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MA Mathematics - 2015 GATE Paper (Practice Test) - Question 1

**Q 1- 5 Carry one mark each.**

**Q.**

Choose the appropriate word/ pharase , out of the four options given below, to complete the following sentence:

apparent lifelessness ___________________ dormant life.

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 2

Fill in te blank with the correct idiom/ pharase.

That boy from the worn was a ___________________ in the sleepy village.

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 3

Choose the statement where underlined word is used correctly.

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 4

Tanya is older than Eric.

Cliff is polder than Tanya

Eric is older than cliff.

If the first two statements are true, then the third statement is :

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 5

Five teams have to compete in a league, with every team playing every other team exactly once, before going to the next round. How many matches will have to be held to completer the league round of matches?

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 6

**Q No 6 - 10 carry Two marks each**

**Q.**

Select the appropriate option in place of underlined part of the sentnce.

__" Increased productivity necessary "__ reflects greater efforts made by the employee.

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 7

Given below are two statements followed by two conclusions. Assuming these statements to be true, decide which one logically follows.

Statements:

I. No manager is leader.

II All leaders are executives.

**Conclusions:**

I No manager is an executive

II No executive is a manager.

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 8

In the given figure angle is a right angle, PS:QS= 3:1, RT: QT=5:2 and PU: UR = 1:1. if area of triangle QTS is 20 cm^{2}. then the area of triangle PQR in cm^{2 }is ________________.

**(Important : you should answer only the numeric value)**

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 9

Right triangle PQR is to be constructed in the xy- plane so that the right angle is at P and line PR is parallel to the x-axis. The x and y coordinates of P,Q, and R are to be integers that satisfy the inequalities -4x5 and 6 y 16. How many diffrent triangles could be constructed with these properties ?

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 10

A coin is tossed thrice. Let X be the event that head occurs in each of the first two tosses. Let Y be the event that a tail occurs on the third toss. Let Z be the event that two tails occur in three tosses. Based on the above information, which one of the following statements is TRUE?

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 11

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

**Q.**

Let T : R^{4} → R^{4} be a linear map defined by

Then the rank of T is equal to _________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 12

Let M be a 3 x 3 matrix and suppose that 1, 2 and 3 are the eigenvalues of M. If

for some scalar α 0, then α is equal to ___________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 13

Let M be a 3 x 3 singular matrix and suppose that 2 and 3 are eigenvalues of M. Then the number

of linearly independent eigenvectors of M_{3} + 2 M +I_{3 }is equal to _________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 14

Let M be a 3 x 3 matrix such that and suppose that for

some . Then | α| is equal to _______

**(Important : you should answer only the numeric value)**

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 15

Let be defined by

Then the function *f* is

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 16

Consider the power series

The radius of convergence of the series is equal to __________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 17

Let C={zC : |z-i|=2} . then is equal to ____________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 18

Let is equal to ___________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 19

Let the random variable X have the distribution function

Then P(2X<4) is equal to ___________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 20

Let X be a random variable having the distribution function

Then E(X) is equal to _________

**(Important : you should answer only the numeric value)**

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 21

In an experiment, a fair die is rolled until two sixes are obtained in succession. The probability that

the experiment will end in the fifth trial is equal to

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 22

Let x_{1} = 2.2, x_{2} = 4.3, x_{3} = 3.1, x_{4} = 4.5, x_{5} = 1.1 and x_{6} = 5.7 be the observed values of a

random sample of size 6 from a U(θ - 1, θ + 4) distribution, where θ ∈ (0, ∞) is unknown. Then

a maximum likelihood estimate of θ is equal to

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 23

Let be the open unit disc in with boundary is the solution of the Dirichlet problem

then u(1/2,0) is equal to

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 24

Let c ∈ Z_{3} be such that is a field. Then c is equal to __________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 26

Let be defined by is equal to __________

**(Important : you should answer only the numeric value)**

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 27

Let _{1}be the usual topology on R. Let _{2} be the topology on R generated by

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 28

Let X be a connected topological space such that there exists a non-constant continuous function

f : X → R, where R is equipped with the usual topology. Let f(X) = { f(x): x ∈ X}. Then

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 29

Let d_{1} and d_{2} denote the usual metric and the discrete metric on R, respectively. Let be defined by f(x) = x, x ∈ R. Then

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 30

If the trapezoidal rule with single interval [0, 1] is exact for approximating the integral

then the value of c is equal to ________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 31

Suppose that the Newton-Raphson method is applied to the equation with an

initial approximation x_{o} sufficiently close to zero. Then, for the root x = 0, the order of convergence of the method is equal to _________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 32

The minimum possible order of a homogeneous linear ordinary differential equation with real constant coefficients having x^{2 }sin (x) as a solution is equal to _______

**(Important : you should answer only the numeric value)**

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 33

The Lagrangian of a system in terms of polar coordinates (r,θ) is given by

where m is the mass, g is the acceleration due to gravity and denotes the derivative of s with

respect to time. Then the equations of motion are

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 34

If y(x) satisfies the initial value problem

then y(2) is equal to __________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 35

It is known that Bessel functions J_{n}(x). for n 0, satisfy the identity

for all t > 0 and xR. The value of is equal to _________

**(Important : you should answer only the numeric value)**

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 36

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

**Q.**

Let X and Y be two random variables having the joint probability density function

Then the conditional probability is equal to

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 37

Let Ω = (0,1] be the sample space and let be a probability function defined by

Then P({1/2}) is equal to __________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 38

Let X_{1}, X_{2} and X_{3} be independent and identically distributed random variables with E(X_{1}) =0 and is defined through the conditional expectation

then is equal to __________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 39

Let X ∼ Poisson(λ), where λ> 0 is unknown. If δ(X) is the unbiased estimator of is equal to ___________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 40

Let X_{1,}, … , X_{n} be a random sample from N(μ,1) distribution, where For testing the null

hypothesis H_{o }:μ=0 against the alternative hypothesis consider the critical region

R

where c is some real constant. If the critical region R has size 0.025 and power 0.7054, then the

value of the sample size n is equal to ___________

**(Important : you should answer only the numeric value)**

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 41

Let X and Y be independently distributed central chi-squared random variables with degrees of

freedom m(3) and n(3) respectively. If E(X/Y) =3 and m+n =14 then E(Y/X ) is equal to

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 42

Let X_{1}, X_{2}, … be a sequence of independent and identically distributed random variables with for n= 1, 2,…,, then is equal to __________

**(Important : you should answer only the numeric value)**

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 43

Let be a solution of the initial value problem

Then *f(1)* is equal to

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 44

Let be the solution of the initial value problem

Then u(2,2) is equal to ________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 45

Let be a subspace of the Euclidean space R^{4}. Then the

square of the distance from the point (1,1,1,1) to the subspace W is equal to _______

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 46

Let T : R^{4} → R^{4 }be a linear map such that the null space of T is

and the rank of (T-4I_{4}) is 3. If the minimal polynomial of T is x(x-4)^{α}. then α is equal to _______

**(Important : you should answer only the numeric value)**

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 47

Let M be an invertible Hermitian matrix and let x,y R be such that x^{2 }<4y. then

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 48

Let Then the number

of elements in the center of the group G is equal to

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 49

The number of ring homomorphisms from is equal to __________

**(Important : you should answer only the numeric value)**

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 50

Let and be two polynomials in

Q [x]. Then, over Q,

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 51

Consider the linear programming problem

Then the maximum value of the objective function is equal to ______

**(Important : you should answer only the numeric value)**

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 52

Let . Under the usual metric on R^{2},

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 54

Let V be a closed subspace of be given by f(x) =x and g(x) = x^{2}. if ^{ } and pg is the orthogonal projection of g on V, then

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 55

Let p(x) be the polynomial of degree at most 3 that passes through the points (-2,12), (-1,1), (0.2) and (2,-8). Then the coefficient of x^{3 } in p(x) is equal to _____________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 56

If, for some α,β R the integration formula

holds for all polynomials p(x) of degree at most 3, then the value of

3(α-β)^{2 }is equal to _____.

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 57

Let y(t) be a continuous function on (0, ∞) whose Laplace transform exists. If y(t) satisfies

then y(1) is equal to _______

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 58

Consider the initial value problem

if y(x) → 0 as x→ 0^{+, }then α is equal to ______________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 60

Consider the unit sphere and the unit normal vector

at each point (x,y,z) on S. The value of the surface integral

is equal to _______

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 61

Let

Define Then the minimum value of f on D is equal to ________

**(Important : you should answer only the numeric value)**

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 62

Let Then there exists a non-constant analytic function f on D such that for all n = 2, 3, 4, …

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 63

Let be the Laurent series expansion of in the annulus 3/2 <|z| <5 . Then a_{1}/a_{2 }is equal to _________

**(Important : you should answer only the numeric value)**

*Answer can only contain numeric values

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 64

The value of is equal to __________

**(Important : you should answer only the numeric value)**

MA Mathematics - 2015 GATE Paper (Practice Test) - Question 65

Suppose that among all continuously differentiable functions y(x), x R with y(0) =0 and y(1) = 1/2, the function y_{0}(x) minimizes the functional

then y_{o }(1/2) is equal to

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