Download Q. 1 – Q. 5 carry one mark each.

GATE 2016
General Aptitude - GA Set-4
Q. 1 – Q. 5 carry one mark each.
Q.1
An apple costs Rs. 10. An onion costs Rs. 8.
Select the most suitable sentence with respect to grammar and usage.
(A) The price of an apple is greater than an onion.
(B) The price of an apple is more than onion.
(C) The price of an apple is greater than that of an onion.
(D) Apples are more costlier than onions.
Q.2
The Buddha said, “Holding on to anger is like grasping a hot coal with the intent of throwing it at
someone else; you are the one who gets burnt.”
Select the word below which is closest in meaning to the word underlined above.
(A) burning
Q.3
Q.4
(C) clutching
(D) flinging
M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-inlaw of M. How is P related to M?
(A) P is the son-in-law of M.
(B) P is the grandchild of M.
(C) P is the daughter-in law of M.
(D) P is the grandfather of M.
The number that least fits this set: (324, 441, 97 and 64) is ________.
(A) 324
Q.5
(B) igniting
(B) 441
(C) 97
(D) 64
It takes 10 s and 15 s, respectively, for two trains travelling at different constant speeds to
completely pass a telegraph post. The length of the first train is 120 m and that of the second train is
150 m. The magnitude of the difference in the speeds of the two trains (in m/s) is ____________.
(A) 2.0
(B) 10.0
(C) 12.0
(D) 22.0
1/3
GATE 2016
General Ap
ptitude - GA Set-44
Q. 6 – Q. 10 carry tw
wo marks each.
Q.6
Thee velocity V of a vehiclle along a sttraight line is
i measured in m/s and plotted as shown
s
with
resppect to timee in seconds. At the endd of the 7 seconds,
s
how
w much willl the odomeeter reading
incrrease by (in m)?
(A)) 0
Q.7
(B) 3
(C) 4
(D) 5
Thee overwhelm
ming numberr of people innfected with
h rabies in Inndia has beenn flagged by
y the World
Heaalth Organizzation as a soource of conncern. It is esstimated thatt inoculatingg 70% of petts and stray
doggs against rabbies can leadd to a significcant reductio
on in the num
mber of peoplle infected with
w rabies.
Whhich of the foollowing can be logicallyy inferred from
m the above sentences?
(A)) The numbber of people in India infeected with raabies is high.
(B)) The number of people in other partts of the world who are innfected with rabies is low
w.
(C)) Rabies cann be eradicated in India by
b vaccinatin
ng 70% of strray dogs.
r
worldw
wide.
(D)) Stray dogss are the main source of rabies
Q.8
A flat
f is shared by four firstt year underggraduate stud
dents. They agreed
a
to alloow the oldestt of them to
enjoy some exttra space in the flat. Maanu is two months
m
olderr than Sravann, who is th
hree months
youunger than Trrideep. Pavaan is one monnth older than
n Sravan. Who
W should occcupy the ex
xtra space in
the flat?
(A)) Manu
Q.9
(B) Sravan
(C) Trideep
(D) Pavan
Finnd the area boounded by thhe lines 3x+22y=14, 2x-3y
y=5 in the firrst quadrant.
(A)) 14.95
(B) 15.25
(C) 15.70
(D) 20.35
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GATE 2016
Q.10
General Aptitude - GA Set-4
A straight line is fit to a data set (ln x, y). This line intercepts the abscissa at ln x = 0.1 and has a
slope of −0.02. What is the value of y at x = 5 from the fit?
(A) −0.030
(B) −0.014
(C) 0.014
(D) 0.030
END OF THE QUESTION PAPER
3/3
GATE 2016
Mathematics - MA
ListofSymbols,NotationsandData
i.i.d.:independentandidenticallydistributed
,
∶Normaldistributionwithmean andvariance
, ∈
∞, ∞ ,
0
∶Expectedvalue(mean)oftherandomvariable Φ
√
∞
:thegreatestintegerlessthanorequalto ∶Setofintegers
∶Setofintegersmodulon
∶Setofrealnumbers
∶Setofcomplexnumbers
∶n‐dimensionalEuclideanspace
Usualmetricdon
isgivenby
,
,…,
,
,
,…,
∑
/
ℓ ∶Normedlinearspaceofallsquare‐summablerealsequences
0,1 ∶Setofallrealvaluedcontinuousfunctionsontheinterval 0,1 0,1 ∶ ∗
,
∈
:
1 ∶ConjugatetransposeofthematrixM
∶TransposeofthematrixM
Id:Identitymatrixofappropriateorder
∶RangespaceofM
∶NullspaceofM
:OrthogonalcomplementofthesubspaceW
MA
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Mathematics - MA
Q.1–Q.25carryonemarkeach.
Q.1
Let
, ,
(P):
beabasisof
,
,
(Q):
,
.ConsiderthefollowingstatementsPandQ:
isabasisof
2
,
3
.
isabasisof
.
WhichoftheabovestatementsholdTRUE?
(A)bothPandQ
(B)onlyP
(C)onlyQ
(D)NeitherPnorQ
Q.2
ConsiderthefollowingstatementsPandQ:
(P):If 1
1
1
1 1
2 4 ,thenMissingular.
3 9
(Q):LetSbeadiagonalizablematrix.IfTisamatrixsuchthatS+5T=Id,thenTis
diagonalizable.
WhichoftheabovestatementsholdTRUE?
(A)bothPandQ
(B)onlyP
(C)onlyQ
(D)NeitherPnorQ
Q.3
ConsiderthefollowingstatementsPandQ:
(P):IfMisan complexmatrix,then
∗
.
(Q):Thereexistsaunitarymatrixwithaneigenvalueλsuchthat|λ|<1.
WhichoftheabovestatementsholdTRUE?
(A)bothPandQ
(B)onlyP
(C)onlyQ
(D)NeitherPnorQ
MA
2/16
GATE 2016
Q.4
Mathematics - MA
ConsiderarealvectorspaceV ofdimensionn and anon‐zerolineartransformation
∶ → .Ifdimension
and
, forsome ∈ \ 0 ,then
whichofthefollowingstatementsisTRUE?
(A)determinant
| |
(B)Thereexistsanon‐trivialsubspace
ofV suchthat
0forall ∈
(C)Tisinvertible
(D) istheonlyeigenvalueofT
Q.5
Let 0, 1 ⋃ 2, 3 and ∶
→
beastrictlyincreasingfunctionsuchthat
isconnected.WhichofthefollowingstatementsisTRUE?
(A) hasexactlyonediscontinuity
(B) hasexactlytwodiscontinuities
(C) hasinfinitelymanydiscontinuities
(D) iscontinuous
Q.6
Let
1and
4,
2. Then,
1
lim
1
→∞
⋯
1
isequalto_____________________
Q.7
Maximum
Q.8
Let , , ,
∶
,
∈
0,1
isequalto_________________
∈ suchthat
0. Then,theCauchyproblem
,
,
0on , ∈ ,
0
hasauniquesolutionif
(A) 0
(B)
0
(C) 0
(D)
0
MA
3/16
GATE 2016
Q.9
Mathematics - MA
Let
, bethed'Alembert's solutionofthe initialvalueproblemforthewave
equation
,0 ,
0
,0 ,
wherecisapositiverealnumberand , aresmoothoddfunctions.Then,
0,1 is
equalto___________
Q.10 Lettheprobabilitydensityfunctionofarandomvariable X be
1
2
0
2
1
1
2
otherwise.
1
0
Then,thevalueofcisequalto________________________
Q.11 LetVbethesetofallsolutionsof theequation ′′
0
′
0satisfying
1 ,where , arepositiverealnumbers.Then,dimension(V)isequalto
_____________________
′
Q.12 Let ′′
0,
∈
∞ , ∞ , where
continuousfunctions.If
2
and
are
and
2
aretwolinearlyindependentsolutionsoftheaboveequation,then
|4 0 2 1 |isequalto____________________
Q.13 Let
betheLegendrepolynomialofdegree and
,wherek
isanon‐negativeinteger.ConsiderthefollowingstatementsPandQ:
(P): 0if .
(Q): 0if isanoddinteger.
WhichoftheabovestatementsholdTRUE?
(A)bothPandQ
(B)only P
(C)onlyQ
(D)NeitherPnorQ
MA
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GATE 2016
Mathematics - MA
Q.14 ConsiderthefollowingstatementsPandQ:
(P):
′′
′
solutionsnear
′′
(Q):
3
0.
′
solutionsnear
0 hastwolinearlyindependentFrobeniusseries
0hastwolinearlyindependentFrobeniusseries
0.
WhichoftheabovestatementsholdTRUE?
(A)bothPandQ
(B)only P
(C)onlyQ
(D)NeitherPnorQ
Q.15 Letthepolynomial
interpolates
at
overtheinterval
beapproximatedbyapolynomialofdegree
2,which
1, 0 and1.Then,themaximumabsoluteinterpolationerror
1, 1 isequalto______________________
Q.16 Let
lim
beasequenceofdistinctpointsin
0,1
∈
∶ | |
1 with
0.ConsiderthefollowingstatementsPandQ:
→∞
(P):Thereexistsauniqueanalyticfunctionfon
0,1 suchthat
for
alln.
(Q):Thereexistsananalyticfunctionfon
and
0,1 suchthat
0ifniseven
1ifnisodd.
WhichoftheabovestatementsholdTRUE?
(A)bothPandQ
(B)only P
(C)onlyQ
(D)NeitherPnorQ
Q.17 Let
,
beatopologicalspacewiththecofinitetopology.Everyinfinitesubset of
is
(A)CompactbutNOTconnected
(B)Bothcompactandconnected
(C)NOTcompactbutconnected
(D)Neithercompactnorconnected
MA
5/16
GATE 2016
Mathematics - MA
Q.18 Let
∶
∈ ,
∈
→ 0 and
∶
⋯
0 .
isequalto_______________________
Then,dimension
Q.19 Consider
, ‖⋅‖∞ ,where‖ ,
definedby
,
‖∞
maximum | |, | | . Let ∶ → be
and thenormpreservinglinearextensionof to
, ‖⋅‖∞ .Then, 1,1,1 isequalto__________________________________
Q.20
|
∶ 0,1 → 0,1 iscalledashrinkingmapif|
,
|
∈ 0,1 andacontractionifthereexistsan
|
|
|forall ,
|
|forall
1 suchthat
∈ 0,1 .
WhichofthefollowingstatementsisTRUEforthefunction
?
(A) isbothashrinkingmapandacontraction
(B) isashrinkingmapbutNOTacontraction
(C) isNOTashrinkingmapbutacontraction
(D) isNeitherashrinkingmapnoracontraction
Q.21 Let bethesetofall
realmatriceswiththeusualnormtopology.Considerthe
followingstatementsPandQ:
(P):Thesetofallsymmetricpositivedefinitematricesin isconnected.
(Q):Thesetofallinvertiblematricesin iscompact.
WhichoftheabovestatementsholdTRUE?
(A)bothPandQ
(B)only P
(C)onlyQ
(D)NeitherPnorQ
MA
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GATE 2016
Mathematics - MA
Q.22 Let
,
,
,…,
functionfor0
bearandomsamplefromthefollowingprobabilitydensity
∞, 0
1, 1
; ,
Γ
;
0
otherwise.
Here and areunknownparameters.WhichofthefollowingstatementsisTRUE?
(A)Maximumlikelihoodestimatorofonly exists
(B)Maximumlikelihoodestimatorofonly exists
(C)Maximumlikelihoodestimators ofboth and
(D)MaximumlikelihoodestimatorofNeither
exist
nor
exists
Q.23 SupposeXandYaretworandomvariablessuchthat
variableforall ,
∈
isanormalrandom
.ConsiderthefollowingstatementsP,Q,RandS:
(P):Xisastandardnormalrandomvariable.
(Q):TheconditionaldistributionofXgivenYisnormal.
(R):TheconditionaldistributionofXgiven (S): isnormal.
hasmean0.
WhichoftheabovestatementsALWAYSholdTRUE?
(A)bothPandQ
(B)bothQandR
(C)bothQandS
(D)bothPandS
Q.24 ConsiderthefollowingstatementsPandQ:
(P):If isanormalsubgroupoforder4ofthesymmetricgroup ,then
is
abelian.
(Q):If
1,
,
,
isthequaterniongroup,then
1,1 isabelian.
WhichoftheabovestatementsholdTRUE?
(A)bothPandQ
(B)only P
(C)onlyQ
(D)NeitherPnorQ
Q.25 Let beafieldoforder32.Thenthenumberofnon‐zerosolutions ,
theequation
∈
of
0 isequalto_______________________________
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Mathematics - MA
Q.26–Q.55carrytwomarkseach.
Q.26 Let
∈
∶ | | 2 beorientedinthecounter‐clockwisedirection.Let
1
2 1
.
Then,thevalueof isequalto__________________________
Q.27 Let
,
2
harmonicconjugate.If 0,0
beaharmonicfunctionand
1,then|
,
its
1,1 | isequalto_______________
Q.28 Let bethetriangularpathconnectingthepoints(0,0),(2,2)and(0,2)inthecounter‐
clockwisedirectionin
.Then
6 isequalto_____________________
Q.29 Letybethesolutionof
| |,
∈
1
0.
′
Then 1 isequalto
(A) (B)
2
(C)2
(D)2
2
Q.30 LetXbearandomvariablewiththefollowingcumulativedistributionfunction:
0
Then
0
1
3
2
4
1 0
1
2 1
1.
1 isequalto___________________
MA
8/16
GATE 2016
Mathematics - MA
Q.31 Letγbethecurvewhichpassesthrough(0,1)andintersectseachcurveofthefamily
orthogonally.Thenγalsopassesthroughthepoint
(A) √2, 0 (B) 0, √2 (C) 1,1 (D)
1,1
Q.32 Let
∑∞
betheFourierseriesofthe
2 periodicfunctiondefinedby
4
∞
,
. Then
∞
isequalto__________________________
Q.33 Let
beacontinuousfunctionon 0, ∞ . If
1
4
4
,
isequalto________________________
then
Q.34 Let
∑
and
.Then,
isequalto
(A)ln 10 1
(B)ln 10
1
(C)ln 10 (D)ln 10
Q.35 Forany ,
∈ \
0,1 , let
,
distance
in imum ,
,
0,1 ∶
,
∈
0,1 .
Then,|| 3,4 ||isequalto____________________
Q.36
Let
and
.Then
′
√
′
√
is
equalto_______________________________________
MA
9/16
GATE 2016
Q.37
Mathematics - MA
Let bearealmatrixwitheigenvalues1,0and3.Iftheeigenvectors
correspondingto1and0are 1,1,1 and 1, 1,0 respectively,thenthevalueof
3 isequalto_________________
Q.38
Let 1
0
0
1 0
1 1 and
0 1
!
1
⋯.If
!
,then
isequalto________________________
Q.39 Lettheintegral
, where
0
2
4
2
4.
ConsiderthefollowingstatementsPandQ:
(P):If isthevalueoftheintegralobtainedbythecompositetrapezoidalrulewith
twoequalsub‐intervals,then isexact.
(Q):If isthevalueoftheintegralobtainedbythecompositetrapezoidalrulewith
threeequalsub‐intervals,then isexact.
WhichoftheabovestatementsholdTRUE?
(A)bothPandQ
(B)only P
(C)onlyQ
(D)NeitherPnorQ
Q.40 Thedifferencebetweentheleast twoeigenvaluesoftheboundaryvalueproblem
′′
0
0,
0,
0
′
0,
isequalto______________________________
Q.41 Thenumberofrootsoftheequation
cos
0 intheinterval
,
isequal
to______________
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10/16
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Mathematics - MA
,
Q.42 Forthefixedpointiteration
0, 1, 2, … …, considerthefollowing
statementsPandQ:
(P):If
1
(Q):If
√2
all
thenthefixedpointiterationconvergesto2forall
∈ 1, 100 .
thenthefixedpointiterationconvergesto2for
∈ 0, 100 .
WhichoftheabovestatementsholdTRUE?
(A)bothPandQ
(B)only P
(C)onlyQ
(D)NeitherPnorQ
Q.43 Let ∶ ℓ → ℓ bedefinedby
,
,⋯,
,⋯
,
,⋯,
,⋯
.
Then
(A)‖ ‖
1
(B)‖ ‖
2butbounded
(C)1
‖ ‖
2
(D)‖ ‖isunbounded
Q.44 Minimize
2 subjectto
2
0, 3
2
0.
Then,theminimumvalueof isequalto_________________________
Q.45 Maximize
11
subjectto
10
2
2
, ,
1
2
0.
Then,themaximumvalueof isequalto_________________________
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Mathematics - MA
Q.46 Let
,
,
, …beasequenceofi.i.d.randomvariableswithmean1.IfNisa
geometricrandomvariablewiththeprobabilitymassfunction
1,2,3, …anditisindependentofthe 's,then
⋯
;
isequalto
_____________________
Q.47 Let
beanexponentialrandomvariablewithmean1 and
variablewithmean2andvariance2.If
and
agammarandom
areindependentlydistributed,then
isequalto_________________________
Q.48 Let
,
,
, …beasequenceofi.i.d.uniform 0,1 randomvariables.Then,the value
of
lim
→∞
ln 1
⋯ ln 1
isequalto____________________
Q.49 LetXbeastandardnormalrandomvariable.Then,
Φ
Φ
Φ
Φ
(A)
(C)
Φ
(B)
(D)
Φ
Φ
Φ
0 |
|
1 isequalto
Q.50 Let
,
,
where
:
,…,
0, 0
1, bearandomsamplefromtheprobabilitydensityfunction
1
0
2
;
0
otherwise,
1areparameters.Considerthefollowingtestingproblem:
1versus
:
0, 2.
WhichofthefollowingstatementsisTRUE?
(A)UniformlyMostPowerfultestdoesNOTexist
(B)UniformlyMostPowerfultestisoftheform∑
, forsome0
∞
(C)UniformlyMostPowerfultestisoftheform∑
, forsome0
∞
(D)UniformlyMostPowerfultestisoftheform
some0
∑
,for
∞
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Q.51 Let
,
,
, …beasequenceofi.i.d.
lim
→∞
, 1 randomvariables.Then,
√
2
|
| isequalto_____________________________
Q.52 Let
,
,
,…,
Maximum
bearandomsamplefromuniform 1,
,
,
,…,
, forsome
1.If
, thentheUMVUEof is
(A)
(B)
(C)
(D)
1,
Q.53 Let
2 bearandomsamplefromaPoissonrandom
variablewithmean ,where ∈ 1, 2 . Then,themaximumlikelihoodestimatorof isequalto____________________
Q.54 Theremainderwhen98!isdividedby101isequalto____________________________
Q.55 Let beagroupwhosepresentationis
, |
,
.
Then isisomorphicto
(A)
(B)
(C)
(D)
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ENDOFTHEQUESTIONPAPER
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Mathematics - MA
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Mathematics - MA
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