GATE 2016

General Aptitude - GA Set-4

Q. 1 Q. 5 carry one mark each.

Q.1

An apple costs Rs. 10. An onion costs Rs. 8.

Select the most suitable sentence with respect to grammar and usage.
(A) The price of an apple is greater than an onion.
(B) The price of an apple is more than onion.
(C) The price of an apple is greater than that of an onion.
(D) Apples are more costlier than onions.

Q.2

The Buddha said, Holding on to anger is like grasping a hot coal with the intent of throwing it at
someone else; you are the one who gets burnt.
Select the word below which is closest in meaning to the word underlined above.
(A) burning

Q.3

Q.4

(C) clutching

(D) flinging

M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-inlaw of M. How is P related to M?
(A) P is the son-in-law of M.

(B) P is the grandchild of M.

(C) P is the daughter-in law of M.

(D) P is the grandfather of M.

The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324

Q.5

(B) igniting

(B) 441

(C) 97

(D) 64

It takes 10 s and 15 s, respectively, for two trains travelling at different constant speeds to
completely pass a telegraph post. The length of the first train is 120 m and that of the second train is
150 m. The magnitude of the difference in the speeds of the two trains (in m/s) is ____________.
(A) 2.0

(B) 10.0

(C) 12.0

(D) 22.0

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GATE 2016

General Ap
ptitude - GA Set-44

Q. 6 Q. 10 carry tw
wo marks each.
Q.6

Thee velocity V of a vehiclle along a sttraight line is

i measured in m/s and plotted as shown
s
with
resppect to timee in seconds. At the endd of the 7 seconds,
s
how
w much willl the odomeeter reading
incrrease by (in m)?

(A)) 0
Q.7

(B) 3

(C) 4

(D) 5

Thee overwhelm
ming numberr of people innfected with
h rabies in Inndia has beenn flagged by
y the World
Heaalth Organizzation as a soource of conncern. It is esstimated thatt inoculatingg 70% of petts and stray
doggs against rabbies can leadd to a significcant reductio
on in the num
mber of peoplle infected with
w rabies.
Whhich of the foollowing can be logicallyy inferred from
m the above sentences?
(A)) The numbber of people in India infeected with raabies is high.
(B)) The number of people in other partts of the world who are innfected with rabies is low
w.
(C)) Rabies cann be eradicated in India by
b vaccinatin
ng 70% of strray dogs.
r
worldw
wide.
(D)) Stray dogss are the main source of rabies

Q.8

A flat
f is shared by four firstt year underggraduate stud
dents. They agreed
a
to alloow the oldestt of them to
enjoy some exttra space in the flat. Maanu is two months
m
olderr than Sravann, who is th
hree months
youunger than Trrideep. Pavaan is one monnth older than
n Sravan. Who
W should occcupy the ex
xtra space in
the flat?
(A)) Manu

Q.9

(B) Sravan

(C) Trideep

(D) Pavan

Finnd the area boounded by thhe lines 3x+22y=14, 2x-3y

y=5 in the firrst quadrant.
(A)) 14.95

(B) 15.25

(C) 15.70

(D) 20.35
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GATE 2016

Q.10

General Aptitude - GA Set-4

A straight line is fit to a data set (ln x, y). This line intercepts the abscissa at ln x = 0.1 and has a
slope of 0.02. What is the value of y at x = 5 from the fit?
(A) 0.030

(B) 0.014

(C) 0.014

(D) 0.030

END OF THE QUESTION PAPER

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GATE 2016

Mathematics - MA

List of Symbols, Notations and Data

i.i.d. : independent and identically distributed

,

Normal distribution with mean and variance

, ,

Expected value (mean) of the random variable

: the greatest integer less than or equal to

Set of integers
Set of integers modulo n
Set of real numbers
Set of complex numbers
n dimensional Euclidean space
Usual metric d on

is given by

,,

,,

Normed linear space of all squaresummable real sequences

0,1 Set of all real valued continuous functions on the interval 0,1
0,1

Conjugate transpose of the matrix M

Transpose of the matrix M

Id : Identity matrix of appropriate order

Range space of M
Null space of M
: Orthogonal complement of the subspace W

MA

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GATE 2016

Mathematics - MA

Q. 1 Q. 25 carry one mark each.

Q.1

Let

, ,

(P) :

be a basis of
,

(Q) :

. Consider the following statements P and Q:

is a basis of

is a basis of

Which of the above statements hold TRUE?

(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Q.2

Consider the following statements P and Q:

(P) : If

1
1
1

1 1
2 4 , then M is singular .
3 9

(Q) : Let S be a diagonalizable matrix. If T is a matrix such that S + 5 T = Id, then T is

diagonalizable.

Which of the above statements hold TRUE?
(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Q.3

Consider the following statements P and Q:

(P) : If M is an

complex matrix, then

(Q) : There exists a unitary matrix with an eigenvalue such that || <1.

Which of the above statements hold TRUE?
(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

MA

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GATE 2016

Q.4

Mathematics - MA

Consider a real vector space V of dimension n and a nonzero linear transformation

. If dimension

and

, for some \ 0 , then

which of the following statements is TRUE?

(A) determinant

| |

(B) There exists a nontrivial subspace

of V such that

0 for all

(C) T is invertible
(D) is the only eigenvalue of T

Q.5

Let

0, 1 2, 3 and

be a strictly increasing function such that

is connected. Which of the following statements is TRUE?

(A) has exactly one discontinuity
(B) has exactly two discontinuities
(C) has infinitely many discontinuities
(D) is continuous

Q.6

Let

1 and

4,

2. Then,
1

lim

is equal to _____________________

Q.7

Maximum

Q.8

Let , , ,

0,1

is equal to _________________

such that

0. Then, the Cauchy problem

,
0 on

, ,

0

has a unique solution if

(A)

(B)

(C)

(D)

MA

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GATE 2016

Q.9

Mathematics - MA

Let

, be the d'Alembert's solution of the initial value problem for the wave

equation

,0

0
,0

where c is a positive real number and , are smooth odd functions. Then,

0,1 is

equal to ___________

Q.10 Let the probability density function of a random variable X be
1
2

1
1
2
otherwise.

1
0

Then, the value of c is equal to ________________________

Q.11 Let V be the set of all solutions of the equation
0

0 satisfying

1 , where , are positive real numbers. Then, dimension(V ) is equal to

_____________________

Q.12 Let

0,

, , where

continuous functions. If

and

are

and

are two linearly independent solutions of the above equation, then

| 4 0

2 1 | is equal to ____________________

Q.13 Let

be the Legendre polynomial of degree and

, where k

is a nonnegative integer. Consider the following statements P and Q:

(P) :

0 if

(Q) :

0 if

is an odd integer.

Which of the above statements hold TRUE?
(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

MA

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GATE 2016

Mathematics - MA

Q.14 Consider the following statements P and Q:

(P) :

solutions near

(Q) :

0.

solutions near

0 has two linearly independent Frobenius series

0 has two linearly independent Frobenius series

0.

Which of the above statements hold TRUE?
(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Q.15 Let the polynomial
interpolates

at

over the interval

be approximated by a polynomial of degree

2, which

1, 0 and 1. Then, the maximum absolute interpolation error

1, 1 is equal to ______________________

Q.16 Let
lim

be a sequence of distinct points in

0,1

| |

1 with

0. Consider the following statements P and Q:

(P) : There exists a unique analytic function f on

0,1 such that

for

all n.

(Q) : There exists an analytic function f on

and

0,1 such that

0 if n is even

1 if n is odd.

Which of the above statements hold TRUE?
(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Q.17 Let

be a topological space with the cofinite topology. Every infinite subset of

is
(A) Compact but NOT connected
(B) Both compact and connected
(C) NOT compact but connected
(D) Neither compact nor connected

MA

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GATE 2016

Mathematics - MA

Q.18 Let

0 and

0 .

is equal to _______________________

Then, dimension

Q.19 Consider

, , where ,

defined by

maximum | |, | | . Let

be

and the norm preserving linear extension of to

, . Then, 1,1,1 is equal to __________________________________

Q.20

0,1 0,1 is called a shrinking map if |

,
|

0,1 and a contraction if there exists an

| for all ,

| for all

1 such that

0,1 .

Which of the following statements is TRUE for the function

(A) is both a shrinking map and a contraction

(B) is a shrinking map but NOT a contraction
(C) is NOT a shrinking map but a contraction
(D) is Neither a shrinking map nor a contraction

Q.21 Let be the set of all

real matrices with the usual norm topology. Consider the

following statements P and Q:

(P) : The set of all symmetric positive definite matrices in is connected.
(Q) : The set of all invertible matrices in is compact.

Which of the above statements hold TRUE?
(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

MA

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GATE 2016

Mathematics - MA

Q.22 Let

,,

function for 0

be a random sample from the following probability density

, 0

1,
1

; ,

;
0

otherwise.

Here and are unknown parameters. Which of the following statements is TRUE?
(A) Maximum likelihood estimator of only exists
(B) Maximum likelihood estimator of only exists
(C) Maximum likelihood estimators of both and
(D) Maximum likelihood estimator of Neither

exist

nor

exists

Q.23 Suppose X and Y are two random variables such that
variable for all ,

is a normal random

. Consider the following statements P, Q, R and S:

(P) : X is a standard normal random variable.

(Q) : The conditional distribution of X given Y is normal.
(R) : The conditional distribution of X given
(S) :

is normal.

has mean 0.

Which of the above statements ALWAYS hold TRUE?
(A) both P and Q

(B) both Q and R

(C) both Q and S

(D) both P and S

Q.24 Consider the following statements P and Q:
(P) : If is a normal subgroup of order 4 of the symmetric group , then

is

abelian.

(Q) : If

1,

is the quaternion group, then

1,1 is abelian.

Which of the above statements hold TRUE?

(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Q.25 Let be a field of order 32. Then the number of nonzero solutions ,
the equation

of

0 is equal to _______________________________

MA

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GATE 2016

Mathematics - MA

Q. 26 Q. 55 carry two marks each.

Q.26 Let

| |

2 be oriented in the counterclockwise direction. Let

1

2

Then, the value of is equal to __________________________

Q.27 Let

harmonic conjugate. If 0,0

be a harmonic function and

1, then |

its

1,1 | is equal to _______________

Q.28 Let be the triangular path connecting the points (0,0), (2,2) and (0,2) in the counter
clockwise direction in

. Then

is equal to _____________________

Q.29 Let y be the solution of
| |,

1
0.

Then 1 is equal to
(A)

(B)

(C) 2

(D) 2

Q.30 Let X be a random variable with the following cumulative distribution function:
0

Then

0
1
3

2
4
1

0
1
2
1
1.

1 is equal to ___________________

MA

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GATE 2016

Mathematics - MA

Q.31 Let be the curve which passes through (0,1) and intersects each curve of the family

orthogonally. Then also passes through the point

(A) 2, 0

(B) 0, 2

(C) 1,1

(D)

1,1

Q.32 Let

be the Fourier series of the

2 periodic function defined by

. Then

is equal to __________________________

Q.33 Let

be a continuous function on 0, . If
1

is equal to ________________________

then

Q.34 Let

and

. Then,

is equal to

(A) ln 10

(B) ln 10

(C) ln 10

(D) ln 10

Q.35 For any ,

0,1 , let

distance

in imum

0,1

0,1 .

Then, || 3,4 || is equal to ____________________

Q.36

Let

and

. Then

is

equal to _______________________________________

MA

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GATE 2016

Q.37

Mathematics - MA

Let

be a real matrix with eigenvalues 1, 0 and 3. If the eigenvectors

corresponding to 1 and 0 are 1,1,1 and 1, 1,0 respectively, then the value of
3 is equal to _________________

Q.38

Let

1
0
0

1 0
1 1 and
0 1

. If

, then

is equal to ________________________

Q.39 Let the integral

, where

0
2

2

4.

Consider the following statements P and Q:

(P) : If is the value of the integral obtained by the composite trapezoidal rule with

two equal subintervals, then is exact.

(Q) : If is the value of the integral obtained by the composite trapezoidal rule with

three equal subintervals, then is exact.

Which of the above statements hold TRUE?
(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Q.40 The difference between the least two eigenvalues of the boundary value problem

0,
0,

0,

is equal to ______________________________

Q.41 The number of roots of the equation

cos

0 in the interval

is equal

to ______________

MA

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GATE 2016

Mathematics - MA

Q.42 For the fixed point iteration

0, 1, 2, , consider the following

statements P and Q:
(P) : If

(Q) : If

all

then the fixed point iteration converges to 2 for all

1, 100 .

then the fixed point iteration converges to 2 for

0, 100 .

Which of the above statements hold TRUE?
(A) both P and Q

(B) only P

(C) only Q

(D) Neither P nor Q

Q.43 Let be defined by
,

,,

,,

Then
(A)

(B)

2 but bounded

(C) 1

(D) is unbounded

Q.44 Minimize

2 subject to
2

0,

3
2
0.

Then, the minimum value of is equal to _________________________

Q.45 Maximize

11

subject to
10
2
2

, ,

1
2
0.

Then, the maximum value of is equal to _________________________

MA

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GATE 2016

Mathematics - MA

Q.46 Let

, be a sequence of i.i.d. random variables with mean 1. If N is a

geometric random variable with the probability mass function

1,2,3, and it is independent of the 's, then

is equal to

_____________________

Q.47 Let

be an exponential random variable with mean 1 and

variable with mean 2 and variance 2. If

and

a gamma random

are independently distributed, then

is equal to _________________________

Q.48 Let

, be a sequence of i.i.d. uniform 0,1 random variables. Then, the value

of
lim

ln 1

ln 1

is equal to ____________________

Q.49 Let X be a standard normal random variable. Then,

(A)
(C)

(B)

(D)

0 |

1 is equal to

Q.50 Let

where
:

,,

0, 0
1,

be a random sample from the probability density function

1
0

0

otherwise,

1 are parameters. Consider the following testing problem:

1 versus

0,

2.

Which of the following statements is TRUE?
(A) Uniformly Most Powerful test does NOT exist
(B) Uniformly Most Powerful test is of the form

, for some 0

(C) Uniformly Most Powerful test is of the form

, for some 0

(D) Uniformly Most Powerful test is of the form

some 0

, for

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GATE 2016

Mathematics - MA

Q.51 Let

, be a sequence of i.i.d.
lim

, 1 random variables. Then,

is equal to _____________________________

Q.52 Let

,,

Maximum

be a random sample from uniform 1,

,

,,

, for some

1. If

, then the UMVUE of is

(A)

(B)

(C)

(D)

1,

Q.53 Let

2 be a random sample from a Poisson random

variable with mean , where 1, 2 . Then, the maximum likelihood estimator of

is equal to ____________________

Q.54 The remainder when 98! is divided by 101 is equal to ____________________________

Q.55 Let be a group whose presentation is
, |

Then is isomorphic to

(A)

(B)

(C)

(D)

MA

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GATE 2016

Mathematics - MA

END OF THE QUESTION PAPER

MA

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GATE 2016

MA

Mathematics - MA

15/16

GATE 2016

Mathematics - MA

MA

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