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CALCUTTAMATHEMATICALSOCIETY AE374,SaltLake(SectorI),Kolkata700064 MathematicalAbilityTest:SampleQuestion ------------------------------------------------------------------------------------------------ Trytoanswerasmanyquestionsasyoucan SectionA(MCQType)/FullMarks40 Allquestionscarryequalmarks Putatick(√)markagainstyouranswer (Nonegativemarkingforwronganswer) Allquestionscarryequalmarks ---------------------------------------------------------------- 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. .....................xxxxxxxxxxxxxxxxxx.......................