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MA Mathematics - 2014 GATE Paper (Practice Test) - Question 1

**Q. 1 – Q. 5 carry one mark each.**

**Q.**

**A student is required to demonstrate a high level of comprehension of the subject, especially in the
social sciences.
The word closest in meaning to comprehension is**

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 2

Choose the most appropriate word from the options given below to complete the following

sentence.

One of his biggest ______ was his ability to forgive.

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 3

Rajan was not happy that Sajan decided to do the project on his own. On observing his

unhappiness, Sajan explained to Rajan that he preferred to work independently.

Which one of the statements below is logically valid and can be inferred from the above sentences?

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 4

If y = 5x^{2} + 3, then the tangent at x = 0, y = 3

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 5

A foundry has a fixed daily cost of Rs 50,000 whenever it operates and a variable cost of Rs 800Q,

where Q is the daily production in tonnes. What is the cost of production in Rs per tonne for a daily

production of 100 tonnes?

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

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 6

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

Q.

Find the odd one in the following group: ALRVX, EPVZB, ITZDF, OYEIK

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 7

Anuj, Bhola, Chandan, Dilip, Eswar and Faisal live on different floors in a six-storeyed building

(the ground floor is numbered 1, the floor above it 2, and so on). Anuj lives on an even-numbered

floor. Bhola does not live on an odd numbered floor. Chandan does not live on any of the floors

below Faisal’s floor. Dilip does not live on floor number 2. Eswar does not live on a floor

immediately above or immediately below Bhola. Faisal lives three floors above Dilip. Which of the

following floor-person combinations is correct?

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 8

The smallest angle of a triangle is equal to two thirds of the smallest angle of a quadrilateral. The

ratio between the angles of the quadrilateral is 3:4:5:6. The largest angle of the triangle is twice its

smallest angle. What is the sum, in degrees, of the second largest angle of the triangle and the

largest angle of the quadrilateral?

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

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 9

One percent of the people of country X are taller than 6 ft. Two percent of the people of country Y

are taller than 6 ft. There are thrice as many people in country X as in country Y. Taking both

countries together, what is the percentage of people taller than 6 ft?

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 10

The monthly rainfall chart based on 50 years of rainfall in Agra is shown in the following figure.

Which of the following are true? (k percentile is the value such that k percent of the data fall below

that value)

(i) On average, it rains more in July than in December

(ii) Every year, the amount of rainfall in August is more than that in January

(iii) July rainfall can be estimated with better confidence than February rainfall

(iv) In August, there is at least 500 mm of rainfall

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 11

**Q. 1 – Q. 25 carry one mark each.**

**Q.**

The function is differentiable at

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 12

The radius of convergence of the power series is _____________

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 13

Let E_{1} and E_{2} be two non empty subsets of a normed linear space X and let

Then which of the following statements is FALSE:

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 14

Let y(x) be the solution to the initial value problem subject to y(1.2) 2. Using the Euler method with the step size h = 0.05, the approximate value of ??(1.3), correct to two

decimal places, is _____________________

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 15

Let α ∈ R. If αx is the polynomial which interpolates the function f (x) = sinπ x on [−1,1]at all the zeroes of the polynomial 4x^{3 }− 3x , then α is ___________

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 16

If u(x,t) is the D’Alembert’s solution to the wave equation with

the condition u(x,0) = 0 is _________________

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 17

The solution to the integral equation

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 18

The general solution to the ordinary differential equation in

terms of Bessel’s functions, ??_{v}(x), is

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 19

The inverse Laplace transform of

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 20

If X_{1} , X_{2} is a random sample of size 2 from an *N *(0,1) population, then follows

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 21

Let be a random variable. Then the value of E[max{Z,0}] is

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 22

The number of non-isomorphic groups of order 10 is ___________

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 23

Let a,b,c,d be real numbers with a < c < d < b. Consider the ring C[a,b] with pointwise

addition and multiplication. If then

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 24

Let ?? be a ring. If R[x]is a principal ideal domain, then R is necessarily a

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 25

Consider the group homomorphism given by φ ( A) = trace(A) . The kernel of φ

is isomorphic to which of the following groups?

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 26

Let X be a set with at least two elements. Let τ and τ′ be two topologies on X such that Which of the following conditions is necessary for the identity function id : to be

continuous?

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 27

Let be such that det(A− I ) = 0 , where I denotes the 3×3 identity matrix. If the

trace(A) =13 and det(A) = 32, then the sum of squares of the eigenvalues of A is ______

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 28

Let V denote the vector space . Then

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 29

Let V be a real inner product space of dimension 10 . Let x, y∈V be non-zero vectors such that

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 30

Consider the following linear programming problem:

Minimize x_{1} + x_{2}

Subject to:

2x_{1} + x_{2} ≥ 8

2x_{1} + 5x_{2} ≥ 10

x_{1}, x_{2} ≥ 0

The optimal value to this problem is _________________________

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 31

Let

be a periodic function of period 2π. The coefficient of sin 3x in the Fourier series expansion of f(x) on the interval [−π, π] is ________________________

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 32

For the sequence of functions

consider the following quantities expressed in terms of Lebesgue integrals

Which of the following is TRUE?

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 33

Which of the following statements about the spaces is TRUE ?

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 35

Let d_{1}, d_{2} and d_{3} be metrics on a set X with at least two elements. Which of the following is NOT

a metric on X?

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 36

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

**Q.**

**Let ** and let ?? be a smooth curve lying in Ω with initial point −1 + 2??

and final point 1 + 2??. The value of

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 37

If ? ? C with a <1, then the value of

where Γ is the simple closed curve |??| = 1 taken with the positive orientation, is _________

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 38

Consider C[−1,1] equipped with the supremum norm given by Define a linear functional T on C[−1,1]by

Then the value of T is _______

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 39

Consider the vector space C[0,1] over R. Consider the following statements:

**P**: If the set { *tf _{1}, t^{2}f_{2}, t^{3}f_{3}*} is linearly independent, then the set {

independent, where

**Q: **If F: C[0,1] → R is given by then F is a linear map

Which of the above statements hold TRUE?

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 40

Using the Newton-Raphson method with the initial guess x^{(0)} = 6, the approximate value of the

real root of x log_{10} x = 4.77 , after the second iteration, is ____________________

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 41

Let the following discrete data be obtained from a curve ?? = ??(??):

Let S be the solid of revolution obtained by rotating the above curve about the ??-axis between x = 0 and ?? = 1 and let V denote its volume. The approximate value of V, obtained using

Simpson’s 1/3 rule, is ______________

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 42

The integral surface of the first order partial differential equation

passing through the curve x^{2} + y^{2} = 2x, z = 0 is

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 43

The boundary value problem, ? is converted into the

integral equation

Then g(2/3) is

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 44

If ??1(??) = ?? is a solution to the differential equation then its

general solution is

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 45

The solution to the initial value problem is

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 46

The time to failure, in months, of light bulbs manufactured at two plants A and B obey the exponential distribution with means 6 and 2 months respectively. Plant B produces four times as many bulbs as plant A does. Bulbs from these plants are indistinguishable. They are mixed and sold together. Given that a bulb purchased at random is working after 12 months, the probability that it was manufactured at plant A is _____

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 47

Let X , Y be continuous random variables with joint density function

The value of E[X +Y ]is ____________________

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 48

Let be the subspace of R, where R is equipped with the usual topology. Which

of the following is FALSE?

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 49

Let A matrix P such that P^{−1}XP is a diagonal matrix, is

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 50

Using the Gauss-Seidel iteration method with the initial guess ,the second approximation for the solution to the system of equations

2x_{1}-x_{2}=7

-x_{1}+2x_{2}-x_{3}=1

-x_{2}+2x_{3}=1

is

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 51

The fourth order Runge-Kutta method given by

is used to solve the initial value problem

If u(1) = 1 is obtained by taking the step size h = 1, then the value of 4 K is ______________

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 52

A particle P of mass m moves along the cycloid x = (θ − sin θ) and ?? = (1 + cos θ),

0 ≤ θ ≤ 2??. Let g denote the acceleration due to gravity. Neglecting the frictional force, the

Lagrangian associated with the motion of the particle P is:

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 53

Suppose that ?? is a population random variable with probability density function

where θ is a parameter. In order to test the null hypothesis H_{0}: θ = 2, against the alternative

hypothesis H_{1}: θ = 3, the following test is used: Reject the null hypothesis if X1 ≥ 1/2 and accept otherwise, where X_{1} is a random sample of size 1 drawn from the above population. Then the power of the test is _____

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 54

Suppose that x_{1}, x_{2},…, x_{n} is a random sample of size n drawn from a population with probability

density function

where θ is a parameter such that θ > 0. The maximum likelihood estimator of θ is

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 55

Let F be a vector field defined on Let be defined by ??(t) = (8 cos 2πt, 17 sin 2πt) and ??(t) = (26 cos 2πt, −10 sin 2πt).

then m is _________________________

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 56

Let g: R3 → R3 be defined by g(x, y, z) = (3y + 4z, 2x − 3z, x + 3y) and let

S = {(x, y, z) ∈ R3 : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1 }. If

then α is _____________________

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 57

Let T_{1}, T_{2} : R^{5} → R^{3} be linear transformations such that rank(T_{1}) = 3 and nullity(T_{2}) = 3. Let

T_{3} : R^{3} → R^{3} be a linear transformation such that T_{3} ° T, = T_{2}. Then rank(T3) is __________

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 58

Let F_{3} be the field of 3 elements and let F_{3} × F_{3} be the vector space over F_{3}. The number of

distinct linearly dependent sets of the form {u, v}, where *u, v* ∈ F_{3} × F_{3} {(0,0)} and *u ≠ v*

is _____________

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 59

Let F_{125} be the field of 125 elements. The number of non-zero elements α ∈ F_{125} such that

α^{5} = α is _______________________

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 60

The value of where R is the region in the first quadrant bounded by the curves

y = x^{2}, y + x = 2 and x = 0 is ______________

*Answer can only contain numeric values

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 61

Consider the heat equation

with the boundary conditions u(0, t) = 0, u(π, t) = 0 for t > 0, and the initial condition

is ___________________

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 62

Consider the partial order in R^{2} given by the relation (x_{1}, y_{1}) < (x_{2}, y_{2}) EITHER if x_{1} < x_{2} OR

if x_{1} = x_{2} and y_{1} < y_{2}. Then in the order topology on R2 defined by the above order

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 63

Consider the following linear programming problem:

Minimize: x^{1} + x^{2} + 2x^{3}

Subject to

The dual to this problem is:

Maximize: 4y_{1} +5y_{2} + 6y_{3}

Subject to

and further subject to:

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 64

Let X = C^{1}[0,1]. For each f ∈ X , define

Which of the following statements is TRUE?

MA Mathematics - 2014 GATE Paper (Practice Test) - Question 65

If the power series converges at 5i and diverges at −3i, then the power series

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