education news iitjee 2012 mathematics

IITJEE 2012PAPER - MATHEMATICS

JAGRANJOSH 1

IITJEE 2012MATHEMATICS PAPER& ANALYSISIITJEE is one of the toughestexaminations in the world. In the year 2012 thisexam wasconductedin two parts,firstwasPAPER1andsecondwas PAPER2.There were 20 questions of eachsubjectin a paperandeach paper consistsof60questions in total. Paper1was of 210 marks and Paper2was of 198marks.This whole booklet is veryusefulfrom exam point of view. This booklet consistsofthreedifferentparts.

First part consists of the analysis of the MATHEMATICS part in Paper1andPaper2.Analysis is veryhelpful in manyaspects; itwillhelpyou inunderstanding: Importanceofchapters. Typeofquestionsasked. Trendofthepapers.

Secondpart contains all the questions asked in mathematics in Paper1and Paper2with the answers. This Part will help you inunderstandingthedifficultyleveloftheexam.

Thirdandthe last part contain the markingscheme andtype of questionsasked inthequestionpaper.


JAGRANJOSH 2

PART 1ANALYSISMATHEMATICS 2012 ANALYSIS

PART 1 PART 2Subject / Topic No of

QuestionsTotalMarks

Subject / Topic No ofQuestions

TotalMarks

ApplicationofDerivatives

2 8 Definite Integration 3 9

Hyperbola 1 4 3D Geometry 2 7TrigonometricEquations 1 4 Matrices &Determinant 2 7Parabola 1 4 Probability 2 7Probability 1 4 Permutation and

Combination2 6

Definite Integration 1 4 Circle 2 6Vector 1 4 Applicationof

Derivatives1 4

Differential Equations 1 4 Continuity andDerivability

1 4

FundamentalsofMathematics

1 4 Functions 1 4

ComplexNumber 1 3 Vectors 1 3Circle 1 3 Sequenceand

Progression1 3

Functions 1 3 Properties of Triangle 1 3Continuity &Derivability 1 3 Limits 1 3Permutation &Combination

1 3Grand Total

20 66

IndefiniteIntegration 1 3Ellipse 1 3Limits 1 33D Coordinate Geometry 1 3Matrices &Determinants 1 3Grand Total 20 70


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GRAPHICAL ANALYSIS

2111111111111111111

0 0.5 1 1.5 2 2.5

Application ofDerivativesHyperbola

Trigonometric EquationsParabola

ProbabilityDefiniteIntegration

VectorDifferential Equations

Fundamentals ofMathematicsComplexNumber

CircleFunctions

Continuity& DerivabilityPermutation & Combination

Indefinite IntegrationEllipseLimits

3DCoordinate GeometryMatrices&Determinants

IITJEEMATHEMATICS2012(PAPER1)


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3

2

2

2

2

2

1

1

1

1

1

1

1

0 0.5 1 1.5 2 2.5 3 3.5

DefiniteIntegration

3DGeometry

Matrices&Determinant

Probability

Permutation andCombination

Circle

Application ofDerivatives

Continuity andDerivability

Functions

Vectors

SequenceandProgression

Properties ofTriangle

Limits

IITJEEMATHEMATICS2012(PAPER2)


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PART 2IITJEE2012PAPER1MATHEMATICS [Code8]

1. If lim 2+ +1 + 1 = 4, then:(A)a =1, b =4 (B) a=1,b = 4(C) a=2,b = 3 (D) a=2,b =3

Ans. (B)

2. Let P = [aij] bea 3 3 matrixand let Q = [bij],where bij = 2i+jaij for 1 i,j 3.If the determinant of Pis 2,then the determinant of the matrix Qis:(A)210 (B) 211(C) 212 (D) 213

Ans. (D)

3. Thelocusof themidpoint of thechord of contact of tangentsdrawnfrompointslyingon the straightline 4x 5y =20to thecirclex2 +y2 =9is:(A)20(x2 +y2) 36x +45y=0 (B) 20(x2 +y2) +36x 45y =0

(C) 36(x2 +y2) 20x + 45y =0 (D) 36(x2 +y2) +20x 45y =0

Ans. (A)

4. Thetotalnumberof ways in which 5ballsof different colourscan bedistributed among3personssothat each person gets at least one ball is:(A)75 (B) 150(C) 210 (D) 243

Ans. (B)5. Theintegral sec 2x(sec x + tan x)9/2 dx equals(forsome arbitrary constant K):(A) 1(sec x +tan x)11/2 { 111 17 (sec x + tan x)2} + K (B) 1(sec x +tan x)11/2 { 111 17 (sec x + tan x)2} + K(C) 1(sec x +tan x)11/2 { 111 + 17 (sec x + tan x)2} + K (D) 1(sec x +tan x)11/2 { 111 + 17 (sec x + tan x)2} + KAns. (C)


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6. Thepoint Pistheintersection of thestraight linejoiningthepointsQ(2,3,5)and R(1,1,4) with theplane 5x 4y z =1.IfSisthefoot of theperpendiculardrawnfromthepoint T(2,1,4)to QR,then thelength of the linesegment PS is(A) 12 (B) 2(C) 2 (D) 22Ans. (A)

7. Let () = 2 , 00, = 0 ,X IR then f is:(A)Differentiable both at x =0 and at x = 2(B) Differentiable at x =0 but not differentiableat x =2(C)Not differentiableat x =0but differentiableat x = 2(D) Differentiable neitherat x =0 norat x = 2

Ans. (B)

8. Let z bea complexnumbersuch that theimaginarypart of z isnonzero and a =zz +z +1 isreal.Then acannot take thevalue:(A)1 (B) 13(C) 12 (D) 34Ans. (D)

9. Theellipse E1: x29 + y24 = 1 isinscribed in arectangleRwhosesidesare parallelto thecoordinateaxes.AnotherellipseE2 passingthrough thepoint (0,4)circumscribestherectangleR. Theeccentricityof theellipseE2 is(A) 22 (B) 32(C)12 (D) 34Ans. (C)

10. The function f: [0, 3] [1,29],defined byf(x) =2x3 15x2 + 36x +1, is:(A)oneoneand onto (B) onto but not oneone(C) oneonebut not onto (D) neither oneonenoronto

Ans. (B)


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11. Tangents are drawn to the hyperbola 29 24 = 1 ,parallelto the straight line2x y =1.Thepointsofcontact of thetangentson thehyperbola are:(A) 922 , 12 (B) 922 , 12(C) (33,22 ) (D) (33, 22 )Ans. (A, B)

12. Let , [0, 2] be such that 2cos (1sin ) sin2 2 + 2 1, tan(2 ) > 0 and1 < 32 . Then cannot satisfy(A)0 < < 2 (B) 2 < < 43(C)43 < < 32 (D) 32 < < 2Ans. (A, C,D)

13. If y(x) satisfies thedifferential equation y ytanx = 2x secx and y(0)=0,then:(A) 4 = 282 (B) 4 = 218(C) 3 = 29 (D) 3 = 43 + 2233Ans. (A, D)

14. Aship isfitted with threeenginesE1,E2 and E3.Theenginesfunction independently of each otherwithrespective probabilities 12 , 14 and 14 . For the ship to be operational at least two of its engines mustfunction.Let Xdenotes theeventthat theship isoperationaland let X1,X2 and X3 denoterespectivelytheevents that theengines E1,E2 and E3 are functioning.Which of thefollowingis(are) true?(A)[1|] = 316 (B) P[Exactly two enginesof theship arefunctioning|X] =78(C) [|2] = 516 (D) [|1] = 716Ans. (B,D)

15. Let S bethe areaof theregion enclosed by = 2 , = 0, = 0 and = 1. Then:(A)S 1 (B) S 1 1(C) S 14 (1 + 1) (D) S 12 + 1 (1 12)Ans. (A, B,D)


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16.If , and are unit vectorssatisfying| |2 + | |2 + | |2 = 9, then |2 + 5 + 5| isAns. (3)

17. Let f: IR IR be defined as () = ||+ |2 1| .The total numberof pointsat which f attainseitheralocal maximumor alocalminimumis

Ans. (5)

18. Let S be thefocusof theparabolay2 =8xand let PQbethecommon chord of thecirclex2 +y2 2x 4y= 0 and thegiven parabola.The area of thetrianglePQSis

Ans. (4)

19. Let p(x) bearealpolynomialof least degree which hasalocalmaximumat x =1and alocalminimumat x = 3.Ifp (1)=6and p (3)=2, then p (0)is

Ans. (9)20. The value of 6 + log3/2 1324 1324 1324 132 isAns. (4)


JAGRANJOSH 9

IITJEE2012PAPER-2MATHEMATICS [Code 8]

1. Let 1,2,3, .. be in harmonic progression with 1 = 5 and 20 = 25.The least positive integer forwhich < 0(A)22 (B) 23(C) 24 (D) 25

Ans. (D)

2. The equation of a plane passing through the line of intersection of the planes + 2 + 3 =2 + = 3 and at adistance 23 fromthepoint (3, 1, 1) is(A) 5 11 + = 17 (B) 2 + = 32 1(C) + + = 3 (D) 2 = 1 2Ans. (A)

3. Let PQR be atriangleof area D with = 2, = 72 and = 52, where a, b, and c are the lenghtsof thesidesof the triangleoppositeto theanglesat P, Qand Rrespectively. Then2 22+ 2 equals(A) 34 (B) 454(C) 342 (D) 4542Ans. (C)

4. If and are vectors such that + = 29 and (2 + 3+ 4) = (2+ 3+ 4) , then apossible value of ( + ) (7+ 2+ 3 is(A)0 (B) 3(C) 4 (D) 8

Ans. (D)


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5. IfPisa33matrixsuch that PT =2P +I,wherePT isthetransposeof Pand Iisthe33identitymatrix,then thereexists acolumn matrix = 000 such that(A) = 000 (B) = (C) = 2 (D) = Ans. (D)

6. Let () and () be the roots of the equation 1 + 3 121 + 1 + 1 + 6 1 = 0 where > 1. Then and are(A) 52 1 (B) 12 1(C) 72 2 (D) 92 3Ans. (B)

7. FourfairdiceD1,D2,D3 and D4,eachhaving sixfacesnumbered1,2,3,4,5,and 6,arerolledsimultaneously.Theprobability thatD4 showsanumberappearingon oneofD1,D2 andD3 is(A) 91216 (B) 108216(C)125216 (D) 127216Ans. (A)8. The value of the integral (A)0 (B)22 4(C)22 + 4 (D)22Ans. (B)


JAGRANJOSH 11

Paragraph for Questions 9and 10:Atangent PT is drawn to thecircle2 + 2 = 4 at the point ( 3, 1) .Astraight lineL,perpendicular toPT isa tangent to thecircle( 3)2 + 2 = 1.9. A possibleequation of is(A) 3 = 1 (B) + 3 = 1(C) 3 = 1 (D) + 3 = 5Ans. (A)

10. A common tangent of thetwo circlesis(A) = 4 (B) = 2(C) + 3 = 4 (D) + 22 = 6Ans. (D)

Paragraph for Questions 11 and 12:Let () = (1 )22 + 2 for all , and let for all (1,).11. Consider thestatements:P :There existssome such that () + 2 = 2(1 + 2)Q :There existssome such that 2() + 1 = 2(1 + ) Then(A)both P and Q are true (B) P istrue and Q isfalse(C)P isfalseand Q is true (D) both P and Q are false

Ans. (C)

12. Which of thefollowingistrue?(A)gisincreasingon (1,)(B) gisdecreasing on (1,)(C) gisincreasingon (1,2)and decreasingon (2,)(D) gisdecreasingon (1,2)and increasingon (2,)

Ans. (B)

Paragraph for Questions 13and 14:Let an denote thenumberof all ndigit positive integers formed bythe digits 0, 1 orboth such that noconsecutive digitsin themare 0.Let bn =the numberof such ndigit integersending with digit 1 and cn =the number of such ndigit integers ending with digit 0.


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13. The value of b6 is(A)7 (B) 8(C) 9 (D) 11

Ans. (A)

14. Which of thefollowingiscorrect?(A)17 = 16 + 15 (B) 17 16 + 15(C)17 17 + 16 (D) 17 = 17 + 16Ans. (A)

15. Foreveryinteger n, let an and bn bereal numbers.Let function begiven by() = + , [2, 2 + 1] + , (2 1,2) ,forall integers n.If f iscontinuous,then which of thefollowing hold(s) for alln?(A)1 1 = 0 (B) = 1(C)an bn+1 = 1 (D) an1 bn = 1Ans. (B,D)

16. If the straight linesx12 = y+1k = z2 and x15 = y+12 = zk are corplanar,then theplane(s) containing thesetwo lines is(are)(A)y + 2z = 1 (B) y + z = 1(C)y z = 1 (D) y 2z = 1Ans. (B,C)17. If the adjoint of a3 3matrixP is 1 4 42 1 71 1 3,then the possiblevalue(S) of thedeterminant of P is(are)(A)2 (B) 1(C) 1 (D) 2

Ans. (A, D)

18. Let f: (1, 1) IR besuch that f(cos4) = 22sec 2 for 0, 4 4 , 2. Then thevalue(s) of f 13 is(are)(A)1 32 (B) 1 +32(C)123 (D) 1 +23Ans. (A, B)


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19. Let X and Y be two eventssuch that P(X|Y) = 12 , P(Y|X) = 13 and P(X Y) = 16 . Which of thefollowingis(are) correct?(A)P(X Y) = 23 (B) X and Y are independent(C)X and Y are not independent (D) P(Xc Y) = 13Ans. (A, B)

20. If f(x) = et2(t 2)(t 3)dtx0 for allx (0,), then(A)f hasalocalmaximum at x = 2 (B) f isdecreasingon (2, 3)(C) there existssomec (0,) such that f (c) = 0 (D) f hasa localat x = 3Ans. (A, B,C, D)


JAGRANJOSH 14

PART 3MARKINGSCHEMEMARKING SCHEMEFORIITJEE 2012 Eachquestionpaperwas consistingof3

parts(Chemistry,MathematicsandPhysics).Eachpartwas consistingofthreeSections.

DescriptionNo of

Questionsper

SubjectMarking Scheme Total Marks

Paper ISingleChoiceQuestions(SectionI)

10 (3,1) 30

Multiple ChoiceQuestions(SectionII)

5 (4,0) 20

Integer TypeQuestions(SectionIII)

5 (4,0) 20

Total 2070

TotalnoofQuestions/TotalMarks

60/210

Paper IISingleChoiceQuestions(SectionI)

8 (3,1) 24

Multiple ChoiceQuestions(SectionII)

6 (4,0) 24

ComprehensionBased(SectionIII)

6 (3,1) 18

Total 20 66 TotalnoofQuestions/TotalMarks

60/198

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