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This mock test of MA Mathematics - 2014 GATE Paper (Practice Test) for GATE helps you for every GATE entrance exam.
This contains 65 Multiple Choice Questions for GATE MA Mathematics - 2014 GATE Paper (Practice Test) (mcq) to study with solutions a complete question bank.
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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**

Solution:

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.

Solution:

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?

Solution:

QUESTION: 4

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

Solution:

*Answer can only contain numeric values

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)**

Solution:

QUESTION: 6

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

Q.

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

Solution:

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?

Solution:

*Answer can only contain numeric values

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)**

Solution:

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?

Solution:

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

Solution:

QUESTION: 11

**Q. 1 – Q. 25 carry one mark each.**

**Q.**

The function is differentiable at

Solution:

*Answer can only contain numeric values

QUESTION: 12

The radius of convergence of the power series is _____________

Solution:

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:

Solution:

*Answer can only contain numeric values

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 _____________________

Solution:

*Answer can only contain numeric values

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 ___________

Solution:

*Answer can only contain numeric values

QUESTION: 16

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

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

Solution:

QUESTION: 17

The solution to the integral equation

Solution:

QUESTION: 18

The general solution to the ordinary differential equation in

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

Solution:

QUESTION: 19

The inverse Laplace transform of

Solution:

QUESTION: 20

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

Solution:

QUESTION: 21

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

Solution:

*Answer can only contain numeric values

QUESTION: 22

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

Solution:

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

Solution:

QUESTION: 24

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

Solution:

QUESTION: 25

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

is isomorphic to which of the following groups?

Solution:

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?

Solution:

*Answer can only contain numeric values

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 ______

Solution:

QUESTION: 28

Let V denote the vector space . Then

Solution:

*Answer can only contain numeric values

QUESTION: 29

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

Solution:

*Answer can only contain numeric values

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 _________________________

Solution:

*Answer can only contain numeric values

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 ________________________

Solution:

QUESTION: 32

For the sequence of functions

consider the following quantities expressed in terms of Lebesgue integrals

Which of the following is TRUE?

Solution:

QUESTION: 33

Which of the following statements about the spaces is TRUE ?

Solution:

QUESTION: 34

The maximum modulus of is

Solution:

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?

Solution:

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

Solution:

*Answer can only contain numeric values

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 _________

Solution:

*Answer can only contain numeric values

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 _______

Solution:

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?

Solution:

*Answer can only contain numeric values

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 ____________________

Solution:

*Answer can only contain numeric values

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 ______________

Solution:

QUESTION: 42

The integral surface of the first order partial differential equation

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

Solution:

*Answer can only contain numeric values

QUESTION: 43

The boundary value problem, ? is converted into the

integral equation

Then g(2/3) is

Solution:

QUESTION: 44

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

general solution is

Solution:

QUESTION: 45

The solution to the initial value problem is

Solution:

*Answer can only contain numeric values

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 _____

Solution:

*Answer can only contain numeric values

QUESTION: 47

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

The value of E[X +Y ]is ____________________

Solution:

QUESTION: 48

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

of the following is FALSE?

Solution:

QUESTION: 49

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

Solution:

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

Solution:

*Answer can only contain numeric values

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 ______________

Solution:

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:

Solution:

*Answer can only contain numeric values

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 _____

Solution:

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

Solution:

*Answer can only contain numeric values

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 _________________________

Solution:

*Answer can only contain numeric values

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 _____________________

Solution:

*Answer can only contain numeric values

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 __________

Solution:

*Answer can only contain numeric values

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 _____________

Solution:

*Answer can only contain numeric values

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 _______________________

Solution:

*Answer can only contain numeric values

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 ______________

Solution:

*Answer can only contain numeric values

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 ___________________

Solution:

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

Solution:

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:

Solution:

QUESTION: 64

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

Which of the following statements is TRUE?

Solution:

QUESTION: 65

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

Solution:

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