Download Graduate Aptitude Test in Engineering (GATE) 2015 MA Mathematics Question Paper
MA 1/9
List of Symbols, Notations and Data
, :?Binomial distribution with ? trials and success probability ; ? ? 1,2, ? ?and?? ?0, 1
, :?Uniform distribution on the interval , ,? ??
, :??Normal distribution with mean ? and variance ,? ?, ? ,0
?? Probability of the event ?
Poisson :?Poisson distribution with mean , 0
:?Expected value (mean) of the random variable
If ? 0,1 ,??then 1.96 ?0.975? and 0.54 ?0.7054
?? Set of integers
???Set of rational numbers
?? Set of real numbers
? Set of complex numbers
:?The cyclic group of order ?
[ ] : Polynomial ring over the field
0, 1 ??Set of all real valued continuous functions on the interval 0, 1
0, 1 ??Set of all real valued continuously differentiable functions on the interval 0, 1
?
??Normed space of all square-summable real sequences
0, 1 :?Space of all square-Lebesgue integrable real valued functions on the interval 0, 1
0, 1 ,??
:? The space 0, 1 ?with ? ?
? |? |
?
?
0, 1 ,??
:? The space 0, 1 ?with ? ?
sup ?| |??? ? 0, 1 ?
:? The orthogonal complement of ? in an inner product space
?? ? -dimensional Euclidean space
Usual metric ? on ??is given by ,
,?,
, ,
,?, ? ? ?
?
:??The ? identity matrix ( ?? the identity matrix when order is NOT specified)
??The order of the element ? of a group
FirstRanker.com - FirstRanker's Choice
GATE 2015 MATHEMATICS ? MA
MA 1/9
List of Symbols, Notations and Data
, :?Binomial distribution with ? trials and success probability ; ? ? 1,2, ? ?and?? ?0, 1
, :?Uniform distribution on the interval , ,? ??
, :??Normal distribution with mean ? and variance ,? ?, ? ,0
?? Probability of the event ?
Poisson :?Poisson distribution with mean , 0
:?Expected value (mean) of the random variable
If ? 0,1 ,??then 1.96 ?0.975? and 0.54 ?0.7054
?? Set of integers
???Set of rational numbers
?? Set of real numbers
? Set of complex numbers
:?The cyclic group of order ?
[ ] : Polynomial ring over the field
0, 1 ??Set of all real valued continuous functions on the interval 0, 1
0, 1 ??Set of all real valued continuously differentiable functions on the interval 0, 1
?
??Normed space of all square-summable real sequences
0, 1 :?Space of all square-Lebesgue integrable real valued functions on the interval 0, 1
0, 1 ,??
:? The space 0, 1 ?with ? ?
? |? |
?
?
0, 1 ,??
:? The space 0, 1 ?with ? ?
sup ?| |??? ? 0, 1 ?
:? The orthogonal complement of ? in an inner product space
?? ? -dimensional Euclidean space
Usual metric ? on ??is given by ,
,?,
, ,
,?, ? ? ?
?
:??The ? identity matrix ( ?? the identity matrix when order is NOT specified)
??The order of the element ? of a group
GATE 2015 MATHEMATICS ? MA
MA 2/9
Q. 1 ? Q. 25 carry one mark each.
Q.1 Let ?? ?? ?be a linear map defined by
,,,,2 3,22, .
Then the rank of is equal to _________
Q.2 Let be a 33 matrix and suppose that 1, 2 and 3 are the eigenvalues of . If
11
for some scalar 0,? then is equal to ___________
Q.3 Let be a 33 singular matrix and suppose that 2 and 3 are eigenvalues of . Then the number
of linearly independent eigenvectors of 2 is equal to __________
Q.4
Let be a 33 matrix such that
2
1
0
6
3
0
and suppose that ?
1
1/2
0
? for
some , , ? . Then |?? | is equal to _______
Q.5 Let : 0, ? ? ? be defined by
sin
.
Then the function is
(A) uniformly continuous on 0, 1 but NOT on 0, ?
(B) uniformly continuous on 0, ? but NOT on 0, 1
(C) uniformly continuous on both 0, 1 and 0, ?
(D) neither uniformly continuous on 0, 1 nor uniformly continuous on 0, ?
Q.6
Consider the power series ? ,
where if iseven
if is odd.
The radius of convergence of the series is equal to __________
Q.7
Let ?? ? ? ? | |2 .? Then
?
is equal to ____________
Q.8
Let ?~?5,
and ?~? 0,1 . Then
is equal to ___________
FirstRanker.com - FirstRanker's Choice
GATE 2015 MATHEMATICS ? MA
MA 1/9
List of Symbols, Notations and Data
, :?Binomial distribution with ? trials and success probability ; ? ? 1,2, ? ?and?? ?0, 1
, :?Uniform distribution on the interval , ,? ??
, :??Normal distribution with mean ? and variance ,? ?, ? ,0
?? Probability of the event ?
Poisson :?Poisson distribution with mean , 0
:?Expected value (mean) of the random variable
If ? 0,1 ,??then 1.96 ?0.975? and 0.54 ?0.7054
?? Set of integers
???Set of rational numbers
?? Set of real numbers
? Set of complex numbers
:?The cyclic group of order ?
[ ] : Polynomial ring over the field
0, 1 ??Set of all real valued continuous functions on the interval 0, 1
0, 1 ??Set of all real valued continuously differentiable functions on the interval 0, 1
?
??Normed space of all square-summable real sequences
0, 1 :?Space of all square-Lebesgue integrable real valued functions on the interval 0, 1
0, 1 ,??
:? The space 0, 1 ?with ? ?
? |? |
?
?
0, 1 ,??
:? The space 0, 1 ?with ? ?
sup ?| |??? ? 0, 1 ?
:? The orthogonal complement of ? in an inner product space
?? ? -dimensional Euclidean space
Usual metric ? on ??is given by ,
,?,
, ,
,?, ? ? ?
?
:??The ? identity matrix ( ?? the identity matrix when order is NOT specified)
??The order of the element ? of a group
GATE 2015 MATHEMATICS ? MA
MA 2/9
Q. 1 ? Q. 25 carry one mark each.
Q.1 Let ?? ?? ?be a linear map defined by
,,,,2 3,22, .
Then the rank of is equal to _________
Q.2 Let be a 33 matrix and suppose that 1, 2 and 3 are the eigenvalues of . If
11
for some scalar 0,? then is equal to ___________
Q.3 Let be a 33 singular matrix and suppose that 2 and 3 are eigenvalues of . Then the number
of linearly independent eigenvectors of 2 is equal to __________
Q.4
Let be a 33 matrix such that
2
1
0
6
3
0
and suppose that ?
1
1/2
0
? for
some , , ? . Then |?? | is equal to _______
Q.5 Let : 0, ? ? ? be defined by
sin
.
Then the function is
(A) uniformly continuous on 0, 1 but NOT on 0, ?
(B) uniformly continuous on 0, ? but NOT on 0, 1
(C) uniformly continuous on both 0, 1 and 0, ?
(D) neither uniformly continuous on 0, 1 nor uniformly continuous on 0, ?
Q.6
Consider the power series ? ,
where if iseven
if is odd.
The radius of convergence of the series is equal to __________
Q.7
Let ?? ? ? ? | |2 .? Then
?
is equal to ____________
Q.8
Let ?~?5,
and ?~? 0,1 . Then
is equal to ___________
GATE 2015 MATHEMATICS ? MA
MA 3/9
Q.9 Let the random variable ? have the distribution function
?
0?????????????? ???if?????? 0 ?????
2
??????????? ???????if??0? 1
3
5
??????????? ???????if??1? 2?
1
2
?
8
?????????? if??2? 3
1if3.
Then 2 4 is equal to ___________
Q.10 Let X be a random variable having the distribution function
?
0if 0
1
4
??????????? if??0? 1
1
3
??????????? if??1? 2?
1
2
??????????? if??2? 11
3
1if
11
3
.
Then is equal to _________
Q.11 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
(A)
(B)
(C)
(D)
Q.12 Let 2.2, 4.3, 3.1, 4.5, 1.1and 5.7 be the observed values of a
random sample of size 6 from a 1, 4 distribution, where ?0,? is unknown. Then
a maximum likelihood estimate of is equal to
(A) 1.8 (B) 2.3 (C) 3.1 (D) 3.6
Q.13 Let ?? , ?? |?
? 1 be the open unit disc in with boundary ?. If , is
the solution of the Dirichlet problem
?
0???????????in???
, 12 on ?,
then ,0 is equal to
(A) 1
(B)
(C)
(D) 1
FirstRanker.com - FirstRanker's Choice
GATE 2015 MATHEMATICS ? MA
MA 1/9
List of Symbols, Notations and Data
, :?Binomial distribution with ? trials and success probability ; ? ? 1,2, ? ?and?? ?0, 1
, :?Uniform distribution on the interval , ,? ??
, :??Normal distribution with mean ? and variance ,? ?, ? ,0
?? Probability of the event ?
Poisson :?Poisson distribution with mean , 0
:?Expected value (mean) of the random variable
If ? 0,1 ,??then 1.96 ?0.975? and 0.54 ?0.7054
?? Set of integers
???Set of rational numbers
?? Set of real numbers
? Set of complex numbers
:?The cyclic group of order ?
[ ] : Polynomial ring over the field
0, 1 ??Set of all real valued continuous functions on the interval 0, 1
0, 1 ??Set of all real valued continuously differentiable functions on the interval 0, 1
?
??Normed space of all square-summable real sequences
0, 1 :?Space of all square-Lebesgue integrable real valued functions on the interval 0, 1
0, 1 ,??
:? The space 0, 1 ?with ? ?
? |? |
?
?
0, 1 ,??
:? The space 0, 1 ?with ? ?
sup ?| |??? ? 0, 1 ?
:? The orthogonal complement of ? in an inner product space
?? ? -dimensional Euclidean space
Usual metric ? on ??is given by ,
,?,
, ,
,?, ? ? ?
?
:??The ? identity matrix ( ?? the identity matrix when order is NOT specified)
??The order of the element ? of a group
GATE 2015 MATHEMATICS ? MA
MA 2/9
Q. 1 ? Q. 25 carry one mark each.
Q.1 Let ?? ?? ?be a linear map defined by
,,,,2 3,22, .
Then the rank of is equal to _________
Q.2 Let be a 33 matrix and suppose that 1, 2 and 3 are the eigenvalues of . If
11
for some scalar 0,? then is equal to ___________
Q.3 Let be a 33 singular matrix and suppose that 2 and 3 are eigenvalues of . Then the number
of linearly independent eigenvectors of 2 is equal to __________
Q.4
Let be a 33 matrix such that
2
1
0
6
3
0
and suppose that ?
1
1/2
0
? for
some , , ? . Then |?? | is equal to _______
Q.5 Let : 0, ? ? ? be defined by
sin
.
Then the function is
(A) uniformly continuous on 0, 1 but NOT on 0, ?
(B) uniformly continuous on 0, ? but NOT on 0, 1
(C) uniformly continuous on both 0, 1 and 0, ?
(D) neither uniformly continuous on 0, 1 nor uniformly continuous on 0, ?
Q.6
Consider the power series ? ,
where if iseven
if is odd.
The radius of convergence of the series is equal to __________
Q.7
Let ?? ? ? ? | |2 .? Then
?
is equal to ____________
Q.8
Let ?~?5,
and ?~? 0,1 . Then
is equal to ___________
GATE 2015 MATHEMATICS ? MA
MA 3/9
Q.9 Let the random variable ? have the distribution function
?
0?????????????? ???if?????? 0 ?????
2
??????????? ???????if??0? 1
3
5
??????????? ???????if??1? 2?
1
2
?
8
?????????? if??2? 3
1if3.
Then 2 4 is equal to ___________
Q.10 Let X be a random variable having the distribution function
?
0if 0
1
4
??????????? if??0? 1
1
3
??????????? if??1? 2?
1
2
??????????? if??2? 11
3
1if
11
3
.
Then is equal to _________
Q.11 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
(A)
(B)
(C)
(D)
Q.12 Let 2.2, 4.3, 3.1, 4.5, 1.1and 5.7 be the observed values of a
random sample of size 6 from a 1, 4 distribution, where ?0,? is unknown. Then
a maximum likelihood estimate of is equal to
(A) 1.8 (B) 2.3 (C) 3.1 (D) 3.6
Q.13 Let ?? , ?? |?
? 1 be the open unit disc in with boundary ?. If , is
the solution of the Dirichlet problem
?
0???????????in???
, 12 on ?,
then ,0 is equal to
(A) 1
(B)
(C)
(D) 1
GATE 2015 MATHEMATICS ? MA
MA 4/9
Q.14
Let ? ??be such that
? ?? ?
is a field. Then is equal to __________
Q.15 Let 0, 1, ? 0, 1 ,??
and 0, 1 ,??
. Then ? is
(A) dense in ? but NOT in ?
(B) dense in ? but NOT in ?
(C) dense in both and ?
(D) neither dense in ? nor dense in
Q.16
Let ?? ? 0, 1 ,??
?? ?be defined by 2
for all ? 0, 1 . Then ? ?
is equal to __________
Q.17 Let be the usual topology on . Let be the topology on generated by
, ? ??? . Then the set ? ?4 sin
1??? ?
is
(A) closed in ,
?but NOT in ,
(B) closed in ,
?but NOT in ,
(C) closed in both ,
?and ,
(D) neither closed in ,
?nor closed in ,
Q.18 Let ? be a connected topological space such that there exists a non-constant continuous function
? ? , where is equipped with the usual topology. Let :? ? . Then
(A) ? is countable but ?is uncountable
(B) is countable but ?? is uncountable
(C) both ?and ?? are countable
(D) both and ? are uncountable
Q.19 Let ? and ?denote the usual metric and the discrete metric on , respectively.
Let ? ,
? ,
be defined by , ? . Then
(A) is continuous but
? is NOT continuous
(B)
is continuous but ?is NOT continuous
(C) both ? and
are continuous
(D) neither ?nor
?is continuous
Q.20 If the trapezoidal rule with single interval 0, 1 is exact for approximating the integral
??,
then the value of is equal to ________
Q.21
Suppose that the Newton-Raphson method is applied to the equation 2
?1
0 with an
initial approximation ? sufficiently close to zero. Then, for the root 0,? the order of
convergence of the method is equal to _________
FirstRanker.com - FirstRanker's Choice
GATE 2015 MATHEMATICS ? MA
MA 1/9
List of Symbols, Notations and Data
, :?Binomial distribution with ? trials and success probability ; ? ? 1,2, ? ?and?? ?0, 1
, :?Uniform distribution on the interval , ,? ??
, :??Normal distribution with mean ? and variance ,? ?, ? ,0
?? Probability of the event ?
Poisson :?Poisson distribution with mean , 0
:?Expected value (mean) of the random variable
If ? 0,1 ,??then 1.96 ?0.975? and 0.54 ?0.7054
?? Set of integers
???Set of rational numbers
?? Set of real numbers
? Set of complex numbers
:?The cyclic group of order ?
[ ] : Polynomial ring over the field
0, 1 ??Set of all real valued continuous functions on the interval 0, 1
0, 1 ??Set of all real valued continuously differentiable functions on the interval 0, 1
?
??Normed space of all square-summable real sequences
0, 1 :?Space of all square-Lebesgue integrable real valued functions on the interval 0, 1
0, 1 ,??
:? The space 0, 1 ?with ? ?
? |? |
?
?
0, 1 ,??
:? The space 0, 1 ?with ? ?
sup ?| |??? ? 0, 1 ?
:? The orthogonal complement of ? in an inner product space
?? ? -dimensional Euclidean space
Usual metric ? on ??is given by ,
,?,
, ,
,?, ? ? ?
?
:??The ? identity matrix ( ?? the identity matrix when order is NOT specified)
??The order of the element ? of a group
GATE 2015 MATHEMATICS ? MA
MA 2/9
Q. 1 ? Q. 25 carry one mark each.
Q.1 Let ?? ?? ?be a linear map defined by
,,,,2 3,22, .
Then the rank of is equal to _________
Q.2 Let be a 33 matrix and suppose that 1, 2 and 3 are the eigenvalues of . If
11
for some scalar 0,? then is equal to ___________
Q.3 Let be a 33 singular matrix and suppose that 2 and 3 are eigenvalues of . Then the number
of linearly independent eigenvectors of 2 is equal to __________
Q.4
Let be a 33 matrix such that
2
1
0
6
3
0
and suppose that ?
1
1/2
0
? for
some , , ? . Then |?? | is equal to _______
Q.5 Let : 0, ? ? ? be defined by
sin
.
Then the function is
(A) uniformly continuous on 0, 1 but NOT on 0, ?
(B) uniformly continuous on 0, ? but NOT on 0, 1
(C) uniformly continuous on both 0, 1 and 0, ?
(D) neither uniformly continuous on 0, 1 nor uniformly continuous on 0, ?
Q.6
Consider the power series ? ,
where if iseven
if is odd.
The radius of convergence of the series is equal to __________
Q.7
Let ?? ? ? ? | |2 .? Then
?
is equal to ____________
Q.8
Let ?~?5,
and ?~? 0,1 . Then
is equal to ___________
GATE 2015 MATHEMATICS ? MA
MA 3/9
Q.9 Let the random variable ? have the distribution function
?
0?????????????? ???if?????? 0 ?????
2
??????????? ???????if??0? 1
3
5
??????????? ???????if??1? 2?
1
2
?
8
?????????? if??2? 3
1if3.
Then 2 4 is equal to ___________
Q.10 Let X be a random variable having the distribution function
?
0if 0
1
4
??????????? if??0? 1
1
3
??????????? if??1? 2?
1
2
??????????? if??2? 11
3
1if
11
3
.
Then is equal to _________
Q.11 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
(A)
(B)
(C)
(D)
Q.12 Let 2.2, 4.3, 3.1, 4.5, 1.1and 5.7 be the observed values of a
random sample of size 6 from a 1, 4 distribution, where ?0,? is unknown. Then
a maximum likelihood estimate of is equal to
(A) 1.8 (B) 2.3 (C) 3.1 (D) 3.6
Q.13 Let ?? , ?? |?
? 1 be the open unit disc in with boundary ?. If , is
the solution of the Dirichlet problem
?
0???????????in???
, 12 on ?,
then ,0 is equal to
(A) 1
(B)
(C)
(D) 1
GATE 2015 MATHEMATICS ? MA
MA 4/9
Q.14
Let ? ??be such that
? ?? ?
is a field. Then is equal to __________
Q.15 Let 0, 1, ? 0, 1 ,??
and 0, 1 ,??
. Then ? is
(A) dense in ? but NOT in ?
(B) dense in ? but NOT in ?
(C) dense in both and ?
(D) neither dense in ? nor dense in
Q.16
Let ?? ? 0, 1 ,??
?? ?be defined by 2
for all ? 0, 1 . Then ? ?
is equal to __________
Q.17 Let be the usual topology on . Let be the topology on generated by
, ? ??? . Then the set ? ?4 sin
1??? ?
is
(A) closed in ,
?but NOT in ,
(B) closed in ,
?but NOT in ,
(C) closed in both ,
?and ,
(D) neither closed in ,
?nor closed in ,
Q.18 Let ? be a connected topological space such that there exists a non-constant continuous function
? ? , where is equipped with the usual topology. Let :? ? . Then
(A) ? is countable but ?is uncountable
(B) is countable but ?? is uncountable
(C) both ?and ?? are countable
(D) both and ? are uncountable
Q.19 Let ? and ?denote the usual metric and the discrete metric on , respectively.
Let ? ,
? ,
be defined by , ? . Then
(A) is continuous but
? is NOT continuous
(B)
is continuous but ?is NOT continuous
(C) both ? and
are continuous
(D) neither ?nor
?is continuous
Q.20 If the trapezoidal rule with single interval 0, 1 is exact for approximating the integral
??,
then the value of is equal to ________
Q.21
Suppose that the Newton-Raphson method is applied to the equation 2
?1
0 with an
initial approximation ? sufficiently close to zero. Then, for the root 0,? the order of
convergence of the method is equal to _________
GATE 2015 MATHEMATICS ? MA
MA 5/9
Q.22 The minimum possible order of a homogeneous linear ordinary differential equation with real
constant coefficients having sin as a solution is equal to __________
Q.23 The Lagrangian of a system in terms of polar coordinates , is given by
?
1
2
?? 1
2
1 cos,
where ? is the mass, ? is the acceleration due to gravity and denotes the derivative of with
respect to time. Then the equations of motion are
(A) 2???1 ?cos,
sin
(B) 2?? ?1 ?cos,
sin
(C) 2?? ?1 ?cos,
sin
(D) 2???1 ?cos,
sin
Q.24 If ?satisfies the initial value problem
, 1 2,
then 2 is equal to __________
Q.25 It is known that Bessel functions , for 0, satisfy the identity
??
???
?
1 ?
?
for all 0? and ? .??The value of 2 ?
is equal to _________
Q. 26 ? Q. 55 carry two marks each.
Q.26 Let ? and ? be two random variables having the joint probability density function
, 2 ???if?0 1
0otherwise.
Then the conditional probability
| is equal to
(A)
(B)
(C)
(D)
Q.27 Let ?0,1 be the sample space and let ? be a probability function defined by
0, 2
if 0 1
2
?????if??
1
2
x1.
Then
is equal to __________
FirstRanker.com - FirstRanker's Choice
GATE 2015 MATHEMATICS ? MA
MA 1/9
List of Symbols, Notations and Data
, :?Binomial distribution with ? trials and success probability ; ? ? 1,2, ? ?and?? ?0, 1
, :?Uniform distribution on the interval , ,? ??
, :??Normal distribution with mean ? and variance ,? ?, ? ,0
?? Probability of the event ?
Poisson :?Poisson distribution with mean , 0
:?Expected value (mean) of the random variable
If ? 0,1 ,??then 1.96 ?0.975? and 0.54 ?0.7054
?? Set of integers
???Set of rational numbers
?? Set of real numbers
? Set of complex numbers
:?The cyclic group of order ?
[ ] : Polynomial ring over the field
0, 1 ??Set of all real valued continuous functions on the interval 0, 1
0, 1 ??Set of all real valued continuously differentiable functions on the interval 0, 1
?
??Normed space of all square-summable real sequences
0, 1 :?Space of all square-Lebesgue integrable real valued functions on the interval 0, 1
0, 1 ,??
:? The space 0, 1 ?with ? ?
? |? |
?
?
0, 1 ,??
:? The space 0, 1 ?with ? ?
sup ?| |??? ? 0, 1 ?
:? The orthogonal complement of ? in an inner product space
?? ? -dimensional Euclidean space
Usual metric ? on ??is given by ,
,?,
, ,
,?, ? ? ?
?
:??The ? identity matrix ( ?? the identity matrix when order is NOT specified)
??The order of the element ? of a group
GATE 2015 MATHEMATICS ? MA
MA 2/9
Q. 1 ? Q. 25 carry one mark each.
Q.1 Let ?? ?? ?be a linear map defined by
,,,,2 3,22, .
Then the rank of is equal to _________
Q.2 Let be a 33 matrix and suppose that 1, 2 and 3 are the eigenvalues of . If
11
for some scalar 0,? then is equal to ___________
Q.3 Let be a 33 singular matrix and suppose that 2 and 3 are eigenvalues of . Then the number
of linearly independent eigenvectors of 2 is equal to __________
Q.4
Let be a 33 matrix such that
2
1
0
6
3
0
and suppose that ?
1
1/2
0
? for
some , , ? . Then |?? | is equal to _______
Q.5 Let : 0, ? ? ? be defined by
sin
.
Then the function is
(A) uniformly continuous on 0, 1 but NOT on 0, ?
(B) uniformly continuous on 0, ? but NOT on 0, 1
(C) uniformly continuous on both 0, 1 and 0, ?
(D) neither uniformly continuous on 0, 1 nor uniformly continuous on 0, ?
Q.6
Consider the power series ? ,
where if iseven
if is odd.
The radius of convergence of the series is equal to __________
Q.7
Let ?? ? ? ? | |2 .? Then
?
is equal to ____________
Q.8
Let ?~?5,
and ?~? 0,1 . Then
is equal to ___________
GATE 2015 MATHEMATICS ? MA
MA 3/9
Q.9 Let the random variable ? have the distribution function
?
0?????????????? ???if?????? 0 ?????
2
??????????? ???????if??0? 1
3
5
??????????? ???????if??1? 2?
1
2
?
8
?????????? if??2? 3
1if3.
Then 2 4 is equal to ___________
Q.10 Let X be a random variable having the distribution function
?
0if 0
1
4
??????????? if??0? 1
1
3
??????????? if??1? 2?
1
2
??????????? if??2? 11
3
1if
11
3
.
Then is equal to _________
Q.11 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
(A)
(B)
(C)
(D)
Q.12 Let 2.2, 4.3, 3.1, 4.5, 1.1and 5.7 be the observed values of a
random sample of size 6 from a 1, 4 distribution, where ?0,? is unknown. Then
a maximum likelihood estimate of is equal to
(A) 1.8 (B) 2.3 (C) 3.1 (D) 3.6
Q.13 Let ?? , ?? |?
? 1 be the open unit disc in with boundary ?. If , is
the solution of the Dirichlet problem
?
0???????????in???
, 12 on ?,
then ,0 is equal to
(A) 1
(B)
(C)
(D) 1
GATE 2015 MATHEMATICS ? MA
MA 4/9
Q.14
Let ? ??be such that
? ?? ?
is a field. Then is equal to __________
Q.15 Let 0, 1, ? 0, 1 ,??
and 0, 1 ,??
. Then ? is
(A) dense in ? but NOT in ?
(B) dense in ? but NOT in ?
(C) dense in both and ?
(D) neither dense in ? nor dense in
Q.16
Let ?? ? 0, 1 ,??
?? ?be defined by 2
for all ? 0, 1 . Then ? ?
is equal to __________
Q.17 Let be the usual topology on . Let be the topology on generated by
, ? ??? . Then the set ? ?4 sin
1??? ?
is
(A) closed in ,
?but NOT in ,
(B) closed in ,
?but NOT in ,
(C) closed in both ,
?and ,
(D) neither closed in ,
?nor closed in ,
Q.18 Let ? be a connected topological space such that there exists a non-constant continuous function
? ? , where is equipped with the usual topology. Let :? ? . Then
(A) ? is countable but ?is uncountable
(B) is countable but ?? is uncountable
(C) both ?and ?? are countable
(D) both and ? are uncountable
Q.19 Let ? and ?denote the usual metric and the discrete metric on , respectively.
Let ? ,
? ,
be defined by , ? . Then
(A) is continuous but
? is NOT continuous
(B)
is continuous but ?is NOT continuous
(C) both ? and
are continuous
(D) neither ?nor
?is continuous
Q.20 If the trapezoidal rule with single interval 0, 1 is exact for approximating the integral
??,
then the value of is equal to ________
Q.21
Suppose that the Newton-Raphson method is applied to the equation 2
?1
0 with an
initial approximation ? sufficiently close to zero. Then, for the root 0,? the order of
convergence of the method is equal to _________
GATE 2015 MATHEMATICS ? MA
MA 5/9
Q.22 The minimum possible order of a homogeneous linear ordinary differential equation with real
constant coefficients having sin as a solution is equal to __________
Q.23 The Lagrangian of a system in terms of polar coordinates , is given by
?
1
2
?? 1
2
1 cos,
where ? is the mass, ? is the acceleration due to gravity and denotes the derivative of with
respect to time. Then the equations of motion are
(A) 2???1 ?cos,
sin
(B) 2?? ?1 ?cos,
sin
(C) 2?? ?1 ?cos,
sin
(D) 2???1 ?cos,
sin
Q.24 If ?satisfies the initial value problem
, 1 2,
then 2 is equal to __________
Q.25 It is known that Bessel functions , for 0, satisfy the identity
??
???
?
1 ?
?
for all 0? and ? .??The value of 2 ?
is equal to _________
Q. 26 ? Q. 55 carry two marks each.
Q.26 Let ? and ? be two random variables having the joint probability density function
, 2 ???if?0 1
0otherwise.
Then the conditional probability
| is equal to
(A)
(B)
(C)
(D)
Q.27 Let ?0,1 be the sample space and let ? be a probability function defined by
0, 2
if 0 1
2
?????if??
1
2
x1.
Then
is equal to __________
GATE 2015 MATHEMATICS ? MA
MA 6/9
Q.28 Let ,
?and?
be independent and identically distributed random variables with 0 and
?
. If ? 0, ? ?? 0, ? is defined through the conditional expectation
?? ? ?
,??? 0 ,
then ? is equal to __________
Q.29 Let ? Poisson , where 0 is unknown. If is the unbiased estimator of
?
3
21 ,?then ? is equal to ___________
Q.30
Let ,?, be a random sample from , 1 distribution, where ?0,
. For testing the null
hypothesis : 0 against the alternative hypothesis : , consider the critical region
? ,
,?,
??
,
where ? is some real constant. If the critical region has size 0.025 and power 0.7054, then the
value of the sample size n is equal to ___________
Q.31 Let ? and ? be independently distributed central chi-squared random variables with degrees of
freedom ? 3? and ? 3 , respectively. If 3 and 14, then is equal to
(A)
(B)
(C)
(D)
Q.32 Let ,
,? be a sequence of independent and identically distributed random variables with
?1 ?
? and ?2 . If ? ,for1,2,?,
then
lim
? 1.8 is equal to __________
Q.33 Let , 2 cos2 , , ? , be a solution of the initial value problem
2
???????
, 0 cos .
Then 1 is equal to
(A)
(B)
(C)
(D)
Q.34 Let , ,? ? , 0,? be the solution of the initial value problem
?????? , 0??????
, 0 ?1.
Then 2,2 is equal to ________
FirstRanker.com - FirstRanker's Choice
GATE 2015 MATHEMATICS ? MA
MA 1/9
List of Symbols, Notations and Data
, :?Binomial distribution with ? trials and success probability ; ? ? 1,2, ? ?and?? ?0, 1
, :?Uniform distribution on the interval , ,? ??
, :??Normal distribution with mean ? and variance ,? ?, ? ,0
?? Probability of the event ?
Poisson :?Poisson distribution with mean , 0
:?Expected value (mean) of the random variable
If ? 0,1 ,??then 1.96 ?0.975? and 0.54 ?0.7054
?? Set of integers
???Set of rational numbers
?? Set of real numbers
? Set of complex numbers
:?The cyclic group of order ?
[ ] : Polynomial ring over the field
0, 1 ??Set of all real valued continuous functions on the interval 0, 1
0, 1 ??Set of all real valued continuously differentiable functions on the interval 0, 1
?
??Normed space of all square-summable real sequences
0, 1 :?Space of all square-Lebesgue integrable real valued functions on the interval 0, 1
0, 1 ,??
:? The space 0, 1 ?with ? ?
? |? |
?
?
0, 1 ,??
:? The space 0, 1 ?with ? ?
sup ?| |??? ? 0, 1 ?
:? The orthogonal complement of ? in an inner product space
?? ? -dimensional Euclidean space
Usual metric ? on ??is given by ,
,?,
, ,
,?, ? ? ?
?
:??The ? identity matrix ( ?? the identity matrix when order is NOT specified)
??The order of the element ? of a group
GATE 2015 MATHEMATICS ? MA
MA 2/9
Q. 1 ? Q. 25 carry one mark each.
Q.1 Let ?? ?? ?be a linear map defined by
,,,,2 3,22, .
Then the rank of is equal to _________
Q.2 Let be a 33 matrix and suppose that 1, 2 and 3 are the eigenvalues of . If
11
for some scalar 0,? then is equal to ___________
Q.3 Let be a 33 singular matrix and suppose that 2 and 3 are eigenvalues of . Then the number
of linearly independent eigenvectors of 2 is equal to __________
Q.4
Let be a 33 matrix such that
2
1
0
6
3
0
and suppose that ?
1
1/2
0
? for
some , , ? . Then |?? | is equal to _______
Q.5 Let : 0, ? ? ? be defined by
sin
.
Then the function is
(A) uniformly continuous on 0, 1 but NOT on 0, ?
(B) uniformly continuous on 0, ? but NOT on 0, 1
(C) uniformly continuous on both 0, 1 and 0, ?
(D) neither uniformly continuous on 0, 1 nor uniformly continuous on 0, ?
Q.6
Consider the power series ? ,
where if iseven
if is odd.
The radius of convergence of the series is equal to __________
Q.7
Let ?? ? ? ? | |2 .? Then
?
is equal to ____________
Q.8
Let ?~?5,
and ?~? 0,1 . Then
is equal to ___________
GATE 2015 MATHEMATICS ? MA
MA 3/9
Q.9 Let the random variable ? have the distribution function
?
0?????????????? ???if?????? 0 ?????
2
??????????? ???????if??0? 1
3
5
??????????? ???????if??1? 2?
1
2
?
8
?????????? if??2? 3
1if3.
Then 2 4 is equal to ___________
Q.10 Let X be a random variable having the distribution function
?
0if 0
1
4
??????????? if??0? 1
1
3
??????????? if??1? 2?
1
2
??????????? if??2? 11
3
1if
11
3
.
Then is equal to _________
Q.11 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
(A)
(B)
(C)
(D)
Q.12 Let 2.2, 4.3, 3.1, 4.5, 1.1and 5.7 be the observed values of a
random sample of size 6 from a 1, 4 distribution, where ?0,? is unknown. Then
a maximum likelihood estimate of is equal to
(A) 1.8 (B) 2.3 (C) 3.1 (D) 3.6
Q.13 Let ?? , ?? |?
? 1 be the open unit disc in with boundary ?. If , is
the solution of the Dirichlet problem
?
0???????????in???
, 12 on ?,
then ,0 is equal to
(A) 1
(B)
(C)
(D) 1
GATE 2015 MATHEMATICS ? MA
MA 4/9
Q.14
Let ? ??be such that
? ?? ?
is a field. Then is equal to __________
Q.15 Let 0, 1, ? 0, 1 ,??
and 0, 1 ,??
. Then ? is
(A) dense in ? but NOT in ?
(B) dense in ? but NOT in ?
(C) dense in both and ?
(D) neither dense in ? nor dense in
Q.16
Let ?? ? 0, 1 ,??
?? ?be defined by 2
for all ? 0, 1 . Then ? ?
is equal to __________
Q.17 Let be the usual topology on . Let be the topology on generated by
, ? ??? . Then the set ? ?4 sin
1??? ?
is
(A) closed in ,
?but NOT in ,
(B) closed in ,
?but NOT in ,
(C) closed in both ,
?and ,
(D) neither closed in ,
?nor closed in ,
Q.18 Let ? be a connected topological space such that there exists a non-constant continuous function
? ? , where is equipped with the usual topology. Let :? ? . Then
(A) ? is countable but ?is uncountable
(B) is countable but ?? is uncountable
(C) both ?and ?? are countable
(D) both and ? are uncountable
Q.19 Let ? and ?denote the usual metric and the discrete metric on , respectively.
Let ? ,
? ,
be defined by , ? . Then
(A) is continuous but
? is NOT continuous
(B)
is continuous but ?is NOT continuous
(C) both ? and
are continuous
(D) neither ?nor
?is continuous
Q.20 If the trapezoidal rule with single interval 0, 1 is exact for approximating the integral
??,
then the value of is equal to ________
Q.21
Suppose that the Newton-Raphson method is applied to the equation 2
?1
0 with an
initial approximation ? sufficiently close to zero. Then, for the root 0,? the order of
convergence of the method is equal to _________
GATE 2015 MATHEMATICS ? MA
MA 5/9
Q.22 The minimum possible order of a homogeneous linear ordinary differential equation with real
constant coefficients having sin as a solution is equal to __________
Q.23 The Lagrangian of a system in terms of polar coordinates , is given by
?
1
2
?? 1
2
1 cos,
where ? is the mass, ? is the acceleration due to gravity and denotes the derivative of with
respect to time. Then the equations of motion are
(A) 2???1 ?cos,
sin
(B) 2?? ?1 ?cos,
sin
(C) 2?? ?1 ?cos,
sin
(D) 2???1 ?cos,
sin
Q.24 If ?satisfies the initial value problem
, 1 2,
then 2 is equal to __________
Q.25 It is known that Bessel functions , for 0, satisfy the identity
??
???
?
1 ?
?
for all 0? and ? .??The value of 2 ?
is equal to _________
Q. 26 ? Q. 55 carry two marks each.
Q.26 Let ? and ? be two random variables having the joint probability density function
, 2 ???if?0 1
0otherwise.
Then the conditional probability
| is equal to
(A)
(B)
(C)
(D)
Q.27 Let ?0,1 be the sample space and let ? be a probability function defined by
0, 2
if 0 1
2
?????if??
1
2
x1.
Then
is equal to __________
GATE 2015 MATHEMATICS ? MA
MA 6/9
Q.28 Let ,
?and?
be independent and identically distributed random variables with 0 and
?
. If ? 0, ? ?? 0, ? is defined through the conditional expectation
?? ? ?
,??? 0 ,
then ? is equal to __________
Q.29 Let ? Poisson , where 0 is unknown. If is the unbiased estimator of
?
3
21 ,?then ? is equal to ___________
Q.30
Let ,?, be a random sample from , 1 distribution, where ?0,
. For testing the null
hypothesis : 0 against the alternative hypothesis : , consider the critical region
? ,
,?,
??
,
where ? is some real constant. If the critical region has size 0.025 and power 0.7054, then the
value of the sample size n is equal to ___________
Q.31 Let ? and ? be independently distributed central chi-squared random variables with degrees of
freedom ? 3? and ? 3 , respectively. If 3 and 14, then is equal to
(A)
(B)
(C)
(D)
Q.32 Let ,
,? be a sequence of independent and identically distributed random variables with
?1 ?
? and ?2 . If ? ,for1,2,?,
then
lim
? 1.8 is equal to __________
Q.33 Let , 2 cos2 , , ? , be a solution of the initial value problem
2
???????
, 0 cos .
Then 1 is equal to
(A)
(B)
(C)
(D)
Q.34 Let , ,? ? , 0,? be the solution of the initial value problem
?????? , 0??????
, 0 ?1.
Then 2,2 is equal to ________
GATE 2015 MATHEMATICS ? MA
MA 7/9
Q.35
Let Span? ?
? 0,0,1,1 ,
? 1, 1,0,0 be a subspace of the Euclidean space . Then the
square of the distance from the point 1,1,1,1 to the subspace is equal to ________
Q.36 Let ?? ?? ?be a linear map such that the null space of is
,,, ? : 0
and the rank of 4?
? is 3. If the minimal polynomial of is 4 ,??
then ? is equal to _______
Q.37 Let ? be an invertible Hermitian matrix and let , ? be such that 4.? Then
(A) both ???? and are singular
(B) ?
?? is singular but is non-singular
(C) ?
??? is non-singular but is singular
(D) both ???? and are non-singular
Q.38 Let ?,,
,
,,, , with 4, 2 and .??Then the number
of elements in the center of the group is equal to
(A) 1 (B) 2 (C) 4 (D) 8
Q.39 The number of ring homomorphisms from
to is equal to __________
Q.40 Let ?9? 10? ?5? 15 and
2 be two polynomials in
?? .????Then, over ?,
(A) ?and are both irreducible
(B) ? is reducible but ?is irreducible
(C) ? is irreducible but ?is reducible
(D) ? and are both reducible
Q.41 Consider the linear programming problem
Maximize 3 9,
???????????????????
?subject?to???????????????????2? 2
????????????????????????????????????????3? 0
2? 3? 10?
, 0.
Then the maximum value of the objective function is equal to ______
Q.42
Let ?,sin
? 0 1?? and ? 0,0 . Under the usual metric on ,
(A) ? is closed but ? is NOT closed
(B) ? is closed but ? is NOT closed
(C) both ? and ? are closed
(D) neither ? nor is closed
FirstRanker.com - FirstRanker's Choice
GATE 2015 MATHEMATICS ? MA
MA 1/9
List of Symbols, Notations and Data
, :?Binomial distribution with ? trials and success probability ; ? ? 1,2, ? ?and?? ?0, 1
, :?Uniform distribution on the interval , ,? ??
, :??Normal distribution with mean ? and variance ,? ?, ? ,0
?? Probability of the event ?
Poisson :?Poisson distribution with mean , 0
:?Expected value (mean) of the random variable
If ? 0,1 ,??then 1.96 ?0.975? and 0.54 ?0.7054
?? Set of integers
???Set of rational numbers
?? Set of real numbers
? Set of complex numbers
:?The cyclic group of order ?
[ ] : Polynomial ring over the field
0, 1 ??Set of all real valued continuous functions on the interval 0, 1
0, 1 ??Set of all real valued continuously differentiable functions on the interval 0, 1
?
??Normed space of all square-summable real sequences
0, 1 :?Space of all square-Lebesgue integrable real valued functions on the interval 0, 1
0, 1 ,??
:? The space 0, 1 ?with ? ?
? |? |
?
?
0, 1 ,??
:? The space 0, 1 ?with ? ?
sup ?| |??? ? 0, 1 ?
:? The orthogonal complement of ? in an inner product space
?? ? -dimensional Euclidean space
Usual metric ? on ??is given by ,
,?,
, ,
,?, ? ? ?
?
:??The ? identity matrix ( ?? the identity matrix when order is NOT specified)
??The order of the element ? of a group
GATE 2015 MATHEMATICS ? MA
MA 2/9
Q. 1 ? Q. 25 carry one mark each.
Q.1 Let ?? ?? ?be a linear map defined by
,,,,2 3,22, .
Then the rank of is equal to _________
Q.2 Let be a 33 matrix and suppose that 1, 2 and 3 are the eigenvalues of . If
11
for some scalar 0,? then is equal to ___________
Q.3 Let be a 33 singular matrix and suppose that 2 and 3 are eigenvalues of . Then the number
of linearly independent eigenvectors of 2 is equal to __________
Q.4
Let be a 33 matrix such that
2
1
0
6
3
0
and suppose that ?
1
1/2
0
? for
some , , ? . Then |?? | is equal to _______
Q.5 Let : 0, ? ? ? be defined by
sin
.
Then the function is
(A) uniformly continuous on 0, 1 but NOT on 0, ?
(B) uniformly continuous on 0, ? but NOT on 0, 1
(C) uniformly continuous on both 0, 1 and 0, ?
(D) neither uniformly continuous on 0, 1 nor uniformly continuous on 0, ?
Q.6
Consider the power series ? ,
where if iseven
if is odd.
The radius of convergence of the series is equal to __________
Q.7
Let ?? ? ? ? | |2 .? Then
?
is equal to ____________
Q.8
Let ?~?5,
and ?~? 0,1 . Then
is equal to ___________
GATE 2015 MATHEMATICS ? MA
MA 3/9
Q.9 Let the random variable ? have the distribution function
?
0?????????????? ???if?????? 0 ?????
2
??????????? ???????if??0? 1
3
5
??????????? ???????if??1? 2?
1
2
?
8
?????????? if??2? 3
1if3.
Then 2 4 is equal to ___________
Q.10 Let X be a random variable having the distribution function
?
0if 0
1
4
??????????? if??0? 1
1
3
??????????? if??1? 2?
1
2
??????????? if??2? 11
3
1if
11
3
.
Then is equal to _________
Q.11 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
(A)
(B)
(C)
(D)
Q.12 Let 2.2, 4.3, 3.1, 4.5, 1.1and 5.7 be the observed values of a
random sample of size 6 from a 1, 4 distribution, where ?0,? is unknown. Then
a maximum likelihood estimate of is equal to
(A) 1.8 (B) 2.3 (C) 3.1 (D) 3.6
Q.13 Let ?? , ?? |?
? 1 be the open unit disc in with boundary ?. If , is
the solution of the Dirichlet problem
?
0???????????in???
, 12 on ?,
then ,0 is equal to
(A) 1
(B)
(C)
(D) 1
GATE 2015 MATHEMATICS ? MA
MA 4/9
Q.14
Let ? ??be such that
? ?? ?
is a field. Then is equal to __________
Q.15 Let 0, 1, ? 0, 1 ,??
and 0, 1 ,??
. Then ? is
(A) dense in ? but NOT in ?
(B) dense in ? but NOT in ?
(C) dense in both and ?
(D) neither dense in ? nor dense in
Q.16
Let ?? ? 0, 1 ,??
?? ?be defined by 2
for all ? 0, 1 . Then ? ?
is equal to __________
Q.17 Let be the usual topology on . Let be the topology on generated by
, ? ??? . Then the set ? ?4 sin
1??? ?
is
(A) closed in ,
?but NOT in ,
(B) closed in ,
?but NOT in ,
(C) closed in both ,
?and ,
(D) neither closed in ,
?nor closed in ,
Q.18 Let ? be a connected topological space such that there exists a non-constant continuous function
? ? , where is equipped with the usual topology. Let :? ? . Then
(A) ? is countable but ?is uncountable
(B) is countable but ?? is uncountable
(C) both ?and ?? are countable
(D) both and ? are uncountable
Q.19 Let ? and ?denote the usual metric and the discrete metric on , respectively.
Let ? ,
? ,
be defined by , ? . Then
(A) is continuous but
? is NOT continuous
(B)
is continuous but ?is NOT continuous
(C) both ? and
are continuous
(D) neither ?nor
?is continuous
Q.20 If the trapezoidal rule with single interval 0, 1 is exact for approximating the integral
??,
then the value of is equal to ________
Q.21
Suppose that the Newton-Raphson method is applied to the equation 2
?1
0 with an
initial approximation ? sufficiently close to zero. Then, for the root 0,? the order of
convergence of the method is equal to _________
GATE 2015 MATHEMATICS ? MA
MA 5/9
Q.22 The minimum possible order of a homogeneous linear ordinary differential equation with real
constant coefficients having sin as a solution is equal to __________
Q.23 The Lagrangian of a system in terms of polar coordinates , is given by
?
1
2
?? 1
2
1 cos,
where ? is the mass, ? is the acceleration due to gravity and denotes the derivative of with
respect to time. Then the equations of motion are
(A) 2???1 ?cos,
sin
(B) 2?? ?1 ?cos,
sin
(C) 2?? ?1 ?cos,
sin
(D) 2???1 ?cos,
sin
Q.24 If ?satisfies the initial value problem
, 1 2,
then 2 is equal to __________
Q.25 It is known that Bessel functions , for 0, satisfy the identity
??
???
?
1 ?
?
for all 0? and ? .??The value of 2 ?
is equal to _________
Q. 26 ? Q. 55 carry two marks each.
Q.26 Let ? and ? be two random variables having the joint probability density function
, 2 ???if?0 1
0otherwise.
Then the conditional probability
| is equal to
(A)
(B)
(C)
(D)
Q.27 Let ?0,1 be the sample space and let ? be a probability function defined by
0, 2
if 0 1
2
?????if??
1
2
x1.
Then
is equal to __________
GATE 2015 MATHEMATICS ? MA
MA 6/9
Q.28 Let ,
?and?
be independent and identically distributed random variables with 0 and
?
. If ? 0, ? ?? 0, ? is defined through the conditional expectation
?? ? ?
,??? 0 ,
then ? is equal to __________
Q.29 Let ? Poisson , where 0 is unknown. If is the unbiased estimator of
?
3
21 ,?then ? is equal to ___________
Q.30
Let ,?, be a random sample from , 1 distribution, where ?0,
. For testing the null
hypothesis : 0 against the alternative hypothesis : , consider the critical region
? ,
,?,
??
,
where ? is some real constant. If the critical region has size 0.025 and power 0.7054, then the
value of the sample size n is equal to ___________
Q.31 Let ? and ? be independently distributed central chi-squared random variables with degrees of
freedom ? 3? and ? 3 , respectively. If 3 and 14, then is equal to
(A)
(B)
(C)
(D)
Q.32 Let ,
,? be a sequence of independent and identically distributed random variables with
?1 ?
? and ?2 . If ? ,for1,2,?,
then
lim
? 1.8 is equal to __________
Q.33 Let , 2 cos2 , , ? , be a solution of the initial value problem
2
???????
, 0 cos .
Then 1 is equal to
(A)
(B)
(C)
(D)
Q.34 Let , ,? ? , 0,? be the solution of the initial value problem
?????? , 0??????
, 0 ?1.
Then 2,2 is equal to ________
GATE 2015 MATHEMATICS ? MA
MA 7/9
Q.35
Let Span? ?
? 0,0,1,1 ,
? 1, 1,0,0 be a subspace of the Euclidean space . Then the
square of the distance from the point 1,1,1,1 to the subspace is equal to ________
Q.36 Let ?? ?? ?be a linear map such that the null space of is
,,, ? : 0
and the rank of 4?
? is 3. If the minimal polynomial of is 4 ,??
then ? is equal to _______
Q.37 Let ? be an invertible Hermitian matrix and let , ? be such that 4.? Then
(A) both ???? and are singular
(B) ?
?? is singular but is non-singular
(C) ?
??? is non-singular but is singular
(D) both ???? and are non-singular
Q.38 Let ?,,
,
,,, , with 4, 2 and .??Then the number
of elements in the center of the group is equal to
(A) 1 (B) 2 (C) 4 (D) 8
Q.39 The number of ring homomorphisms from
to is equal to __________
Q.40 Let ?9? 10? ?5? 15 and
2 be two polynomials in
?? .????Then, over ?,
(A) ?and are both irreducible
(B) ? is reducible but ?is irreducible
(C) ? is irreducible but ?is reducible
(D) ? and are both reducible
Q.41 Consider the linear programming problem
Maximize 3 9,
???????????????????
?subject?to???????????????????2? 2
????????????????????????????????????????3? 0
2? 3? 10?
, 0.
Then the maximum value of the objective function is equal to ______
Q.42
Let ?,sin
? 0 1?? and ? 0,0 . Under the usual metric on ,
(A) ? is closed but ? is NOT closed
(B) ? is closed but ? is NOT closed
(C) both ? and ? are closed
(D) neither ? nor is closed
GATE 2015 MATHEMATICS ? MA
MA 8/9
Q.43
Let ? ?? ?
:??
1 . Then
(A) is bounded (B) is closed
(C) is a subspace (D) has an interior point
Q.44 Let ? be a closed subspace of 0, 1 and let , ? 0, 1 be given by ? and
?
. If Span? ? ? and is the orthogonal projection of ? on ,? then
,? 0, 1? , is
(A)
(B)
(C)
(D)
Q.45 Let ?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 in is equal to _________
Q.46 If, for some , ? ,? the integration formula
holds for all polynomials of degree at most 3, then the value of 3? is equal to _____
Q.47 Let be a continuous function on 0, ? whose Laplace transform exists. If ?satisfies
1 cos ,
then 1 is equal to _______
Q.48 Consider the initial value problem
?
60, 1, 1 6.
If ?0? as ? 0
,? then ? is equal to __________
Q.49 Define , : 0,1? ? ?by
? ? sin
and 1
?.
Then
(A) is continuous but is NOT continuous
(B) is continuous but is NOT continuous
(C) both and are continuous
(D) neither nor is continuous
Q.50 Consider the unit sphere ,, ? :
1 and the unit normal vector
,, ? at each point ,, on . The value of the surface integral
?
2? ?sin
2?
sin
????
is equal to _______
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GATE 2015 MATHEMATICS ? MA
MA 1/9
List of Symbols, Notations and Data
, :?Binomial distribution with ? trials and success probability ; ? ? 1,2, ? ?and?? ?0, 1
, :?Uniform distribution on the interval , ,? ??
, :??Normal distribution with mean ? and variance ,? ?, ? ,0
?? Probability of the event ?
Poisson :?Poisson distribution with mean , 0
:?Expected value (mean) of the random variable
If ? 0,1 ,??then 1.96 ?0.975? and 0.54 ?0.7054
?? Set of integers
???Set of rational numbers
?? Set of real numbers
? Set of complex numbers
:?The cyclic group of order ?
[ ] : Polynomial ring over the field
0, 1 ??Set of all real valued continuous functions on the interval 0, 1
0, 1 ??Set of all real valued continuously differentiable functions on the interval 0, 1
?
??Normed space of all square-summable real sequences
0, 1 :?Space of all square-Lebesgue integrable real valued functions on the interval 0, 1
0, 1 ,??
:? The space 0, 1 ?with ? ?
? |? |
?
?
0, 1 ,??
:? The space 0, 1 ?with ? ?
sup ?| |??? ? 0, 1 ?
:? The orthogonal complement of ? in an inner product space
?? ? -dimensional Euclidean space
Usual metric ? on ??is given by ,
,?,
, ,
,?, ? ? ?
?
:??The ? identity matrix ( ?? the identity matrix when order is NOT specified)
??The order of the element ? of a group
GATE 2015 MATHEMATICS ? MA
MA 2/9
Q. 1 ? Q. 25 carry one mark each.
Q.1 Let ?? ?? ?be a linear map defined by
,,,,2 3,22, .
Then the rank of is equal to _________
Q.2 Let be a 33 matrix and suppose that 1, 2 and 3 are the eigenvalues of . If
11
for some scalar 0,? then is equal to ___________
Q.3 Let be a 33 singular matrix and suppose that 2 and 3 are eigenvalues of . Then the number
of linearly independent eigenvectors of 2 is equal to __________
Q.4
Let be a 33 matrix such that
2
1
0
6
3
0
and suppose that ?
1
1/2
0
? for
some , , ? . Then |?? | is equal to _______
Q.5 Let : 0, ? ? ? be defined by
sin
.
Then the function is
(A) uniformly continuous on 0, 1 but NOT on 0, ?
(B) uniformly continuous on 0, ? but NOT on 0, 1
(C) uniformly continuous on both 0, 1 and 0, ?
(D) neither uniformly continuous on 0, 1 nor uniformly continuous on 0, ?
Q.6
Consider the power series ? ,
where if iseven
if is odd.
The radius of convergence of the series is equal to __________
Q.7
Let ?? ? ? ? | |2 .? Then
?
is equal to ____________
Q.8
Let ?~?5,
and ?~? 0,1 . Then
is equal to ___________
GATE 2015 MATHEMATICS ? MA
MA 3/9
Q.9 Let the random variable ? have the distribution function
?
0?????????????? ???if?????? 0 ?????
2
??????????? ???????if??0? 1
3
5
??????????? ???????if??1? 2?
1
2
?
8
?????????? if??2? 3
1if3.
Then 2 4 is equal to ___________
Q.10 Let X be a random variable having the distribution function
?
0if 0
1
4
??????????? if??0? 1
1
3
??????????? if??1? 2?
1
2
??????????? if??2? 11
3
1if
11
3
.
Then is equal to _________
Q.11 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
(A)
(B)
(C)
(D)
Q.12 Let 2.2, 4.3, 3.1, 4.5, 1.1and 5.7 be the observed values of a
random sample of size 6 from a 1, 4 distribution, where ?0,? is unknown. Then
a maximum likelihood estimate of is equal to
(A) 1.8 (B) 2.3 (C) 3.1 (D) 3.6
Q.13 Let ?? , ?? |?
? 1 be the open unit disc in with boundary ?. If , is
the solution of the Dirichlet problem
?
0???????????in???
, 12 on ?,
then ,0 is equal to
(A) 1
(B)
(C)
(D) 1
GATE 2015 MATHEMATICS ? MA
MA 4/9
Q.14
Let ? ??be such that
? ?? ?
is a field. Then is equal to __________
Q.15 Let 0, 1, ? 0, 1 ,??
and 0, 1 ,??
. Then ? is
(A) dense in ? but NOT in ?
(B) dense in ? but NOT in ?
(C) dense in both and ?
(D) neither dense in ? nor dense in
Q.16
Let ?? ? 0, 1 ,??
?? ?be defined by 2
for all ? 0, 1 . Then ? ?
is equal to __________
Q.17 Let be the usual topology on . Let be the topology on generated by
, ? ??? . Then the set ? ?4 sin
1??? ?
is
(A) closed in ,
?but NOT in ,
(B) closed in ,
?but NOT in ,
(C) closed in both ,
?and ,
(D) neither closed in ,
?nor closed in ,
Q.18 Let ? be a connected topological space such that there exists a non-constant continuous function
? ? , where is equipped with the usual topology. Let :? ? . Then
(A) ? is countable but ?is uncountable
(B) is countable but ?? is uncountable
(C) both ?and ?? are countable
(D) both and ? are uncountable
Q.19 Let ? and ?denote the usual metric and the discrete metric on , respectively.
Let ? ,
? ,
be defined by , ? . Then
(A) is continuous but
? is NOT continuous
(B)
is continuous but ?is NOT continuous
(C) both ? and
are continuous
(D) neither ?nor
?is continuous
Q.20 If the trapezoidal rule with single interval 0, 1 is exact for approximating the integral
??,
then the value of is equal to ________
Q.21
Suppose that the Newton-Raphson method is applied to the equation 2
?1
0 with an
initial approximation ? sufficiently close to zero. Then, for the root 0,? the order of
convergence of the method is equal to _________
GATE 2015 MATHEMATICS ? MA
MA 5/9
Q.22 The minimum possible order of a homogeneous linear ordinary differential equation with real
constant coefficients having sin as a solution is equal to __________
Q.23 The Lagrangian of a system in terms of polar coordinates , is given by
?
1
2
?? 1
2
1 cos,
where ? is the mass, ? is the acceleration due to gravity and denotes the derivative of with
respect to time. Then the equations of motion are
(A) 2???1 ?cos,
sin
(B) 2?? ?1 ?cos,
sin
(C) 2?? ?1 ?cos,
sin
(D) 2???1 ?cos,
sin
Q.24 If ?satisfies the initial value problem
, 1 2,
then 2 is equal to __________
Q.25 It is known that Bessel functions , for 0, satisfy the identity
??
???
?
1 ?
?
for all 0? and ? .??The value of 2 ?
is equal to _________
Q. 26 ? Q. 55 carry two marks each.
Q.26 Let ? and ? be two random variables having the joint probability density function
, 2 ???if?0 1
0otherwise.
Then the conditional probability
| is equal to
(A)
(B)
(C)
(D)
Q.27 Let ?0,1 be the sample space and let ? be a probability function defined by
0, 2
if 0 1
2
?????if??
1
2
x1.
Then
is equal to __________
GATE 2015 MATHEMATICS ? MA
MA 6/9
Q.28 Let ,
?and?
be independent and identically distributed random variables with 0 and
?
. If ? 0, ? ?? 0, ? is defined through the conditional expectation
?? ? ?
,??? 0 ,
then ? is equal to __________
Q.29 Let ? Poisson , where 0 is unknown. If is the unbiased estimator of
?
3
21 ,?then ? is equal to ___________
Q.30
Let ,?, be a random sample from , 1 distribution, where ?0,
. For testing the null
hypothesis : 0 against the alternative hypothesis : , consider the critical region
? ,
,?,
??
,
where ? is some real constant. If the critical region has size 0.025 and power 0.7054, then the
value of the sample size n is equal to ___________
Q.31 Let ? and ? be independently distributed central chi-squared random variables with degrees of
freedom ? 3? and ? 3 , respectively. If 3 and 14, then is equal to
(A)
(B)
(C)
(D)
Q.32 Let ,
,? be a sequence of independent and identically distributed random variables with
?1 ?
? and ?2 . If ? ,for1,2,?,
then
lim
? 1.8 is equal to __________
Q.33 Let , 2 cos2 , , ? , be a solution of the initial value problem
2
???????
, 0 cos .
Then 1 is equal to
(A)
(B)
(C)
(D)
Q.34 Let , ,? ? , 0,? be the solution of the initial value problem
?????? , 0??????
, 0 ?1.
Then 2,2 is equal to ________
GATE 2015 MATHEMATICS ? MA
MA 7/9
Q.35
Let Span? ?
? 0,0,1,1 ,
? 1, 1,0,0 be a subspace of the Euclidean space . Then the
square of the distance from the point 1,1,1,1 to the subspace is equal to ________
Q.36 Let ?? ?? ?be a linear map such that the null space of is
,,, ? : 0
and the rank of 4?
? is 3. If the minimal polynomial of is 4 ,??
then ? is equal to _______
Q.37 Let ? be an invertible Hermitian matrix and let , ? be such that 4.? Then
(A) both ???? and are singular
(B) ?
?? is singular but is non-singular
(C) ?
??? is non-singular but is singular
(D) both ???? and are non-singular
Q.38 Let ?,,
,
,,, , with 4, 2 and .??Then the number
of elements in the center of the group is equal to
(A) 1 (B) 2 (C) 4 (D) 8
Q.39 The number of ring homomorphisms from
to is equal to __________
Q.40 Let ?9? 10? ?5? 15 and
2 be two polynomials in
?? .????Then, over ?,
(A) ?and are both irreducible
(B) ? is reducible but ?is irreducible
(C) ? is irreducible but ?is reducible
(D) ? and are both reducible
Q.41 Consider the linear programming problem
Maximize 3 9,
???????????????????
?subject?to???????????????????2? 2
????????????????????????????????????????3? 0
2? 3? 10?
, 0.
Then the maximum value of the objective function is equal to ______
Q.42
Let ?,sin
? 0 1?? and ? 0,0 . Under the usual metric on ,
(A) ? is closed but ? is NOT closed
(B) ? is closed but ? is NOT closed
(C) both ? and ? are closed
(D) neither ? nor is closed
GATE 2015 MATHEMATICS ? MA
MA 8/9
Q.43
Let ? ?? ?
:??
1 . Then
(A) is bounded (B) is closed
(C) is a subspace (D) has an interior point
Q.44 Let ? be a closed subspace of 0, 1 and let , ? 0, 1 be given by ? and
?
. If Span? ? ? and is the orthogonal projection of ? on ,? then
,? 0, 1? , is
(A)
(B)
(C)
(D)
Q.45 Let ?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 in is equal to _________
Q.46 If, for some , ? ,? the integration formula
holds for all polynomials of degree at most 3, then the value of 3? is equal to _____
Q.47 Let be a continuous function on 0, ? whose Laplace transform exists. If ?satisfies
1 cos ,
then 1 is equal to _______
Q.48 Consider the initial value problem
?
60, 1, 1 6.
If ?0? as ? 0
,? then ? is equal to __________
Q.49 Define , : 0,1? ? ?by
? ? sin
and 1
?.
Then
(A) is continuous but is NOT continuous
(B) is continuous but is NOT continuous
(C) both and are continuous
(D) neither nor is continuous
Q.50 Consider the unit sphere ,, ? :
1 and the unit normal vector
,, ? at each point ,, on . The value of the surface integral
?
2? ?sin
2?
sin
????
is equal to _______
GATE 2015 MATHEMATICS ? MA
MA 9/9
Q.51 Let , ? : 1 1000, 1 1000 . Define
, ? 2
500
500
.
Then the minimum value of ? on is equal to ________
Q.52 Let ? ? ? | | ?1? .? Then there exists a non-constant analytic function ?on ? such that
for all 2, 3, 4, ?
(A) ?
?0
(B) 0
(C) 1?
0 (D) 0
Q.53
Let ?
?be the Laurent series expansion of
?in the annulus
|| 5.? Then
???is equal to _________
Q.54
The value of
? ???
? ?
| |
is equal to __________
Q.55 Suppose that among all continuously differentiable functions ,? ,?
with 0 ?0? and 1 ?
, the function minimizes the functional
? 1.
?
Then ? is equal to
(A) 0
(B)
(C)
(D)
END OF THE QUESTION PAPER
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This post was last modified on 19 December 2019