IITJEE2009PAPERIITJEE 2009 MATHEMATICS PAPER& ANALYSISIITJEE is one of the toughestexaminations in the world. In the year 2009 thisexam wasconductedin two parts,firstwasPAPER1andsecondwas PAPER2.There were 20 questions in PAPER1 and 19 questions in PAPER2 of eachsubject. Eachpaperwas consistingof 60 and57 questions respectively.Paper1and Paper2 both were of 240 marks. This whole booklet is very useful fromexam pointofview.This bookletconsists ofthreedifferentparts.

First part consists of the analysis of the MATHEMATICS part in Paper1andPaper2.Analysis is veryhelpful in manyaspects; it willhelpyou inunderstanding:

Importanceofchapters. Typeofquestions asked. Trendofthepapers.

Secondpart contains themarkingschemeandtypeofquestionsaskedintheQuestionPapers.

Thirdandthe last part contain all the questionsasked in MATHEMATICSin Paper1 and Paper2 with the answers. This part will help you inunderstandingthe difficultyleveloftheexam.

IITJEE2009PAPERPART 1ANALYSISMATHEMATICS 2009 ANALYSIS

PAPER 1 PAPER 2Topic Noof

QuestionsTopic No of

QuestionsAlgebra 7 CoordinateGeometry 5

Vectors,3D,Matrices&Determinants 5 Trigonometry 4

IntegralCalculus 3 Algebra 3DifferentialCalculus 2 DifferentialCalculus 3

Trigonometry 2 Vectors,3D,Matrices&Determinants 2CoordinateGeometry 1 IntegralCalculus 2

TotalQuestions/TotalMarks 20/80 Total Questions/TotalMarks 19/80

IITJEE2009PAPERGRAPHICAL ANALYSIS

Algebra35%

Vectors, 3D,Matrices&Determinants

25%

Integral Calculus15%

Differential Calculus10%

Trigonometry10%

CoordinateGeometry

5%

JEEMATHEMATICS2009(PAPER1)

CoordinateGeometry

26%

Trigonometry21%Algebra

16%

Differential Calculus16%

Vectors, 3D,Matrices&Determinants

10%

Integral Calculus11%

JEEMATHEMATICS2009(PAPER2)

IITJEE2009PAPERPART 2MARKINGSCHEME

MARKING SCHEMEFORIITJEE 2009Eachquestionpaperwas consisting of3parts(Chemistry,MathematicsandPhysics).Eachpartwasconsisting offourSections.

Description No ofQuestions

perSubject

Marking Scheme Total Marks

Paper ISingleCorrectAnswerType(SectionI)

8 (3,1) 24

MultipleCorrectAnswer(s) Type(SectionII)

4 (4,1) 16

Paragraph Type(SectionIII) 6 (41) 24MatrixMatchType(SectionIV)

2 (8,0) 16

Total 20 80 TotalnoofQuestions/TotalMarks 60/240Paper IISingleCorrectAnswerType(SectionI)

4 (3,1) 12

Multiple ChoiceCorrect Type(SectionII)

5 (4,1) 20

MatrixMatchType(SectionIII)

2 (8,0) 16

Integer Type(SectionIV) 8 (4,1) 32

Total 19 80 TotalnoofQuestions/TotalMarks 57/240

IITJEE2009PAPERPART 3QUESTIONPAPERIITJEE2009 PAPER 1[MATHEMATICS]

SECTIONI Single CorrectChoiceTypeThissection contains8multiplechoicequestions. Each question has4choices(A),(B),(C) and(D) foritsanswer,out of which ONLYONE is correct.

1. Let z = x +iy beacomplexnumber where x and yare integers.Then theareaof therectanglewhoseverticesare therootsof theequation zz3 + zz3 = 350 is(A)48 (B) 32(C) 40 (D) 80

Ans. (A)

2. If a ,b ,c and d are unit vectorssuch that (a b ).(c d ) =1and a . c =12,then(A)a ,b ,c are noncoplanar (B) b ,c,d are noncoplanar(C)b ,d are nonparallel (D) a ,d are paralleland b ,c areparallelAns. (C)

3. Thelinepassingthrough theextremityA of themajoraxis and extremityB of theminor axisof theellipsex2 +9y2 =9meetsitsauxiliarycircleat thepoint M.Then theareaof thetrianglewith verticesat A,M and theorigin Ois(A)3110 (B) 2910(C)2110 (D) 2710Ans. (D)

4. Let z = cos+ i sin. Then thevalueof Im (z2m1)15m=1 at = 20 is(A) 1sin 20 (B) 13sin 20(C) 12sin 20 (D) 14sin 20Ans. (D)

IITJEE2009PAPER5. Let P(3,2,6)beapoint in spaceand Qbeapoint on theline r =(i j + 2k) +(3i + j + 5k).Then thevalueof forwhich the vector PQ isparallelto theplane x 4y+ 3z=1is(A)14 (B) 14(C)18 (D) 18Ans. (A)

6. Thenumberof seven digit integers,with sumof thedigitsequalto 10 and formed byusingthe digits1, 2and 3only, is(A)55 (B) 66(C) 77 (D) 88Ans. (C)

7. Let f bea nonnegativefunction defined on theinterval[0,1].If1 (f (t))2dt = f(t)dt, 0 x 1, and f(0) = 0, thenx0x0(A)f 12 < 12 and f 13 > 13 (B) f 12 > 12 and f 13 > 13(C)f 12 < 12 and f 13 < 13 (D) f 12 > 12 and f 13 < 13Ans. (C)

8. Tangentsdrawnfromthepoint P(1,8)to thecirclex2 +y2 6x 4y 11 =0touch thecircleat the points Aand B.Theequation of thecircumcircleof thetriangle PAB is(A)x2 +y2 + 4x 6y+ 19 =0 (B) x2 +y2 4x 10y+19 =0(C) x2 +y2 2x+6y 29 =0 (D) x2 +y2 6x4y+19=0Ans. (B)

SECTIONIIMultiple Correct ChoiceTypeThissection contains4multiplechoicequestions. Each question has4choices(A),(B),(C) and(D) foritsanswer, out of which ONE OR MORE is/are correct.

9. In atriangleABCwith fixed baseBC,thevertex Amovessuch that cosB +cosC =4sin2 A2.Ifa,b and c denote the lengths of the sides of the triangle opposite to the angles A, B and C,respectively, then(A)b +c =4a (B) b +c =2a(C) locusof point Aisan ellipse (D) locusof point A isa pairof straight lines

Ans. (B,C)

IITJEE2009PAPER10. If sin 4x2 + cos 4x3 = 15 , then(A)tan2 x =23 (B) sin 8x8 + cos 8x27 = 1125(C) tan2 x = 13 (D) sin 8x8 + cos 8x27 = 2125Ans. (A, B)

11. Let L =limx0 a a2 x2 x24x4 , a > 0. If L is finite, then(A)a =2 (B) a=1(C)L = 164 (D) L = 132Ans. (A, C)

12. Areaof theregion bounded bythe curvey =ex and linesx = 0and y= e is(A)e 1 (B) In(e + 1 y)dye1(C) e exdx10 (D) In ydye1Ans. (B,C, D)

SECTIONIII Comprehension TypeThissection contains2groupsof questions. Each group has3multiplechoicequestionsbasedon aparagraph.Each question has4choices (A), (B), (C) and (D) for its answer,out of whichONLYONE iscorrect.

Paragraph for question Nos. 13 to15:A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tossesrequired.

13. The probabilitythat X=3 equals(A) 25216 (B) 2536(C) 536 (D) 125216Ans. (A)

IITJEE2009PAPER14. The probabilitythat X> 3 equals(A)125216 (B) 2536(C) 536 (D) 25216Ans. (B)

15. The conditionalprobabilitythat X > 6given X>3equals(A)125216 (B) 25216(C) 536 (D) 2536Ans. (D)

Paragraph for question Nos. 16 to 18:Let A betheset of all33symmetricmatricesallof whoseentriesareeither0or1.Fiveofthese entries are 1and fourof them are0.

16. The numberof matricesin A is(A)12 (B) 6(C) 9 (D) 3

Ans.(A)17. Thenumber of matricesAin A forwhich thesystem of linear equationsA xyz = 100 has aunique solution, is(A)lessthan 4 (B) at least 4but lessthan 7(C) atleast 7 but lessthan 10 (D) at least 10

Ans. (B)18. The number of matricesA in A forwhich the system of linear equations A xyz = 100 isinconsistent,is(A)0 (B) more than 2(C) 2 (D) 1

Ans. (B)

IITJEE2009PAPERSECTION IVMatrix Match Type

This section contains 2 questions. Each question contains statements given in two columns,which have to bematched. The statement in Column I are labelled A,B, C and D,while thestatementsin Column II are labelled p,q,r,sand t.Anygiven statement in Column I can havecorrect matching with ONE OR MORE statement (s) in Column II. The appropriate bubblescorrespondingto the answersto these questions haveto bedarkened:

19.Match the conicsin Column I with the statements/expressionsin Column II.Column I Column II

(A)Circle (p) Thelocusof thepoint (h,k) forwhich thelinehx +ky= 1touchesthe circlex2 +y2 =4

(B) parabola (q) Pointsz in thecomplexplanesatisfying|z +2| |z 2| = 3

(C) Ellipse (r) Points of the conic have parametricrepresentation x = 3 1 t21+ t2 , y = 2t1+ t2

(D) Hyperbola (s) Theeccentricityof theconic liesin theinterval1x

IITJEE2009PAPERIITJEE2009PAPER 2 [MATHEMATICS]

SECTIONI Single CorrectChoiceTypeThissection contains4multiplechoicequestions. Each question has4choices(A),(B),(C) and(D) foritsanswer, out of which ONLYONE is correct.

1. Thenormalat apoint Pon the ellipsex2+4y2 = 16 meetsthexaxis at Q.If M isthemidpointof theline segment PQ,then thelocusof M intersectsthelatusrectumsof thegiven ellipseatthe points(A) 532 , 27 (B) 532 , 194 (C)32 , 17 (D) 3,2 347 Ans: (C)

2. Thelocusof theorthocentre of the triangleformed bythe lines(1+p)x py+p(1 +p) =0,(1+q)x qy+q(1 +q) =0and y= 0, where p q, is(A)a hyperbola (B) aparabola(C) an ellipse (D) astraight line

Ans: (D)

3. A line with positivedirectioncosines passes through the point P(2, 1,2)and makesequalangleswith the coordinateaxes.Thelinemeetstheplane2x+y+z =9at point Q. Thelengthof the line segment PQ equals(A)1 (B) 2(C)3 (D) 2Ans. (C)

4.If the sumof first n termsof an A.P.iscn2,then thesumof squaresof these n termsis(A) 421 26 (B) 42+1 23(C)421 23 (D) 42+1 26Ans: (C)

IITJEE2009PAPERSECTIONII Multiple Correct ChoiceType

Thissection contains5multiplechoicequestions. Each question has4choices(A),(B),(C) and(D) foritsanswer, out of which ONE OR MOREis/are correct.

5. Thetangent PT and thenormalPN to theparabolay2 =4ax at apoint Pon it meet itsaxis atpointsT and N,respectively.The locusof thecentroid of thetriangle PTN isaparabola whose(A)vertexis23 , 0 (B) directrixisx = 0(C) latusrectumis 23 (D) focusis (a,0)Ans. (A, D)

6.For function f(x) =x cos1,x 1,(A)forat least one x in interval[1,),f(x +2) f(x) 2(D) f(x) isstrictly decreasingin theinterval[1,)

Ans. (B,C, D)

7.For 0 2, thesolution (s) of 6 1cosm ec 0 + (1)4 cosec0 + 4 =24 is(are)(A)4 (B) 6(C) 12 (D) 512Ans: (C,D)

8. An ellipseintersectsthehyperbola2x2 2y2 =1orthogonally. Theeccentricityof theellipseisreciprocal of that of the hyperbola. If the axesof theellipse are along the coordinates axes,then(A)equationof ellipseis x2 +2y2 =2 (B) thefociof ellipseare ( 1,0)(C) equation of ellipse isx2 +2y2 =4 (D) the fociof ellipseare( 2 ,0)Ans. (A, B)

IITJEE2009PAPER9.If In = sin (1+) sin dx,n =0,1,2then(A)In =In+2 (B)

10

1m

I 2m+1 = 10 (C)

10

1m

I 2m = 0 (D) In =In+1Ans. (A,B,C)

SECTION IIIMatrix Match TypeThis section contains 2 questions. Each question contains statements given in two columns,which have to be matched. The statement in Column I are labelled A,B, C and D,while thestatementsin Column II are labelled p,q,r,sand t.Anygiven statement in Column Ican havecorrect matching with ONE OR MORE statement (s) in Column II. The appropriate bubblescorrespondingto the answersto these questions haveto bedarkened:

10.Match the statements/expressionsin Column I with thevaluesgiven in Column II.Column I Column II

(A) Thenumber of solutionsof the (p) 1equationxesinx cosx =0in the interval 0, 2

(B) Value(s) of k forwhich theplaneskx +4y+z = 0, 4x +ky+ 2z = 0 (q) 2and 2x+ 2y+z =0intersect in astraight line

(C) Value(s) of k forwhich |x 1| +|x 2|+|x + 1|+ |x +2| (r)3=4khas integersolution(s)

(D) Ify =y+ 1and y(0)=1 then value(s) of y (ln2) (s) 4(t) 5

Ans. (A) (p)(B) (q,s)(C) (q,r,s,t) (D) (r)

11.Match the statements/expressionsin Column Iwith the values given in Column II.Column I Column II(A)Root(s) of theexpression 2sin2+ sin22=2 (p) 6(B) Pointsof discontinuityof the functionf(x) =6 cos 3 , (q) 4where [y] denotesthe largest integerlessthan or equalto y(C) Volumeof theparallelopiped with itsedgesrepresented by (r)the vectors +,+2 and +,+(D) Anglebetween vectors and where , and are (s) 2unit vectorssatisfying + +3 =0 (t) Ans. (A) (q, s) (B) (p,r,s, t) (C) (t)(D) (r)

IITJEE2009PAPERSECTION IV Integer Answer Type

Thissection contains8questions. Theanswerto each of thequestionsisasingledigit integer,ranging from0 to 9. Theappropriatebubblesbelow the respectivequestion numbers in theORShaveto bedarkened.Forexample,if the correct answersto question numbersX,Y,ZandW(say) are 6,0,9and 2, respectively:

12. Let f:R Rbeacontinuousfunction which satisfiesf(x) = ()0 dt.Then thevalueof f(ln5)isAns. (0)

13. Thecentresof two circlesC1 and C2 each of unit radiusareat adistanceof 6unitsfromeachother.Let P be themidpoint of the linesegment joining thecentresof C1 and C2 and CbeacircletouchingcirclesC1 and C2 externally. Ifacommon tangent to C1 and Cpassingthrough Pisalso acommon tangent to C2 and C,then theradiusof thecircle Cis

Ans. (8)

14. Thesmallest valueof k,forwhich both therootsof theequation x2 8kx +16(k2 k +1) =0are real, distinct and havevalues at least 4,is

Ans. (2)

15. Themaximumvalueof thefunction f(x) =2x3 15 x2 +36x 48 on theset A={x|x2 +20 9x|} is

Ans. (7)

16. Let ABCand ABC betwo noncongruent triangleswith sidesAB =4,AC= AC= 22 andangleB = 300.Theabsolutevalueof thedifferencebetween theareasof these trianglesis

Ans. (4)

17.If the function f(x) =x3 +ex/2 and g(x) =f1(x),then thevalueof g (1)is

Ans. (2)

18. Let p(x) bea polynomialof degree4havingextremum at x =1,2 and lim0x1 + ()2 =2

.Then the value of p(2)is

Ans. (0)

IITJEE2009PAPER19. Let (x, y, z) be points with integer coordinates satisfying the system of homogeneousequations:3x y z = 0 3x+z = 03x+2y+z =0.Then thenumber of such pointsfor which x2 +y2 +z2 100 is

Ans. (7)