Download GATE (Graduate Aptitude Test in Engineering) Last 10 Years 2020, 2019, 2018, 2017, 2016, 2015, 2014, 2013, 2012, 2011 and 2010 MA Mathematics Question Papers With Solutions And Answer Keys
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
GATE 2016 Mathematics - MA
MA 2/16
Q. 1 ? Q. 25 carry one mark each.
Q. 1 Let , , be a b as is of . Conside r th e f ollowin g statements P and Q:
(P ) : ,, is a bas is of .
(Q ) : , 2 , 3 is a basis of
.
Wh i ch of th e ab ove s t atements h o l d TRUE?
(A) b o th P a nd Q ( B ) only P
(C ) only Q ( D) N either P n or Q
Q. 2 Consider the following stateme nts P an d Q:
(P ) : If 11 1
12 4
13 9
, then M is s in gular .
(Q ) : Let S b e a diagon a l iza b le matrix. If T is a matrix suc h t hat S + 5 T = Id , then T is
diagonalizable.
Which of the above s tate ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n ly Q ( D) Neither P nor Q
Q. 3 Consider the following stateme nts P an d Q:
(P ) : If M i s an complex matrix, then ?
.
(Q ) : There e xis t s a unitary ma trix with an e igen value ? su ch t h a t | ?| <1.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n l y Q ( D) Neither P n or Q
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
GATE 2016 Mathematics - MA
MA 2/16
Q. 1 ? Q. 25 carry one mark each.
Q. 1 Let , , be a b as is of . Conside r th e f ollowin g statements P and Q:
(P ) : ,, is a bas is of .
(Q ) : , 2 , 3 is a basis of
.
Wh i ch of th e ab ove s t atements h o l d TRUE?
(A) b o th P a nd Q ( B ) only P
(C ) only Q ( D) N either P n or Q
Q. 2 Consider the following stateme nts P an d Q:
(P ) : If 11 1
12 4
13 9
, then M is s in gular .
(Q ) : Let S b e a diagon a l iza b le matrix. If T is a matrix suc h t hat S + 5 T = Id , then T is
diagonalizable.
Which of the above s tate ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n ly Q ( D) Neither P nor Q
Q. 3 Consider the following stateme nts P an d Q:
(P ) : If M i s an complex matrix, then ?
.
(Q ) : There e xis t s a unitary ma trix with an e igen value ? su ch t h a t | ?| <1.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n l y Q ( D) Neither P n or Q
GATE 2016 Mathematics - MA
MA 3/16
Q. 4 Consider a r eal vec t or s p a ce V of d imen sion n and a no n ?zero line ar tra ns formation
? ? . If dim e ns ion and , f or some ? \0 , t hen
wh ic h of th e follo wing stateme nts is TRUE?
(A) determina n t | |
(B) Th e re exists a non?tri v ial s u b s p a ce of V such th a t 0 f o r all ? (C) T is invertib le
(D) is th e on ly e igenvalue of T
Q. 5 Let 0, 1 ?2, 3 a nd ? ? be a strictly increasing f u nction such that
is conn ected. Wh i ch of the foll owing statements is TRUE?
(A) h as e xactly one discontinuity
(B) has exactly two di s continu i ti es
(C) h as inf in itely man y d iscontinu i ties
(D) is conti nuou s
Q. 6 Let 1 and
4, 2.Then,
lim
? ?
1
1
? 1
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 7 Maximu m ? , ? 0 ,1 is equal to _ _ _ _ ___ __ ___ __ ___
Q. 8 Let , , , ? su ch that
0 . T h e n, the C auch y pr o blem
,,?,
, 0 o n 0
has a u nique solution if
(A) 0 (B) 0
(C) 0 ( D ) 0
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
GATE 2016 Mathematics - MA
MA 2/16
Q. 1 ? Q. 25 carry one mark each.
Q. 1 Let , , be a b as is of . Conside r th e f ollowin g statements P and Q:
(P ) : ,, is a bas is of .
(Q ) : , 2 , 3 is a basis of
.
Wh i ch of th e ab ove s t atements h o l d TRUE?
(A) b o th P a nd Q ( B ) only P
(C ) only Q ( D) N either P n or Q
Q. 2 Consider the following stateme nts P an d Q:
(P ) : If 11 1
12 4
13 9
, then M is s in gular .
(Q ) : Let S b e a diagon a l iza b le matrix. If T is a matrix suc h t hat S + 5 T = Id , then T is
diagonalizable.
Which of the above s tate ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n ly Q ( D) Neither P nor Q
Q. 3 Consider the following stateme nts P an d Q:
(P ) : If M i s an complex matrix, then ?
.
(Q ) : There e xis t s a unitary ma trix with an e igen value ? su ch t h a t | ?| <1.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n l y Q ( D) Neither P n or Q
GATE 2016 Mathematics - MA
MA 3/16
Q. 4 Consider a r eal vec t or s p a ce V of d imen sion n and a no n ?zero line ar tra ns formation
? ? . If dim e ns ion and , f or some ? \0 , t hen
wh ic h of th e follo wing stateme nts is TRUE?
(A) determina n t | |
(B) Th e re exists a non?tri v ial s u b s p a ce of V such th a t 0 f o r all ? (C) T is invertib le
(D) is th e on ly e igenvalue of T
Q. 5 Let 0, 1 ?2, 3 a nd ? ? be a strictly increasing f u nction such that
is conn ected. Wh i ch of the foll owing statements is TRUE?
(A) h as e xactly one discontinuity
(B) has exactly two di s continu i ti es
(C) h as inf in itely man y d iscontinu i ties
(D) is conti nuou s
Q. 6 Let 1 and
4, 2.Then,
lim
? ?
1
1
? 1
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 7 Maximu m ? , ? 0 ,1 is equal to _ _ _ _ ___ __ ___ __ ___
Q. 8 Let , , , ? su ch that
0 . T h e n, the C auch y pr o blem
,,?,
, 0 o n 0
has a u nique solution if
(A) 0 (B) 0
(C) 0 ( D ) 0
GATE 2016 Mathematics - MA
MA 4/16
Q. 9 Let , be the d ' A lembert's s olu tion of th e in itial va lu e pro b le m for th e w ave
eq uatio n
0
, 0 ,
, 0 ,
wh ere c is a positi v e real numb e r a nd , are smo o th odd func t ions. Th e n, 0,1 is
eq ual to _ ___ __ ___ __
Q. 10 L e t th e pro babi l ity density func tion of a r a ndo m variable X be
0 1
2
2 1 1
2
1
0otherwise.
Then, the v a lue of c i s eq ual to ___ __ ___ ___ __ __ ___ ___ ___
Q. 11 Let V be th e set of all so l u tio n s o f the eq uatio n ?? ? 0 satisfyin g
0 1 , whe r e , are positive r ea l numbers. Th e n, di m ension(V ) is equal to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 12 Let ?? ? 0, ? ? , ? ,where and are
c o ntinuous func t i o ns. If
2 and 2 are two lin e arly indepe ndent s olu t ions o f t he ab o ve eq u a t ion , then
| 4 0 2 1 | is e qual to __ __ ___ ___ ___ __ __ ___
Q. 13
Let b e th e Legendre pol y nomial of degree and
, where k
i s a n on ?neg ative inte ger. Consi der the fo llowi ng stat e ment s P and Q:
(P ) : 0 i f .
(Q) : 0 if is a n odd intege r.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
GATE 2016 Mathematics - MA
MA 2/16
Q. 1 ? Q. 25 carry one mark each.
Q. 1 Let , , be a b as is of . Conside r th e f ollowin g statements P and Q:
(P ) : ,, is a bas is of .
(Q ) : , 2 , 3 is a basis of
.
Wh i ch of th e ab ove s t atements h o l d TRUE?
(A) b o th P a nd Q ( B ) only P
(C ) only Q ( D) N either P n or Q
Q. 2 Consider the following stateme nts P an d Q:
(P ) : If 11 1
12 4
13 9
, then M is s in gular .
(Q ) : Let S b e a diagon a l iza b le matrix. If T is a matrix suc h t hat S + 5 T = Id , then T is
diagonalizable.
Which of the above s tate ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n ly Q ( D) Neither P nor Q
Q. 3 Consider the following stateme nts P an d Q:
(P ) : If M i s an complex matrix, then ?
.
(Q ) : There e xis t s a unitary ma trix with an e igen value ? su ch t h a t | ?| <1.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n l y Q ( D) Neither P n or Q
GATE 2016 Mathematics - MA
MA 3/16
Q. 4 Consider a r eal vec t or s p a ce V of d imen sion n and a no n ?zero line ar tra ns formation
? ? . If dim e ns ion and , f or some ? \0 , t hen
wh ic h of th e follo wing stateme nts is TRUE?
(A) determina n t | |
(B) Th e re exists a non?tri v ial s u b s p a ce of V such th a t 0 f o r all ? (C) T is invertib le
(D) is th e on ly e igenvalue of T
Q. 5 Let 0, 1 ?2, 3 a nd ? ? be a strictly increasing f u nction such that
is conn ected. Wh i ch of the foll owing statements is TRUE?
(A) h as e xactly one discontinuity
(B) has exactly two di s continu i ti es
(C) h as inf in itely man y d iscontinu i ties
(D) is conti nuou s
Q. 6 Let 1 and
4, 2.Then,
lim
? ?
1
1
? 1
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 7 Maximu m ? , ? 0 ,1 is equal to _ _ _ _ ___ __ ___ __ ___
Q. 8 Let , , , ? su ch that
0 . T h e n, the C auch y pr o blem
,,?,
, 0 o n 0
has a u nique solution if
(A) 0 (B) 0
(C) 0 ( D ) 0
GATE 2016 Mathematics - MA
MA 4/16
Q. 9 Let , be the d ' A lembert's s olu tion of th e in itial va lu e pro b le m for th e w ave
eq uatio n
0
, 0 ,
, 0 ,
wh ere c is a positi v e real numb e r a nd , are smo o th odd func t ions. Th e n, 0,1 is
eq ual to _ ___ __ ___ __
Q. 10 L e t th e pro babi l ity density func tion of a r a ndo m variable X be
0 1
2
2 1 1
2
1
0otherwise.
Then, the v a lue of c i s eq ual to ___ __ ___ ___ __ __ ___ ___ ___
Q. 11 Let V be th e set of all so l u tio n s o f the eq uatio n ?? ? 0 satisfyin g
0 1 , whe r e , are positive r ea l numbers. Th e n, di m ension(V ) is equal to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 12 Let ?? ? 0, ? ? , ? ,where and are
c o ntinuous func t i o ns. If
2 and 2 are two lin e arly indepe ndent s olu t ions o f t he ab o ve eq u a t ion , then
| 4 0 2 1 | is e qual to __ __ ___ ___ ___ __ __ ___
Q. 13
Let b e th e Legendre pol y nomial of degree and
, where k
i s a n on ?neg ative inte ger. Consi der the fo llowi ng stat e ment s P and Q:
(P ) : 0 i f .
(Q) : 0 if is a n odd intege r.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 5/16
Q. 14 Consider the following stateme nts P an d Q:
(P) :
?? ?
0 has two linearly i ndepende nt Fro beni us series
solutions near 0.
(Q) :
?? 3 ? 0 has two li n early independe n t F r obenius series
solutions near 0.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 15 L e t th e polynomial
be app r o x i m ated b y a p o ly no mi al o f degre e 2, which
i n ter p olates
at 1, 0 and 1. T h e n, th e m a x imum abs olute interpol a t i o n er r or
ove r th e inter val 1 , 1 is equal to __ __ ___ ___ __ ___ __ _ _ _ __
Q. 16 Let b e a se q u ence o f dist inc t points in 0, 1 ? ? | | 1 with
lim
? ?
0 . Consider the following stateme nts P an d Q :
(P ) : Th e r e exists a u niq u e analytic functi o n f on 0,1 suc h t ha t for
a ll n .
(Q ) : Th e r e exists an an alytic functi o n f on 0, 1 such that 0 if n i s e v en
and 1 if n is o d d.
Which o f the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 17 Let , b e a to pologic a l space with the cofinite to p o logy. Every infi n i t e subset of
i s
(A) Compac t but NOT co nnected
(B) Both compa c t a nd connected
(C) NOT compa c t b u t c o nne c t e d
(D) Ne i t h e r c o mp act no r c o nnected
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
GATE 2016 Mathematics - MA
MA 2/16
Q. 1 ? Q. 25 carry one mark each.
Q. 1 Let , , be a b as is of . Conside r th e f ollowin g statements P and Q:
(P ) : ,, is a bas is of .
(Q ) : , 2 , 3 is a basis of
.
Wh i ch of th e ab ove s t atements h o l d TRUE?
(A) b o th P a nd Q ( B ) only P
(C ) only Q ( D) N either P n or Q
Q. 2 Consider the following stateme nts P an d Q:
(P ) : If 11 1
12 4
13 9
, then M is s in gular .
(Q ) : Let S b e a diagon a l iza b le matrix. If T is a matrix suc h t hat S + 5 T = Id , then T is
diagonalizable.
Which of the above s tate ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n ly Q ( D) Neither P nor Q
Q. 3 Consider the following stateme nts P an d Q:
(P ) : If M i s an complex matrix, then ?
.
(Q ) : There e xis t s a unitary ma trix with an e igen value ? su ch t h a t | ?| <1.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n l y Q ( D) Neither P n or Q
GATE 2016 Mathematics - MA
MA 3/16
Q. 4 Consider a r eal vec t or s p a ce V of d imen sion n and a no n ?zero line ar tra ns formation
? ? . If dim e ns ion and , f or some ? \0 , t hen
wh ic h of th e follo wing stateme nts is TRUE?
(A) determina n t | |
(B) Th e re exists a non?tri v ial s u b s p a ce of V such th a t 0 f o r all ? (C) T is invertib le
(D) is th e on ly e igenvalue of T
Q. 5 Let 0, 1 ?2, 3 a nd ? ? be a strictly increasing f u nction such that
is conn ected. Wh i ch of the foll owing statements is TRUE?
(A) h as e xactly one discontinuity
(B) has exactly two di s continu i ti es
(C) h as inf in itely man y d iscontinu i ties
(D) is conti nuou s
Q. 6 Let 1 and
4, 2.Then,
lim
? ?
1
1
? 1
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 7 Maximu m ? , ? 0 ,1 is equal to _ _ _ _ ___ __ ___ __ ___
Q. 8 Let , , , ? su ch that
0 . T h e n, the C auch y pr o blem
,,?,
, 0 o n 0
has a u nique solution if
(A) 0 (B) 0
(C) 0 ( D ) 0
GATE 2016 Mathematics - MA
MA 4/16
Q. 9 Let , be the d ' A lembert's s olu tion of th e in itial va lu e pro b le m for th e w ave
eq uatio n
0
, 0 ,
, 0 ,
wh ere c is a positi v e real numb e r a nd , are smo o th odd func t ions. Th e n, 0,1 is
eq ual to _ ___ __ ___ __
Q. 10 L e t th e pro babi l ity density func tion of a r a ndo m variable X be
0 1
2
2 1 1
2
1
0otherwise.
Then, the v a lue of c i s eq ual to ___ __ ___ ___ __ __ ___ ___ ___
Q. 11 Let V be th e set of all so l u tio n s o f the eq uatio n ?? ? 0 satisfyin g
0 1 , whe r e , are positive r ea l numbers. Th e n, di m ension(V ) is equal to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 12 Let ?? ? 0, ? ? , ? ,where and are
c o ntinuous func t i o ns. If
2 and 2 are two lin e arly indepe ndent s olu t ions o f t he ab o ve eq u a t ion , then
| 4 0 2 1 | is e qual to __ __ ___ ___ ___ __ __ ___
Q. 13
Let b e th e Legendre pol y nomial of degree and
, where k
i s a n on ?neg ative inte ger. Consi der the fo llowi ng stat e ment s P and Q:
(P ) : 0 i f .
(Q) : 0 if is a n odd intege r.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 5/16
Q. 14 Consider the following stateme nts P an d Q:
(P) :
?? ?
0 has two linearly i ndepende nt Fro beni us series
solutions near 0.
(Q) :
?? 3 ? 0 has two li n early independe n t F r obenius series
solutions near 0.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 15 L e t th e polynomial
be app r o x i m ated b y a p o ly no mi al o f degre e 2, which
i n ter p olates
at 1, 0 and 1. T h e n, th e m a x imum abs olute interpol a t i o n er r or
ove r th e inter val 1 , 1 is equal to __ __ ___ ___ __ ___ __ _ _ _ __
Q. 16 Let b e a se q u ence o f dist inc t points in 0, 1 ? ? | | 1 with
lim
? ?
0 . Consider the following stateme nts P an d Q :
(P ) : Th e r e exists a u niq u e analytic functi o n f on 0,1 suc h t ha t for
a ll n .
(Q ) : Th e r e exists an an alytic functi o n f on 0, 1 such that 0 if n i s e v en
and 1 if n is o d d.
Which o f the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 17 Let , b e a to pologic a l space with the cofinite to p o logy. Every infi n i t e subset of
i s
(A) Compac t but NOT co nnected
(B) Both compa c t a nd connected
(C) NOT compa c t b u t c o nne c t e d
(D) Ne i t h e r c o mp act no r c o nnected
GATE 2016 Mathematics - MA
MA 6/16
Q. 18 Let ?
? , ?0 and ?
?
?
0 .
T hen, dim ens ion
is equal to __ ___ __ ___ ___ __ ____ _ ___
Q. 19 Consider , ? ? ?
?
, wher e ? , ?
?
maximum | | , | | . Let ?
? be
def i n e d by ,
an d the no r m pre s e rvi ng l ine a r e x tension o f to
, ? ? ?
?
. Th en, 1, 1,1 is e qual to __ ___ __ ___ ___ __ ____ _ ___ ___ __ ____ _ _
Q. 20 ? 0, 1 ? 0, 1 is called a sh rinking map if | | | | for all
, ? 0,1 and a contr a ction if there exists an 1 such tha t
| | | | f or all , ? 0,1 .
Which of the following stateme nts is TRU E f o r the function
?
(A) is both a s h rinking map and a c o ntractio n
(B) is a s hrinking m ap but NO T a c ontra c tion
(C) i s NOT a shrinking map but a c o ntractio n
(D) is Ne i t her a shri nki n g m a p no r a c ontrac ti on
Q. 21 Let be t he s et of all r eal ma t rices with the usual norm t o p ology. Con side r the
following sta t emen ts P and Q:
(P ) : T h e s e t of all sym m e tric p ositive defin i te matrices in is conne cte d .
(Q ) : T h e s e t of all inver tibl e m atric es in i s c om pact .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
GATE 2016 Mathematics - MA
MA 2/16
Q. 1 ? Q. 25 carry one mark each.
Q. 1 Let , , be a b as is of . Conside r th e f ollowin g statements P and Q:
(P ) : ,, is a bas is of .
(Q ) : , 2 , 3 is a basis of
.
Wh i ch of th e ab ove s t atements h o l d TRUE?
(A) b o th P a nd Q ( B ) only P
(C ) only Q ( D) N either P n or Q
Q. 2 Consider the following stateme nts P an d Q:
(P ) : If 11 1
12 4
13 9
, then M is s in gular .
(Q ) : Let S b e a diagon a l iza b le matrix. If T is a matrix suc h t hat S + 5 T = Id , then T is
diagonalizable.
Which of the above s tate ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n ly Q ( D) Neither P nor Q
Q. 3 Consider the following stateme nts P an d Q:
(P ) : If M i s an complex matrix, then ?
.
(Q ) : There e xis t s a unitary ma trix with an e igen value ? su ch t h a t | ?| <1.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n l y Q ( D) Neither P n or Q
GATE 2016 Mathematics - MA
MA 3/16
Q. 4 Consider a r eal vec t or s p a ce V of d imen sion n and a no n ?zero line ar tra ns formation
? ? . If dim e ns ion and , f or some ? \0 , t hen
wh ic h of th e follo wing stateme nts is TRUE?
(A) determina n t | |
(B) Th e re exists a non?tri v ial s u b s p a ce of V such th a t 0 f o r all ? (C) T is invertib le
(D) is th e on ly e igenvalue of T
Q. 5 Let 0, 1 ?2, 3 a nd ? ? be a strictly increasing f u nction such that
is conn ected. Wh i ch of the foll owing statements is TRUE?
(A) h as e xactly one discontinuity
(B) has exactly two di s continu i ti es
(C) h as inf in itely man y d iscontinu i ties
(D) is conti nuou s
Q. 6 Let 1 and
4, 2.Then,
lim
? ?
1
1
? 1
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 7 Maximu m ? , ? 0 ,1 is equal to _ _ _ _ ___ __ ___ __ ___
Q. 8 Let , , , ? su ch that
0 . T h e n, the C auch y pr o blem
,,?,
, 0 o n 0
has a u nique solution if
(A) 0 (B) 0
(C) 0 ( D ) 0
GATE 2016 Mathematics - MA
MA 4/16
Q. 9 Let , be the d ' A lembert's s olu tion of th e in itial va lu e pro b le m for th e w ave
eq uatio n
0
, 0 ,
, 0 ,
wh ere c is a positi v e real numb e r a nd , are smo o th odd func t ions. Th e n, 0,1 is
eq ual to _ ___ __ ___ __
Q. 10 L e t th e pro babi l ity density func tion of a r a ndo m variable X be
0 1
2
2 1 1
2
1
0otherwise.
Then, the v a lue of c i s eq ual to ___ __ ___ ___ __ __ ___ ___ ___
Q. 11 Let V be th e set of all so l u tio n s o f the eq uatio n ?? ? 0 satisfyin g
0 1 , whe r e , are positive r ea l numbers. Th e n, di m ension(V ) is equal to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 12 Let ?? ? 0, ? ? , ? ,where and are
c o ntinuous func t i o ns. If
2 and 2 are two lin e arly indepe ndent s olu t ions o f t he ab o ve eq u a t ion , then
| 4 0 2 1 | is e qual to __ __ ___ ___ ___ __ __ ___
Q. 13
Let b e th e Legendre pol y nomial of degree and
, where k
i s a n on ?neg ative inte ger. Consi der the fo llowi ng stat e ment s P and Q:
(P ) : 0 i f .
(Q) : 0 if is a n odd intege r.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 5/16
Q. 14 Consider the following stateme nts P an d Q:
(P) :
?? ?
0 has two linearly i ndepende nt Fro beni us series
solutions near 0.
(Q) :
?? 3 ? 0 has two li n early independe n t F r obenius series
solutions near 0.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 15 L e t th e polynomial
be app r o x i m ated b y a p o ly no mi al o f degre e 2, which
i n ter p olates
at 1, 0 and 1. T h e n, th e m a x imum abs olute interpol a t i o n er r or
ove r th e inter val 1 , 1 is equal to __ __ ___ ___ __ ___ __ _ _ _ __
Q. 16 Let b e a se q u ence o f dist inc t points in 0, 1 ? ? | | 1 with
lim
? ?
0 . Consider the following stateme nts P an d Q :
(P ) : Th e r e exists a u niq u e analytic functi o n f on 0,1 suc h t ha t for
a ll n .
(Q ) : Th e r e exists an an alytic functi o n f on 0, 1 such that 0 if n i s e v en
and 1 if n is o d d.
Which o f the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 17 Let , b e a to pologic a l space with the cofinite to p o logy. Every infi n i t e subset of
i s
(A) Compac t but NOT co nnected
(B) Both compa c t a nd connected
(C) NOT compa c t b u t c o nne c t e d
(D) Ne i t h e r c o mp act no r c o nnected
GATE 2016 Mathematics - MA
MA 6/16
Q. 18 Let ?
? , ?0 and ?
?
?
0 .
T hen, dim ens ion
is equal to __ ___ __ ___ ___ __ ____ _ ___
Q. 19 Consider , ? ? ?
?
, wher e ? , ?
?
maximum | | , | | . Let ?
? be
def i n e d by ,
an d the no r m pre s e rvi ng l ine a r e x tension o f to
, ? ? ?
?
. Th en, 1, 1,1 is e qual to __ ___ __ ___ ___ __ ____ _ ___ ___ __ ____ _ _
Q. 20 ? 0, 1 ? 0, 1 is called a sh rinking map if | | | | for all
, ? 0,1 and a contr a ction if there exists an 1 such tha t
| | | | f or all , ? 0,1 .
Which of the following stateme nts is TRU E f o r the function
?
(A) is both a s h rinking map and a c o ntractio n
(B) is a s hrinking m ap but NO T a c ontra c tion
(C) i s NOT a shrinking map but a c o ntractio n
(D) is Ne i t her a shri nki n g m a p no r a c ontrac ti on
Q. 21 Let be t he s et of all r eal ma t rices with the usual norm t o p ology. Con side r the
following sta t emen ts P and Q:
(P ) : T h e s e t of all sym m e tric p ositive defin i te matrices in is conne cte d .
(Q ) : T h e s e t of all inver tibl e m atric es in i s c om pact .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 7/16
Q. 22 Let ,
,
,?,
be a r and o m samp le f rom the followin g pr o bability d ensity
func ti on for 0 ?,0 1,
; , 1
? ; 0otherwise.
He re a nd are un kn own parame ters. Which of th e f ollowin g st at em ent s i s TRU E?
(A) Max i mum likel i hood e stima t or of on ly e xists
(B) Maximum li kelihood esti m ator of o n l y exis ts
(C) Maximum likeli h o o d estimator s o f bo th and e xist
(D) Ma ximu m likeli h o od estimator of Neither nor e xists
Q. 23 Suppose X an d Y are tw o rando m variables suc h th a t is a norma l random
variab le for al l , ? . Conside r the f ollowin g statements P, Q , R an d S :
(P ) : X is a stan dard n o r mal ran d om v ariable.
(Q) : Th e conditi o na l distrib u ti on of X g ive n Y is n o rmal.
(R) : Th e co nditio nal distributio n of X given is n o rmal.
(S) : h a s m e a n 0.
Which of the above state me nts A LWAYS hold T R U E?
(A) both P and Q ( B) b oth Q a nd R
(C) b o th Q a nd S ( D) b oth P a nd S
Q.24 Co nsi d er t h e foll owing st a t e m e nts P and Q :
( P ) : I f i s a no rmal subgroup o f o r d er 4 o f the s y mmetric gro up
, the n
is
a beli a n .
(Q ) : If 1 , , , is the qua ternion group, then
1, 1
i s ab el ian.
Whic h of th e abo ve st at e ments hol d T RUE?
(A ) both P a nd Q (B) onl y P
(C ) onl y Q ( D) Neith er P nor Q
Q.25 Let b e a fiel d of order 3 2. T hen t he num ber o f no n? z e ro so luti ons , ? of
the eq ua tio n
0 is equal to __ ___ __ ___ _ _ ___ ___ __ ____ _ ___ __ _
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
GATE 2016 Mathematics - MA
MA 2/16
Q. 1 ? Q. 25 carry one mark each.
Q. 1 Let , , be a b as is of . Conside r th e f ollowin g statements P and Q:
(P ) : ,, is a bas is of .
(Q ) : , 2 , 3 is a basis of
.
Wh i ch of th e ab ove s t atements h o l d TRUE?
(A) b o th P a nd Q ( B ) only P
(C ) only Q ( D) N either P n or Q
Q. 2 Consider the following stateme nts P an d Q:
(P ) : If 11 1
12 4
13 9
, then M is s in gular .
(Q ) : Let S b e a diagon a l iza b le matrix. If T is a matrix suc h t hat S + 5 T = Id , then T is
diagonalizable.
Which of the above s tate ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n ly Q ( D) Neither P nor Q
Q. 3 Consider the following stateme nts P an d Q:
(P ) : If M i s an complex matrix, then ?
.
(Q ) : There e xis t s a unitary ma trix with an e igen value ? su ch t h a t | ?| <1.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n l y Q ( D) Neither P n or Q
GATE 2016 Mathematics - MA
MA 3/16
Q. 4 Consider a r eal vec t or s p a ce V of d imen sion n and a no n ?zero line ar tra ns formation
? ? . If dim e ns ion and , f or some ? \0 , t hen
wh ic h of th e follo wing stateme nts is TRUE?
(A) determina n t | |
(B) Th e re exists a non?tri v ial s u b s p a ce of V such th a t 0 f o r all ? (C) T is invertib le
(D) is th e on ly e igenvalue of T
Q. 5 Let 0, 1 ?2, 3 a nd ? ? be a strictly increasing f u nction such that
is conn ected. Wh i ch of the foll owing statements is TRUE?
(A) h as e xactly one discontinuity
(B) has exactly two di s continu i ti es
(C) h as inf in itely man y d iscontinu i ties
(D) is conti nuou s
Q. 6 Let 1 and
4, 2.Then,
lim
? ?
1
1
? 1
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 7 Maximu m ? , ? 0 ,1 is equal to _ _ _ _ ___ __ ___ __ ___
Q. 8 Let , , , ? su ch that
0 . T h e n, the C auch y pr o blem
,,?,
, 0 o n 0
has a u nique solution if
(A) 0 (B) 0
(C) 0 ( D ) 0
GATE 2016 Mathematics - MA
MA 4/16
Q. 9 Let , be the d ' A lembert's s olu tion of th e in itial va lu e pro b le m for th e w ave
eq uatio n
0
, 0 ,
, 0 ,
wh ere c is a positi v e real numb e r a nd , are smo o th odd func t ions. Th e n, 0,1 is
eq ual to _ ___ __ ___ __
Q. 10 L e t th e pro babi l ity density func tion of a r a ndo m variable X be
0 1
2
2 1 1
2
1
0otherwise.
Then, the v a lue of c i s eq ual to ___ __ ___ ___ __ __ ___ ___ ___
Q. 11 Let V be th e set of all so l u tio n s o f the eq uatio n ?? ? 0 satisfyin g
0 1 , whe r e , are positive r ea l numbers. Th e n, di m ension(V ) is equal to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 12 Let ?? ? 0, ? ? , ? ,where and are
c o ntinuous func t i o ns. If
2 and 2 are two lin e arly indepe ndent s olu t ions o f t he ab o ve eq u a t ion , then
| 4 0 2 1 | is e qual to __ __ ___ ___ ___ __ __ ___
Q. 13
Let b e th e Legendre pol y nomial of degree and
, where k
i s a n on ?neg ative inte ger. Consi der the fo llowi ng stat e ment s P and Q:
(P ) : 0 i f .
(Q) : 0 if is a n odd intege r.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 5/16
Q. 14 Consider the following stateme nts P an d Q:
(P) :
?? ?
0 has two linearly i ndepende nt Fro beni us series
solutions near 0.
(Q) :
?? 3 ? 0 has two li n early independe n t F r obenius series
solutions near 0.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 15 L e t th e polynomial
be app r o x i m ated b y a p o ly no mi al o f degre e 2, which
i n ter p olates
at 1, 0 and 1. T h e n, th e m a x imum abs olute interpol a t i o n er r or
ove r th e inter val 1 , 1 is equal to __ __ ___ ___ __ ___ __ _ _ _ __
Q. 16 Let b e a se q u ence o f dist inc t points in 0, 1 ? ? | | 1 with
lim
? ?
0 . Consider the following stateme nts P an d Q :
(P ) : Th e r e exists a u niq u e analytic functi o n f on 0,1 suc h t ha t for
a ll n .
(Q ) : Th e r e exists an an alytic functi o n f on 0, 1 such that 0 if n i s e v en
and 1 if n is o d d.
Which o f the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 17 Let , b e a to pologic a l space with the cofinite to p o logy. Every infi n i t e subset of
i s
(A) Compac t but NOT co nnected
(B) Both compa c t a nd connected
(C) NOT compa c t b u t c o nne c t e d
(D) Ne i t h e r c o mp act no r c o nnected
GATE 2016 Mathematics - MA
MA 6/16
Q. 18 Let ?
? , ?0 and ?
?
?
0 .
T hen, dim ens ion
is equal to __ ___ __ ___ ___ __ ____ _ ___
Q. 19 Consider , ? ? ?
?
, wher e ? , ?
?
maximum | | , | | . Let ?
? be
def i n e d by ,
an d the no r m pre s e rvi ng l ine a r e x tension o f to
, ? ? ?
?
. Th en, 1, 1,1 is e qual to __ ___ __ ___ ___ __ ____ _ ___ ___ __ ____ _ _
Q. 20 ? 0, 1 ? 0, 1 is called a sh rinking map if | | | | for all
, ? 0,1 and a contr a ction if there exists an 1 such tha t
| | | | f or all , ? 0,1 .
Which of the following stateme nts is TRU E f o r the function
?
(A) is both a s h rinking map and a c o ntractio n
(B) is a s hrinking m ap but NO T a c ontra c tion
(C) i s NOT a shrinking map but a c o ntractio n
(D) is Ne i t her a shri nki n g m a p no r a c ontrac ti on
Q. 21 Let be t he s et of all r eal ma t rices with the usual norm t o p ology. Con side r the
following sta t emen ts P and Q:
(P ) : T h e s e t of all sym m e tric p ositive defin i te matrices in is conne cte d .
(Q ) : T h e s e t of all inver tibl e m atric es in i s c om pact .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 7/16
Q. 22 Let ,
,
,?,
be a r and o m samp le f rom the followin g pr o bability d ensity
func ti on for 0 ?,0 1,
; , 1
? ; 0otherwise.
He re a nd are un kn own parame ters. Which of th e f ollowin g st at em ent s i s TRU E?
(A) Max i mum likel i hood e stima t or of on ly e xists
(B) Maximum li kelihood esti m ator of o n l y exis ts
(C) Maximum likeli h o o d estimator s o f bo th and e xist
(D) Ma ximu m likeli h o od estimator of Neither nor e xists
Q. 23 Suppose X an d Y are tw o rando m variables suc h th a t is a norma l random
variab le for al l , ? . Conside r the f ollowin g statements P, Q , R an d S :
(P ) : X is a stan dard n o r mal ran d om v ariable.
(Q) : Th e conditi o na l distrib u ti on of X g ive n Y is n o rmal.
(R) : Th e co nditio nal distributio n of X given is n o rmal.
(S) : h a s m e a n 0.
Which of the above state me nts A LWAYS hold T R U E?
(A) both P and Q ( B) b oth Q a nd R
(C) b o th Q a nd S ( D) b oth P a nd S
Q.24 Co nsi d er t h e foll owing st a t e m e nts P and Q :
( P ) : I f i s a no rmal subgroup o f o r d er 4 o f the s y mmetric gro up
, the n
is
a beli a n .
(Q ) : If 1 , , , is the qua ternion group, then
1, 1
i s ab el ian.
Whic h of th e abo ve st at e ments hol d T RUE?
(A ) both P a nd Q (B) onl y P
(C ) onl y Q ( D) Neith er P nor Q
Q.25 Let b e a fiel d of order 3 2. T hen t he num ber o f no n? z e ro so luti ons , ? of
the eq ua tio n
0 is equal to __ ___ __ ___ _ _ ___ ___ __ ____ _ ___ __ _
GATE 2016 Mathematics - MA
MA 8/16
Q. 26 ? Q. 55 carry two marks each.
Q. 26 Let ? ? || 2 be oriente d in the counter?clockwise direction . Let
1
2 1
.
Then, the v a lue of i s equal to ___ ___ __ ___ ___ __ _ _ _ __ ___ __
Q. 27 Let , 2 be a h a rmonic fu n ctio n and , i t s
harmon ic c onju gate. If 0,0 1 , th en | 1 , 1 | is eq u al to _ __ ___ ___ __ ___ _
Q. 28 L e t b e th e triangula r path co nnecting the poin ts (0,0), (2,2) and (0,2) in the counter?
clockwise direction in
. T h en
6
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 29 Let y b e th e solutio n of
? | |,? 1 0.
Then 1 i s equ a l to
(A)
(B)
2
(C) 2
(D) 2 2
Q. 30 Let X b e a r ando m v ari able with th e f oll owi ng c um ul ati ve d i stri b u ti o n functio n :
00
0 1
2
3
4
1
2
1
1 1.
Then 1 is equal to __ ___ ___ __ ___ __ ____
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
GATE 2016 Mathematics - MA
MA 2/16
Q. 1 ? Q. 25 carry one mark each.
Q. 1 Let , , be a b as is of . Conside r th e f ollowin g statements P and Q:
(P ) : ,, is a bas is of .
(Q ) : , 2 , 3 is a basis of
.
Wh i ch of th e ab ove s t atements h o l d TRUE?
(A) b o th P a nd Q ( B ) only P
(C ) only Q ( D) N either P n or Q
Q. 2 Consider the following stateme nts P an d Q:
(P ) : If 11 1
12 4
13 9
, then M is s in gular .
(Q ) : Let S b e a diagon a l iza b le matrix. If T is a matrix suc h t hat S + 5 T = Id , then T is
diagonalizable.
Which of the above s tate ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n ly Q ( D) Neither P nor Q
Q. 3 Consider the following stateme nts P an d Q:
(P ) : If M i s an complex matrix, then ?
.
(Q ) : There e xis t s a unitary ma trix with an e igen value ? su ch t h a t | ?| <1.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n l y Q ( D) Neither P n or Q
GATE 2016 Mathematics - MA
MA 3/16
Q. 4 Consider a r eal vec t or s p a ce V of d imen sion n and a no n ?zero line ar tra ns formation
? ? . If dim e ns ion and , f or some ? \0 , t hen
wh ic h of th e follo wing stateme nts is TRUE?
(A) determina n t | |
(B) Th e re exists a non?tri v ial s u b s p a ce of V such th a t 0 f o r all ? (C) T is invertib le
(D) is th e on ly e igenvalue of T
Q. 5 Let 0, 1 ?2, 3 a nd ? ? be a strictly increasing f u nction such that
is conn ected. Wh i ch of the foll owing statements is TRUE?
(A) h as e xactly one discontinuity
(B) has exactly two di s continu i ti es
(C) h as inf in itely man y d iscontinu i ties
(D) is conti nuou s
Q. 6 Let 1 and
4, 2.Then,
lim
? ?
1
1
? 1
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 7 Maximu m ? , ? 0 ,1 is equal to _ _ _ _ ___ __ ___ __ ___
Q. 8 Let , , , ? su ch that
0 . T h e n, the C auch y pr o blem
,,?,
, 0 o n 0
has a u nique solution if
(A) 0 (B) 0
(C) 0 ( D ) 0
GATE 2016 Mathematics - MA
MA 4/16
Q. 9 Let , be the d ' A lembert's s olu tion of th e in itial va lu e pro b le m for th e w ave
eq uatio n
0
, 0 ,
, 0 ,
wh ere c is a positi v e real numb e r a nd , are smo o th odd func t ions. Th e n, 0,1 is
eq ual to _ ___ __ ___ __
Q. 10 L e t th e pro babi l ity density func tion of a r a ndo m variable X be
0 1
2
2 1 1
2
1
0otherwise.
Then, the v a lue of c i s eq ual to ___ __ ___ ___ __ __ ___ ___ ___
Q. 11 Let V be th e set of all so l u tio n s o f the eq uatio n ?? ? 0 satisfyin g
0 1 , whe r e , are positive r ea l numbers. Th e n, di m ension(V ) is equal to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 12 Let ?? ? 0, ? ? , ? ,where and are
c o ntinuous func t i o ns. If
2 and 2 are two lin e arly indepe ndent s olu t ions o f t he ab o ve eq u a t ion , then
| 4 0 2 1 | is e qual to __ __ ___ ___ ___ __ __ ___
Q. 13
Let b e th e Legendre pol y nomial of degree and
, where k
i s a n on ?neg ative inte ger. Consi der the fo llowi ng stat e ment s P and Q:
(P ) : 0 i f .
(Q) : 0 if is a n odd intege r.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 5/16
Q. 14 Consider the following stateme nts P an d Q:
(P) :
?? ?
0 has two linearly i ndepende nt Fro beni us series
solutions near 0.
(Q) :
?? 3 ? 0 has two li n early independe n t F r obenius series
solutions near 0.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 15 L e t th e polynomial
be app r o x i m ated b y a p o ly no mi al o f degre e 2, which
i n ter p olates
at 1, 0 and 1. T h e n, th e m a x imum abs olute interpol a t i o n er r or
ove r th e inter val 1 , 1 is equal to __ __ ___ ___ __ ___ __ _ _ _ __
Q. 16 Let b e a se q u ence o f dist inc t points in 0, 1 ? ? | | 1 with
lim
? ?
0 . Consider the following stateme nts P an d Q :
(P ) : Th e r e exists a u niq u e analytic functi o n f on 0,1 suc h t ha t for
a ll n .
(Q ) : Th e r e exists an an alytic functi o n f on 0, 1 such that 0 if n i s e v en
and 1 if n is o d d.
Which o f the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 17 Let , b e a to pologic a l space with the cofinite to p o logy. Every infi n i t e subset of
i s
(A) Compac t but NOT co nnected
(B) Both compa c t a nd connected
(C) NOT compa c t b u t c o nne c t e d
(D) Ne i t h e r c o mp act no r c o nnected
GATE 2016 Mathematics - MA
MA 6/16
Q. 18 Let ?
? , ?0 and ?
?
?
0 .
T hen, dim ens ion
is equal to __ ___ __ ___ ___ __ ____ _ ___
Q. 19 Consider , ? ? ?
?
, wher e ? , ?
?
maximum | | , | | . Let ?
? be
def i n e d by ,
an d the no r m pre s e rvi ng l ine a r e x tension o f to
, ? ? ?
?
. Th en, 1, 1,1 is e qual to __ ___ __ ___ ___ __ ____ _ ___ ___ __ ____ _ _
Q. 20 ? 0, 1 ? 0, 1 is called a sh rinking map if | | | | for all
, ? 0,1 and a contr a ction if there exists an 1 such tha t
| | | | f or all , ? 0,1 .
Which of the following stateme nts is TRU E f o r the function
?
(A) is both a s h rinking map and a c o ntractio n
(B) is a s hrinking m ap but NO T a c ontra c tion
(C) i s NOT a shrinking map but a c o ntractio n
(D) is Ne i t her a shri nki n g m a p no r a c ontrac ti on
Q. 21 Let be t he s et of all r eal ma t rices with the usual norm t o p ology. Con side r the
following sta t emen ts P and Q:
(P ) : T h e s e t of all sym m e tric p ositive defin i te matrices in is conne cte d .
(Q ) : T h e s e t of all inver tibl e m atric es in i s c om pact .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 7/16
Q. 22 Let ,
,
,?,
be a r and o m samp le f rom the followin g pr o bability d ensity
func ti on for 0 ?,0 1,
; , 1
? ; 0otherwise.
He re a nd are un kn own parame ters. Which of th e f ollowin g st at em ent s i s TRU E?
(A) Max i mum likel i hood e stima t or of on ly e xists
(B) Maximum li kelihood esti m ator of o n l y exis ts
(C) Maximum likeli h o o d estimator s o f bo th and e xist
(D) Ma ximu m likeli h o od estimator of Neither nor e xists
Q. 23 Suppose X an d Y are tw o rando m variables suc h th a t is a norma l random
variab le for al l , ? . Conside r the f ollowin g statements P, Q , R an d S :
(P ) : X is a stan dard n o r mal ran d om v ariable.
(Q) : Th e conditi o na l distrib u ti on of X g ive n Y is n o rmal.
(R) : Th e co nditio nal distributio n of X given is n o rmal.
(S) : h a s m e a n 0.
Which of the above state me nts A LWAYS hold T R U E?
(A) both P and Q ( B) b oth Q a nd R
(C) b o th Q a nd S ( D) b oth P a nd S
Q.24 Co nsi d er t h e foll owing st a t e m e nts P and Q :
( P ) : I f i s a no rmal subgroup o f o r d er 4 o f the s y mmetric gro up
, the n
is
a beli a n .
(Q ) : If 1 , , , is the qua ternion group, then
1, 1
i s ab el ian.
Whic h of th e abo ve st at e ments hol d T RUE?
(A ) both P a nd Q (B) onl y P
(C ) onl y Q ( D) Neith er P nor Q
Q.25 Let b e a fiel d of order 3 2. T hen t he num ber o f no n? z e ro so luti ons , ? of
the eq ua tio n
0 is equal to __ ___ __ ___ _ _ ___ ___ __ ____ _ ___ __ _
GATE 2016 Mathematics - MA
MA 8/16
Q. 26 ? Q. 55 carry two marks each.
Q. 26 Let ? ? || 2 be oriente d in the counter?clockwise direction . Let
1
2 1
.
Then, the v a lue of i s equal to ___ ___ __ ___ ___ __ _ _ _ __ ___ __
Q. 27 Let , 2 be a h a rmonic fu n ctio n and , i t s
harmon ic c onju gate. If 0,0 1 , th en | 1 , 1 | is eq u al to _ __ ___ ___ __ ___ _
Q. 28 L e t b e th e triangula r path co nnecting the poin ts (0,0), (2,2) and (0,2) in the counter?
clockwise direction in
. T h en
6
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 29 Let y b e th e solutio n of
? | |,? 1 0.
Then 1 i s equ a l to
(A)
(B)
2
(C) 2
(D) 2 2
Q. 30 Let X b e a r ando m v ari able with th e f oll owi ng c um ul ati ve d i stri b u ti o n functio n :
00
0 1
2
3
4
1
2
1
1 1.
Then 1 is equal to __ ___ ___ __ ___ __ ____
GATE 2016 Mathematics - MA
MA 9/16
Q. 31 Let ? b e the curve which passes thr o ugh ( 0 ,1 ) and i n tersects ea c h c u r v e of the f amil y
orthogon a l ly. Th en ? a l so passes th rough the poin t
(A) ? 2,0 (B) 0, ? 2
(C) 1,1 ( D ) 1,1
Q. 32 Let ?
?
be the Fourier s e rie s of th e
2 periodic fu nction defined by 4, . Then
?
?
i s eq u al to ___ __ ___ ___ ___ __ __ ___ ___ __
Q. 33 L e t b e a con t inu o us f u nction on 0, ? . If
1 4 4 ,
the n
is equal to __ _ _ ___ _ _ ___ ___ __ ____ _ __
Q. 34
Let
?
and
. Th e n,
is e q u a l to
(A) ln 10 1 (B) l n 10 1
(C) l n 10
(D) ln10
Q. 35 F o r any , ? \ 0,1
, let
, distance , ,0,1
inimum ? ,
?0,1
.
Then , || 3,4 || i s equal to ___ __ ___ __ ___ ___ ___ _
Q. 36
Let
an d
.
Th e n ?
?
?
? is
eq ual to ___ __ ___ ___ __ __ ___ ___ __ ___ __ ___ ___ __ ___
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
GATE 2016 Mathematics - MA
MA 2/16
Q. 1 ? Q. 25 carry one mark each.
Q. 1 Let , , be a b as is of . Conside r th e f ollowin g statements P and Q:
(P ) : ,, is a bas is of .
(Q ) : , 2 , 3 is a basis of
.
Wh i ch of th e ab ove s t atements h o l d TRUE?
(A) b o th P a nd Q ( B ) only P
(C ) only Q ( D) N either P n or Q
Q. 2 Consider the following stateme nts P an d Q:
(P ) : If 11 1
12 4
13 9
, then M is s in gular .
(Q ) : Let S b e a diagon a l iza b le matrix. If T is a matrix suc h t hat S + 5 T = Id , then T is
diagonalizable.
Which of the above s tate ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n ly Q ( D) Neither P nor Q
Q. 3 Consider the following stateme nts P an d Q:
(P ) : If M i s an complex matrix, then ?
.
(Q ) : There e xis t s a unitary ma trix with an e igen value ? su ch t h a t | ?| <1.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n l y Q ( D) Neither P n or Q
GATE 2016 Mathematics - MA
MA 3/16
Q. 4 Consider a r eal vec t or s p a ce V of d imen sion n and a no n ?zero line ar tra ns formation
? ? . If dim e ns ion and , f or some ? \0 , t hen
wh ic h of th e follo wing stateme nts is TRUE?
(A) determina n t | |
(B) Th e re exists a non?tri v ial s u b s p a ce of V such th a t 0 f o r all ? (C) T is invertib le
(D) is th e on ly e igenvalue of T
Q. 5 Let 0, 1 ?2, 3 a nd ? ? be a strictly increasing f u nction such that
is conn ected. Wh i ch of the foll owing statements is TRUE?
(A) h as e xactly one discontinuity
(B) has exactly two di s continu i ti es
(C) h as inf in itely man y d iscontinu i ties
(D) is conti nuou s
Q. 6 Let 1 and
4, 2.Then,
lim
? ?
1
1
? 1
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 7 Maximu m ? , ? 0 ,1 is equal to _ _ _ _ ___ __ ___ __ ___
Q. 8 Let , , , ? su ch that
0 . T h e n, the C auch y pr o blem
,,?,
, 0 o n 0
has a u nique solution if
(A) 0 (B) 0
(C) 0 ( D ) 0
GATE 2016 Mathematics - MA
MA 4/16
Q. 9 Let , be the d ' A lembert's s olu tion of th e in itial va lu e pro b le m for th e w ave
eq uatio n
0
, 0 ,
, 0 ,
wh ere c is a positi v e real numb e r a nd , are smo o th odd func t ions. Th e n, 0,1 is
eq ual to _ ___ __ ___ __
Q. 10 L e t th e pro babi l ity density func tion of a r a ndo m variable X be
0 1
2
2 1 1
2
1
0otherwise.
Then, the v a lue of c i s eq ual to ___ __ ___ ___ __ __ ___ ___ ___
Q. 11 Let V be th e set of all so l u tio n s o f the eq uatio n ?? ? 0 satisfyin g
0 1 , whe r e , are positive r ea l numbers. Th e n, di m ension(V ) is equal to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 12 Let ?? ? 0, ? ? , ? ,where and are
c o ntinuous func t i o ns. If
2 and 2 are two lin e arly indepe ndent s olu t ions o f t he ab o ve eq u a t ion , then
| 4 0 2 1 | is e qual to __ __ ___ ___ ___ __ __ ___
Q. 13
Let b e th e Legendre pol y nomial of degree and
, where k
i s a n on ?neg ative inte ger. Consi der the fo llowi ng stat e ment s P and Q:
(P ) : 0 i f .
(Q) : 0 if is a n odd intege r.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 5/16
Q. 14 Consider the following stateme nts P an d Q:
(P) :
?? ?
0 has two linearly i ndepende nt Fro beni us series
solutions near 0.
(Q) :
?? 3 ? 0 has two li n early independe n t F r obenius series
solutions near 0.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 15 L e t th e polynomial
be app r o x i m ated b y a p o ly no mi al o f degre e 2, which
i n ter p olates
at 1, 0 and 1. T h e n, th e m a x imum abs olute interpol a t i o n er r or
ove r th e inter val 1 , 1 is equal to __ __ ___ ___ __ ___ __ _ _ _ __
Q. 16 Let b e a se q u ence o f dist inc t points in 0, 1 ? ? | | 1 with
lim
? ?
0 . Consider the following stateme nts P an d Q :
(P ) : Th e r e exists a u niq u e analytic functi o n f on 0,1 suc h t ha t for
a ll n .
(Q ) : Th e r e exists an an alytic functi o n f on 0, 1 such that 0 if n i s e v en
and 1 if n is o d d.
Which o f the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 17 Let , b e a to pologic a l space with the cofinite to p o logy. Every infi n i t e subset of
i s
(A) Compac t but NOT co nnected
(B) Both compa c t a nd connected
(C) NOT compa c t b u t c o nne c t e d
(D) Ne i t h e r c o mp act no r c o nnected
GATE 2016 Mathematics - MA
MA 6/16
Q. 18 Let ?
? , ?0 and ?
?
?
0 .
T hen, dim ens ion
is equal to __ ___ __ ___ ___ __ ____ _ ___
Q. 19 Consider , ? ? ?
?
, wher e ? , ?
?
maximum | | , | | . Let ?
? be
def i n e d by ,
an d the no r m pre s e rvi ng l ine a r e x tension o f to
, ? ? ?
?
. Th en, 1, 1,1 is e qual to __ ___ __ ___ ___ __ ____ _ ___ ___ __ ____ _ _
Q. 20 ? 0, 1 ? 0, 1 is called a sh rinking map if | | | | for all
, ? 0,1 and a contr a ction if there exists an 1 such tha t
| | | | f or all , ? 0,1 .
Which of the following stateme nts is TRU E f o r the function
?
(A) is both a s h rinking map and a c o ntractio n
(B) is a s hrinking m ap but NO T a c ontra c tion
(C) i s NOT a shrinking map but a c o ntractio n
(D) is Ne i t her a shri nki n g m a p no r a c ontrac ti on
Q. 21 Let be t he s et of all r eal ma t rices with the usual norm t o p ology. Con side r the
following sta t emen ts P and Q:
(P ) : T h e s e t of all sym m e tric p ositive defin i te matrices in is conne cte d .
(Q ) : T h e s e t of all inver tibl e m atric es in i s c om pact .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 7/16
Q. 22 Let ,
,
,?,
be a r and o m samp le f rom the followin g pr o bability d ensity
func ti on for 0 ?,0 1,
; , 1
? ; 0otherwise.
He re a nd are un kn own parame ters. Which of th e f ollowin g st at em ent s i s TRU E?
(A) Max i mum likel i hood e stima t or of on ly e xists
(B) Maximum li kelihood esti m ator of o n l y exis ts
(C) Maximum likeli h o o d estimator s o f bo th and e xist
(D) Ma ximu m likeli h o od estimator of Neither nor e xists
Q. 23 Suppose X an d Y are tw o rando m variables suc h th a t is a norma l random
variab le for al l , ? . Conside r the f ollowin g statements P, Q , R an d S :
(P ) : X is a stan dard n o r mal ran d om v ariable.
(Q) : Th e conditi o na l distrib u ti on of X g ive n Y is n o rmal.
(R) : Th e co nditio nal distributio n of X given is n o rmal.
(S) : h a s m e a n 0.
Which of the above state me nts A LWAYS hold T R U E?
(A) both P and Q ( B) b oth Q a nd R
(C) b o th Q a nd S ( D) b oth P a nd S
Q.24 Co nsi d er t h e foll owing st a t e m e nts P and Q :
( P ) : I f i s a no rmal subgroup o f o r d er 4 o f the s y mmetric gro up
, the n
is
a beli a n .
(Q ) : If 1 , , , is the qua ternion group, then
1, 1
i s ab el ian.
Whic h of th e abo ve st at e ments hol d T RUE?
(A ) both P a nd Q (B) onl y P
(C ) onl y Q ( D) Neith er P nor Q
Q.25 Let b e a fiel d of order 3 2. T hen t he num ber o f no n? z e ro so luti ons , ? of
the eq ua tio n
0 is equal to __ ___ __ ___ _ _ ___ ___ __ ____ _ ___ __ _
GATE 2016 Mathematics - MA
MA 8/16
Q. 26 ? Q. 55 carry two marks each.
Q. 26 Let ? ? || 2 be oriente d in the counter?clockwise direction . Let
1
2 1
.
Then, the v a lue of i s equal to ___ ___ __ ___ ___ __ _ _ _ __ ___ __
Q. 27 Let , 2 be a h a rmonic fu n ctio n and , i t s
harmon ic c onju gate. If 0,0 1 , th en | 1 , 1 | is eq u al to _ __ ___ ___ __ ___ _
Q. 28 L e t b e th e triangula r path co nnecting the poin ts (0,0), (2,2) and (0,2) in the counter?
clockwise direction in
. T h en
6
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 29 Let y b e th e solutio n of
? | |,? 1 0.
Then 1 i s equ a l to
(A)
(B)
2
(C) 2
(D) 2 2
Q. 30 Let X b e a r ando m v ari able with th e f oll owi ng c um ul ati ve d i stri b u ti o n functio n :
00
0 1
2
3
4
1
2
1
1 1.
Then 1 is equal to __ ___ ___ __ ___ __ ____
GATE 2016 Mathematics - MA
MA 9/16
Q. 31 Let ? b e the curve which passes thr o ugh ( 0 ,1 ) and i n tersects ea c h c u r v e of the f amil y
orthogon a l ly. Th en ? a l so passes th rough the poin t
(A) ? 2,0 (B) 0, ? 2
(C) 1,1 ( D ) 1,1
Q. 32 Let ?
?
be the Fourier s e rie s of th e
2 periodic fu nction defined by 4, . Then
?
?
i s eq u al to ___ __ ___ ___ ___ __ __ ___ ___ __
Q. 33 L e t b e a con t inu o us f u nction on 0, ? . If
1 4 4 ,
the n
is equal to __ _ _ ___ _ _ ___ ___ __ ____ _ __
Q. 34
Let
?
and
. Th e n,
is e q u a l to
(A) ln 10 1 (B) l n 10 1
(C) l n 10
(D) ln10
Q. 35 F o r any , ? \ 0,1
, let
, distance , ,0,1
inimum ? ,
?0,1
.
Then , || 3,4 || i s equal to ___ __ ___ __ ___ ___ ___ _
Q. 36
Let
an d
.
Th e n ?
?
?
? is
eq ual to ___ __ ___ ___ __ __ ___ ___ __ ___ __ ___ ___ __ ___
GATE 2016 Mathematics - MA
MA 10/16
Q. 37
Let b e a real matr i x with e igenvalu es 1 , 0 a nd 3. If th e e i ge nve ctors
c o rres p ond i ng to 1 and 0 a r e 1,1,1
a nd 1, 1 ,0 re s p e ctivel y, th e n th e value o f
3 i s eq ual to _ ___ __ ___ ____ _ ___
Q. 38
Let 11 0
01 1
00 1
a nd
!
!
? . If
, th e n
1
i s eq u al to ___ __ ___ ___ ___ __ __ ___ ___
Q. 39
L e t th e inte gral , w h e r e 02
4 2 4.
Consider the following stateme nts P an d Q:
(P ) : If
is the va l ue of the integr al o b t ained b y the c om po site tra pez oidal rule with
t wo equal s u b?i n terval s , then
is exac t.
(Q ) : If is the va l ue of the integr al o b t ained b y the c om po site tra pez oidal rule with
thr ee eq ual su b ? interva ls, t he n is e x act.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 40 The diff eren ce bet ween t he least two e i genval u e s of th e b ou nda r y valu e p roblem
?? 0, 0 0 0, ? 0,
i s eq u al to _ ___ ___ __ ____ _ ___ __ ___ ____ _ ___
Q. 41
The numb er of r o ots of the equa tion
cos 0 in the i n terval ,
i s equal
to __ ___ __ _____ __
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
GATE 2016 Mathematics - MA
MA 2/16
Q. 1 ? Q. 25 carry one mark each.
Q. 1 Let , , be a b as is of . Conside r th e f ollowin g statements P and Q:
(P ) : ,, is a bas is of .
(Q ) : , 2 , 3 is a basis of
.
Wh i ch of th e ab ove s t atements h o l d TRUE?
(A) b o th P a nd Q ( B ) only P
(C ) only Q ( D) N either P n or Q
Q. 2 Consider the following stateme nts P an d Q:
(P ) : If 11 1
12 4
13 9
, then M is s in gular .
(Q ) : Let S b e a diagon a l iza b le matrix. If T is a matrix suc h t hat S + 5 T = Id , then T is
diagonalizable.
Which of the above s tate ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n ly Q ( D) Neither P nor Q
Q. 3 Consider the following stateme nts P an d Q:
(P ) : If M i s an complex matrix, then ?
.
(Q ) : There e xis t s a unitary ma trix with an e igen value ? su ch t h a t | ?| <1.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n l y Q ( D) Neither P n or Q
GATE 2016 Mathematics - MA
MA 3/16
Q. 4 Consider a r eal vec t or s p a ce V of d imen sion n and a no n ?zero line ar tra ns formation
? ? . If dim e ns ion and , f or some ? \0 , t hen
wh ic h of th e follo wing stateme nts is TRUE?
(A) determina n t | |
(B) Th e re exists a non?tri v ial s u b s p a ce of V such th a t 0 f o r all ? (C) T is invertib le
(D) is th e on ly e igenvalue of T
Q. 5 Let 0, 1 ?2, 3 a nd ? ? be a strictly increasing f u nction such that
is conn ected. Wh i ch of the foll owing statements is TRUE?
(A) h as e xactly one discontinuity
(B) has exactly two di s continu i ti es
(C) h as inf in itely man y d iscontinu i ties
(D) is conti nuou s
Q. 6 Let 1 and
4, 2.Then,
lim
? ?
1
1
? 1
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 7 Maximu m ? , ? 0 ,1 is equal to _ _ _ _ ___ __ ___ __ ___
Q. 8 Let , , , ? su ch that
0 . T h e n, the C auch y pr o blem
,,?,
, 0 o n 0
has a u nique solution if
(A) 0 (B) 0
(C) 0 ( D ) 0
GATE 2016 Mathematics - MA
MA 4/16
Q. 9 Let , be the d ' A lembert's s olu tion of th e in itial va lu e pro b le m for th e w ave
eq uatio n
0
, 0 ,
, 0 ,
wh ere c is a positi v e real numb e r a nd , are smo o th odd func t ions. Th e n, 0,1 is
eq ual to _ ___ __ ___ __
Q. 10 L e t th e pro babi l ity density func tion of a r a ndo m variable X be
0 1
2
2 1 1
2
1
0otherwise.
Then, the v a lue of c i s eq ual to ___ __ ___ ___ __ __ ___ ___ ___
Q. 11 Let V be th e set of all so l u tio n s o f the eq uatio n ?? ? 0 satisfyin g
0 1 , whe r e , are positive r ea l numbers. Th e n, di m ension(V ) is equal to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 12 Let ?? ? 0, ? ? , ? ,where and are
c o ntinuous func t i o ns. If
2 and 2 are two lin e arly indepe ndent s olu t ions o f t he ab o ve eq u a t ion , then
| 4 0 2 1 | is e qual to __ __ ___ ___ ___ __ __ ___
Q. 13
Let b e th e Legendre pol y nomial of degree and
, where k
i s a n on ?neg ative inte ger. Consi der the fo llowi ng stat e ment s P and Q:
(P ) : 0 i f .
(Q) : 0 if is a n odd intege r.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 5/16
Q. 14 Consider the following stateme nts P an d Q:
(P) :
?? ?
0 has two linearly i ndepende nt Fro beni us series
solutions near 0.
(Q) :
?? 3 ? 0 has two li n early independe n t F r obenius series
solutions near 0.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 15 L e t th e polynomial
be app r o x i m ated b y a p o ly no mi al o f degre e 2, which
i n ter p olates
at 1, 0 and 1. T h e n, th e m a x imum abs olute interpol a t i o n er r or
ove r th e inter val 1 , 1 is equal to __ __ ___ ___ __ ___ __ _ _ _ __
Q. 16 Let b e a se q u ence o f dist inc t points in 0, 1 ? ? | | 1 with
lim
? ?
0 . Consider the following stateme nts P an d Q :
(P ) : Th e r e exists a u niq u e analytic functi o n f on 0,1 suc h t ha t for
a ll n .
(Q ) : Th e r e exists an an alytic functi o n f on 0, 1 such that 0 if n i s e v en
and 1 if n is o d d.
Which o f the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 17 Let , b e a to pologic a l space with the cofinite to p o logy. Every infi n i t e subset of
i s
(A) Compac t but NOT co nnected
(B) Both compa c t a nd connected
(C) NOT compa c t b u t c o nne c t e d
(D) Ne i t h e r c o mp act no r c o nnected
GATE 2016 Mathematics - MA
MA 6/16
Q. 18 Let ?
? , ?0 and ?
?
?
0 .
T hen, dim ens ion
is equal to __ ___ __ ___ ___ __ ____ _ ___
Q. 19 Consider , ? ? ?
?
, wher e ? , ?
?
maximum | | , | | . Let ?
? be
def i n e d by ,
an d the no r m pre s e rvi ng l ine a r e x tension o f to
, ? ? ?
?
. Th en, 1, 1,1 is e qual to __ ___ __ ___ ___ __ ____ _ ___ ___ __ ____ _ _
Q. 20 ? 0, 1 ? 0, 1 is called a sh rinking map if | | | | for all
, ? 0,1 and a contr a ction if there exists an 1 such tha t
| | | | f or all , ? 0,1 .
Which of the following stateme nts is TRU E f o r the function
?
(A) is both a s h rinking map and a c o ntractio n
(B) is a s hrinking m ap but NO T a c ontra c tion
(C) i s NOT a shrinking map but a c o ntractio n
(D) is Ne i t her a shri nki n g m a p no r a c ontrac ti on
Q. 21 Let be t he s et of all r eal ma t rices with the usual norm t o p ology. Con side r the
following sta t emen ts P and Q:
(P ) : T h e s e t of all sym m e tric p ositive defin i te matrices in is conne cte d .
(Q ) : T h e s e t of all inver tibl e m atric es in i s c om pact .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 7/16
Q. 22 Let ,
,
,?,
be a r and o m samp le f rom the followin g pr o bability d ensity
func ti on for 0 ?,0 1,
; , 1
? ; 0otherwise.
He re a nd are un kn own parame ters. Which of th e f ollowin g st at em ent s i s TRU E?
(A) Max i mum likel i hood e stima t or of on ly e xists
(B) Maximum li kelihood esti m ator of o n l y exis ts
(C) Maximum likeli h o o d estimator s o f bo th and e xist
(D) Ma ximu m likeli h o od estimator of Neither nor e xists
Q. 23 Suppose X an d Y are tw o rando m variables suc h th a t is a norma l random
variab le for al l , ? . Conside r the f ollowin g statements P, Q , R an d S :
(P ) : X is a stan dard n o r mal ran d om v ariable.
(Q) : Th e conditi o na l distrib u ti on of X g ive n Y is n o rmal.
(R) : Th e co nditio nal distributio n of X given is n o rmal.
(S) : h a s m e a n 0.
Which of the above state me nts A LWAYS hold T R U E?
(A) both P and Q ( B) b oth Q a nd R
(C) b o th Q a nd S ( D) b oth P a nd S
Q.24 Co nsi d er t h e foll owing st a t e m e nts P and Q :
( P ) : I f i s a no rmal subgroup o f o r d er 4 o f the s y mmetric gro up
, the n
is
a beli a n .
(Q ) : If 1 , , , is the qua ternion group, then
1, 1
i s ab el ian.
Whic h of th e abo ve st at e ments hol d T RUE?
(A ) both P a nd Q (B) onl y P
(C ) onl y Q ( D) Neith er P nor Q
Q.25 Let b e a fiel d of order 3 2. T hen t he num ber o f no n? z e ro so luti ons , ? of
the eq ua tio n
0 is equal to __ ___ __ ___ _ _ ___ ___ __ ____ _ ___ __ _
GATE 2016 Mathematics - MA
MA 8/16
Q. 26 ? Q. 55 carry two marks each.
Q. 26 Let ? ? || 2 be oriente d in the counter?clockwise direction . Let
1
2 1
.
Then, the v a lue of i s equal to ___ ___ __ ___ ___ __ _ _ _ __ ___ __
Q. 27 Let , 2 be a h a rmonic fu n ctio n and , i t s
harmon ic c onju gate. If 0,0 1 , th en | 1 , 1 | is eq u al to _ __ ___ ___ __ ___ _
Q. 28 L e t b e th e triangula r path co nnecting the poin ts (0,0), (2,2) and (0,2) in the counter?
clockwise direction in
. T h en
6
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 29 Let y b e th e solutio n of
? | |,? 1 0.
Then 1 i s equ a l to
(A)
(B)
2
(C) 2
(D) 2 2
Q. 30 Let X b e a r ando m v ari able with th e f oll owi ng c um ul ati ve d i stri b u ti o n functio n :
00
0 1
2
3
4
1
2
1
1 1.
Then 1 is equal to __ ___ ___ __ ___ __ ____
GATE 2016 Mathematics - MA
MA 9/16
Q. 31 Let ? b e the curve which passes thr o ugh ( 0 ,1 ) and i n tersects ea c h c u r v e of the f amil y
orthogon a l ly. Th en ? a l so passes th rough the poin t
(A) ? 2,0 (B) 0, ? 2
(C) 1,1 ( D ) 1,1
Q. 32 Let ?
?
be the Fourier s e rie s of th e
2 periodic fu nction defined by 4, . Then
?
?
i s eq u al to ___ __ ___ ___ ___ __ __ ___ ___ __
Q. 33 L e t b e a con t inu o us f u nction on 0, ? . If
1 4 4 ,
the n
is equal to __ _ _ ___ _ _ ___ ___ __ ____ _ __
Q. 34
Let
?
and
. Th e n,
is e q u a l to
(A) ln 10 1 (B) l n 10 1
(C) l n 10
(D) ln10
Q. 35 F o r any , ? \ 0,1
, let
, distance , ,0,1
inimum ? ,
?0,1
.
Then , || 3,4 || i s equal to ___ __ ___ __ ___ ___ ___ _
Q. 36
Let
an d
.
Th e n ?
?
?
? is
eq ual to ___ __ ___ ___ __ __ ___ ___ __ ___ __ ___ ___ __ ___
GATE 2016 Mathematics - MA
MA 10/16
Q. 37
Let b e a real matr i x with e igenvalu es 1 , 0 a nd 3. If th e e i ge nve ctors
c o rres p ond i ng to 1 and 0 a r e 1,1,1
a nd 1, 1 ,0 re s p e ctivel y, th e n th e value o f
3 i s eq ual to _ ___ __ ___ ____ _ ___
Q. 38
Let 11 0
01 1
00 1
a nd
!
!
? . If
, th e n
1
i s eq u al to ___ __ ___ ___ ___ __ __ ___ ___
Q. 39
L e t th e inte gral , w h e r e 02
4 2 4.
Consider the following stateme nts P an d Q:
(P ) : If
is the va l ue of the integr al o b t ained b y the c om po site tra pez oidal rule with
t wo equal s u b?i n terval s , then
is exac t.
(Q ) : If is the va l ue of the integr al o b t ained b y the c om po site tra pez oidal rule with
thr ee eq ual su b ? interva ls, t he n is e x act.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 40 The diff eren ce bet ween t he least two e i genval u e s of th e b ou nda r y valu e p roblem
?? 0, 0 0 0, ? 0,
i s eq u al to _ ___ ___ __ ____ _ ___ __ ___ ____ _ ___
Q. 41
The numb er of r o ots of the equa tion
cos 0 in the i n terval ,
i s equal
to __ ___ __ _____ __
GATE 2016 Mathematics - MA
MA 11/16
Q. 42 F o r th e fixed po int i t e r ation , 0,1,2,??, consider the following
statements P and Q :
(P ) : If 1
then the fixe d poin t iteration c on verges t o 2 for all
? 1, 10 0 .
(Q ) : If ?2 the n th e fixed p o i nt iter ation converg es t o 2 fo r
a ll
? 0, 10 0 .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neither P nor Q
Q. 43 Let ? ?
? ?
be d e fined b y
,
,? , ,?
,
,?,
,? .
Then
(A) ? ? 1
(B) ? ? 2 but bounded
(C) 1 ? ? 2
(D) ? ? is unbo unded
Q. 44 Minimi z e 2 subject to
2 3
2
0, 0.
Then, the m i nimum value of is equal to _ __ ___ __ _ _ _ ___ __ ___ _ _ _ __ _
Q. 45 Maximi ze 1 1 su b ject to
10 1
2 2 2
, , 0.
Then, the m a ximu m val u e of is equal to __ _ _ ___ _ _ _ _ _ ___ __ ____ _ ___
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
GATE 2016 Mathematics - MA
MA 2/16
Q. 1 ? Q. 25 carry one mark each.
Q. 1 Let , , be a b as is of . Conside r th e f ollowin g statements P and Q:
(P ) : ,, is a bas is of .
(Q ) : , 2 , 3 is a basis of
.
Wh i ch of th e ab ove s t atements h o l d TRUE?
(A) b o th P a nd Q ( B ) only P
(C ) only Q ( D) N either P n or Q
Q. 2 Consider the following stateme nts P an d Q:
(P ) : If 11 1
12 4
13 9
, then M is s in gular .
(Q ) : Let S b e a diagon a l iza b le matrix. If T is a matrix suc h t hat S + 5 T = Id , then T is
diagonalizable.
Which of the above s tate ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n ly Q ( D) Neither P nor Q
Q. 3 Consider the following stateme nts P an d Q:
(P ) : If M i s an complex matrix, then ?
.
(Q ) : There e xis t s a unitary ma trix with an e igen value ? su ch t h a t | ?| <1.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n l y Q ( D) Neither P n or Q
GATE 2016 Mathematics - MA
MA 3/16
Q. 4 Consider a r eal vec t or s p a ce V of d imen sion n and a no n ?zero line ar tra ns formation
? ? . If dim e ns ion and , f or some ? \0 , t hen
wh ic h of th e follo wing stateme nts is TRUE?
(A) determina n t | |
(B) Th e re exists a non?tri v ial s u b s p a ce of V such th a t 0 f o r all ? (C) T is invertib le
(D) is th e on ly e igenvalue of T
Q. 5 Let 0, 1 ?2, 3 a nd ? ? be a strictly increasing f u nction such that
is conn ected. Wh i ch of the foll owing statements is TRUE?
(A) h as e xactly one discontinuity
(B) has exactly two di s continu i ti es
(C) h as inf in itely man y d iscontinu i ties
(D) is conti nuou s
Q. 6 Let 1 and
4, 2.Then,
lim
? ?
1
1
? 1
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 7 Maximu m ? , ? 0 ,1 is equal to _ _ _ _ ___ __ ___ __ ___
Q. 8 Let , , , ? su ch that
0 . T h e n, the C auch y pr o blem
,,?,
, 0 o n 0
has a u nique solution if
(A) 0 (B) 0
(C) 0 ( D ) 0
GATE 2016 Mathematics - MA
MA 4/16
Q. 9 Let , be the d ' A lembert's s olu tion of th e in itial va lu e pro b le m for th e w ave
eq uatio n
0
, 0 ,
, 0 ,
wh ere c is a positi v e real numb e r a nd , are smo o th odd func t ions. Th e n, 0,1 is
eq ual to _ ___ __ ___ __
Q. 10 L e t th e pro babi l ity density func tion of a r a ndo m variable X be
0 1
2
2 1 1
2
1
0otherwise.
Then, the v a lue of c i s eq ual to ___ __ ___ ___ __ __ ___ ___ ___
Q. 11 Let V be th e set of all so l u tio n s o f the eq uatio n ?? ? 0 satisfyin g
0 1 , whe r e , are positive r ea l numbers. Th e n, di m ension(V ) is equal to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 12 Let ?? ? 0, ? ? , ? ,where and are
c o ntinuous func t i o ns. If
2 and 2 are two lin e arly indepe ndent s olu t ions o f t he ab o ve eq u a t ion , then
| 4 0 2 1 | is e qual to __ __ ___ ___ ___ __ __ ___
Q. 13
Let b e th e Legendre pol y nomial of degree and
, where k
i s a n on ?neg ative inte ger. Consi der the fo llowi ng stat e ment s P and Q:
(P ) : 0 i f .
(Q) : 0 if is a n odd intege r.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 5/16
Q. 14 Consider the following stateme nts P an d Q:
(P) :
?? ?
0 has two linearly i ndepende nt Fro beni us series
solutions near 0.
(Q) :
?? 3 ? 0 has two li n early independe n t F r obenius series
solutions near 0.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 15 L e t th e polynomial
be app r o x i m ated b y a p o ly no mi al o f degre e 2, which
i n ter p olates
at 1, 0 and 1. T h e n, th e m a x imum abs olute interpol a t i o n er r or
ove r th e inter val 1 , 1 is equal to __ __ ___ ___ __ ___ __ _ _ _ __
Q. 16 Let b e a se q u ence o f dist inc t points in 0, 1 ? ? | | 1 with
lim
? ?
0 . Consider the following stateme nts P an d Q :
(P ) : Th e r e exists a u niq u e analytic functi o n f on 0,1 suc h t ha t for
a ll n .
(Q ) : Th e r e exists an an alytic functi o n f on 0, 1 such that 0 if n i s e v en
and 1 if n is o d d.
Which o f the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 17 Let , b e a to pologic a l space with the cofinite to p o logy. Every infi n i t e subset of
i s
(A) Compac t but NOT co nnected
(B) Both compa c t a nd connected
(C) NOT compa c t b u t c o nne c t e d
(D) Ne i t h e r c o mp act no r c o nnected
GATE 2016 Mathematics - MA
MA 6/16
Q. 18 Let ?
? , ?0 and ?
?
?
0 .
T hen, dim ens ion
is equal to __ ___ __ ___ ___ __ ____ _ ___
Q. 19 Consider , ? ? ?
?
, wher e ? , ?
?
maximum | | , | | . Let ?
? be
def i n e d by ,
an d the no r m pre s e rvi ng l ine a r e x tension o f to
, ? ? ?
?
. Th en, 1, 1,1 is e qual to __ ___ __ ___ ___ __ ____ _ ___ ___ __ ____ _ _
Q. 20 ? 0, 1 ? 0, 1 is called a sh rinking map if | | | | for all
, ? 0,1 and a contr a ction if there exists an 1 such tha t
| | | | f or all , ? 0,1 .
Which of the following stateme nts is TRU E f o r the function
?
(A) is both a s h rinking map and a c o ntractio n
(B) is a s hrinking m ap but NO T a c ontra c tion
(C) i s NOT a shrinking map but a c o ntractio n
(D) is Ne i t her a shri nki n g m a p no r a c ontrac ti on
Q. 21 Let be t he s et of all r eal ma t rices with the usual norm t o p ology. Con side r the
following sta t emen ts P and Q:
(P ) : T h e s e t of all sym m e tric p ositive defin i te matrices in is conne cte d .
(Q ) : T h e s e t of all inver tibl e m atric es in i s c om pact .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 7/16
Q. 22 Let ,
,
,?,
be a r and o m samp le f rom the followin g pr o bability d ensity
func ti on for 0 ?,0 1,
; , 1
? ; 0otherwise.
He re a nd are un kn own parame ters. Which of th e f ollowin g st at em ent s i s TRU E?
(A) Max i mum likel i hood e stima t or of on ly e xists
(B) Maximum li kelihood esti m ator of o n l y exis ts
(C) Maximum likeli h o o d estimator s o f bo th and e xist
(D) Ma ximu m likeli h o od estimator of Neither nor e xists
Q. 23 Suppose X an d Y are tw o rando m variables suc h th a t is a norma l random
variab le for al l , ? . Conside r the f ollowin g statements P, Q , R an d S :
(P ) : X is a stan dard n o r mal ran d om v ariable.
(Q) : Th e conditi o na l distrib u ti on of X g ive n Y is n o rmal.
(R) : Th e co nditio nal distributio n of X given is n o rmal.
(S) : h a s m e a n 0.
Which of the above state me nts A LWAYS hold T R U E?
(A) both P and Q ( B) b oth Q a nd R
(C) b o th Q a nd S ( D) b oth P a nd S
Q.24 Co nsi d er t h e foll owing st a t e m e nts P and Q :
( P ) : I f i s a no rmal subgroup o f o r d er 4 o f the s y mmetric gro up
, the n
is
a beli a n .
(Q ) : If 1 , , , is the qua ternion group, then
1, 1
i s ab el ian.
Whic h of th e abo ve st at e ments hol d T RUE?
(A ) both P a nd Q (B) onl y P
(C ) onl y Q ( D) Neith er P nor Q
Q.25 Let b e a fiel d of order 3 2. T hen t he num ber o f no n? z e ro so luti ons , ? of
the eq ua tio n
0 is equal to __ ___ __ ___ _ _ ___ ___ __ ____ _ ___ __ _
GATE 2016 Mathematics - MA
MA 8/16
Q. 26 ? Q. 55 carry two marks each.
Q. 26 Let ? ? || 2 be oriente d in the counter?clockwise direction . Let
1
2 1
.
Then, the v a lue of i s equal to ___ ___ __ ___ ___ __ _ _ _ __ ___ __
Q. 27 Let , 2 be a h a rmonic fu n ctio n and , i t s
harmon ic c onju gate. If 0,0 1 , th en | 1 , 1 | is eq u al to _ __ ___ ___ __ ___ _
Q. 28 L e t b e th e triangula r path co nnecting the poin ts (0,0), (2,2) and (0,2) in the counter?
clockwise direction in
. T h en
6
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 29 Let y b e th e solutio n of
? | |,? 1 0.
Then 1 i s equ a l to
(A)
(B)
2
(C) 2
(D) 2 2
Q. 30 Let X b e a r ando m v ari able with th e f oll owi ng c um ul ati ve d i stri b u ti o n functio n :
00
0 1
2
3
4
1
2
1
1 1.
Then 1 is equal to __ ___ ___ __ ___ __ ____
GATE 2016 Mathematics - MA
MA 9/16
Q. 31 Let ? b e the curve which passes thr o ugh ( 0 ,1 ) and i n tersects ea c h c u r v e of the f amil y
orthogon a l ly. Th en ? a l so passes th rough the poin t
(A) ? 2,0 (B) 0, ? 2
(C) 1,1 ( D ) 1,1
Q. 32 Let ?
?
be the Fourier s e rie s of th e
2 periodic fu nction defined by 4, . Then
?
?
i s eq u al to ___ __ ___ ___ ___ __ __ ___ ___ __
Q. 33 L e t b e a con t inu o us f u nction on 0, ? . If
1 4 4 ,
the n
is equal to __ _ _ ___ _ _ ___ ___ __ ____ _ __
Q. 34
Let
?
and
. Th e n,
is e q u a l to
(A) ln 10 1 (B) l n 10 1
(C) l n 10
(D) ln10
Q. 35 F o r any , ? \ 0,1
, let
, distance , ,0,1
inimum ? ,
?0,1
.
Then , || 3,4 || i s equal to ___ __ ___ __ ___ ___ ___ _
Q. 36
Let
an d
.
Th e n ?
?
?
? is
eq ual to ___ __ ___ ___ __ __ ___ ___ __ ___ __ ___ ___ __ ___
GATE 2016 Mathematics - MA
MA 10/16
Q. 37
Let b e a real matr i x with e igenvalu es 1 , 0 a nd 3. If th e e i ge nve ctors
c o rres p ond i ng to 1 and 0 a r e 1,1,1
a nd 1, 1 ,0 re s p e ctivel y, th e n th e value o f
3 i s eq ual to _ ___ __ ___ ____ _ ___
Q. 38
Let 11 0
01 1
00 1
a nd
!
!
? . If
, th e n
1
i s eq u al to ___ __ ___ ___ ___ __ __ ___ ___
Q. 39
L e t th e inte gral , w h e r e 02
4 2 4.
Consider the following stateme nts P an d Q:
(P ) : If
is the va l ue of the integr al o b t ained b y the c om po site tra pez oidal rule with
t wo equal s u b?i n terval s , then
is exac t.
(Q ) : If is the va l ue of the integr al o b t ained b y the c om po site tra pez oidal rule with
thr ee eq ual su b ? interva ls, t he n is e x act.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 40 The diff eren ce bet ween t he least two e i genval u e s of th e b ou nda r y valu e p roblem
?? 0, 0 0 0, ? 0,
i s eq u al to _ ___ ___ __ ____ _ ___ __ ___ ____ _ ___
Q. 41
The numb er of r o ots of the equa tion
cos 0 in the i n terval ,
i s equal
to __ ___ __ _____ __
GATE 2016 Mathematics - MA
MA 11/16
Q. 42 F o r th e fixed po int i t e r ation , 0,1,2,??, consider the following
statements P and Q :
(P ) : If 1
then the fixe d poin t iteration c on verges t o 2 for all
? 1, 10 0 .
(Q ) : If ?2 the n th e fixed p o i nt iter ation converg es t o 2 fo r
a ll
? 0, 10 0 .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neither P nor Q
Q. 43 Let ? ?
? ?
be d e fined b y
,
,? , ,?
,
,?,
,? .
Then
(A) ? ? 1
(B) ? ? 2 but bounded
(C) 1 ? ? 2
(D) ? ? is unbo unded
Q. 44 Minimi z e 2 subject to
2 3
2
0, 0.
Then, the m i nimum value of is equal to _ __ ___ __ _ _ _ ___ __ ___ _ _ _ __ _
Q. 45 Maximi ze 1 1 su b ject to
10 1
2 2 2
, , 0.
Then, the m a ximu m val u e of is equal to __ _ _ ___ _ _ _ _ _ ___ __ ____ _ ___
GATE 2016 Mathematics - MA
MA 12/16
Q. 46 Let ,
,
, ? be a seque nce of i.i.d. ran dom va ria b les with mea n 1 . If N is a
geometric r a ndom vari a b l e wi th the p ro babi lity mass function
;
1, 2, 3, ? and i t is ind ep end ent of th e 's, th en
? is eq u al to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 47 Let be an expon ential r andom va r i ab le wi t h mea n 1 a n d a ga mma r and o m
variab le wi t h me an 2 and varianc e 2. If an d
a re i ndepe ndentl y distr ib uted, the n
is e q ual to __ ___ __ ___ ___ __ __ ___ ___ __
Q. 48 Let
,
,
, ? be a seq uenc e of i .i. d . uniform 0,1 random vari a b l es. Th en, th e v al ue
of
lim
? ?
ln1 ? ln1
is equal to __ _ _ ___ ___ __ _ _ _ __ ___
Q. 49
Let X b e a s t andar d nor mal rando m variable. Then, 0 | | 1 is e qual to
(A)
?
?
(B)
? ?
(C)
?
?
(D)
?
?
Q. 50 Let ,
,
,?,
b e a ra n dom samp le fro m the p r o babi l ity density func tion
1 2 ;0
0 otherwise,
wh ere 0, 0 1 are parameters. Consid er the f ollowin g testing pro blem:
: 1 , 1 versus
: 0, 2.
Which of the following statements is T RU E?
(A) Un if ormly Most P ower ful test does NOT e x ist
(B) U n ifor mly Mo st Powe r f u l tes t is of the fo r m ?,
for some 0 ?
(C) U nifo r ml y Most Powerful tes t is of the for m ?,
for some 0 ?
(D) Unifor mly Most Powerful tes t i s o f th e f o r m ?
,
for
some 0
?
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
GATE 2016 Mathematics - MA
MA 2/16
Q. 1 ? Q. 25 carry one mark each.
Q. 1 Let , , be a b as is of . Conside r th e f ollowin g statements P and Q:
(P ) : ,, is a bas is of .
(Q ) : , 2 , 3 is a basis of
.
Wh i ch of th e ab ove s t atements h o l d TRUE?
(A) b o th P a nd Q ( B ) only P
(C ) only Q ( D) N either P n or Q
Q. 2 Consider the following stateme nts P an d Q:
(P ) : If 11 1
12 4
13 9
, then M is s in gular .
(Q ) : Let S b e a diagon a l iza b le matrix. If T is a matrix suc h t hat S + 5 T = Id , then T is
diagonalizable.
Which of the above s tate ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n ly Q ( D) Neither P nor Q
Q. 3 Consider the following stateme nts P an d Q:
(P ) : If M i s an complex matrix, then ?
.
(Q ) : There e xis t s a unitary ma trix with an e igen value ? su ch t h a t | ?| <1.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n l y Q ( D) Neither P n or Q
GATE 2016 Mathematics - MA
MA 3/16
Q. 4 Consider a r eal vec t or s p a ce V of d imen sion n and a no n ?zero line ar tra ns formation
? ? . If dim e ns ion and , f or some ? \0 , t hen
wh ic h of th e follo wing stateme nts is TRUE?
(A) determina n t | |
(B) Th e re exists a non?tri v ial s u b s p a ce of V such th a t 0 f o r all ? (C) T is invertib le
(D) is th e on ly e igenvalue of T
Q. 5 Let 0, 1 ?2, 3 a nd ? ? be a strictly increasing f u nction such that
is conn ected. Wh i ch of the foll owing statements is TRUE?
(A) h as e xactly one discontinuity
(B) has exactly two di s continu i ti es
(C) h as inf in itely man y d iscontinu i ties
(D) is conti nuou s
Q. 6 Let 1 and
4, 2.Then,
lim
? ?
1
1
? 1
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 7 Maximu m ? , ? 0 ,1 is equal to _ _ _ _ ___ __ ___ __ ___
Q. 8 Let , , , ? su ch that
0 . T h e n, the C auch y pr o blem
,,?,
, 0 o n 0
has a u nique solution if
(A) 0 (B) 0
(C) 0 ( D ) 0
GATE 2016 Mathematics - MA
MA 4/16
Q. 9 Let , be the d ' A lembert's s olu tion of th e in itial va lu e pro b le m for th e w ave
eq uatio n
0
, 0 ,
, 0 ,
wh ere c is a positi v e real numb e r a nd , are smo o th odd func t ions. Th e n, 0,1 is
eq ual to _ ___ __ ___ __
Q. 10 L e t th e pro babi l ity density func tion of a r a ndo m variable X be
0 1
2
2 1 1
2
1
0otherwise.
Then, the v a lue of c i s eq ual to ___ __ ___ ___ __ __ ___ ___ ___
Q. 11 Let V be th e set of all so l u tio n s o f the eq uatio n ?? ? 0 satisfyin g
0 1 , whe r e , are positive r ea l numbers. Th e n, di m ension(V ) is equal to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 12 Let ?? ? 0, ? ? , ? ,where and are
c o ntinuous func t i o ns. If
2 and 2 are two lin e arly indepe ndent s olu t ions o f t he ab o ve eq u a t ion , then
| 4 0 2 1 | is e qual to __ __ ___ ___ ___ __ __ ___
Q. 13
Let b e th e Legendre pol y nomial of degree and
, where k
i s a n on ?neg ative inte ger. Consi der the fo llowi ng stat e ment s P and Q:
(P ) : 0 i f .
(Q) : 0 if is a n odd intege r.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 5/16
Q. 14 Consider the following stateme nts P an d Q:
(P) :
?? ?
0 has two linearly i ndepende nt Fro beni us series
solutions near 0.
(Q) :
?? 3 ? 0 has two li n early independe n t F r obenius series
solutions near 0.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 15 L e t th e polynomial
be app r o x i m ated b y a p o ly no mi al o f degre e 2, which
i n ter p olates
at 1, 0 and 1. T h e n, th e m a x imum abs olute interpol a t i o n er r or
ove r th e inter val 1 , 1 is equal to __ __ ___ ___ __ ___ __ _ _ _ __
Q. 16 Let b e a se q u ence o f dist inc t points in 0, 1 ? ? | | 1 with
lim
? ?
0 . Consider the following stateme nts P an d Q :
(P ) : Th e r e exists a u niq u e analytic functi o n f on 0,1 suc h t ha t for
a ll n .
(Q ) : Th e r e exists an an alytic functi o n f on 0, 1 such that 0 if n i s e v en
and 1 if n is o d d.
Which o f the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 17 Let , b e a to pologic a l space with the cofinite to p o logy. Every infi n i t e subset of
i s
(A) Compac t but NOT co nnected
(B) Both compa c t a nd connected
(C) NOT compa c t b u t c o nne c t e d
(D) Ne i t h e r c o mp act no r c o nnected
GATE 2016 Mathematics - MA
MA 6/16
Q. 18 Let ?
? , ?0 and ?
?
?
0 .
T hen, dim ens ion
is equal to __ ___ __ ___ ___ __ ____ _ ___
Q. 19 Consider , ? ? ?
?
, wher e ? , ?
?
maximum | | , | | . Let ?
? be
def i n e d by ,
an d the no r m pre s e rvi ng l ine a r e x tension o f to
, ? ? ?
?
. Th en, 1, 1,1 is e qual to __ ___ __ ___ ___ __ ____ _ ___ ___ __ ____ _ _
Q. 20 ? 0, 1 ? 0, 1 is called a sh rinking map if | | | | for all
, ? 0,1 and a contr a ction if there exists an 1 such tha t
| | | | f or all , ? 0,1 .
Which of the following stateme nts is TRU E f o r the function
?
(A) is both a s h rinking map and a c o ntractio n
(B) is a s hrinking m ap but NO T a c ontra c tion
(C) i s NOT a shrinking map but a c o ntractio n
(D) is Ne i t her a shri nki n g m a p no r a c ontrac ti on
Q. 21 Let be t he s et of all r eal ma t rices with the usual norm t o p ology. Con side r the
following sta t emen ts P and Q:
(P ) : T h e s e t of all sym m e tric p ositive defin i te matrices in is conne cte d .
(Q ) : T h e s e t of all inver tibl e m atric es in i s c om pact .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 7/16
Q. 22 Let ,
,
,?,
be a r and o m samp le f rom the followin g pr o bability d ensity
func ti on for 0 ?,0 1,
; , 1
? ; 0otherwise.
He re a nd are un kn own parame ters. Which of th e f ollowin g st at em ent s i s TRU E?
(A) Max i mum likel i hood e stima t or of on ly e xists
(B) Maximum li kelihood esti m ator of o n l y exis ts
(C) Maximum likeli h o o d estimator s o f bo th and e xist
(D) Ma ximu m likeli h o od estimator of Neither nor e xists
Q. 23 Suppose X an d Y are tw o rando m variables suc h th a t is a norma l random
variab le for al l , ? . Conside r the f ollowin g statements P, Q , R an d S :
(P ) : X is a stan dard n o r mal ran d om v ariable.
(Q) : Th e conditi o na l distrib u ti on of X g ive n Y is n o rmal.
(R) : Th e co nditio nal distributio n of X given is n o rmal.
(S) : h a s m e a n 0.
Which of the above state me nts A LWAYS hold T R U E?
(A) both P and Q ( B) b oth Q a nd R
(C) b o th Q a nd S ( D) b oth P a nd S
Q.24 Co nsi d er t h e foll owing st a t e m e nts P and Q :
( P ) : I f i s a no rmal subgroup o f o r d er 4 o f the s y mmetric gro up
, the n
is
a beli a n .
(Q ) : If 1 , , , is the qua ternion group, then
1, 1
i s ab el ian.
Whic h of th e abo ve st at e ments hol d T RUE?
(A ) both P a nd Q (B) onl y P
(C ) onl y Q ( D) Neith er P nor Q
Q.25 Let b e a fiel d of order 3 2. T hen t he num ber o f no n? z e ro so luti ons , ? of
the eq ua tio n
0 is equal to __ ___ __ ___ _ _ ___ ___ __ ____ _ ___ __ _
GATE 2016 Mathematics - MA
MA 8/16
Q. 26 ? Q. 55 carry two marks each.
Q. 26 Let ? ? || 2 be oriente d in the counter?clockwise direction . Let
1
2 1
.
Then, the v a lue of i s equal to ___ ___ __ ___ ___ __ _ _ _ __ ___ __
Q. 27 Let , 2 be a h a rmonic fu n ctio n and , i t s
harmon ic c onju gate. If 0,0 1 , th en | 1 , 1 | is eq u al to _ __ ___ ___ __ ___ _
Q. 28 L e t b e th e triangula r path co nnecting the poin ts (0,0), (2,2) and (0,2) in the counter?
clockwise direction in
. T h en
6
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 29 Let y b e th e solutio n of
? | |,? 1 0.
Then 1 i s equ a l to
(A)
(B)
2
(C) 2
(D) 2 2
Q. 30 Let X b e a r ando m v ari able with th e f oll owi ng c um ul ati ve d i stri b u ti o n functio n :
00
0 1
2
3
4
1
2
1
1 1.
Then 1 is equal to __ ___ ___ __ ___ __ ____
GATE 2016 Mathematics - MA
MA 9/16
Q. 31 Let ? b e the curve which passes thr o ugh ( 0 ,1 ) and i n tersects ea c h c u r v e of the f amil y
orthogon a l ly. Th en ? a l so passes th rough the poin t
(A) ? 2,0 (B) 0, ? 2
(C) 1,1 ( D ) 1,1
Q. 32 Let ?
?
be the Fourier s e rie s of th e
2 periodic fu nction defined by 4, . Then
?
?
i s eq u al to ___ __ ___ ___ ___ __ __ ___ ___ __
Q. 33 L e t b e a con t inu o us f u nction on 0, ? . If
1 4 4 ,
the n
is equal to __ _ _ ___ _ _ ___ ___ __ ____ _ __
Q. 34
Let
?
and
. Th e n,
is e q u a l to
(A) ln 10 1 (B) l n 10 1
(C) l n 10
(D) ln10
Q. 35 F o r any , ? \ 0,1
, let
, distance , ,0,1
inimum ? ,
?0,1
.
Then , || 3,4 || i s equal to ___ __ ___ __ ___ ___ ___ _
Q. 36
Let
an d
.
Th e n ?
?
?
? is
eq ual to ___ __ ___ ___ __ __ ___ ___ __ ___ __ ___ ___ __ ___
GATE 2016 Mathematics - MA
MA 10/16
Q. 37
Let b e a real matr i x with e igenvalu es 1 , 0 a nd 3. If th e e i ge nve ctors
c o rres p ond i ng to 1 and 0 a r e 1,1,1
a nd 1, 1 ,0 re s p e ctivel y, th e n th e value o f
3 i s eq ual to _ ___ __ ___ ____ _ ___
Q. 38
Let 11 0
01 1
00 1
a nd
!
!
? . If
, th e n
1
i s eq u al to ___ __ ___ ___ ___ __ __ ___ ___
Q. 39
L e t th e inte gral , w h e r e 02
4 2 4.
Consider the following stateme nts P an d Q:
(P ) : If
is the va l ue of the integr al o b t ained b y the c om po site tra pez oidal rule with
t wo equal s u b?i n terval s , then
is exac t.
(Q ) : If is the va l ue of the integr al o b t ained b y the c om po site tra pez oidal rule with
thr ee eq ual su b ? interva ls, t he n is e x act.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 40 The diff eren ce bet ween t he least two e i genval u e s of th e b ou nda r y valu e p roblem
?? 0, 0 0 0, ? 0,
i s eq u al to _ ___ ___ __ ____ _ ___ __ ___ ____ _ ___
Q. 41
The numb er of r o ots of the equa tion
cos 0 in the i n terval ,
i s equal
to __ ___ __ _____ __
GATE 2016 Mathematics - MA
MA 11/16
Q. 42 F o r th e fixed po int i t e r ation , 0,1,2,??, consider the following
statements P and Q :
(P ) : If 1
then the fixe d poin t iteration c on verges t o 2 for all
? 1, 10 0 .
(Q ) : If ?2 the n th e fixed p o i nt iter ation converg es t o 2 fo r
a ll
? 0, 10 0 .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neither P nor Q
Q. 43 Let ? ?
? ?
be d e fined b y
,
,? , ,?
,
,?,
,? .
Then
(A) ? ? 1
(B) ? ? 2 but bounded
(C) 1 ? ? 2
(D) ? ? is unbo unded
Q. 44 Minimi z e 2 subject to
2 3
2
0, 0.
Then, the m i nimum value of is equal to _ __ ___ __ _ _ _ ___ __ ___ _ _ _ __ _
Q. 45 Maximi ze 1 1 su b ject to
10 1
2 2 2
, , 0.
Then, the m a ximu m val u e of is equal to __ _ _ ___ _ _ _ _ _ ___ __ ____ _ ___
GATE 2016 Mathematics - MA
MA 12/16
Q. 46 Let ,
,
, ? be a seque nce of i.i.d. ran dom va ria b les with mea n 1 . If N is a
geometric r a ndom vari a b l e wi th the p ro babi lity mass function
;
1, 2, 3, ? and i t is ind ep end ent of th e 's, th en
? is eq u al to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 47 Let be an expon ential r andom va r i ab le wi t h mea n 1 a n d a ga mma r and o m
variab le wi t h me an 2 and varianc e 2. If an d
a re i ndepe ndentl y distr ib uted, the n
is e q ual to __ ___ __ ___ ___ __ __ ___ ___ __
Q. 48 Let
,
,
, ? be a seq uenc e of i .i. d . uniform 0,1 random vari a b l es. Th en, th e v al ue
of
lim
? ?
ln1 ? ln1
is equal to __ _ _ ___ ___ __ _ _ _ __ ___
Q. 49
Let X b e a s t andar d nor mal rando m variable. Then, 0 | | 1 is e qual to
(A)
?
?
(B)
? ?
(C)
?
?
(D)
?
?
Q. 50 Let ,
,
,?,
b e a ra n dom samp le fro m the p r o babi l ity density func tion
1 2 ;0
0 otherwise,
wh ere 0, 0 1 are parameters. Consid er the f ollowin g testing pro blem:
: 1 , 1 versus
: 0, 2.
Which of the following statements is T RU E?
(A) Un if ormly Most P ower ful test does NOT e x ist
(B) U n ifor mly Mo st Powe r f u l tes t is of the fo r m ?,
for some 0 ?
(C) U nifo r ml y Most Powerful tes t is of the for m ?,
for some 0 ?
(D) Unifor mly Most Powerful tes t i s o f th e f o r m ?
,
for
some 0
?
GATE 2016 Mathematics - MA
MA 13/16
Q.51 Let
,
,
, ? be a sequen c e of i .i.d . , 1 rando m variables. Then,
lim
? ?
? 2
| |
is e q u al to __ ___ __ ___ ____ _ ___ __ ___ ____ _ _
Q.52 Let
,
,
,?, be a rand o m samp le from un ifor m 1, , f or som e 1. If
Maximum ,
,
,? , , the n t he U MV UE of i s
(A )
(B)
(C )
(D)
Q. 53 Let
1,
2 b e a ra ndo m sample fro m a Po i sson rando m
variab le wi t h me an , wh e re ? 1, 2 . T hen, t he m axi mu m li kelihood e sti m ato r of
i s eq u al to ___ __ ___ ___ ___ __ __ __
Q. 54 The remainder when 9 8 ! is divi ded b y 101 is e q u a l to __ ___ ___ __ ___ __ ___ ____ _ ___ __
Q. 55 L e t b e a gr o u p whose p r esentati o n is
, |
, .
Then is isomorp hic to
(A) (B)
(C) (D)
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
GATE 2016 Mathematics - MA
MA 2/16
Q. 1 ? Q. 25 carry one mark each.
Q. 1 Let , , be a b as is of . Conside r th e f ollowin g statements P and Q:
(P ) : ,, is a bas is of .
(Q ) : , 2 , 3 is a basis of
.
Wh i ch of th e ab ove s t atements h o l d TRUE?
(A) b o th P a nd Q ( B ) only P
(C ) only Q ( D) N either P n or Q
Q. 2 Consider the following stateme nts P an d Q:
(P ) : If 11 1
12 4
13 9
, then M is s in gular .
(Q ) : Let S b e a diagon a l iza b le matrix. If T is a matrix suc h t hat S + 5 T = Id , then T is
diagonalizable.
Which of the above s tate ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n ly Q ( D) Neither P nor Q
Q. 3 Consider the following stateme nts P an d Q:
(P ) : If M i s an complex matrix, then ?
.
(Q ) : There e xis t s a unitary ma trix with an e igen value ? su ch t h a t | ?| <1.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n l y Q ( D) Neither P n or Q
GATE 2016 Mathematics - MA
MA 3/16
Q. 4 Consider a r eal vec t or s p a ce V of d imen sion n and a no n ?zero line ar tra ns formation
? ? . If dim e ns ion and , f or some ? \0 , t hen
wh ic h of th e follo wing stateme nts is TRUE?
(A) determina n t | |
(B) Th e re exists a non?tri v ial s u b s p a ce of V such th a t 0 f o r all ? (C) T is invertib le
(D) is th e on ly e igenvalue of T
Q. 5 Let 0, 1 ?2, 3 a nd ? ? be a strictly increasing f u nction such that
is conn ected. Wh i ch of the foll owing statements is TRUE?
(A) h as e xactly one discontinuity
(B) has exactly two di s continu i ti es
(C) h as inf in itely man y d iscontinu i ties
(D) is conti nuou s
Q. 6 Let 1 and
4, 2.Then,
lim
? ?
1
1
? 1
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 7 Maximu m ? , ? 0 ,1 is equal to _ _ _ _ ___ __ ___ __ ___
Q. 8 Let , , , ? su ch that
0 . T h e n, the C auch y pr o blem
,,?,
, 0 o n 0
has a u nique solution if
(A) 0 (B) 0
(C) 0 ( D ) 0
GATE 2016 Mathematics - MA
MA 4/16
Q. 9 Let , be the d ' A lembert's s olu tion of th e in itial va lu e pro b le m for th e w ave
eq uatio n
0
, 0 ,
, 0 ,
wh ere c is a positi v e real numb e r a nd , are smo o th odd func t ions. Th e n, 0,1 is
eq ual to _ ___ __ ___ __
Q. 10 L e t th e pro babi l ity density func tion of a r a ndo m variable X be
0 1
2
2 1 1
2
1
0otherwise.
Then, the v a lue of c i s eq ual to ___ __ ___ ___ __ __ ___ ___ ___
Q. 11 Let V be th e set of all so l u tio n s o f the eq uatio n ?? ? 0 satisfyin g
0 1 , whe r e , are positive r ea l numbers. Th e n, di m ension(V ) is equal to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 12 Let ?? ? 0, ? ? , ? ,where and are
c o ntinuous func t i o ns. If
2 and 2 are two lin e arly indepe ndent s olu t ions o f t he ab o ve eq u a t ion , then
| 4 0 2 1 | is e qual to __ __ ___ ___ ___ __ __ ___
Q. 13
Let b e th e Legendre pol y nomial of degree and
, where k
i s a n on ?neg ative inte ger. Consi der the fo llowi ng stat e ment s P and Q:
(P ) : 0 i f .
(Q) : 0 if is a n odd intege r.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 5/16
Q. 14 Consider the following stateme nts P an d Q:
(P) :
?? ?
0 has two linearly i ndepende nt Fro beni us series
solutions near 0.
(Q) :
?? 3 ? 0 has two li n early independe n t F r obenius series
solutions near 0.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 15 L e t th e polynomial
be app r o x i m ated b y a p o ly no mi al o f degre e 2, which
i n ter p olates
at 1, 0 and 1. T h e n, th e m a x imum abs olute interpol a t i o n er r or
ove r th e inter val 1 , 1 is equal to __ __ ___ ___ __ ___ __ _ _ _ __
Q. 16 Let b e a se q u ence o f dist inc t points in 0, 1 ? ? | | 1 with
lim
? ?
0 . Consider the following stateme nts P an d Q :
(P ) : Th e r e exists a u niq u e analytic functi o n f on 0,1 suc h t ha t for
a ll n .
(Q ) : Th e r e exists an an alytic functi o n f on 0, 1 such that 0 if n i s e v en
and 1 if n is o d d.
Which o f the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 17 Let , b e a to pologic a l space with the cofinite to p o logy. Every infi n i t e subset of
i s
(A) Compac t but NOT co nnected
(B) Both compa c t a nd connected
(C) NOT compa c t b u t c o nne c t e d
(D) Ne i t h e r c o mp act no r c o nnected
GATE 2016 Mathematics - MA
MA 6/16
Q. 18 Let ?
? , ?0 and ?
?
?
0 .
T hen, dim ens ion
is equal to __ ___ __ ___ ___ __ ____ _ ___
Q. 19 Consider , ? ? ?
?
, wher e ? , ?
?
maximum | | , | | . Let ?
? be
def i n e d by ,
an d the no r m pre s e rvi ng l ine a r e x tension o f to
, ? ? ?
?
. Th en, 1, 1,1 is e qual to __ ___ __ ___ ___ __ ____ _ ___ ___ __ ____ _ _
Q. 20 ? 0, 1 ? 0, 1 is called a sh rinking map if | | | | for all
, ? 0,1 and a contr a ction if there exists an 1 such tha t
| | | | f or all , ? 0,1 .
Which of the following stateme nts is TRU E f o r the function
?
(A) is both a s h rinking map and a c o ntractio n
(B) is a s hrinking m ap but NO T a c ontra c tion
(C) i s NOT a shrinking map but a c o ntractio n
(D) is Ne i t her a shri nki n g m a p no r a c ontrac ti on
Q. 21 Let be t he s et of all r eal ma t rices with the usual norm t o p ology. Con side r the
following sta t emen ts P and Q:
(P ) : T h e s e t of all sym m e tric p ositive defin i te matrices in is conne cte d .
(Q ) : T h e s e t of all inver tibl e m atric es in i s c om pact .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 7/16
Q. 22 Let ,
,
,?,
be a r and o m samp le f rom the followin g pr o bability d ensity
func ti on for 0 ?,0 1,
; , 1
? ; 0otherwise.
He re a nd are un kn own parame ters. Which of th e f ollowin g st at em ent s i s TRU E?
(A) Max i mum likel i hood e stima t or of on ly e xists
(B) Maximum li kelihood esti m ator of o n l y exis ts
(C) Maximum likeli h o o d estimator s o f bo th and e xist
(D) Ma ximu m likeli h o od estimator of Neither nor e xists
Q. 23 Suppose X an d Y are tw o rando m variables suc h th a t is a norma l random
variab le for al l , ? . Conside r the f ollowin g statements P, Q , R an d S :
(P ) : X is a stan dard n o r mal ran d om v ariable.
(Q) : Th e conditi o na l distrib u ti on of X g ive n Y is n o rmal.
(R) : Th e co nditio nal distributio n of X given is n o rmal.
(S) : h a s m e a n 0.
Which of the above state me nts A LWAYS hold T R U E?
(A) both P and Q ( B) b oth Q a nd R
(C) b o th Q a nd S ( D) b oth P a nd S
Q.24 Co nsi d er t h e foll owing st a t e m e nts P and Q :
( P ) : I f i s a no rmal subgroup o f o r d er 4 o f the s y mmetric gro up
, the n
is
a beli a n .
(Q ) : If 1 , , , is the qua ternion group, then
1, 1
i s ab el ian.
Whic h of th e abo ve st at e ments hol d T RUE?
(A ) both P a nd Q (B) onl y P
(C ) onl y Q ( D) Neith er P nor Q
Q.25 Let b e a fiel d of order 3 2. T hen t he num ber o f no n? z e ro so luti ons , ? of
the eq ua tio n
0 is equal to __ ___ __ ___ _ _ ___ ___ __ ____ _ ___ __ _
GATE 2016 Mathematics - MA
MA 8/16
Q. 26 ? Q. 55 carry two marks each.
Q. 26 Let ? ? || 2 be oriente d in the counter?clockwise direction . Let
1
2 1
.
Then, the v a lue of i s equal to ___ ___ __ ___ ___ __ _ _ _ __ ___ __
Q. 27 Let , 2 be a h a rmonic fu n ctio n and , i t s
harmon ic c onju gate. If 0,0 1 , th en | 1 , 1 | is eq u al to _ __ ___ ___ __ ___ _
Q. 28 L e t b e th e triangula r path co nnecting the poin ts (0,0), (2,2) and (0,2) in the counter?
clockwise direction in
. T h en
6
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 29 Let y b e th e solutio n of
? | |,? 1 0.
Then 1 i s equ a l to
(A)
(B)
2
(C) 2
(D) 2 2
Q. 30 Let X b e a r ando m v ari able with th e f oll owi ng c um ul ati ve d i stri b u ti o n functio n :
00
0 1
2
3
4
1
2
1
1 1.
Then 1 is equal to __ ___ ___ __ ___ __ ____
GATE 2016 Mathematics - MA
MA 9/16
Q. 31 Let ? b e the curve which passes thr o ugh ( 0 ,1 ) and i n tersects ea c h c u r v e of the f amil y
orthogon a l ly. Th en ? a l so passes th rough the poin t
(A) ? 2,0 (B) 0, ? 2
(C) 1,1 ( D ) 1,1
Q. 32 Let ?
?
be the Fourier s e rie s of th e
2 periodic fu nction defined by 4, . Then
?
?
i s eq u al to ___ __ ___ ___ ___ __ __ ___ ___ __
Q. 33 L e t b e a con t inu o us f u nction on 0, ? . If
1 4 4 ,
the n
is equal to __ _ _ ___ _ _ ___ ___ __ ____ _ __
Q. 34
Let
?
and
. Th e n,
is e q u a l to
(A) ln 10 1 (B) l n 10 1
(C) l n 10
(D) ln10
Q. 35 F o r any , ? \ 0,1
, let
, distance , ,0,1
inimum ? ,
?0,1
.
Then , || 3,4 || i s equal to ___ __ ___ __ ___ ___ ___ _
Q. 36
Let
an d
.
Th e n ?
?
?
? is
eq ual to ___ __ ___ ___ __ __ ___ ___ __ ___ __ ___ ___ __ ___
GATE 2016 Mathematics - MA
MA 10/16
Q. 37
Let b e a real matr i x with e igenvalu es 1 , 0 a nd 3. If th e e i ge nve ctors
c o rres p ond i ng to 1 and 0 a r e 1,1,1
a nd 1, 1 ,0 re s p e ctivel y, th e n th e value o f
3 i s eq ual to _ ___ __ ___ ____ _ ___
Q. 38
Let 11 0
01 1
00 1
a nd
!
!
? . If
, th e n
1
i s eq u al to ___ __ ___ ___ ___ __ __ ___ ___
Q. 39
L e t th e inte gral , w h e r e 02
4 2 4.
Consider the following stateme nts P an d Q:
(P ) : If
is the va l ue of the integr al o b t ained b y the c om po site tra pez oidal rule with
t wo equal s u b?i n terval s , then
is exac t.
(Q ) : If is the va l ue of the integr al o b t ained b y the c om po site tra pez oidal rule with
thr ee eq ual su b ? interva ls, t he n is e x act.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 40 The diff eren ce bet ween t he least two e i genval u e s of th e b ou nda r y valu e p roblem
?? 0, 0 0 0, ? 0,
i s eq u al to _ ___ ___ __ ____ _ ___ __ ___ ____ _ ___
Q. 41
The numb er of r o ots of the equa tion
cos 0 in the i n terval ,
i s equal
to __ ___ __ _____ __
GATE 2016 Mathematics - MA
MA 11/16
Q. 42 F o r th e fixed po int i t e r ation , 0,1,2,??, consider the following
statements P and Q :
(P ) : If 1
then the fixe d poin t iteration c on verges t o 2 for all
? 1, 10 0 .
(Q ) : If ?2 the n th e fixed p o i nt iter ation converg es t o 2 fo r
a ll
? 0, 10 0 .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neither P nor Q
Q. 43 Let ? ?
? ?
be d e fined b y
,
,? , ,?
,
,?,
,? .
Then
(A) ? ? 1
(B) ? ? 2 but bounded
(C) 1 ? ? 2
(D) ? ? is unbo unded
Q. 44 Minimi z e 2 subject to
2 3
2
0, 0.
Then, the m i nimum value of is equal to _ __ ___ __ _ _ _ ___ __ ___ _ _ _ __ _
Q. 45 Maximi ze 1 1 su b ject to
10 1
2 2 2
, , 0.
Then, the m a ximu m val u e of is equal to __ _ _ ___ _ _ _ _ _ ___ __ ____ _ ___
GATE 2016 Mathematics - MA
MA 12/16
Q. 46 Let ,
,
, ? be a seque nce of i.i.d. ran dom va ria b les with mea n 1 . If N is a
geometric r a ndom vari a b l e wi th the p ro babi lity mass function
;
1, 2, 3, ? and i t is ind ep end ent of th e 's, th en
? is eq u al to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 47 Let be an expon ential r andom va r i ab le wi t h mea n 1 a n d a ga mma r and o m
variab le wi t h me an 2 and varianc e 2. If an d
a re i ndepe ndentl y distr ib uted, the n
is e q ual to __ ___ __ ___ ___ __ __ ___ ___ __
Q. 48 Let
,
,
, ? be a seq uenc e of i .i. d . uniform 0,1 random vari a b l es. Th en, th e v al ue
of
lim
? ?
ln1 ? ln1
is equal to __ _ _ ___ ___ __ _ _ _ __ ___
Q. 49
Let X b e a s t andar d nor mal rando m variable. Then, 0 | | 1 is e qual to
(A)
?
?
(B)
? ?
(C)
?
?
(D)
?
?
Q. 50 Let ,
,
,?,
b e a ra n dom samp le fro m the p r o babi l ity density func tion
1 2 ;0
0 otherwise,
wh ere 0, 0 1 are parameters. Consid er the f ollowin g testing pro blem:
: 1 , 1 versus
: 0, 2.
Which of the following statements is T RU E?
(A) Un if ormly Most P ower ful test does NOT e x ist
(B) U n ifor mly Mo st Powe r f u l tes t is of the fo r m ?,
for some 0 ?
(C) U nifo r ml y Most Powerful tes t is of the for m ?,
for some 0 ?
(D) Unifor mly Most Powerful tes t i s o f th e f o r m ?
,
for
some 0
?
GATE 2016 Mathematics - MA
MA 13/16
Q.51 Let
,
,
, ? be a sequen c e of i .i.d . , 1 rando m variables. Then,
lim
? ?
? 2
| |
is e q u al to __ ___ __ ___ ____ _ ___ __ ___ ____ _ _
Q.52 Let
,
,
,?, be a rand o m samp le from un ifor m 1, , f or som e 1. If
Maximum ,
,
,? , , the n t he U MV UE of i s
(A )
(B)
(C )
(D)
Q. 53 Let
1,
2 b e a ra ndo m sample fro m a Po i sson rando m
variab le wi t h me an , wh e re ? 1, 2 . T hen, t he m axi mu m li kelihood e sti m ato r of
i s eq u al to ___ __ ___ ___ ___ __ __ __
Q. 54 The remainder when 9 8 ! is divi ded b y 101 is e q u a l to __ ___ ___ __ ___ __ ___ ____ _ ___ __
Q. 55 L e t b e a gr o u p whose p r esentati o n is
, |
, .
Then is isomorp hic to
(A) (B)
(C) (D)
GATE 2016 Mathematics - MA
MA 14/16
END OF THE QUESTION PAPER
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
GATE 2016 Mathematics - MA
MA 2/16
Q. 1 ? Q. 25 carry one mark each.
Q. 1 Let , , be a b as is of . Conside r th e f ollowin g statements P and Q:
(P ) : ,, is a bas is of .
(Q ) : , 2 , 3 is a basis of
.
Wh i ch of th e ab ove s t atements h o l d TRUE?
(A) b o th P a nd Q ( B ) only P
(C ) only Q ( D) N either P n or Q
Q. 2 Consider the following stateme nts P an d Q:
(P ) : If 11 1
12 4
13 9
, then M is s in gular .
(Q ) : Let S b e a diagon a l iza b le matrix. If T is a matrix suc h t hat S + 5 T = Id , then T is
diagonalizable.
Which of the above s tate ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n ly Q ( D) Neither P nor Q
Q. 3 Consider the following stateme nts P an d Q:
(P ) : If M i s an complex matrix, then ?
.
(Q ) : There e xis t s a unitary ma trix with an e igen value ? su ch t h a t | ?| <1.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n l y Q ( D) Neither P n or Q
GATE 2016 Mathematics - MA
MA 3/16
Q. 4 Consider a r eal vec t or s p a ce V of d imen sion n and a no n ?zero line ar tra ns formation
? ? . If dim e ns ion and , f or some ? \0 , t hen
wh ic h of th e follo wing stateme nts is TRUE?
(A) determina n t | |
(B) Th e re exists a non?tri v ial s u b s p a ce of V such th a t 0 f o r all ? (C) T is invertib le
(D) is th e on ly e igenvalue of T
Q. 5 Let 0, 1 ?2, 3 a nd ? ? be a strictly increasing f u nction such that
is conn ected. Wh i ch of the foll owing statements is TRUE?
(A) h as e xactly one discontinuity
(B) has exactly two di s continu i ti es
(C) h as inf in itely man y d iscontinu i ties
(D) is conti nuou s
Q. 6 Let 1 and
4, 2.Then,
lim
? ?
1
1
? 1
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 7 Maximu m ? , ? 0 ,1 is equal to _ _ _ _ ___ __ ___ __ ___
Q. 8 Let , , , ? su ch that
0 . T h e n, the C auch y pr o blem
,,?,
, 0 o n 0
has a u nique solution if
(A) 0 (B) 0
(C) 0 ( D ) 0
GATE 2016 Mathematics - MA
MA 4/16
Q. 9 Let , be the d ' A lembert's s olu tion of th e in itial va lu e pro b le m for th e w ave
eq uatio n
0
, 0 ,
, 0 ,
wh ere c is a positi v e real numb e r a nd , are smo o th odd func t ions. Th e n, 0,1 is
eq ual to _ ___ __ ___ __
Q. 10 L e t th e pro babi l ity density func tion of a r a ndo m variable X be
0 1
2
2 1 1
2
1
0otherwise.
Then, the v a lue of c i s eq ual to ___ __ ___ ___ __ __ ___ ___ ___
Q. 11 Let V be th e set of all so l u tio n s o f the eq uatio n ?? ? 0 satisfyin g
0 1 , whe r e , are positive r ea l numbers. Th e n, di m ension(V ) is equal to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 12 Let ?? ? 0, ? ? , ? ,where and are
c o ntinuous func t i o ns. If
2 and 2 are two lin e arly indepe ndent s olu t ions o f t he ab o ve eq u a t ion , then
| 4 0 2 1 | is e qual to __ __ ___ ___ ___ __ __ ___
Q. 13
Let b e th e Legendre pol y nomial of degree and
, where k
i s a n on ?neg ative inte ger. Consi der the fo llowi ng stat e ment s P and Q:
(P ) : 0 i f .
(Q) : 0 if is a n odd intege r.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 5/16
Q. 14 Consider the following stateme nts P an d Q:
(P) :
?? ?
0 has two linearly i ndepende nt Fro beni us series
solutions near 0.
(Q) :
?? 3 ? 0 has two li n early independe n t F r obenius series
solutions near 0.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 15 L e t th e polynomial
be app r o x i m ated b y a p o ly no mi al o f degre e 2, which
i n ter p olates
at 1, 0 and 1. T h e n, th e m a x imum abs olute interpol a t i o n er r or
ove r th e inter val 1 , 1 is equal to __ __ ___ ___ __ ___ __ _ _ _ __
Q. 16 Let b e a se q u ence o f dist inc t points in 0, 1 ? ? | | 1 with
lim
? ?
0 . Consider the following stateme nts P an d Q :
(P ) : Th e r e exists a u niq u e analytic functi o n f on 0,1 suc h t ha t for
a ll n .
(Q ) : Th e r e exists an an alytic functi o n f on 0, 1 such that 0 if n i s e v en
and 1 if n is o d d.
Which o f the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 17 Let , b e a to pologic a l space with the cofinite to p o logy. Every infi n i t e subset of
i s
(A) Compac t but NOT co nnected
(B) Both compa c t a nd connected
(C) NOT compa c t b u t c o nne c t e d
(D) Ne i t h e r c o mp act no r c o nnected
GATE 2016 Mathematics - MA
MA 6/16
Q. 18 Let ?
? , ?0 and ?
?
?
0 .
T hen, dim ens ion
is equal to __ ___ __ ___ ___ __ ____ _ ___
Q. 19 Consider , ? ? ?
?
, wher e ? , ?
?
maximum | | , | | . Let ?
? be
def i n e d by ,
an d the no r m pre s e rvi ng l ine a r e x tension o f to
, ? ? ?
?
. Th en, 1, 1,1 is e qual to __ ___ __ ___ ___ __ ____ _ ___ ___ __ ____ _ _
Q. 20 ? 0, 1 ? 0, 1 is called a sh rinking map if | | | | for all
, ? 0,1 and a contr a ction if there exists an 1 such tha t
| | | | f or all , ? 0,1 .
Which of the following stateme nts is TRU E f o r the function
?
(A) is both a s h rinking map and a c o ntractio n
(B) is a s hrinking m ap but NO T a c ontra c tion
(C) i s NOT a shrinking map but a c o ntractio n
(D) is Ne i t her a shri nki n g m a p no r a c ontrac ti on
Q. 21 Let be t he s et of all r eal ma t rices with the usual norm t o p ology. Con side r the
following sta t emen ts P and Q:
(P ) : T h e s e t of all sym m e tric p ositive defin i te matrices in is conne cte d .
(Q ) : T h e s e t of all inver tibl e m atric es in i s c om pact .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 7/16
Q. 22 Let ,
,
,?,
be a r and o m samp le f rom the followin g pr o bability d ensity
func ti on for 0 ?,0 1,
; , 1
? ; 0otherwise.
He re a nd are un kn own parame ters. Which of th e f ollowin g st at em ent s i s TRU E?
(A) Max i mum likel i hood e stima t or of on ly e xists
(B) Maximum li kelihood esti m ator of o n l y exis ts
(C) Maximum likeli h o o d estimator s o f bo th and e xist
(D) Ma ximu m likeli h o od estimator of Neither nor e xists
Q. 23 Suppose X an d Y are tw o rando m variables suc h th a t is a norma l random
variab le for al l , ? . Conside r the f ollowin g statements P, Q , R an d S :
(P ) : X is a stan dard n o r mal ran d om v ariable.
(Q) : Th e conditi o na l distrib u ti on of X g ive n Y is n o rmal.
(R) : Th e co nditio nal distributio n of X given is n o rmal.
(S) : h a s m e a n 0.
Which of the above state me nts A LWAYS hold T R U E?
(A) both P and Q ( B) b oth Q a nd R
(C) b o th Q a nd S ( D) b oth P a nd S
Q.24 Co nsi d er t h e foll owing st a t e m e nts P and Q :
( P ) : I f i s a no rmal subgroup o f o r d er 4 o f the s y mmetric gro up
, the n
is
a beli a n .
(Q ) : If 1 , , , is the qua ternion group, then
1, 1
i s ab el ian.
Whic h of th e abo ve st at e ments hol d T RUE?
(A ) both P a nd Q (B) onl y P
(C ) onl y Q ( D) Neith er P nor Q
Q.25 Let b e a fiel d of order 3 2. T hen t he num ber o f no n? z e ro so luti ons , ? of
the eq ua tio n
0 is equal to __ ___ __ ___ _ _ ___ ___ __ ____ _ ___ __ _
GATE 2016 Mathematics - MA
MA 8/16
Q. 26 ? Q. 55 carry two marks each.
Q. 26 Let ? ? || 2 be oriente d in the counter?clockwise direction . Let
1
2 1
.
Then, the v a lue of i s equal to ___ ___ __ ___ ___ __ _ _ _ __ ___ __
Q. 27 Let , 2 be a h a rmonic fu n ctio n and , i t s
harmon ic c onju gate. If 0,0 1 , th en | 1 , 1 | is eq u al to _ __ ___ ___ __ ___ _
Q. 28 L e t b e th e triangula r path co nnecting the poin ts (0,0), (2,2) and (0,2) in the counter?
clockwise direction in
. T h en
6
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 29 Let y b e th e solutio n of
? | |,? 1 0.
Then 1 i s equ a l to
(A)
(B)
2
(C) 2
(D) 2 2
Q. 30 Let X b e a r ando m v ari able with th e f oll owi ng c um ul ati ve d i stri b u ti o n functio n :
00
0 1
2
3
4
1
2
1
1 1.
Then 1 is equal to __ ___ ___ __ ___ __ ____
GATE 2016 Mathematics - MA
MA 9/16
Q. 31 Let ? b e the curve which passes thr o ugh ( 0 ,1 ) and i n tersects ea c h c u r v e of the f amil y
orthogon a l ly. Th en ? a l so passes th rough the poin t
(A) ? 2,0 (B) 0, ? 2
(C) 1,1 ( D ) 1,1
Q. 32 Let ?
?
be the Fourier s e rie s of th e
2 periodic fu nction defined by 4, . Then
?
?
i s eq u al to ___ __ ___ ___ ___ __ __ ___ ___ __
Q. 33 L e t b e a con t inu o us f u nction on 0, ? . If
1 4 4 ,
the n
is equal to __ _ _ ___ _ _ ___ ___ __ ____ _ __
Q. 34
Let
?
and
. Th e n,
is e q u a l to
(A) ln 10 1 (B) l n 10 1
(C) l n 10
(D) ln10
Q. 35 F o r any , ? \ 0,1
, let
, distance , ,0,1
inimum ? ,
?0,1
.
Then , || 3,4 || i s equal to ___ __ ___ __ ___ ___ ___ _
Q. 36
Let
an d
.
Th e n ?
?
?
? is
eq ual to ___ __ ___ ___ __ __ ___ ___ __ ___ __ ___ ___ __ ___
GATE 2016 Mathematics - MA
MA 10/16
Q. 37
Let b e a real matr i x with e igenvalu es 1 , 0 a nd 3. If th e e i ge nve ctors
c o rres p ond i ng to 1 and 0 a r e 1,1,1
a nd 1, 1 ,0 re s p e ctivel y, th e n th e value o f
3 i s eq ual to _ ___ __ ___ ____ _ ___
Q. 38
Let 11 0
01 1
00 1
a nd
!
!
? . If
, th e n
1
i s eq u al to ___ __ ___ ___ ___ __ __ ___ ___
Q. 39
L e t th e inte gral , w h e r e 02
4 2 4.
Consider the following stateme nts P an d Q:
(P ) : If
is the va l ue of the integr al o b t ained b y the c om po site tra pez oidal rule with
t wo equal s u b?i n terval s , then
is exac t.
(Q ) : If is the va l ue of the integr al o b t ained b y the c om po site tra pez oidal rule with
thr ee eq ual su b ? interva ls, t he n is e x act.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 40 The diff eren ce bet ween t he least two e i genval u e s of th e b ou nda r y valu e p roblem
?? 0, 0 0 0, ? 0,
i s eq u al to _ ___ ___ __ ____ _ ___ __ ___ ____ _ ___
Q. 41
The numb er of r o ots of the equa tion
cos 0 in the i n terval ,
i s equal
to __ ___ __ _____ __
GATE 2016 Mathematics - MA
MA 11/16
Q. 42 F o r th e fixed po int i t e r ation , 0,1,2,??, consider the following
statements P and Q :
(P ) : If 1
then the fixe d poin t iteration c on verges t o 2 for all
? 1, 10 0 .
(Q ) : If ?2 the n th e fixed p o i nt iter ation converg es t o 2 fo r
a ll
? 0, 10 0 .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neither P nor Q
Q. 43 Let ? ?
? ?
be d e fined b y
,
,? , ,?
,
,?,
,? .
Then
(A) ? ? 1
(B) ? ? 2 but bounded
(C) 1 ? ? 2
(D) ? ? is unbo unded
Q. 44 Minimi z e 2 subject to
2 3
2
0, 0.
Then, the m i nimum value of is equal to _ __ ___ __ _ _ _ ___ __ ___ _ _ _ __ _
Q. 45 Maximi ze 1 1 su b ject to
10 1
2 2 2
, , 0.
Then, the m a ximu m val u e of is equal to __ _ _ ___ _ _ _ _ _ ___ __ ____ _ ___
GATE 2016 Mathematics - MA
MA 12/16
Q. 46 Let ,
,
, ? be a seque nce of i.i.d. ran dom va ria b les with mea n 1 . If N is a
geometric r a ndom vari a b l e wi th the p ro babi lity mass function
;
1, 2, 3, ? and i t is ind ep end ent of th e 's, th en
? is eq u al to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 47 Let be an expon ential r andom va r i ab le wi t h mea n 1 a n d a ga mma r and o m
variab le wi t h me an 2 and varianc e 2. If an d
a re i ndepe ndentl y distr ib uted, the n
is e q ual to __ ___ __ ___ ___ __ __ ___ ___ __
Q. 48 Let
,
,
, ? be a seq uenc e of i .i. d . uniform 0,1 random vari a b l es. Th en, th e v al ue
of
lim
? ?
ln1 ? ln1
is equal to __ _ _ ___ ___ __ _ _ _ __ ___
Q. 49
Let X b e a s t andar d nor mal rando m variable. Then, 0 | | 1 is e qual to
(A)
?
?
(B)
? ?
(C)
?
?
(D)
?
?
Q. 50 Let ,
,
,?,
b e a ra n dom samp le fro m the p r o babi l ity density func tion
1 2 ;0
0 otherwise,
wh ere 0, 0 1 are parameters. Consid er the f ollowin g testing pro blem:
: 1 , 1 versus
: 0, 2.
Which of the following statements is T RU E?
(A) Un if ormly Most P ower ful test does NOT e x ist
(B) U n ifor mly Mo st Powe r f u l tes t is of the fo r m ?,
for some 0 ?
(C) U nifo r ml y Most Powerful tes t is of the for m ?,
for some 0 ?
(D) Unifor mly Most Powerful tes t i s o f th e f o r m ?
,
for
some 0
?
GATE 2016 Mathematics - MA
MA 13/16
Q.51 Let
,
,
, ? be a sequen c e of i .i.d . , 1 rando m variables. Then,
lim
? ?
? 2
| |
is e q u al to __ ___ __ ___ ____ _ ___ __ ___ ____ _ _
Q.52 Let
,
,
,?, be a rand o m samp le from un ifor m 1, , f or som e 1. If
Maximum ,
,
,? , , the n t he U MV UE of i s
(A )
(B)
(C )
(D)
Q. 53 Let
1,
2 b e a ra ndo m sample fro m a Po i sson rando m
variab le wi t h me an , wh e re ? 1, 2 . T hen, t he m axi mu m li kelihood e sti m ato r of
i s eq u al to ___ __ ___ ___ ___ __ __ __
Q. 54 The remainder when 9 8 ! is divi ded b y 101 is e q u a l to __ ___ ___ __ ___ __ ___ ____ _ ___ __
Q. 55 L e t b e a gr o u p whose p r esentati o n is
, |
, .
Then is isomorp hic to
(A) (B)
(C) (D)
GATE 2016 Mathematics - MA
MA 14/16
END OF THE QUESTION PAPER
GATE 2016 Mathematics - MA
MA 15/16
FirstRanker.com - FirstRanker's Choice
GATE 2016 General Aptitude - GA Set-4
1/3
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 (B) igniting (C) clutching (D) flinging
Q.3 M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-in-
law 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.
Q.4 The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324 (B) 441 (C) 97 (D) 64
Q.5 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
Q. 6 ? Q.
Q.6 The
resp
incr
(A)
Q.7 The
Hea
dog
Wh
(A)
(B)
(C)
(D)
Q.8 A f
enj
you
the
(A)
Q.9 Fin
(A)
GATE 2016
10 carry tw
e velocity V
pect to time
rease by (in
) 0
e overwhelm
alth Organiz
gs against rab
hich of the fo
) The numb
) The numb
) Rabies can
) Stray dogs
flat is shared
oy some ext
unger than Tr
flat?
) Manu
nd the area bo
) 14.95
wo marks
V of a vehicl
e in seconds
m)?
(B)
ming number
zation as a so
bies can lead
ollowing can
ber of people
er of people
n be eradicat
s are the mai
by four first
tra space in
rideep. Pava
(B)
ounded by th
(B)
each.
le along a st
. At the end
3
r of people in
ource of con
d to a signific
be logically
in India infe
in other part
ed in India b
n source of r
t year underg
the flat. Ma
an is one mon
Sravan
he lines 3x+2
15.25
traight line i
d of the 7 s
(C)
nfected with
ncern. It is es
cant reductio
y inferred from
ected with ra
ts of the wor
by vaccinatin
rabies worldw
graduate stud
anu is two m
nth older than
(C)
2y=14, 2x-3y
(C)
is measured
seconds, how
4
h rabies in In
stimated that
on in the num
m the above
abies is high.
ld who are in
ng 70% of str
wide.
dents. They a
months older
n Sravan. W
Trideep
y=5 in the fir
15.70
in m/s and
w much will
(D)
ndia has been
t inoculating
mber of peopl
sentences?
nfected with
ray dogs.
agreed to allo
r than Sravan
Who should oc
(D)
rst quadrant.
(D)
General Ap
plotted as s
l the odome
5
n flagged by
g 70% of pet
le infected w
rabies is low
ow the oldest
n, who is th
ccupy the ex
Pavan
20.35
ptitude - GA Set-4
2/3
shown with
eter reading
y the World
ts and stray
with rabies.
w.
t of them to
hree months
xtra space in
4
3
GATE 2016 General Aptitude - GA Set-4
3/3
Q.10 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
GATE 2016 Mathematics - MA
MA 1/16
List of S ymbols, Notations a n d Data
i.i.d . : ind epen den t an d iden t ically d istributed
, ? Nor mal d i s t ri but i on w it h mean and varian c e , ? ? , ? , 0
? Expec ted v a lue (mean) of the random variab l e
?
?
?
: th e grea t est integer l e ss than or e q u a l to
? Set of integers
? S et of in teger s m odu l o n
? S et of real numbers
? Set of comp lex numbers
? n ? dime nsion a l Eu clide a n space
Usua l metric d on is give n by ,
,? , , ,
,? , ?
/
?
? Normed lin e ar sp a ce of all s quare ? s ummab l e real sequ e nc es
0,1 ? Se t of all real v alued contin uous f unctions on the interva l 0,1
0,1
? , ? :
1
?
? Conjugate t rans p o se of the ma trix M
? Tr a nspose of the matrix M
Id : Ide n t ity matri x of app r o p riate o r der
? Ran ge sp a ce of M
? Null space of M
: Orthogonal co m p l ement of th e s ubs p ace W
GATE 2016 Mathematics - MA
MA 2/16
Q. 1 ? Q. 25 carry one mark each.
Q. 1 Let , , be a b as is of . Conside r th e f ollowin g statements P and Q:
(P ) : ,, is a bas is of .
(Q ) : , 2 , 3 is a basis of
.
Wh i ch of th e ab ove s t atements h o l d TRUE?
(A) b o th P a nd Q ( B ) only P
(C ) only Q ( D) N either P n or Q
Q. 2 Consider the following stateme nts P an d Q:
(P ) : If 11 1
12 4
13 9
, then M is s in gular .
(Q ) : Let S b e a diagon a l iza b le matrix. If T is a matrix suc h t hat S + 5 T = Id , then T is
diagonalizable.
Which of the above s tate ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n ly Q ( D) Neither P nor Q
Q. 3 Consider the following stateme nts P an d Q:
(P ) : If M i s an complex matrix, then ?
.
(Q ) : There e xis t s a unitary ma trix with an e igen value ? su ch t h a t | ?| <1.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) only P
(C) o n l y Q ( D) Neither P n or Q
GATE 2016 Mathematics - MA
MA 3/16
Q. 4 Consider a r eal vec t or s p a ce V of d imen sion n and a no n ?zero line ar tra ns formation
? ? . If dim e ns ion and , f or some ? \0 , t hen
wh ic h of th e follo wing stateme nts is TRUE?
(A) determina n t | |
(B) Th e re exists a non?tri v ial s u b s p a ce of V such th a t 0 f o r all ? (C) T is invertib le
(D) is th e on ly e igenvalue of T
Q. 5 Let 0, 1 ?2, 3 a nd ? ? be a strictly increasing f u nction such that
is conn ected. Wh i ch of the foll owing statements is TRUE?
(A) h as e xactly one discontinuity
(B) has exactly two di s continu i ti es
(C) h as inf in itely man y d iscontinu i ties
(D) is conti nuou s
Q. 6 Let 1 and
4, 2.Then,
lim
? ?
1
1
? 1
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 7 Maximu m ? , ? 0 ,1 is equal to _ _ _ _ ___ __ ___ __ ___
Q. 8 Let , , , ? su ch that
0 . T h e n, the C auch y pr o blem
,,?,
, 0 o n 0
has a u nique solution if
(A) 0 (B) 0
(C) 0 ( D ) 0
GATE 2016 Mathematics - MA
MA 4/16
Q. 9 Let , be the d ' A lembert's s olu tion of th e in itial va lu e pro b le m for th e w ave
eq uatio n
0
, 0 ,
, 0 ,
wh ere c is a positi v e real numb e r a nd , are smo o th odd func t ions. Th e n, 0,1 is
eq ual to _ ___ __ ___ __
Q. 10 L e t th e pro babi l ity density func tion of a r a ndo m variable X be
0 1
2
2 1 1
2
1
0otherwise.
Then, the v a lue of c i s eq ual to ___ __ ___ ___ __ __ ___ ___ ___
Q. 11 Let V be th e set of all so l u tio n s o f the eq uatio n ?? ? 0 satisfyin g
0 1 , whe r e , are positive r ea l numbers. Th e n, di m ension(V ) is equal to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 12 Let ?? ? 0, ? ? , ? ,where and are
c o ntinuous func t i o ns. If
2 and 2 are two lin e arly indepe ndent s olu t ions o f t he ab o ve eq u a t ion , then
| 4 0 2 1 | is e qual to __ __ ___ ___ ___ __ __ ___
Q. 13
Let b e th e Legendre pol y nomial of degree and
, where k
i s a n on ?neg ative inte ger. Consi der the fo llowi ng stat e ment s P and Q:
(P ) : 0 i f .
(Q) : 0 if is a n odd intege r.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 5/16
Q. 14 Consider the following stateme nts P an d Q:
(P) :
?? ?
0 has two linearly i ndepende nt Fro beni us series
solutions near 0.
(Q) :
?? 3 ? 0 has two li n early independe n t F r obenius series
solutions near 0.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 15 L e t th e polynomial
be app r o x i m ated b y a p o ly no mi al o f degre e 2, which
i n ter p olates
at 1, 0 and 1. T h e n, th e m a x imum abs olute interpol a t i o n er r or
ove r th e inter val 1 , 1 is equal to __ __ ___ ___ __ ___ __ _ _ _ __
Q. 16 Let b e a se q u ence o f dist inc t points in 0, 1 ? ? | | 1 with
lim
? ?
0 . Consider the following stateme nts P an d Q :
(P ) : Th e r e exists a u niq u e analytic functi o n f on 0,1 suc h t ha t for
a ll n .
(Q ) : Th e r e exists an an alytic functi o n f on 0, 1 such that 0 if n i s e v en
and 1 if n is o d d.
Which o f the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 17 Let , b e a to pologic a l space with the cofinite to p o logy. Every infi n i t e subset of
i s
(A) Compac t but NOT co nnected
(B) Both compa c t a nd connected
(C) NOT compa c t b u t c o nne c t e d
(D) Ne i t h e r c o mp act no r c o nnected
GATE 2016 Mathematics - MA
MA 6/16
Q. 18 Let ?
? , ?0 and ?
?
?
0 .
T hen, dim ens ion
is equal to __ ___ __ ___ ___ __ ____ _ ___
Q. 19 Consider , ? ? ?
?
, wher e ? , ?
?
maximum | | , | | . Let ?
? be
def i n e d by ,
an d the no r m pre s e rvi ng l ine a r e x tension o f to
, ? ? ?
?
. Th en, 1, 1,1 is e qual to __ ___ __ ___ ___ __ ____ _ ___ ___ __ ____ _ _
Q. 20 ? 0, 1 ? 0, 1 is called a sh rinking map if | | | | for all
, ? 0,1 and a contr a ction if there exists an 1 such tha t
| | | | f or all , ? 0,1 .
Which of the following stateme nts is TRU E f o r the function
?
(A) is both a s h rinking map and a c o ntractio n
(B) is a s hrinking m ap but NO T a c ontra c tion
(C) i s NOT a shrinking map but a c o ntractio n
(D) is Ne i t her a shri nki n g m a p no r a c ontrac ti on
Q. 21 Let be t he s et of all r eal ma t rices with the usual norm t o p ology. Con side r the
following sta t emen ts P and Q:
(P ) : T h e s e t of all sym m e tric p ositive defin i te matrices in is conne cte d .
(Q ) : T h e s e t of all inver tibl e m atric es in i s c om pact .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
GATE 2016 Mathematics - MA
MA 7/16
Q. 22 Let ,
,
,?,
be a r and o m samp le f rom the followin g pr o bability d ensity
func ti on for 0 ?,0 1,
; , 1
? ; 0otherwise.
He re a nd are un kn own parame ters. Which of th e f ollowin g st at em ent s i s TRU E?
(A) Max i mum likel i hood e stima t or of on ly e xists
(B) Maximum li kelihood esti m ator of o n l y exis ts
(C) Maximum likeli h o o d estimator s o f bo th and e xist
(D) Ma ximu m likeli h o od estimator of Neither nor e xists
Q. 23 Suppose X an d Y are tw o rando m variables suc h th a t is a norma l random
variab le for al l , ? . Conside r the f ollowin g statements P, Q , R an d S :
(P ) : X is a stan dard n o r mal ran d om v ariable.
(Q) : Th e conditi o na l distrib u ti on of X g ive n Y is n o rmal.
(R) : Th e co nditio nal distributio n of X given is n o rmal.
(S) : h a s m e a n 0.
Which of the above state me nts A LWAYS hold T R U E?
(A) both P and Q ( B) b oth Q a nd R
(C) b o th Q a nd S ( D) b oth P a nd S
Q.24 Co nsi d er t h e foll owing st a t e m e nts P and Q :
( P ) : I f i s a no rmal subgroup o f o r d er 4 o f the s y mmetric gro up
, the n
is
a beli a n .
(Q ) : If 1 , , , is the qua ternion group, then
1, 1
i s ab el ian.
Whic h of th e abo ve st at e ments hol d T RUE?
(A ) both P a nd Q (B) onl y P
(C ) onl y Q ( D) Neith er P nor Q
Q.25 Let b e a fiel d of order 3 2. T hen t he num ber o f no n? z e ro so luti ons , ? of
the eq ua tio n
0 is equal to __ ___ __ ___ _ _ ___ ___ __ ____ _ ___ __ _
GATE 2016 Mathematics - MA
MA 8/16
Q. 26 ? Q. 55 carry two marks each.
Q. 26 Let ? ? || 2 be oriente d in the counter?clockwise direction . Let
1
2 1
.
Then, the v a lue of i s equal to ___ ___ __ ___ ___ __ _ _ _ __ ___ __
Q. 27 Let , 2 be a h a rmonic fu n ctio n and , i t s
harmon ic c onju gate. If 0,0 1 , th en | 1 , 1 | is eq u al to _ __ ___ ___ __ ___ _
Q. 28 L e t b e th e triangula r path co nnecting the poin ts (0,0), (2,2) and (0,2) in the counter?
clockwise direction in
. T h en
6
i s eq u al to ___ __ ___ ___ ___ __ __ ___
Q. 29 Let y b e th e solutio n of
? | |,? 1 0.
Then 1 i s equ a l to
(A)
(B)
2
(C) 2
(D) 2 2
Q. 30 Let X b e a r ando m v ari able with th e f oll owi ng c um ul ati ve d i stri b u ti o n functio n :
00
0 1
2
3
4
1
2
1
1 1.
Then 1 is equal to __ ___ ___ __ ___ __ ____
GATE 2016 Mathematics - MA
MA 9/16
Q. 31 Let ? b e the curve which passes thr o ugh ( 0 ,1 ) and i n tersects ea c h c u r v e of the f amil y
orthogon a l ly. Th en ? a l so passes th rough the poin t
(A) ? 2,0 (B) 0, ? 2
(C) 1,1 ( D ) 1,1
Q. 32 Let ?
?
be the Fourier s e rie s of th e
2 periodic fu nction defined by 4, . Then
?
?
i s eq u al to ___ __ ___ ___ ___ __ __ ___ ___ __
Q. 33 L e t b e a con t inu o us f u nction on 0, ? . If
1 4 4 ,
the n
is equal to __ _ _ ___ _ _ ___ ___ __ ____ _ __
Q. 34
Let
?
and
. Th e n,
is e q u a l to
(A) ln 10 1 (B) l n 10 1
(C) l n 10
(D) ln10
Q. 35 F o r any , ? \ 0,1
, let
, distance , ,0,1
inimum ? ,
?0,1
.
Then , || 3,4 || i s equal to ___ __ ___ __ ___ ___ ___ _
Q. 36
Let
an d
.
Th e n ?
?
?
? is
eq ual to ___ __ ___ ___ __ __ ___ ___ __ ___ __ ___ ___ __ ___
GATE 2016 Mathematics - MA
MA 10/16
Q. 37
Let b e a real matr i x with e igenvalu es 1 , 0 a nd 3. If th e e i ge nve ctors
c o rres p ond i ng to 1 and 0 a r e 1,1,1
a nd 1, 1 ,0 re s p e ctivel y, th e n th e value o f
3 i s eq ual to _ ___ __ ___ ____ _ ___
Q. 38
Let 11 0
01 1
00 1
a nd
!
!
? . If
, th e n
1
i s eq u al to ___ __ ___ ___ ___ __ __ ___ ___
Q. 39
L e t th e inte gral , w h e r e 02
4 2 4.
Consider the following stateme nts P an d Q:
(P ) : If
is the va l ue of the integr al o b t ained b y the c om po site tra pez oidal rule with
t wo equal s u b?i n terval s , then
is exac t.
(Q ) : If is the va l ue of the integr al o b t ained b y the c om po site tra pez oidal rule with
thr ee eq ual su b ? interva ls, t he n is e x act.
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neithe r P nor Q
Q. 40 The diff eren ce bet ween t he least two e i genval u e s of th e b ou nda r y valu e p roblem
?? 0, 0 0 0, ? 0,
i s eq u al to _ ___ ___ __ ____ _ ___ __ ___ ____ _ ___
Q. 41
The numb er of r o ots of the equa tion
cos 0 in the i n terval ,
i s equal
to __ ___ __ _____ __
GATE 2016 Mathematics - MA
MA 11/16
Q. 42 F o r th e fixed po int i t e r ation , 0,1,2,??, consider the following
statements P and Q :
(P ) : If 1
then the fixe d poin t iteration c on verges t o 2 for all
? 1, 10 0 .
(Q ) : If ?2 the n th e fixed p o i nt iter ation converg es t o 2 fo r
a ll
? 0, 10 0 .
Which of the above sta te ment s hold TRUE ?
(A) both P and Q ( B) o n l y P
(C) o n l y Q ( D) Neither P nor Q
Q. 43 Let ? ?
? ?
be d e fined b y
,
,? , ,?
,
,?,
,? .
Then
(A) ? ? 1
(B) ? ? 2 but bounded
(C) 1 ? ? 2
(D) ? ? is unbo unded
Q. 44 Minimi z e 2 subject to
2 3
2
0, 0.
Then, the m i nimum value of is equal to _ __ ___ __ _ _ _ ___ __ ___ _ _ _ __ _
Q. 45 Maximi ze 1 1 su b ject to
10 1
2 2 2
, , 0.
Then, the m a ximu m val u e of is equal to __ _ _ ___ _ _ _ _ _ ___ __ ____ _ ___
GATE 2016 Mathematics - MA
MA 12/16
Q. 46 Let ,
,
, ? be a seque nce of i.i.d. ran dom va ria b les with mea n 1 . If N is a
geometric r a ndom vari a b l e wi th the p ro babi lity mass function
;
1, 2, 3, ? and i t is ind ep end ent of th e 's, th en
? is eq u al to
_ _ _ ___ ___ __ _ _ _ __ ___ __
Q. 47 Let be an expon ential r andom va r i ab le wi t h mea n 1 a n d a ga mma r and o m
variab le wi t h me an 2 and varianc e 2. If an d
a re i ndepe ndentl y distr ib uted, the n
is e q ual to __ ___ __ ___ ___ __ __ ___ ___ __
Q. 48 Let
,
,
, ? be a seq uenc e of i .i. d . uniform 0,1 random vari a b l es. Th en, th e v al ue
of
lim
? ?
ln1 ? ln1
is equal to __ _ _ ___ ___ __ _ _ _ __ ___
Q. 49
Let X b e a s t andar d nor mal rando m variable. Then, 0 | | 1 is e qual to
(A)
?
?
(B)
? ?
(C)
?
?
(D)
?
?
Q. 50 Let ,
,
,?,
b e a ra n dom samp le fro m the p r o babi l ity density func tion
1 2 ;0
0 otherwise,
wh ere 0, 0 1 are parameters. Consid er the f ollowin g testing pro blem:
: 1 , 1 versus
: 0, 2.
Which of the following statements is T RU E?
(A) Un if ormly Most P ower ful test does NOT e x ist
(B) U n ifor mly Mo st Powe r f u l tes t is of the fo r m ?,
for some 0 ?
(C) U nifo r ml y Most Powerful tes t is of the for m ?,
for some 0 ?
(D) Unifor mly Most Powerful tes t i s o f th e f o r m ?
,
for
some 0
?
GATE 2016 Mathematics - MA
MA 13/16
Q.51 Let
,
,
, ? be a sequen c e of i .i.d . , 1 rando m variables. Then,
lim
? ?
? 2
| |
is e q u al to __ ___ __ ___ ____ _ ___ __ ___ ____ _ _
Q.52 Let
,
,
,?, be a rand o m samp le from un ifor m 1, , f or som e 1. If
Maximum ,
,
,? , , the n t he U MV UE of i s
(A )
(B)
(C )
(D)
Q. 53 Let
1,
2 b e a ra ndo m sample fro m a Po i sson rando m
variab le wi t h me an , wh e re ? 1, 2 . T hen, t he m axi mu m li kelihood e sti m ato r of
i s eq u al to ___ __ ___ ___ ___ __ __ __
Q. 54 The remainder when 9 8 ! is divi ded b y 101 is e q u a l to __ ___ ___ __ ___ __ ___ ____ _ ___ __
Q. 55 L e t b e a gr o u p whose p r esentati o n is
, |
, .
Then is isomorp hic to
(A) (B)
(C) (D)
GATE 2016 Mathematics - MA
MA 14/16
END OF THE QUESTION PAPER
GATE 2016 Mathematics - MA
MA 15/16
GATE 2016 Mathematics - MA
MA 16/16
FirstRanker.com - FirstRanker's Choice
This post was last modified on 18 December 2019