maths prena classes answer key of maths jee 2015

JEEMAIN2015 (04Apr15) Question&Solutions 1 www.prernaclasses.com

04April2015JEEMAIN2015PARTBMATHEMATICS

31. Thesumofcoefficientsofintegralpowersof x inthebinomialexpansionof(12x)50is

(1)21(350) (2)

21(3501) (3)

21(250+1) (4)

21(350+1)

31. (4) (12x)50

Tr+1=(1)r50Cr(2x)

r=(1)r50Cr2rxr/2

r =0,2,4,.......,50Sumofcoefficients=50C02

0+50C222+50C42

4.......50C5250

2)1()1( 5050 xx - + +

= ,where x =2

21350 +

= .

32. Let f (x) beapolynomial of degree fourhaving extremevalueat x = 1and x = 2. If

3)(

120

=

+

x

xfLimx

,then f (2)isequalto

(1) 4 (2) 0 (3) 4 (4) 8

32. (2) Letf(x)=ax4+bx3+cx2Q 3)(

120

=

+

x

xfLimx

2)(

20 =

x

xfLimx

c =2

f'(x)=4ax3+3bx2+ 2cxf' (1)=4a +3b + 4=0f' (2)=32a +12b + 8=0 a=1/2, b =2 \ f(x)=(1/2) x42x3+ 2x2 \ f (2)=0.

33. Themeanofthedatasetcomprisingof16observationsis16.Ifoneoftheobservationvalued16isdeletedandthreenewobservationsvalued3,4and5areaddedtothedata,thenthemeanofresultantdata,is(1) 16.0 (2) 15.8 (3) 14.0 (4) 16.8

33. (3) =

= = 16

12561616

iix Sumofnewobservations=

= = + + + - =

18

1252)543(16256

iix

\ Meanoftheresultantdata= 0.1418/25218

18

1 = = = i

ix

.

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34. Thesumoffirst8temsoftheseries ......531321

3121

11 333333

+ + + + +

+ + +

+ is

(1) 96 (2) 142 (3) 192 (4) 71

34. (1) =

=

=

+ =

+

9

1

29

12

2

)1(412

)1(

r

r

r

rr

rr

[ ] [ ] 1)10......21(41

10......3241 222222 - + + + = + + + =

=(1/4)384=96.

35. LetO bethevertexand Q beanypointontheparabola x2=8y.Ifthepoint P dividesthelinesegmentOQ internallyintheratio1:3,thenthelocusofP is(1) y2=x (2) y2=2x (3) x2=2y (4) x2=y

35. (3)

(0, 0)

3

O

PQ(22t,t 2 )

IfP :(h, k) h =22t /4, k = t2/4Eliminating t, h2=2k Locus: x2= 2y.

36. Let a and b betherootsofequation x26x 2=0.If an= a n b n,for n 1,thenthevalueof

9

810

22

aaa -

isequalto

(1) 6 (2) 3 (3) 3 (4) 6

36. (2))(2

)(2)(2

299

881010

9

810

b - a

b - a - b - a =

- a

aa3

)(2

66

)(2

)2()2(99

88

99

2828 =

b - a

b b - a a =

b - a

- b b - - a a =

37. If12identicalballsaretobeplacedin3identicalboxes,thentheprobabilitythatoneoftheboxescontainsexactly3ballsis(1) 55(2/3)10 (2) 220(1/3)12 (3) 22(1/3)11 (4) (55/3)(2/3)11

37. (4) 12

93

12

3

2 =

CP

38. Acomplexnumber z issaidtobeunimodularif| z |=1.Suppose z1and z2arecomplexnumbers

suchthat21

21

22zzzz

- -

isunimodularand z2isnotunimodular.Thenthepoint z1liesona

(1) straightlineparallelto y axis (2) circleofradius2(3) circleofradius 2 (4) straightlineparallelto xaxis

38. (2) Q 2212

21 |2||2| zzzz - = - )2)(22()2)(2( 21212121 zzzzzzzz - - = - - | z1|

2(1| z2|2)=4(1| z2|

2) | z1|=2.


39. Theintegral

+ 4/342 )1(xx

dxequals

(1) (x4+1)1/4+ c (2) (x4+1)1/4+ c

(3) cx

x +

+ -

4/1

4

4 1(4) c

x

x +

+ 4/1

4

4 1

39. (3)

+ 4/342 )1(xx

dx

Let1+(1/ x4)= t(4/ x5) dx = dt(1/x5) dx =(1/4) dt =(1/4) (dt / t3/4)=(1/4)(t3/4+1/(3/4)+1) +c =(1/4)(t1/4/(1/4)) +c

=(1+(1/ x4))1/4+c = cx

x +

+ -

4/1

4

4 1

40. Thenumberofpoints,havingbothcoordinatesasintegers,thatlieintheinteriorofthetrianglewithvertices(0,0),(0,41)and(41,0)is(1) 861 (2) 820 (3) 780 (4) 901

40. (3) x +y0

y >0(1,1)(1,2)............(1,39)=39cases(2,1)(2,2)............(2,38)=38cases | | (39,1)

\ requiredno.ofcases=1+2+3...............39=24039

=780.

41. Thedistanceofthepoint(1,0,2)fromthepointofintersectionoftheline

122

41

32 -

= +

= - zyx

andtheplane x y + z=16,is

(1) 8 (2) 213 (3) 13 (4) 14241. (3) Anypointonthegivenlinecanbetakenas(3l +2,4l 1,12l +2)

Forpointofintersection,point(3l +2,4l 1,12l +2)mustlieon x y + z=16 \ 3l +24l +1+12l +2=16 l =1 \ Point (5,3,14)

Reqd.distance= 13)214()03()15( 222 = - + - + - .

42. Theequationoftheplanecontainingtheline2x 5y + z =3, x + y +4z =5andparalleltotheplane x +3y +6z =1,is(1) x +3y +6z =7 (2) x +3y +6z =7(3) 2x +6y +12z =13 (4) 2x +6y +12z =13


42. (2) Foranypoint,thegivenline, z=0 2x 5y =3and x + y =5Solving x =4, y =1 \ Thepointontheline(4,1,0)Reqd.planeparallelto x +3y +6z=1 \ D.Rofnormaltoreqd.plane(1,3,6) \ Eq.ofreqd.plane:1(x 4)+3(y 1)+6(z0)=0 x +3y +6z =7.

43. Thearea(insq.units)oftheregiondescribedby{(x, y): y2 2x and y 4x 1}is(1) 5/65 (2) 15/64 (3) 9/32 (4) 7/32

43. (3)

(1 / 4, 0)

1 A

B -1 / 2

(0, - 1)

Pointofintersectionoflineandparabolahave ycoordinates1/2and1

\ Reqd.area= 32/924

11

2/1

2 =

-

+

-

dyyy

.

44. IfmistheA.M.oftwodistinctrealnumberslandn(l,n>1)andG1,G2andG3arethreegeometricmeansbetween l and n,thenG1

4+2G24+G3

4equals(1) 4lm2n (2) 4lmn2 (3) 4 l2m2n2 (4) 4l2mn

44. (1) G14= l3.n,G2

4= l2n2,G34= ln3

Also,2m = l+ nNow G1

4+2G24+G3

4 =ln (l2+2ln + n2)=ln (l + n)2=4 lm2n.

45. Locusoftheimageofthepoint(2,3)intheline(2x 3y +4)+ k (x 2y +3)=0, k R,isa(1) straightlineparallelto y axis (2) circleofradius 2(3) circleofradius 3 (4) straightlineparallelto xaxis

45. (2) Pointofintersectionoffamilyoflinesare(1,2).Now,equationofanylinepassingthrough(1,2)is y 2=m(x 1).Lettheimagebe(a, b).Now,midpointliesontheline b 1=ma ........(i)

Also, 123

- =

- a - b

m .........(ii)

eliminatingm,weget a 2+ b 22a 4b +3=0Locusis, x2+ y22x 4y +3=0,circlehavingradius 2.

ORLocusisgoingtobeacirclewithcentreatpointofconcurrencyi.e.at(1,2)andcirclemustpassthrough(2,3).Henceradius= 2.


46. Thearea(insq.units)ofthequadrilateralformedbythetangentsattheendpointsofthelatera

rectatotheellipse 159

22 = +

yx,is

(1) 18 (2) 27/2 (3) 27 (4) 27/4

46. (3) B

A(9/2,0)

(0,3)(ae,b2 /a)

(0,0)

a2=9,b2=5Now5=9(1 e2) e =2/3ae =2and b2/ a =5/3Nowtheequationoftangentat(2,5/3)is(2x /9)+(y /3)=1Now,reqd.area=4(1/2)(9/2)3=27.

47. Thenumberofintegersgreaterthan6,000thatcanbeformed,usingthedigits3,5,6,7and8,withoutrepetition,is(1) 192 (2) 120 (3) 72 (4) 216

47. (1) Totalnumberofintegers=Numberof4digitedintegers+Numberof5digitedintegers=34P3+

5P5=192.

48. LetA andBbetwosetscontainingfourandtwoelementsrespectively.ThenthenumberofsubsetsoftheAB,eachhavingatleastthreeelementsis(1) 256 (2) 275 (3) 510 (4) 219

48. (4) NumberofsubsetsofsetsAB,eachhavingatleastthreeelements=8C3+

8C4+.......+8C8=2

88C08C1

8C2=2561828=219.

49. Let

- + = - - -

2111

1

2tantantan

x

xxy ,where| x |


50. Theintegral

+ - +

4

2

22

2

)1236log(log

logdx

xxx

xis

(1) 4 (2) 1 (3) 6 (4) 2

50. (2) 2)6log(log

)6log(log2

4

2

4

2

22

22 = =

- +

- + = dxdx

xx

xxI I =1.

51. Thenegationof~ s (~ r s)isequivalentto(1) s (r ~ s) (2) s (r ~ s) (3) s r (4) s ~ r

51. (3) Negationof~ s (~ r s)is~[~ s (~ r s)] @ ~(~ s) ~(~ r s) @ s (~(~ r) ~ s) @ s (r ~ s) @ (s r) (s ~ s) @ (s r) F [Q s ~ s isfallacy] @ s r.

52. Iftheanglesofelevationofthetopofatowerfromthreecollinearpoints A, B andC,onalineleadingtothefootofthetower,are30,45and60respectively,thentheratio AB :BC is(1) 3: 2 (2) 1: 3 (3) 2:3 (4) 3:1

52. (4)

BA C D

E

xz y

x+y

30 45 60

Fromthefigure,tan30=1/ 3= zyxyx

zyxED

+ + +

= + + .........(1)

SinceBD=ED 3 = + + + yxzyx

2

13 + =

+ zyx

..........(2)

Againtan60= 3=xyx +

x + y = 3x .........(3)

2

133 + =

zx

)13(31

- =

zx

.........(4)

Now2

13 + = +

zy

zx

from(2) 3/1)13(3

12

13 =

- -

+ =

zy

z : y = 3:1


53.xx

xxLimx 4tan

)cos3)(2cos1(0

+ -

isequalto

(1) 3 (2) 2 (3) 1/2 (4) 4

53. (3) Limit= 2/141

42

444.

4tan4

4

sin24

4tan2cos1

)cos3(2

2

000 = =

= -

+ x

x

x

xLim

xxx

LimxLimxxx

.

54. Let ar, b

rand c

r be threenonzerovectorssuch thatno twoof themarecollinearand

acbcbar r r r r r

||||31

)( = .If q istheanglebetweenvectorsbrandc

r,thenavalueofsin q is

(1) 2/3 (2) 2/3 (3) 23/3 (4) 22/3

54. (4) Here acbabcbacr r r v r r v

31

).().( = -

acbbacr v r r r v

+ q =

31

cos).(

Butbrand a

r arenoncollinear

31

cos031

cos - = q = + q 322

sin = q

55. If

- =

ba

A

2

212

221

isamatrixsatisfyingtheequation AAT=9I,where I is33identitymatrix,

thentheorderedpair(a, b)isequalto(1) (2,1) (2) (2,1) (3) (2,1) (4) (2,1)

55. (3) Since AAT=9I a +4+2b =0forIstrow,IIIrdcolumnand2a +22b =0for2ndrow,3rdcolumn \ a =2, b =1.

56. Ifthefunction

< + +

= 53,2

30,1)(

xmx

xxkxg isdifferentiable,thenthevalueof k +m is

(1) 16/5 (2) 10/3 (3) 4 (4) 2

56. (4) Forcontinuityat x =3,3m +2=2kFordifferentiabilityat x =3, k =4m \ k=8/5,m =2/5 k+m =2.


57. Thesetofallvaluesof l forwhichthesystemoflinearequations:2x12x2+x3= lx12x13x2+2x3= lx2x1+2x2 = lx3hasanontrivialsolution(1) isasingleton (2) containstwoelements(3) containsmorethantwoelements (4) isanemptyset

57. (3) Makingcoefficientdeterminantzero,wegetacubic l 3+ l 27l 3=0where(l +3)(l 22l 1)=0 \ Threedistinctvaluesas l =3,1 2.

58. Thenormaltothecurve, x2+2xy 3y2=0,at(1,1)(1) meetsthecurveagaininthesecondquadrant.(2) meetsthecurveagaininthethirdquadrant.(3) meetsthecurveagaininthefourthquadrant.(4) doesnotmeetthecurveagain.

58. (3) Q x2+2xy3y2=0 ........(i)

Diff.w.r.t. x 2x +2y +2xdxdy

6y.dxdy

=0 xyyx

dxdy

- +

= 3

\ Slopeofnormalat(1,1)=2/2=1Equationofnormal: y 1=1(x 1) x + y =2 ..........(ii)From(i)and(ii) x2+2x (2x)3(2 x)2=0 x24x +3=0 (x 1)(x 3)=0 x =1,3and y =1,1.

59. Thenumberofcommontangentstothecircles x2+ y24x6y 12=0andx2+y2+6x +18y +26=0,is(1) 2 (2) 3 (3) 4 (4) 1

59. (2) Q (x 2)2+(y 3)2=52Centre (2,3), r1=5and (x +3)2+(y+9)2=82C2 (3,9), r2=5

\ C1C2= 1314425 = + , r1+r2=13 C1C2=r1+r2.

60. Let y(x)bethesolutionofthedifferentialequation(x log x)dxdy

+ y =2x log x,(x 1).Then y(e)

isequalto(1) 0 (2) 2 (3) 2e (4) e

60. (2) Q 2.log1

= + yxxdx

dy x 1

\ I.F= xee xdx

xx log)log(loglog1

= =

Hence,solutionis y .log(x)= 2.log xdx y .log(x)=2(x log x x)+ kIf x =1then k =2 \ y(e)=2(e e)+2 y(e)=2.

maths prena classes answer key of maths jee 2015

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