Download Sample Question for the Advanced Mathematical Ability Test

CALCUTTAMATHEMATICALSOCIETY
AE374,SaltLake(SectorI),Kolkata700064
MathematicalAbilityTest:SampleQuestion
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SectionA(MCQType)/FullMarks40
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1. Thenumberofrealrootsoftheequation
│1-x│+x2=5is
(A) 1(B)2(C)4(D)0
2. Asquareiscutintotworectanglessuchthatthesumofthe
perimetersofthetworectanglesis48cm;thenthesideofthe
squareis(incm)
(A) 5(B)8(C)6(D)Noneoftheabove
3. Given(x+2)and(2x-1)arefactorsof(2x3+ax2+bx+10)
then(a2+b2)isequalto
(A)338(B)218(C)74(D)noneoftheabove.
4.Thesumoftendistinctrealnumbersis50.Thenthesumof
theirsquaresis:
(A)canbe50(B)is100(C)lessthan250(D)isalways
greaterthan250
5.Letx,ybepositiveintegerssuchthat(x+y)2-2(xy)2=1
Ifz=x+y,thenzequalsto
(A)2(B)3(C)4(D)5
6.ApersonispaidRs150foreachdayheworksandis
finedRs30foreachdayheisabsent.Ifin40dayshis
netearningisRs3300/-thenthenumberofdayshe
wasabsentis
(A)20(B)10(C)15(D)12
7. Atwowheelercompanyincreaseditsproductionofaparticular
brandfrom80000to92610inthreeyears.Therateofgrowthfor
theparticularbrandis
(A) 6%(B)4%(C)4.5%(D)5%
8. AthinksofapositiveintegerwhichBdoublesandCtreblestheB’s
number.FinallyDmultipliesC’snumberby6.Enoticesthatthe
sumofthefournumbersisaperfectsquare.Thesmallestnumber
thatAcouldhavethoughtofis
(A)3(B)2(C)4(D)5
9. RamandRahimindividuallycancompleteaworkin15daysand20
daysrespectively.Theyjointlycompletedanotherworkin30days.
Theirearningratiofromtheworkshouldbe
(A) 5:4(B)4:3(C)2:3(D)noneoftheabove
10.pandqaretwodistinctprimenumbers(p>q)greaterthan5.
thenp2-q2is
(A) alwaysdivisibleby6butnotdivisibleby12
(B) alwaysdivisibleby12butnotdivisibleby24
(C) alwaysdivisibleby24butnotnecessarilydivisibleby48
(D) noneoftheabove.
SectionB(ShortAnswerType)
FullMarks80
Allquestionscarryequalmarks
Giveallrelevantstepsforallanswers.
(1) Howmanypositiveintegerslessthan2015maybewrittenasasum
oftwoconsecutivepositiveintegersandalsocanbewrittenassum
offiveconsecutivepositiveintegers.
(2) Apositiveintegerpcanbewrittenasm2+3n2,wheremandnare
positiveintegers.Anotherpositiveintegerqcanbewrittenas
q=c2+3d2,wherecanddarepositiveintegers.Showthatpqalso
canbewritteninthesameformi.e.pqcanbewrittenas
pq=r2+3s2whererandsarepositiveintegers.
(3) Whichisgreater(31)11or(17)14?Givereasonsforyouranswer
(Hint:replace31by34)
(4) Solveforx,y,z
x+y+z=14
x2+y2+z2=84
xy=z2
(5) Findthesmallestnumberwhichwhendividedby3,5,7,11leaves
remainder2,4,6,1respectively.
(6) Aconewithcircularbaseiscutintotwosectionsbyahorizontal
planeparalleltothebaseinsuchamannerthatthecurvedsurfaces
ofthetwosectionsareofequalarea.Iftheheightoftheoriginal
coneis15cmfindtheheightofthesmallerconeremainedafterthe
cut.
(7) Findthevalueofthesum
cos(π/1000) + cos (2π/1000) + .......... + cos(999π/1000)
(8) Twocirclesofradiir,stoucheachotheratapoint.Acommon
tangent(otherthantheoneatA)touchesthecirclesatPandQ
respectively.ShowthatPQ2=4rs
(9) Iff(x)isapolynomialwithintegercoefficientsandf(1)andf(2)
arebothoddthenprovethatthereexistsnointegernforwhichf(n)
=0
(10) Supposem,nareintegersbothgreaterthan3andm=n2-n.
Showthat(m2-2m)isdivisibleby24.
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