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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 2/33 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 1/16 GATE 2016 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 4/16 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 6/16 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_______________________________ MA 7/16 GATE 2016 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______________ MA 10/16 GATE 2016 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_________________________ MA 11/16 GATE 2016 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 ∞ MA 12/16 GATE 2016 Mathematics - MA 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) MA 13/16 GATE 2016 Mathematics - MA ENDOFTHEQUESTIONPAPER MA 14/16 GATE 2016 MA Mathematics - MA 15/16 GATE 2016 Mathematics - MA MA 16/16