# MA Mathematics - 2014 GATE Paper (Practice Test)

## 65 Questions MCQ Test GATE Past Year Papers for Practice (All Branches) | MA Mathematics - 2014 GATE Paper (Practice Test)

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Attempt MA Mathematics - 2014 GATE Paper (Practice Test) | 65 questions in 180 minutes | Mock test for GATE preparation | Free important questions MCQ to study GATE Past Year Papers for Practice (All Branches) for GATE Exam | Download free PDF with solutions
QUESTION: 1

Solution:
QUESTION: 2

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 = 5x2 + 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

(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   E1 and E2 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 4x3 − 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  X1 , X2 is a random sample of size 2 from an (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
= 0 . Then the dimension of is _________________

Solution:
*Answer can only contain numeric values
QUESTION: 30

Consider the following linear programming problem:
Minimize x1 + x2
Subject to:
2x1 + x2 ≥ 8
2x1 + 5x2 ≥ 10
x1, x2 ≥ 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 d1, d2 and d3 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 {  tf1, t2f2, t3f3} is linearly independent, then the set { f1,f2,f3} is linearly
independent, where f1,f2,f3  ∈C [0,1] and tn represents the polynomial function t→ tn, n ? N

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 log10 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 x2 + y2 = 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−1XP 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

2x1-x2=7

-x1+2x2-x3=1

-x2+2x3=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 H0: θ = 2, against the alternative
hypothesis H1: θ = 3, the following test is used: Reject the null hypothesis if X1 ≥ 1/2 and accept otherwise, where  X1 is a random sample of size 1 drawn from the above population. Then the power of the test is _____

Solution:
QUESTION: 54

Suppose that x1, x2,…, xn 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 T1, T2 : R5 → R3 be linear transformations such that rank(T1) = 3 and nullity(T2) = 3. Let
T3 : R3 → R3 be a linear transformation such that T3 ° T, = T2. Then rank(T3) is __________

Solution:
*Answer can only contain numeric values
QUESTION: 58

Let F3 be the field of 3 elements and let F3 × F3 be the vector space over F3. The number of
distinct linearly dependent sets of the form {u, v}, where u, v ∈ F3 × F3 {(0,0)} and u ≠ v
is _____________

Solution:
*Answer can only contain numeric values
QUESTION: 59

Let F125 be the field of 125 elements. The number of non-zero elements α ∈ F125 such that
α5 = 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 = x2, 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 R2 given by the relation (x1, y1) < (x2, y2) EITHER if x1 < x2 OR
if x1 = x2 and y1 < y2. Then in the order topology on R2 defined by the above order

Solution:
QUESTION: 63

Consider the following linear programming problem:
Minimize: x1 + x2 + 2x3
Subject to The dual to this problem is:
Maximize: 4y1 +5y2 + 6y3
Subject to and further subject to:

Solution:
QUESTION: 64

Let X = C1[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

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