solution of the differential equation

is given by (with an arbitrary positive constant k)

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 16

Let pn(x) be the polynomial solution of the differential equation
with pn(1) = 1 for n = 1; 2; 3; ....If

then n is

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 17

In the permutation group S6; the number of elements of order 8 is

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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) =

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 22

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 23

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 24

Consider the subspaces

of

Then the dimension of W1 + W2 equals .

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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 ____________.

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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 .

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 27

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 28

If the Laplace transform of y(t) is given by Y (s) = L(y(t)) =
then y(0) + y'(0) = .

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 29

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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 .

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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 _____________.

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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 3a2 = .

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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) = ___________-.

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 34

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

Then

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 35

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

Then Yn converges, in probability, to .

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 36

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 37

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 38

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 39

Consider the polynomial p(X) = X4 + 4 in the ring Q[X] of polynomials in the variable X with coefﬁcients in the ﬁeld Q of rational numbers. Then

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 40

Which one of the following statements is true?

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 41

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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?

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 43

If the characteristic curves of the partial differential equation are  where c1 and c2 are constants, then

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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?

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 45

Let X denote R2 end owed with the usual topology. LetY denote Rendowed with the co-ﬁnite topology. If Z is the product topological space Y × Y, then

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 46

Consider Rn 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?

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 47

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 48

Let X1,X2,...,Xn be independent and identically distributed random variables with probability density function given by

Also, let  Then the maximum likelihood estimator of θ is

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 49

Consider the Linear Programming Problem (LPP):

Maximize αx1 + x2

Subject to

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 50

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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 3d2 = .

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 52

Consider the matrix  where I9 is the 9×9 identity matrix and uT is the transpose of u. If λ and µ are two distinct eigenvalues of A, then

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 53

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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 .

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 56

If is a solution of the differential equation

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 57

Let L2([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 L2([0,1]) deﬁned by

where µ denotes the Lebesgue measure on

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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 .

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 59

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

then α + β = _____________.

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 60

If

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 61

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

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 62

Let X1,X2,X3,X4 be independent exponential random variables with mean 1,1/2,1/3,1/4, respectively. Then Y = min(X1,X2,X3,X4) has exponential distribution with mean equal to ______________.

Mathematics - (MA) 2018 GATE Paper (Practice Test) - 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 _______.

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 64

Let X1 and X2 be independent geometric random variables with the same probability mass function given by Then the value of  correct up to three decimal places is

Mathematics - (MA) 2018 GATE Paper (Practice Test) - Question 65

Acertain commodityisproduced bythemanufacturing plants P1 and P2 whosecapacitiesare 6 and 5 units, respectively. The commodity is shipped to markets M1, M2, M3 and M4 whose requirements are 1, 2, 3 and 5 units, respectively. The transportation cost per unit from plant Pi to market Mj is as follows:

Then the optimal cost of transportation is ______________

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