Objective Mathematics

Transcription

1 -0 {Mais & Advace} B.E.(CIVIL), MNIT,JAIPUR(Rajastha) Copyright L.K.Sharma 0.

2 Er. L.K.Sharma a egieerig graduate from NIT, Jaipur (Rajastha), {Gold medalist, Uiversity of Rajastha} is a well kow ame amog the egieerig aspirats for the last 5 years. He has bee hoored with "BHAMASHAH AWARD" two times for the academic ecellece i the state of Rajastha. He is popular amog the studet commuity for possessig the ecellet ability to commuicate the mathematical cocepts i aalytical ad graphical way. He has worked with may coachig istitutes of Delhi ad Kota, {presetly associated with Guidace, Kalu Sarai, New Delhi as seior mathematics faculty}. He has bee a seior mathematics {} faculty at Delhi Public School, RK Puram for five years. He is actively ivolved i the field of olie teachig to the egieerig aspirats ad is associated with iprof Learig Solutios Idia (P) Ltd for last years. As a premium member of (a olie teachig ad learig portal), he has delivered may olie lectures o differet topics of mathematics at ad AIEEE level.{some of the free olie public classes at wiziq ca be accessed at }. Sice last years may egieerig aspirats have got tremedous help with the blog mailtolks.blogspot.com ad with lauch of the site mathematicsgya.weebly.com, egieerig aspirats get the golde opportuity to access the best study/practice material i mathematics at school level ad /AIEEE/BITSAT level. The best part of the site is availability of e-book of OBJECTIVE MATHEMATICS for JEE- 0 authored by Er. L.K.Sharma, complete book with detailed solutios is available for free dowload as the PDF files of differet chapters of JEE-mathematics. Copyright L.K.Sharma 0.

3 Cotets. Quadratic Equatios - 8. Sequeces ad Series 9-6. Comple Numbers Biomial Theorem Permutatio ad Combiatio Probability Matrices Determiats Logarithm Fuctios Limits Cotiuity ad Differetiability Differetiatio Taget ad Normal Rolle's Theorem ad Mea Value Theorem Mootoocity Maima ad Miima Idefiite Itegral Defiite Itegral 4-0. Area Bouded by Curves - 0. Differetial Equatios - 7

4 . Basics of D-Geometry 8-4. Straight Lies Pair of Straight Lies Circles Parabola Ellipse Hyperbola Vectors Dimesioal Geometry Trigoometric Ratios ad Idetities Trigoometric Equatios ad Iequatios Solutio of Triagle Iverse Trigoometric Fuctios 9-5

5 Multiple choice questios with ONE correct aswer : ( Questios No. -5 ). If the equatio = ( + ) is havig eactly three distict real solutios, the ehaustive set of values of '' is give by : (a) 5, 5 (c), 5 (b),, 9 7 (d),, 4 4. Let a, b, c be distict real umbers, the roots of ( a)( b) = a + b + c ab bc ac, are :. If (a) real ad equal (c) real ad uequal 6 0 (b) imagiary (d) real is havig three positive real roots, the ' ' must be : (a) 4 (b) 8 (c) 0 (d) 4. If a, b, c are distict real umbers, the umber of real roots of equatio is/are : ( a)( b) ( b)( c) ( c)( a) ( c a)( c b) ( a b)( a c) ( b c)( b a) (a) (b) 4 (c) fiitely may (d) ifiitely may 5. If a + b + c = 0 ad a + b + c = 0 have a commo root ad a, b, c are i : (a) A.P. (c) H.P. a b c,, a b c 6. If all the roots of equatios (b) G.P. are i A.P., the (d) oe of these 4 ( a )( ) ( a )( ) are imagiary, the rage of 'a' is : (a) (, ] (b) (, ) (c) (, ) (d) (, ) 7. Total umber of itegral solutios of iequatio 4 ( 4) ( ) 5 6 ( 5) (7 ) (a) four (c) two 0 is/are : (b) three (d) oly oe 8. If eactly oe root of 5 + (a + ) + a = 0 lies i the iterval (, ), the (a) a > (b) < a < (c) a > 0 (d) oe of these 9. If both roots of 4 0 p + (5 p +5p 66) = 0 are less tha, the 'p' lies i : 4 (a), 5 4 (c), 5 (b) (, ) (d) (, ) 0. If a + a + a 0 R, the 'a' lies i (a) [, ) (b) (, ] (c) [, ) (d) (, ]. If + a + = 0 ad 4 + a + = 0 have a commo root, the value of 'a' is (a) (b) (c) 0 (d). If + p + is a factor of a + b + c, the (a) a + c + ab = 0 (b) a c + ab = 0 (c) a c ab = 0 (d) a + c ab = 0. If epressio a ( b c ) b ( c a ) c ( a b ) is a perfect square of oe degree polyomial of, the a, b, c are i : (a) A.P. (c) H.P. (b) G.P. (d) oe of these [ ] Mathematics for JEE-0

6 4. The value of for which the quadratic equatio (si ) ( + si ) = 0 has roots whose sum of squares is least, is : (a) 4 (c) (b) (d) 6 5. If cos, si, si are i G.P., the roots of (cot ) 0 are : (a) equal (b) real (c) imagiary (d) greater tha a 6. If belogs to : (a) [, ) (b) (, ) (c) R [, ] (d) (, ) holds R, the 'a' 7. The umber of real solutios of the equatio 4 4 is/are : (a) 0 (b) (c) (d) 4 8. Let, be the roots of quadratic equatio a + b + c = 0, the roots of the equatio a b ( ) + c( ) = 0 are : (a), (c), (b), (d),. If real polyomial f () leaves remaider 5 ad ( + ) whe divided by ( ) ad ( ) respectively, the remaider whe f () is divided by ( )( ) is : (a) (b) + 4 (c) + (d) + 6. Let a R ad equatio + a + = 0 is havig oe of the root as square of the aother root, the 'a' is equal to : (a) / (b) (c) (d) / 4. If the quadratic equatio a ( + ) + b ( + ) 5 = 0 is satisfied for all R, the umber of ordered pairs (a, b) which are possible is/are : (a) 0 (b) (c) fiitely may (d) ifiitely may 5. The smallest value of 'k' for which both the roots of the equatio 8k + 6(k k + ) = 0 are real ad distict ad have values at least 4, is : (a) (b) (c) (d) 6. Let f () = ( k)( k ) be egative for all [, ], where k R, the complete set of values of 'k' belog to : (a), (c), (b) 0, (d), If the equatio 0a b c 0 has equal roots, the : (a) 4 5 b 5a (b) c a b (c) c 6a 0 (d) b 5a c 0 0. If a, b ad c are ot all equal ad, are the roots of a + b + c = 0, the value of ( + + ) ( ) is : (a) zero (c) egative. The equatio (b) positive (d) o-egative (log 5 ) (log ) 4 4 has : (a) eactly two real roots (b) o real root (c) oe irratioal root (d) three ratioal roots ad A, the 7. Let A y : 4 y 50, y N total umber of values of ' ' for which the equatio 0 is havig itegral roots, is equal to : (a) 8 (b) (c) 9 (d) 0 8. Let,, R ad l, l, l form a geometric sequece. If the quadratic equatio of 0 has real roots, the absolute value is ot less tha : (a) 4 (b) (c) (d) [ ] Mathematics for JEE-0

7 9. Let a, b, c R ad f ( ) a b c, where the equatio f ( ) 0 has o real root. If y k 0 is taget to the curve y f ( ), where k, the : (a) a b + c > 0 (b) c 0 R (c) 4a b c 0 (d) a b 4c 0 0. Let a, b, c be the sides of a scalee triagle ad R. If the roots of the equatio ( a b c) ( ab bc ac) 0 are real, the : (a) maimum positive itegral value of is (b) miimum positive itegral value of is (c) values of lies i, (d), 4 /. Let a < b ad a, b are the real roots of equatio 0. If b, the the equatio log b a has (a) oe root i (, a) (b) oe root i ( b, ) (c) oe root i (a, b) (d) o root i (a, b). Let p, q Q ad cos 8 + p + q = 0, the : (a) be a root of the equatio si cos p 0 for all R, where [.] represets the greatest iteger fuctio. (b) Value of log q (c) 8q 4 p 0 (d) si cos p 0 for all R, where [.] represets the greatest iteger fuctio. ad a, b S.. Let S : 5 6 0, R If the equatio 7 4 si( a b) is satisfied for at least oe real value of, the (a) miimum possible value of a + b is / (b) maimum possible value of a + b is 7 / (c) miimum possible value of a + b is / (d) maimum possible value of a + b is / 4. If all the four roots of the bi-quadratic equatio are positive i ature, the : (a) value of is 45 (b) value of is 08 (c) value of 0 (d) value of log0.5 5 log 5 5. Let, equatio be the real roots of the quadratic 0, where a, b R. a b ad, A, the If A : 4 0 ; R which of the followig statemets are icorrect : b (a) a b (b) a (c) a 4 (d) a 4b 0 Followig questios are assertio ad reasoig type questios. Each of these questios cotais two statemets, Statemet (Assertio) ad Statemet (Reaso). Each of these questios has four alterative aswers, oly oe of them is the correct aswer. Select the correct aswer from the give optios : (a) Both Statemet ad Statemet are true ad Statemet is the correct eplaatio of Statemet. (b) Both Statemet ad Statemet are true but Statemet is ot the correct eplaatio of Statemet. 6. Let (c) Statemet is true but Statemet is false. (d) Statemet is false but Statemet is true. a, b, c R, a 0, f ( ) a b c, where b 4 ac. If f () = 0 has, as two real ad distict roots ad f ( k) f ( ) 0,, k R, has eactly oe real root betwee ad, the Statemet : 0 a k Statemet : the values of 'k' do't deped upo the values of ' '. [ ] Mathematics for JEE-0

8 7. Statemet : If a, b, c R, the at least oe of the followig equatios... (), (), () has a real solutio + (a b) + (b c) = 0... () + (b c) + (c a) = 0... () + (c a) + (a b) = 0... () Statemet : The ecessary ad sufficiet coditio for at least oe of the three quadratic equatios, with discrimiat,,, to have real roots is Statemet : If the equatio si ( 6 0) cos ( 6 0) 0 is havig real solutio, the value of ' ' must be log 8 Statemet : [, ]. si ( ) cos ( ) 0 for all 9. Statemet : If equatio ( ) 0 is havig itegral roots, the there eists oly oe itegral value of ' ' 40. Let Statemet : = is the oly itegral solutio of the equatio ( ) 0, if I. f ( ) a b c, a, b, c R ad a 0. Statemet : If f ( ) 0 has distict real roots, the the equatio have real roots f '( ) f ( ). f "( ) 0 ca ever Statemet : If f ( ) 0 has o-real roots, the they occur i cojugate pairs. [ 4 ] Mathematics for JEE-0

9 Comprehesio passage () ( Questios No. - ) Let a, b R {0} ad,, be the roots of the equatio a b b 0. If aswer the followig questios.. The value of b + 9a + 0 is equal to : (a) (b) 5 (c) (d) ( ) ( ) ( ). The miimum value of ( ) to : (a) (c) 8, the (b) 9 (d). The miimum value of a b is equal to : b (a) (c) is equal (b) 4 (d) 8 Comprehesio passage () ( Questios No. 4-6 ) Let, be the roots of equatio a b 0, ad, be the roots of equatio a b.if 0 S : a b 0, R ad f : R S R is a fuctio which is defied as the aswer the followig questio. a b f ( ), a b 4. If,,, R ad, the (a) f ( ) is icreasig i (, ) (b) f () is icreasig i (, ) (c) f () is decreasig i (, ) (d) f () is icreasig i (, ) 5. If,,, R ad, the : (a) f '( ) 0 R {, }. (b) f () has local maima i (, ) ad local miima i (, ). (c) f () has local miima i (, ) ad local maima i (, ). (d) f '( ) 0 R {, } 6. If,,, are the o-real values ad f () is defied R, the : (a) f ' () = 0 has real ad distict roots. (b) f ' () = 0 has real ad equal roots. (c) f ' () = 0 has imagiary roots. (d) othig ca be cocluded i geeral for f ' (). Comprehesio passage () ( Questios No. 7-9 ) Cosider the fuctio f () = ( + m) (m + ) + (8m + ), where m R { } 7. If f () > 0 holds true R, the set of values of 'm' is : (a) (0, ) (b) (, ) (c) (, ) (d) (, 0) 8. The set of values of 'm' for which f () = 0 has at least oe egative root is : (a) (, ) (b) (c), 8, 8 (d), 8 9. The umber of real values of 'm' such that f () = 0 has roots which are i the ratio : is /are : (a) 0 (b) (c) 4 (d) 0. Let, be the roots of the quadratic equatio m ( ) m 0, where m 0 & m, m are two values of m for which m m If P, the value of m m is equal to 4. P is equal to... 7 [ 5 ] Mathematics for JEE-0

10 . Let a, b, c, d be distict real umbers, where the roots of 0 c d = 0 are a ad b. If the roots of 0a b = 0 are c ad d, the value of ( ) 605 a b c d is.... If a, b are comple umbers ad oe of the roots of the equatio + a + b = 0 is purely real where as the a other is purely imagiary, the value of is equal to... ( a) b. If the equatio 4 (a + ) + + (a + ) = 0 is havig at least two distict positive real roots, the the miimum itegral value of parameter 'a' is equal to If the equatios a + b + c + 4d = 0 ad a + b + c = 0 have a o-zero commo root, the the miimum value of ( c bd )( b ac ) is equal to If I ad the roots of quadratic equatio are ratioal i ature, the miimum possible value of is equal to Match the followig colums (I) ad (II) Colum (I) Colum (II) (a) If roots of b + c = 0 are two cosecutive (p) itegers, the (b 4c) is (b) If, 4, the least value of the epressio (q) 0 ( 6 + 7) is : (c) Number of solutios of equatio 4 is /are (r) (d) Miimum value of f ( ) is : (s) 7. Match the followig colums (I) ad (II) Colum (I) Colum (II) (a) If ( ) ( ) R, the (p) (0, 4) belogs to the iterval (b) If sum ad product of the quadratic equatio ( 5 5) ( 4) 0 are both less tha oe, the set of possible values of is (c) If 5 ( ) 69 is always positive the set of is (q) (r), 5 5, (d) If roots of equatio ( a 8a ) a 4a 0 (s) (, ) are opposite i sig, the set of values of a is [ 6 ] Mathematics for JEE-0

11 8. Let f ( ) a b c, a 0, a, b, c R. If colum (I) represets the coditios o a, b, c ad colum (II) correspods to the graph of f ( ), where D ( b 4 ac), the match colums (I) ad (II). Colum (I) Colum (II) (a) a, b, c R ad D > 0 (p) (b) a, c R ad b R, D O (q) (c) a, b, c R ad D O (r) (d) a, b R, c R ad D 0 (s) [ 7 ] Mathematics for JEE-0

12 . (d). (c). (b) 4. (d) 5. (b) 6. (c) 7. (a) 8. (b) 9. (d) 0. (a). (b). (c). (c) 4. (c) 5. (b) 6. (b) 7. (b) 8. (b) 9. (c) 0. (b). (c). (c). (c) 4. (c) 5. (b) 6. (b) 7. (d) 8. (b) 9. (d) 0. (d). (a, b, d). (a, b). (a, d) 4. (c, d) 5. (b, c, d) 6. (b) 7. (c) 8. (d) 9. (c) 40. (b). (c). (d). (a) 4. (a) 5. (b) E 6. (a) 7. (d) 8. (b) 9. (a) 0. ( 4 ). ( ). ( ). ( ) 4. ( 0 ) 5. ( 8 ) 6. (a) s 7. (a) q 8. (a) q (b) p (b) r (b) s (c) r (c) s (c) q, r, s (d) s (d) p (d) p [ 8 ] Mathematics for JEE-0

13 . If sum of '' terms of a sequece is give by S Tr ( )( ), the (a) 4 (c) 5 67 r (b) (d) 4 9 r is equal to : T. Let a, b, c be distict o-zero real umbers such that a, b, c are i harmoic progressio ad a, b, c are i arithmetic progressio, the : (a) b + ac = 0 (b) 4b + ac = 0 (c) b ac = 0 (d) 4b ac = 0. Let a, b, c are i A.P. ad a, b, c are i G.P., if a < b < c ad a + b + c = /, the value of 'a' is : (a) (c) r (b) (d) 4. If a, b, c R, the maimum value of bc ac ab is b c a c a b (a) ( a b c ) (b) abc (c) ( a b c ) (d) abc 5. If the sum of first terms of a A.P. is c, the the sum of squares of these terms is : (a) (c) (4 ) c 6 (4 ) c (b) (d) (4 ) c (4 ) c 6 p q r s 6. Let R {} ad ( l), ( l), ( l), ( l ) be i G.P., the pqr, pqs, prs, qrs are i : (a) A.P. (c) H.P. (b) G.P. (d) A.G.P. 7. Let T, Tr Tr Tr r N ad S... T T T T, the (a) S00 4 (b) S00 (c) S00 8. Let S (d) 0 S r, the (r ) is give by : r r (a) S 8S (b) S4 4 S (c) S 6 S (d) S4 6 S 9. Let { } represets G.P. with commo ratio 'r' such that k k k k 0, the umber of possible values for 'r' is/are : (a) (b) (c) (d) 4 0. Let, y be o-zero real umbers ad the epressio + y 48 4 y 4 is ot less tha 'k', the value of 'k' is equal to : (a) (b) (c) 8 (d) 8. Let 0 A.M.'s ad 0 H.M.'s be iserted i betwee ad. If 'A' be ay A.M. ad 'H' be the correspodig H.M., the H(5 A) is equal to : (a) 6 (b) 0 (c) (d) 8 [ 9 ] Mathematics for JEE-0

14 Sequeces ad Series. Let a, b, c ad the iequality R b ( ( a c) 4 b ) ( a c) 0 holds true for a b c all real value of '', the e, e, e are i : (a) A.P. (b) G.P. (c) H.P. (d) oe of these.. Let 'A ' deotes the sum of terms of a A.P. ad A A A, the A is equal to : (a) 4 (b) 6 (c) 8 (d) 0 0. I a sequece of (4 + ) terms, the first ( + ) terms are i A.P. whose commo differece is, ad the last ( + ) terms are i G.P. whose commo ratio is /. If the middle terms of the A.P. ad G.P. are equal, the the middle term of sequece is : (a). ( ). (b)., (c). (d) ( ).. Let a, a, a,..., a 50 be 50 distict umbers i. 4. If a 0, roots of equatio are i G. P., the : (a) ac db (c) a b c d a b c d (b) a c d b (d) ab 0 cd 5. Let a, b, c be o-zero real umbers ad 4a + 9b + 6c = (ab + 6bc + 4ac), the a, b, c are i : (a) A.P. (c) H.P. (b) G.P. (d) A.G.P. 6. I a set of four umbers, if first three terms are i G.P. ad the last three terms are i A.P. with commo differece 6, the sum of the four umbers, whe the first ad the last terms are equal, is give by : (a) 0 (b) 4 (c) 6 (d) 8 7. Let the real umbers,, be i A.P. ad satisfy the equatio (a) (c) p, p, ( ) p q 0, the : (b) q, 7 (d) q, 7 / 50 r A.P., ad ar a a50 r 5 ( ) ( ), 7 where N, the value of is equal to : (a) 4 (b) (c) 8 (d) 0. Let three umbers be removed from the geometric sequece {a } ad the geometric mea of the remaiig terms is 5 7. If a..., 4 8 the value of '' ca be : (a) 0 (b) 8 (c) 0 (d). Let, y R ad y 6, the the least value of + 4y is equal to : (a) (b) 0 (c) 8 (d) 0 4. Let S... terms ad S lim( S ), 4 8 if S S, the least value of '' is : I ABC, if all the sides are i A.P., the the correspodig e-radii are i : (a) A.P. (b) G.P. (c) H.P. (d) oe of these. 8r 9. Let S, the lim ( S) is equal to : 4 4r r (a) 4 (b) (c) (d) 0 (a) (b) 0 (c) (d) 6 5. Let the sides of a triagle be i arithmetic progressio. If the greatest agle of triagle is double the smallest agle, the the cosie value of the smallest agle is equal to : (a) 8 (c) 4 5 (b) 4 (d) 4 [ 0 ] Mathematics for JEE-0

15 6. If a, b R, where a, A, A, b are i arithmetic progressio, a, G, G, b are i geometric progressio ad a, H, H, b are i harmoic progressio, the which of the followig relatios are correct? (a) (b) (c) (d) G G G G H H H H G G A A A A H H GG ( a b)( b a) H H 9ab A A ( a b)( b a) H H 9ab 7. Let four cosecutive itegers form a icreasig arithmetic progressio ad oe of these umbers is equal to the sum of the squares of the other three umbers, the : (a) the smallest umber is 0. (b) the largest umber is. (c) sum of all the four umbers is. (d) product of all the four umbers is For two distict positive umbers, let A, G, H deote the AM, GM ad HM respectively. For, N, if A ad H has arithmetic, geometric ad harmoic meas as A, G, H respectively, the : (a) A A A A4... (b) G G G G4... (c) H > H > H > H 4 >... (d) G = G = G = G 4 = Let {a } represets the arithmetic sequece for which 0. Let a =, a = ad a = +, the : (a) a a (b) a = 0 (c) a 5 (d) a a a 4... ( ) ad b + a =. If b > a for all > 0, where N, the, possible values of atural umber ' 0 ' ca be : (a) 4 (b) 6 (c) 8 (d) Followig questios are assertio ad reasoig type questios. Each of these questios cotais two statemets, Statemet (Assertio) ad Statemet (Reaso). Each of these questios has four alterative aswers, oly oe of them is the correct aswer. Select the correct aswer from the give optios : (a) Both Statemet ad Statemet are true ad Statemet is the correct eplaatio of Statemet. (b) Both Statemet ad Statemet are true but Statemet is ot the correct eplaatio of Statemet. (c) Statemet is true but Statemet is false. (d) Statemet is false but Statemet is true.. Statemet : Let three positive umbers i geometric progressio represet the sides of a triagle, the the commo ratio of the G.P. ca be si 5 Statemet : the commo ratio of the G.P. i cosideratio lies i betwee si. 0 si 0 ad. Statemet : I a triagle ABC, if cot A, cot B, cot C forms a A.P., the,, also form a b a c b a c A.P. Statemet :,, a b c form a H.P.. Statemet : If [.] ad {.} deote the greatest iteger fuctio ad the fractioal part, the, [], {} ca ever form a geometric progressio for ay positive ratioal value of Statemet :, [ ], { } ca form a G.P. for R 7, oly if si. 0 [ ] Mathematics for JEE-0

16 Sequeces ad Series 4. Statemet : If a, b, c R, the the miimum value of a( b c ) b( c a ) c( a b ) is equal to 6abc Statemet : for a, a, a, a4,... a R, ( AM )( HM ) ( GM ) N {} 5. Statemet : Let S N, the S l( )..., 4 5 Statemet : l ( + ) > l () N [ ] Mathematics for JEE-0

17 Comprehesio passage () ( Questios No. - ) Let V r deote the sum of the first r terms of a arithmetic progressio (A.P.) whose first term is r ad the commo differece is (r ). Let Tr Vr Vr ad Qr Tr Tr for r =,,.... The sum V + V V is : 5. Let Q { a, b, c}, where a < b < c, the the roots of the quadratic equatio a + b + c = 0 are : (a) real (c) real ad equal (b) real ad uequal (d) o-real 6. Sum of all the elemets of set P Q is equal to : (a) 56 (b) (c) 9 (d) 5 (a) ( )( ) (b) ( )( ) (c) ( ) (d) ( ). T r is always : (a) a odd umber (c) a prime umber (b) a eve umber (d) a composite umber. Which oe of the followig is a correct statemet? (a) Q, Q, Q,... are i A.P. with commo differece 5 (b) Q, Q, Q,... are i A.P. with commo differece 6 (c) Q, Q, Q,... are i A.P. with commo differece (d) Q = Q = Q =... Comprehesio passage () ( Questios No. 4-6 ) Let P ad Q be two sets each of which cosistig of three umbers i A.P. ad G.P. respectively. Sum of the elemets of set P is ad product of the elemets of set Q is 8, where the commo differece ad the commo ratio of A.P. ad G.P. are represeted by 'd' ad 'r' respectively. If sum of the squares of the terms of A.P. is 8 times the sum of the terms of G.P., where d = r, ad d, r questios. I, the aswer the followig 4. Total umber of terms i the set of P Q is/are : (a) 0 (b) (c) (d) 7. Let ad y be two real umbers such that the k th mea betwee ad y is equal to the k th mea betwee ad y whe arithmetic meas are placed betwee them i both the situatios. The value of y epressio is equal to... k 8. Let S ad r r ( ) S '..., ( ) ( )( ) 6 S ' the value of is equal to... S 9. Let a A.P. ad a G.P. each has as the first term ad as the secod term, where 0. If sum of ifiite terms of G.P. is 4 ad the sum of first terms of ( ) A.P. ca be writte as, the value of k 'k' is equal to Let sum of the squares of three distict real umber i geometric progressio be S ad their sum is p ( S ). If p R, the total umber of possible itegral values of 'p' is/are.... Let a, b, c, d, e R ad s = a + b + c + d + e, if ( s a)( s b)( s c)( s d)( s e) miimum value of abcde is 4, the value of is... [ ] Mathematics for JEE-0

18 Sequeces ad Series. Match the followig colums (I) ad (II) Colum (I) Colum (II) (a) Let 009, r r ( r ) the sum of all (p) the digits of the umber ' ' is (b) The largest positive term of the harmoic progressio (q) 4 (c) whose first two terms are 5 ad, If I / 4 0 is equal to ta d, where N, ad (r),,... form a A.P., the (s) I I I I I I commo differece of this A.P. is (d) Value of log 9 7. Match the followig colums (I) ad (II). Colum (I) is equal to (t) 6 Colum (II) (a) If p is prime umber ad N, where (p) i arithmetic progressio p log p, the first three smallest possible values of are (b) If a, a, a, a 4, a 5 are five o-zero distict umbers (q) i geometric progressio such that a, a, a are i A.P., a, a, a 4 are i G.P. ad a, a 4, a 5 are i H.P., the a, a, a 5 are (r) i harmoic progressio (c) ta 70º, ta 50º + ta 0º ad ta 0º are (d) If a, b are positive distict real umber ad,, are (s) ot is arithmetic progressio three roots of a b b a such that b a a b ad c, the a, b, c are (t) ot i geometric progressio [ 4 ] Mathematics for JEE-0

19 4. Match the followig colums (I) ad (II). Colum (I) (a) If sum of first positive itegers is 5 times the sum of (p) Colum (II) their squares, the is (b) If, 0, are i G..P., the the value (q) 7 of is 7 (c) If log, log ( 5) ad log are i A.P., the (r) 4 value of is (d) Let S, S, S,... be squares such that for each, (s) 6 legth of side of S equals the legth of diagoal of S. If legth of S is.5 cm, the for which values of is the area of S less tha sq. cm. (t) 5. Match the followig colums (I) ad (II). Colum (I) (a) If altitudes of a triagle are i A.P., the sides of triagle are i (b) Colum (II) (p) A.P. a b a b If b c b c 0 ad, the a, b, c are i (q) G.P. 0 If a a a a a a a, the a4 a a4 a a4 (c) a, a, a, a 4 are i (r) H.P. (d) If (y ), (y a) ad (y z) are i H.P., the ( a), (y a), (z a) are i (s) A.G.P. [ 5 ] Mathematics for JEE-0

20 Sequeces ad Series. (a). (a). (d) 4. (a) 5. (c) E 6. (c) 7. (c) 8. (c) 9. (c) 0. (a). (a). (b). (b) 4. (a) 5. (c) 6. (b) 7. (b) 8. (c) 9. (b) 0. (c). (c). (d). (b) 4. (a) 5. (b) 6. (b, c) 7. (b, c,d) 8. (a, d) 9. (a, c) 0. (b, c). (a). (d). (a) 4. (c) 5. (b). (b). (d). (b) 4. (b) 5. (d) E 6. (b) 7. ( ) 8. ( ) 9. ( 8 ) 0. ( 9 ). ( 5 ). (a) r. (a) s, t 4. (a) q 5. (a) r (b) t (b) q, s (b) r (a) q (c) p (c) p, t (c) p (a) r (d) q (d) r, s, t (d) p, q, r, s (a) q [ 6 ] Mathematics for JEE-0

21 . If A(z ), B(z ) ad C(z ) are the vertices of a equilateral triagle i the clockwise directio, the z z z arg is : z z (a) 4 (c) 6 (b) (d). Let comple umbers z ad z satisfy the coditios z z z + 6i = ad z 4i = i respectively, the miimum value of z z is : (a) 8 (b) 6 (c) 4 (d). For o-zero comple umber 'z', if z i z, the arg ( ) (a) 4 (c) 5 4 (b) 4 (d) 7 4 i z is equal to : 4. If ad are comple umbers, the maimum value of is : (a) (b) (c) (d) 4 5. If,, are the roots of cubic equatio = 0, ' ' is o-real cube root of uity, the is : (a) 8 (b) (c) (d) 6. f (z) is o-real fuctio of comple umber 'z' ad whe f (z) is divided by (z i) ad (z + i) the remaiders are i ad + i respectively, the the remaider whe f (z) is divided by ( z ) is equal to : (a) i z (b) iz i (c) iz i i (d) iz 7. If k k, k N, ad comple 8. If umber 'z' satisfy z z... z, the : (a) z 4 (b) z 4 (c) z 4 (d) z z z z z ad z, the z is equal to : (a) (b) (c) (d) 4 9. A particle P starts from the poit z0 i, where i. It moves first horizotally away from origi by 5 uits ad the vertically away from origi by uits to reach a poit z. From z the particle moves uits i the directio of the vector i j ad the it moves through a agle 90º i aticlockwise directio o a circle with cetre at origi to reach a poit z. The poit z is give by : (a) 6 + 7i (c) 7 + 6i (b) 7 + 6i (d) 6 + 7i 0. Cosider a square OABC, where O is origi ad A(z 0 ), B(z ), C(z ) are i aticlockwise sese, the equatio of circle iscribed i the square is : (a) z z0( i) z0 (b) z ( i ) z 0 z0 (c) z ( i) z0 z0 (d) z ( i) z0 z0 [ 7 ] Mathematics for JEE-0

22 Comple Numbers. If A(z ), B(z ) ad C(z ) are the vertices of a triagle ABC iscribed i the circle z = ad iteral agle bisector of (a) z (c) z 4 zz 4 z z z A meet the circumferece at D(z 4 ), the (b) z 4 (d) z 4 z z z z z z z i. Cetre of the arc represeted by arg z i 4 4 is give by : (a) (5 5 i) (b) (5 5) i (c) (9i 5) (d) (9i 5). If a, b, c are itegers ot all equal ad is cube root of uity ( ), epressio a b c is : (a) 0 (b) (c) the miimum value of the (d) 4. Let z = 0 + 6i ad z = 4 + 6i. If z is ay comple z z umber such that arg z z, the 4 (a) z 7 9 i (b) z 7 9 i (c) z 7 9 i (d) z 7 9 i 5. If A(z ), B(z ) ad C(z ) form a isosceles right agled triagle ad A, the (a) ( z z ) ( z z )( z z ) (b) ( z z ) ( z z )( z z ) (c) ( z z ) ( z z )( z z ) (d) ( z z ) ( z z )( z z ) 7. Let z = 0 ad z i =, the miimum value of z z is : (a) (b) 6 (c) 4 (d) oe of these 8. Area of regio o the comple plae which is bouded by the curve z + i + z i = 8 is : (a) 8 (c) 6 (b) 4 (d) oe of these 9. If z ad w are two o-zero comple umbers such z that zw = ad arg, the zw is equal to: w (a) (b) (c) i (d) i i i i 0. Let e, y e ad z e ad y z 0, the which oe of the followig is ot correct : (a) 0 y z (b) y yz z 0 (c) y z (d) y z yz. Let z iy be a comple umber where ad y are itegers, the the area of the rectagle whose vertices are roots of the equatio ( z ) z z( z ) 50 is : (a) 48 (b) (c) 40 (d) 80. Let z cos i si, the the value of summatio 5 r Im z o at r (a) si (c) o si o is equal to : (b) (d) si 4si o o 6. If comple umber 'z' satisfy z + i = 5, the comple umber havig magitude-wise miimum argumet is : (a) ( 5 i ) (b) (5 i) (c) ( 5 ) i i (d) ( 5 ) i i. Let A( z ), B( z) ad C( z ) form triagle ABC o the argad plae such that ABC is : (a) equilateral (c) isosceles z z i, the z z (b) right agled (d) scalee [ 8 ] Mathematics for JEE-0

23 4. If movig comple umber 'z' satisfy the coditios, 5 z i ad arg( z i ), the area of regio which is represeted by 'z' is : (a) (b) 9. Let A( z), B( z ) ad C(z ) be the vertices of ABC o the comple plae, where the triagle ABC is iscribed i circle z =. If altitude through A meets the circle z = at D ad image of D about BC is E, the (a) comple poit 'E' is z + z + z. (c) (d) z (b) comple poit 'D' is z z 5. A ma walks a distace of uits from the origi towards the orth-east (N 45º E) directio. From there, he walks a distace of 4 uits towards the orthwest (N 45º W) directio to reach a poit P, the the positio of P i the argad plae is : i / 4 (a) e 4i (c) (4 i) e i / 4 (b) ( 4 i) e i (d) ( 4 i) e i 6. Let z r, where r {,,,..., }, be the '' distict roots of the equatio eists some the '' ca be : z r for which (a) 4 (b) 8 r (c) (d) 6 / 4 / 4 r Cr. If there zr ( i ) arg, ( i ) 4 7. Let + i ad + i be the two vertices of a equilateral triagle o the comple plae, the the third verte of triagle ca be give by : (a) ( )i (b) ( )i, (c) comple poit 'E' is ( z z z). z (d) comple poit 'D' is z z 0. Let P, Q, R be three sets of comple umbers as defied below: P z : Re( z( i)) Q z : z i R { z : Im( z) } I the cotet of give sets, which of the followig statemets are correct? (a) umber of elemets i the set P Q R are ifiite. (b) If 'z' be ay poit i P Q R, the z 5 i z i 6 (c) umber of elemets i the set P Q R is oe. (d) umber of elemets i the set P Q are two. (c) ( )i (d) ( )i 8. Let,, be the comple umbers, ad z z 0, where z C. If the quadratic equatio i 'z' is havig (a) both roots real, the. (b) both roots purely imagiary, the. (c) both roots real, the. (d) both roots purely imagiary, the. Followig questios are assertio ad reasoig type questios. Each of these questios cotais two statemets, Statemet (Assertio) ad Statemet (Reaso). Each of these questios has four alterative aswers, oly oe of them is the correct aswer. Select the correct aswer from the give optios : (a) Both Statemet ad Statemet are true ad Statemet is the correct eplaatio of Statemet. (b) Both Statemet ad Statemet are true but Statemet is ot the correct eplaatio of Statemet. (c) Statemet is true but Statemet is false. (d) Statemet is false but Statemet is true. [ 9 ] Mathematics for JEE-0

24 Comple Numbers. Statemet : Let 'z' be the movig comple poit o argad plae for which z i z si arg ( z) 4 the the locus of 'z' is part of a ellipse Statemet : Ellipse is the locus of a poit for which sum of its distaces from two distict fied poits is always costat, where the costat sum is more tha the distace betwee the fied poits.. Statemet : If i l i cos si( ( ) ) i 0, the value of is equal to, Statemet : The umber of poits of itersectio of C ad C is oly oe Statemet : Two o-parallel lies always itersect at oly oe poit i -dimesioal plae. 4. Let z = 5 + 8i ad z satisfy z i, the Statemet : miimum value of iz z is equal to 8 Statemet : maimum value of z is Statemet : cos (cos ),. Let the equatios arg( z 4 i) ad 5 arg( z i) be represeted by the curves C 6 ad C respectively o the comple plae, the 5. Statemet : Let m, N ad the equatios m z 0 ad z 0 is havig oly oe commo root, the m ad must be differet prime umbers Statemet : the commo root for the equatios m z 0 ad z 0 is if m ad are differet prime umbers. [ 0 ] Mathematics for JEE-0

25 Comprehesio passage () ( Questios No. 4-6 ) Comprehesio passage () ( Questios No. - ) Let,,,,... be the th roots of uity, k k the k cos i si, where k = 0,,,, 4...,, further 0 ca be epressed as ( ) ( ) ( k ). Now aswer the followig k questios based o above iformatio z z If comple umber ' z ' satisfy z i ad comple umber ' z ' satisfy z + 4 i =, the aswer the followig questios. 4. Miimum value of z z is : (a) (b) (c) (d) 5 5. If magitude of arg(z ) is miimum the z is : (a) 5 (b) 4 (c) 4 (d) 8 7. Value of k k cos i si k is equal to : (a) 0 (b) i 6 (c) cos. e 6. Value of is equal to : (4 ) k 4 ( 4) (a) (4 ) (b) 4 4 ( 4) (4 ) ( )4 (c) 4 4 ( 4) (d) (4 ) k i 8 (d) cos. e 8. If,,,... forms a polygo o the comple plae, the area of the circle iscribed i the polygo is give by : (a) si (b) cos (c) cos (d) cos 6. Maimum possible value of z is : (a) 5 (b) ( 5) (c) ( 5 ) (d) ( 5 ) Comprehesio passage () ( Questios No. 7-9 ) Let P( z), Q( z ) ad R(z ) represet the vertices of a isosceles triagle PQR o the argad plae, where RQ = PR ad QPR. If icetre of PQR is give by I(z 4 ), the aswer the followig questios. PR PQ 7. The value of PQ PI is equal to : ( z z )( z z) (a) ( z z ) 4 ( z z)( z z) (b) ( z z ) 4 ( z z)( z z ) (c) ( z z ) 4 ( z z)( z z ) (d) ( z z ) 4 8. The value of ( z z) ta.ta is equal to : (a) z z z z z z4 ( )( ) (b) z z z z z z4 ( )( ) (c) z z z z z z4 ( )( ) (d) z z z z z z4 ( )( ) [ ] Mathematics for JEE-0

26 Comple Numbers cos 9. The value of ( z z4 ). is equal to : cos (a) z z z z ( )( ) ( z z)( z z ) (b) ( z z ) 4 ( z z )( z z ) (c) ( z z ) 4 (d) ( z z ) ( z z ). Let movig comple poit A (z 0 ) satisfy the coditio zo i zo 6 i 0, ad comple poits B, C are represeted by + 6i ad i respectively. If the area of triagle ABC is maimum, the three times the i-radius of triagle ABC is.... Let z be ui-modular comple umber, the value of arg z / ( z. z ) / arg( z ) equal to..., where arg( z) 0, 8, is. Let A( z), B( z), C( z ) form a triagle ABC, where 0. Let movig comple umber ' z o ' lies o the curve C o argad plae, where 7 zo i ta 8 arg 5 zo ta i 8 ta ( ). If the curve C o argad plae is represeted by z =, the area of the regio bouded by the curves C ad C is equal to Match the followig Colums (I) ad (II). Colum (I) ABC ACB ( ). ( z z) If cosec k, the value of 'k' ( z z )( z z ) is equal to Let z, z, z be three distict comple umbers, where z z 4, z z ad z z 4 z 4. If 8z z 7z z 64 z z is equal to 'k', the value of k is equal to... 6 Colum (II) (a) Let R ad 'z' be ay comple umber such that (p) z cos z, the miimum value of z is : (b) Let z = + iy, where, y I. Area of the (q) 7 octago whose vertices are the roots of the equatio ( z z) z z 00 is : (c) Let z be comple umber such that (r) 4 ( z z )(4 i) ( i)( z z ) 6i 0, the value of z z is : (s) 6 (d) Let z z z, the miimum value of z z z z z z is : (t) 7 [ ] Mathematics for JEE-0

27 6. Match the followig colums (I) ad (II). Colum (I) Colum (II) 4 (a) The roots of the equatio z z z 0 o the (p) a ellipse comple plae are represeted by the vertices of : (b) If variable comple umber 'z' satisfy the coditio z z z z 4, the locus of z is give by : 4 (c) The roots of the equatio z z z z 0 o the comple plae are represeted by the vertices of : 6 4 (d) The roots of the equatio z z z 0 o the comple plae are represeted by the vertices of : (q) a square (r) a trapezium (s) a heago (t) a equilateral triagle [ ] Mathematics for JEE-0

28 Comple Numbers. (d). (b). (b) 4. (b) 5. (d) E 6. (b) 7. (a) 8. (a) 9. (d) 0. (c). (a). (d). (b) 4. (d) 5. (c) 6. (c) 7. (c) 8. (b) 9. (d) 0. (c). (a). (d). (c) 4. (b) 5. (d) 6. (b,d) 7. (c, d) 8. (b, c) 9. (a, b) 0. (b, c, d). (d). (b). (b) 4. (b) 5. (d). (c). (b). (b) 4. (c) 5. (c) E 6. (b) 7. (a) 8. (c) 9. (a) 0. ( ). ( 4 ). ( ). ( 4 ) 4. ( 6 ) 5. (a) p 6. (a) t (b) s (b) q (c) t (c) r (d) q (d) q [ 4 ] Mathematics for JEE-0

29 7. Let r r r0 ( ) C, the value of. Maimum value of the term idepedet of i the 0 cos epasio of si, (a) 0! (5!) 0! (c) 04(5!) (b) 0! (5!) (d) 0 C8 where R, is :. Sum of the series, 0 C C + 0 C C 0 is equal to : (a) C 0 (b) C (c). C0 (d) C 9. Coefficiet of 5 i the epasio of the product ( + ) 6 ( ) 7 is : (a) 7 (b) 7 (c) 70 (d) If the biomial coefficiets of three cosecutive terms i the epasio of ( + ) are i the ratio : 7 : 4, the value of '' is : (a) (b) 65 (c) 55 (d) Coefficiet of 5 i ( ) ( )... ( ) 0 is : (a) C 6 C 5 (b) C 6 C 6 r r l0 ( ). Cr. is equal to : ( 0 ) r l r0 (a) (b) (c) 0 (d) 8. Coefficiet of 4 i epasio of ( ) is : (a) 605 (b) 80 (c) 990 (d) If Cr, the r 0. r is equal to : r (a) + (b) + + (c) 4( ) (d) 4( + ) 0 r r r 0 7 ( ). Cr... r r r r0 is equal to : (a) (c) i j (b) (d) The value of j. Ci is equal to : (a) ( ) (b) ( ). (c) ( ). (d). (c) C 5 0 C 4 (d) C C 5 6. Let 6 C r a, the sum of the series, r 0 6 a 7a a 5 a a, is equal to :. Let N ad of r ( ). ar is equal to : r0 ( ) a ; the value r0 r r (a) 5 a 8 (b) 70 a 8 (c) 5 a 8 (d) 70 a 8 (a) a (b) a (c) a (d) a [ 5 ] Mathematics for JEE-0

30 Biomial Theorem. Let {, } ad the digits at the uit's place ad te's place of are 9 ad 0 respectively, the ( ) must be divisible by : I (a) 6 (b) 6 (c) 0 (d) 8 4. Let T r deotes the r th term i the epasio of ( + ) ad T is the oly term which is umerically greatest eactly for three atural values of '', the '' ca be: (a) 5 (b) 0 (c) 7 (d) 8 5. Let + = 40, where, N ad the value of Cr. Cr is maimum, the value of '' must r0 be : (a) 5 (b) 5 (c) 0 (d) 6. Value of C (si ) is equal to : 0 (a).cos.si (c).cos.si (b).si.cos (d).si.cos 7. If [.] represets the greatest iteger fuctio ad ( ), the value of to : (a) 0 (b) (c) (d) 8. For atural umber m, if [ ] is equal ( ) m y ( y) a y a y..., ad a = a = 0 the (m, ) ordered pair is : (a) (5, 45) (b) (0, 45) (c) (5, 0) (d) (45, 5) 9. The coefficiet of 8 i, is equal to : (a) 50 (b) 45 (c) 50 (d) 45 r0 ( r ) r 5, where 0. Let T be the term which is idepedet of ' ' i the biomial epasio of / / / the T is equal to : (a) 00 (b) 0 (c) 40 (d) 500 (0). Let a, N, ad the value of a is! greatest, the : (a) 998 (b) 999 (c) = 000 (d) = 00. Let I, (5 ), where is a iteger ad (0, ), the : (a) ( ) is divisible by (b). Let 4. If (4) (c) is divisible by 0 (d) is a odd iteger where + r r A Cr.cos & B Cr.cos, r 0 r0! Cr, the which of the followig r!.( r)! statemets are correct : (a) A B (b) A B 7 (c) B8 (d) A6 7 m m ( ) S, where (, ), the the m correct statemets are : (a) coefficiet of (b) i S is ( S) lim ( ) ( ) p p... (c) coefficiet of i S is r (d) value of r Cr r 0 is 0, [ 6 ] Mathematics for JEE-0

31 5. Let T r deotes the r th term i the biomial epasio of ( + ), where T ad T are equal for at least oe itegral value of, the value of '' ca be : (a) (b) 7 (c) (d) 8 7. Statemet : If the biomial epasio of ( 7) cotais oly two ratioal terms, the value of '' ca be 0 Statemet : The applicable atural values of '' are 6, 8, 0, which are all eve i ature. 8. Statemet : The coefficiet of term cotaiig º i Followig questios are assertio ad reasoig type questios. Each of these questios cotais two statemets, Statemet (Assertio) ad Statemet (Reaso). Each of these questios has four alterative aswers, oly oe of them is the correct aswer. Select the correct aswer from the give optios : (a) Both Statemet ad Statemet are true ad Statemet is the correct eplaatio of Statemet. (b) Both Statemet ad Statemet are true but Statemet is ot the correct eplaatio of Statemet. (c) Statemet is true but Statemet is false. (d) Statemet is false but Statemet is true. 6. Statemet : Total umber of distict terms i the 4 epasio of ( y ) ( y) is 8, Statemet : Total umber of commo terms i the epasio of ( + y ) ad ( + y) 4 are. the epasio of Statemet : umber. is 46 C C is maimum, if is eve atural 9. Let a, b, c deote the sides of a triagle ABC opposite to the vertices A, B ad C respectively, the Statemet : r is r 0 r r Value of C ( a).( b).cos ( r) A r B equal to zero Statemet : I ay triagle ABC, (a cos B + b cos A) = c for all R. 0. Statemet : If 50 C 5 is divisible by (8), where N, the maimum value of ca be Statemet : C (r ) for all N.! r [ 7 ] Mathematics for JEE-0

32 Biomial Theorem 5. Value of S S is equal to : Comprehesio passage () ( Questios No. - ) Let f () = ( + + ) = a 0 + a + a a, ad g() = b 0 + b + b + b b, where bk k, N. Aswer the followig questios based o the give iformatio. (a) (, ) P (b) P(, ) (c) (, ) P (d) (, ) 6 P. If f () = g ( + ), the value of a is equal to : (a) C (b) (c) C (d) C. I f (), if is eve positive iteger, the value of ( a a a a a...) ( a a a a...) is equal to : (a) (b) (c) 0 (d) 4. I f (), if is positive itegral multiple of, the r ( ). ar. C is equal to : r r 0 (a) / (c) / C C (b) C / C (d) C / Comprehesio passage () ( Questios No. 4-6 ) Let m, N ad S ( r), if m r m m m m P( m, ) m!..., m m m m m 6. Value of S S is equal to : (a) P(, ) P(, ) (b) P(, ) + P(, ) (c) P(, ) P(, ) (d) P(, ) + P(, ) 7. If ( + ) = C 0 + C + C +... C, where N, ad r r to... C r Cr 540, the value of is equal 8. Let the biomial coefficiets of the rd, 4 th, 5 th ad 6 th terms i the epasio of ( + ) 00 be a, b, c ad d respectively. If, are relatively prime umbers b ac a ad, the value of ( ) is equal c bd c to... p where q questios. p C q, the aswer the followig 9. Let N ad C 00 C, the umber of possible values of '' is equal to... S 6 4. Value of lim 7 is equal to : (a) 0 (b) /7 (c) /6 (d) / If 0 C C C... C0 4 is equal to (0 ), the value of ' ' is equal to... [ 8 ] Mathematics for JEE-0

33 . Match the followig colums (I) ad (II) Colum (I) Colum (II) (a) (b) C0 C C... upto terms. (p) C C C C0 C C... upto 0 terms. (q) (c) C0 C C... upto terms. (r) 6 (9!) 5 (9!) (d) 0 r 0 r ( ) C (4r ) r. Match the followig colums (I) ad (II). Colum (I) (a) If the sith term i the biomial epasio of (p) 5 log (/5) log5 ( ) the values of '' ca be 7 8 (b) The secod last digit of umber 7 is equal to (s) (t) is 84, (q) ( ) d Colum (II) (r) 4 (c) The coefficiet of 0 i the epasio of ( + ) 8 (s) is ot divisible by (d) The positive iteger which is greater tha ca be 5 0 ( ) (t) [ 9 ] Mathematics for JEE-0

34 Biomial Theorem. (b). (d). (b) 4. (c) 5. (b) E 6. (c) 7. (c) 8. (c) 9. (c) 0. (b). (a). (c). (c) 4. (c) 5. (c) 6. (a) 7. (b) 8. (a) 9. (d) 0. (b). (b, c). (a, c). (b, d) 4. (c, d) 5. (a, b) 6. (a) 7. (c) 8. (b) 9. (d) 0. (b). (c). (a). (b) 4. (b) 5. (b) E 6. (c) 7. ( 8 ) 8. ( ) 9. ( 7 ) 0. ( ). (a) s. (a) q, t (b) p (b) r (c) r (c) p, s (d) t (d) p, r, s [ 0 ] Mathematics for JEE-0

35 6. If the L.C.M. of ' ' ad ' 4 ' is p q r, where. The letters of the word 'GHAJINI' are permuted ad all the permutatios are arraged i a alphabetical order as i a Eglish dictioary, the total umber of words that appear after the word 'GHAJINI' is give by : (a) 09 (b) 009 (c) 09 (d) 09. If Joh is allowed to select at most ( + ) chocolates from a collectio of ( + ) distict chocolates, the total umber of ways by which Joh ca select at least two chocolates are give by : (a) (4) 4. C (b) (4) 4. C (c) (4) C (d) (4) C. The coefficiet of ( ).( ) is (a) C50 C i the epasio of p, q, r are prime umbers ad, I, the the umber of ordered pairs (, ) are : (a) 5 (b) 40 (c) 5 (d) 9 7. Total umber of o-egative itegral solutios of 8 0, is give by : (a) 45 (b) 685 (c) 50 (d) If Mr. ad Mrs. Rustamji arrage a dier party of 0 guests ad they are havig fied seats opposite oe aother o the circular diig table, the total umber of arragemets o the table, if Mr. ad Mrs. Batliwala amog the guests do't wish to sit together, are give by : (a) 48 (8!) (b) 888 (8!) (c) 74 (8!) (d) 64 (8!) 9. If 0 idetical balls are to be placed i idetical boes, the the total umber of ways by which this placemet is possible, if o bo remais empty, is give by : (a) 0 (b) (c) 9 (d) 5 (b) (c) (d) C500 C C498 C C50 C X ad Y are ay five digits umbers, total umber of ways of formig X ad Y with repetitio, so that these umbers ca be added without usig the carryig operatio at ay stage, is equal to : (a) 45(55) 4 (b) 6(55) 4 (c) (55) 5 (d) 5(55) 4 5. A team of four studets is to be selected from a total of studets, total umber of ways i which team ca be selected if two particular studets refuse to be together ad other two particular studets wish to be together oly, is equal to : (a) 6 (b) 8 (c) 0 (d) Total umber of ways by which the word 'HAPPYNEWYEAR' ca by arraged so that all vowels appear together ad all cosoats appear together, is give by : (a) (7!) (b) 6(8!) (c) 8 (7!) (d) (8!). The umber of seve digit itegers, with sum of the digits equal to 0 ad formed by usig the digits, ad oly, is : (a) 55 (b) 66 (c) 77 (d) 88. Let r be a variable vector ad a i j k such that scalar values r. i, r. j ad r. k are positive itegers. If r. a is ot greater tha 0, the total umbers of possible r are give by : (a) 80 (b) 0 (c) 40 (d) 00 [ ] Mathematics for JEE-0

36 Permutatio ad Combiatio. Let three lies L, L, L be give by + y =, 4 6y 5 ad 6 9y 0 respectively. If lie L r cotais r differet poits o it, where r {,, }, the maimum umber of triagles which ca be formed with vertices at the give poits o the lies, are give by : (a) 0 (b) 04 (c) 64 (d) Total umber of ways of selectig two umbers from the set of {,,, 4,..., } so that their sum is divisible by is equal to : (a) (c) (b) (d) 4. Let fuctio ' f ' be defied from set A to set B, where A B {,,, 4}. If f ( ), where A, the total umber of fuctios which are surjective is give by : (a) (b) 0 (c) 9 (d) 8 5. Total umber of five digit umbers that ca be formed, havig the property that every succeedig digit is greater tha the precedig digit, is equal to : (a) 9 P 5 (b) 9 C 4 (c) 0 C 5 (d) 0 P 5 6. A -digit umber is a positive umber with eactly digits. Nie hudred distict -digit umbers are to be formed usig oly the three digits, 5 ad 7. The smallest value of for which this is possible, is : (a) 6 (b) 7 (c) 8 (d) 9 7. Cosider boes which are umbered by cosecutive atural umbers startig with the umber m. If the bo with labelled umber k, k m, cotais k distict books, the total umber of ways by which m books ca be selected from ay oe of the boes, are : (a) Cm (b) m Cm m (c) Cm (d) C 8. Total umber of triplets (, y, z) which ca be formed, selectig, y, z from the set {,,, 4,... 00} such that y z, is equal to : (a) 00 C (b) 0 C (c) 0 C (d) 00 C 9. Total umber of ways i which a group of 0 boys ad girls ca be arraged i a row such that eactly boys sit i betwee girls, is equal to : (a) 440(8!) (b) 70(8!) (c) 0(9!) (d) 80(8!). Total umber of four letters words that ca be formed from the letters of the word 'DPSRKPURAM', is give by (a) 0 C 4.(4!) (b) 90 (c) Coefficiet of 4 i 4!.!. ( ) (d) Coefficiet of 4 6 i. Cosider seve digit umber , where,,... 0, havig the property that 4 is the 7 greatest digit ad digits towards the left ad right of 4 are i decreasig order, the total umber of such umbers i which all digits are distict is give by : (a) 9 C 7. 6 C (b) 9 C. 6 C 4 (c). 9 C 7. 5 C (d). 9 C. 5 C. Cosider yz = 4, where, y, z I, the (a) Total umber of positive itegral solutios for, y, z are 8 (b) Total umber of itegral solutios for, y, z are 90 (c) Total umber of positive itegral solutios for, y, z are 0 (d) Total umber of itegral solutios for, y, z are 0 4. If Cr ( m 8). C r ; the possible value of 'm' ca be : (a) 4 (b) (c) (d) 5 5. Let 0 differet books are to be distributed amog four studets A, B, C ad D. If A ad B get books each C ad D get books each, the total umber of ways of distributio are equal to : (a) 0 C 4 (b) 500 0! (c) 600 (d) (!) (!) 6 [ ] Mathematics for JEE-0

37 Followig questios are assertio ad reasoig type questios. Each of these questios cotais two statemets, Statemet (Assertio) ad Statemet (Reaso). Each of these questios has four alterative aswers, oly oe of them is the correct aswer. Select the correct aswer from the give optios : (a) Both Statemet ad Statemet are true ad Statemet is the correct eplaatio of Statemet. (b) Both Statemet ad Statemet are true but Statemet is ot the correct eplaatio of Statemet. (c) Statemet is true but Statemet is false. (d) Statemet is false but Statemet is true. 8. Statemet : Total umber of polyomials of the form + a + b + c which are divisible by +, where a, b, c {,,,...0} must be 0 Statemet : value of 'b' ca be selected i 0 ways from the set of first 0 atural umber ad a = c =. a b 9. Statemet : If a, b N ad 7.5, where ad 7 is havig ad 5 positive divisors respectively, the the umber of positive divisors of 5 is 6 6. Statemet : If, m I, the always a itegral value ( m)! N m (!).( m!) is Statemet : 'N' represets the total umbers of ways of equal distributio of (m) distict objects amog 'm' persos. 7. Statemet : From a group of 5 teachers ad 5 studets, if a team of 5 persos is to be formed havig at least two teachers the total umber of ways be which team ca be formed is give by 5 C 8. C {i.e., selectio of teachers from 5 ad more persos from remaiig 8} Statemet II : The team may have 5 teachers, or 4 teachers ad studet, or teachers ad studets, or teachers ad studets. Statemet : Sum of all the positive divisors of a b ( ).( ), where a, b N, is equal to a b ( )( ) umbers., provided ad are the prime 0. Statemet : Let A, A..., A 0 be thirty sets each with five elemets ad B, B,..., B be sets each with three elemets such that 0 A Bi S. If i i i each elemet of S belogs to eactly te of the A i s ad eactly ie of the B j s, the the value of is 45 Statemet : Ai ( Ai ), where ( A) i i represet the umber of elemets of set A. [ ] Mathematics for JEE-0

38 Permutatio ad Combiatio Comprehesio passage () ( Questios No. - ) Cosider the letters of the word 'MATHEMATICS', some of them are idetical ad some are distict. Letters are classified as repeatig ad o-repeatig, such as {M, A, T} is repeatig set of letters ad {H, E, I, C, S} is o-repeatig set of letters, aswer the followig questios based o give iformatio.. Total umbers of words, takig all letters at a time, such that at least oe repeatig letter is at odd positio i each word is give by (a) 9! 8 (c) 9! 4 (b)! 8! 9! (d) 8 4. Total umber or words, takig all letters at a time, i which o vowel is together, is give by 7! 8 4! (a). C 4 (!)! 8 (c) 7!. C 4 4!! (b) 7! 8 4!. C4.!! (d) 7! 8 4!. C4. 8!. Total umber of words, takig all letters at a time, such that each word cotais both M's together ad both T's together but both A's are ot together, is give by (a) 8 C.7!! 0! (b) 8 4 (c) 6(6!) (d) 9(7!) Comprehesio passage () ( Questios No. 4-6 ) Let B, B ad B are three differet boes which cotais y, y ad y distict balls respectively, where. y i {,, }, yi 0 ad i y y If total umber of ways by which Joh ca select eactly balls from the boes is 'N ' ad he is ot allowed to select two balls from the same bo, the aswer the followig questios 4. If y = 4, the value of N is equal to : (a) 90 (b) (c) 40 (d) 9 5. If N assumes its maimum value, the which oe of the followig is correct : (a) y = y = 5 (b) y = y = 8 (c) y = 8 (d) y = 6 6. Maimum value of N is equal to : (a) (b) 40 (c) (d) 0 Comprehesio passage () ( Questios No. 7-9 ) Let A = {,,, 4,..., } be the set of first atural umbers, where S A. If the umber of elemets i set S is represeted by ( S) ad the least umber i the set S is deoted by S mi, the aswer the followig questios. 7. If ay of the subset S of set A is havig ( S) = r, where r, the maimum value of S mi which ca occur is equal to : (a) r (b) r (c) r + (d) r + 8. The umber of subsets 'S' with S mi = m ad ( S) = r, is equal to : m (a) m C r (b) m Cr (c) Cr (d) C r m 9. Let ( S) = r ad S mi = m, where r m, the sum of all the S mi for possible subsets 'S' is equal to : m (a) m C r m (b) C r (c) ( ) m m Cr r Cr m m (d) m Cr Cr [ 4 ] Mathematics for JEE-0

39 0. Let 'N' triagles ca be formed by joiig the vertices of a regular decago i which o two cosecutive N vertices are selected, the value of is equal to Let i C umber of ways four tickets ca be selected from 5 tickets umbered from to 5 so that o two cosecutive umbered tickets are selected, the the value of is equal to.... Let all the letters of the word SACHHABACHHA be arraged i a matri of order 4, ad at least oe of the row of matri is havig all the idetical elemets. If the total umber of arragemets are 'N', the least prime umber dividig the umber 'N' is equal to.... Let P() deotes the sum of the eve digits of the umber '', for eample : P(859) 8 0, the value of of P( r) is equal to r 4. Let 6 people are to be arraged aroud a regular octagoal frame such that people ca either sit at the corer or at the mid of the side. If the umber of ways i which the arragemet is possible is (5!), the value of ' ' is equal to Cosider a set 'A' cotaiig 8 differet elemets from which a subset 'P' is chose ad the set A is recostructed by replacig the elemets of P. From set A if aother subset Q is chose, the match the followig colums for the umber of ways of choosig P ad Q i colum (II) with the coditios i colum (I) Colum (I) Colum (II) (a) P Q cotais eactly oe elemet (p) 656 (b) Q is subset of P (q) 4 (c) P Q cotais eactly oe elemet (r) 56 (d) P Q A (s) 7496 (t) Cosider all possible permutatios of the letters of the word ENDEANOEL. Match the statemets i colum I with the statemets i colum II. Colum (I) Colum (II) (a) The umber of permutatios cotaiig the word ENDEA (p) 0 (b) The umber of permutatios i which the letter E occurs (q) 40 i the first ad the last positios (r) 840 (c) The umber of permutatios i which oe of the letters D, L, N occurs i the last five positios (s) 50 (d) The umber of permutatios i which the letters A, E, O (t) 40 occur oly i odd positios 6 [ 5 ] Mathematics for JEE-0

40 Permutatio ad Combiatio. (c). (d). (d) 4. (b) 5. (a) E 6. (c) 7. (d) 8. (c) 9. (d) 0. (d). (c). (b). (b) 4. (c) 5. (b) 6. (b) 7. (d) 8. (b) 9. (a) 0. (b). (b, d). (a, d). (c, d) 4. (a, c, d) 5. (b, d) 6. (c) 7. (d) 8. (c) 9. (b) 0. (b). (b). (a). (a) 4. (d) 5. (c) E 6. (c) 7. (c) 8. (d) 9. (a) 0. ( 5 ). ( 8 ). ( ). ( ) 4. ( ) 5. (a) s 6. (a) p (b) p (b) s (c) q (c) q (d) p (d) q [ 6 ] Mathematics for JEE-0

41 6. Let 'A' ad 'B' be two evets such that P(A) = 0.70,. Let A, B, C be pair-wise idepedet evets, where P( A B C) 0 ad P(C) > 0, the equal to : (a) P( A) P( B) (b) P( A) P( B) (c) P( A) P( B) (d) P( A) P( B) A B P is C. If three idetical dice are rolled, the probability that the same umber appears o each of them is : (a) 6 (c) 8 (b) 6 (d) 8. If A, B, C are three mutually idepedet evets, where P( A B C) P( A B C) ad P( A C) P( A B C), the P( AC B) is equal to : (a) (c) 6 (b) 5 6 (d) 4 P(B) = 0.40 ad P( A B) 0.5, the is equal to : (a) 0.0 (b) 0.5 (c) 0.40 (d) B P A B 7. Three umbers are chose at radom without replacemet from {,,,..., 0}. Probability that the miimum of the chose umber is or their maimum is 7, is give by : (a) /0 (b) /40 (c) /50 (d) 7/40 8. If a, b, c, d {0, }, the the probability that system of equatios a + by = ; c + dy = 4 is havig uique solutio is give by : (a) 5 8 (b) 8 (c) (d) 9. For a studet to qualify, he must pass at least two out of the three eams. The probability that he will pass the first eam is, if he fails i oe of the eams the the probability of his passig i the et eam is 4 otherwise it remais the same. The probability that studet will pass the eam is : 4. A ubiased die is throw ad the umber show o the die is put for 'p' i the equatio + p + = 0, probability of the equatio to have real roots is : (a) 4 5 (c) 4 (b) 8 (d) 4 (a) (c) (b) (d) 4 5. Miimum umber of times a fair coi must be tossed so that the probability of gettig at least oe head is at least 0.95 is (a) 4 (b) 5 (c) 6 (d) Let eight players P, P, P,... P 8 be paired radomly i each roud for a kock-out touramet. If the player P i wis if i > j, the the probability that player P 6 reaches the fial roud is : (a) 5 (c) 0 7 (b) 8 5 (d) oe of these [ 7 ] Mathematics for JEE-0

42 . Let Joh appears i the eams of physics, chemistry ad mathematics ad his respective probability of passig the eams is p, c ad m. If Joh has 80% chace of passig i at least oe of the three eams, 55% chace of passig i at least two eams, ad 5% chace of passig i eactly two of the eams, the p + c + m is equal to : (a) 0 (c) 7 0 (b) 8 (d) 45. Let oe hudred idetical cois, each with probability 'p' of showig up head are tossed oce. If 0 < p < ad the probability that head turs up o 50 cois is equal to the probability that head turs up o 5 cois, the the value of 'p' is : (a) 50 0 (c) 5 0 (b) 49 0 (d) 5 0. I a set of four bulbs it is kow that eactly two of them are defective. If the bulbs are tested oe by oe i radom order till both the defective bulbs are idetified, the the probability that oly two tests are eeded is give by : (a) 6 (c) 4 (b) (d) 4. Let faces of a ubiased die are red, faces are yellow ad face is gree. If the die is tossed three times, the the probability that the colors red, yellow ad gree appear i the first, secod ad the third tosses respectively is : (a) 8 (c) 7 6 Probability (b) 6 (d) 9 5. Let oe Idia ad four America me ad their wives are to be seated radomly aroud a circular table. If each America ma is seated adjacet to his wife, the the probability that Idia ma is also seated adjacet to his wife is give by : (a) 5 (b) 6. Let 'A' ad 'B' be two idepedet evets. The probability that both A ad B happe is ad the probability that either A or B happe is, the P( A) 4 P( B) may be (a) or 0 (b) 7 or 0 7 (c) 0 or (d) 7 or 7. A ur cotais white ad black balls, a ball is draw at radom, if it is white it is ot replaced ito the ur, otherwise it is dropped alog with oe aother ball of same color. The process is repeated, probability that the third draw ball is black, is : (a) 60 (c) 9 0 (b) 4 60 (d) oe of these 8. A eperimet has te equally likely outcomes. Let A ad B be two o-empty evets of the eperimet. If A cosists of 4 outcomes, the umber of outcomes that B must have so that A ad B are idepedet, is : (a), 4 or 8 (b), 6 or 9 (c) 4 or 8 (d) 5 or 0 9. A fair die is tossed repeatedly util a si is obtaied, if 'k' deotes the umber of tosses required, the the coditioal probability that 'k' is ot less tha si whe it is give that 'k' is greater tha, is equal to : (a) 5 6 (c) 5 6 (b) 5 6 (d) A bo cotai 5 cois, 8 of which are fair ad the rest are biased. The probability of gettig a head o fair coi ad biased coi is ad respectively. If a coi is draw radomly from the bo ad tossed twice, first time it shows head ad the secod time it shows tail, the the probability that the coi draw is fair, is give by : (a) 5 8 (b) 9 6 (c) 5 (d) (c) 8 (d) [ 8 ] Mathematics for JEE-0

43 . A perso goes to office either by car, scooter, bus or trai probability of which beig,, ad 7 respectively. Probability that he reaches office late, if he takes car, scooter, bus or trai is,, ad 9 respectively. If it is give that he reached office i time the the probability that he travelled by car is : 6. For two evets A ad B, if B P A A P( A) P, the the correct statemets are : B 4 7 (a) P( A B) (b) P( A B) 8 8, (a) 7 (c) 7 (b) 7 (d) 4 7. Let 'K' be the itegral values of for which the iequatio < 0 holds. If three fair dice are rolled together, the the probability that the sum of the umbers appearig o the dice is K, is give by : (a) 4 6 (c) 4 (b) 5 4 (d) 6. Let set 'S' cotais all the matrices of order i which all the etries are either 0 or. If a matri is selected radomly from set 'S' ad it is foud that it cotais eactly five of the etries as, the the probability that the matri is symmetric, is give by : (a) 6 56 (c) (b) 8 (d) Let two positive real umbers '' ad 'y' are chose radomly, where 0, ad y 0, A (c) P B 4 B (d) P A 7. Let A, B, C be three idepedet evets, where P( A) P( B) 4 P( C), the : (a) probability of occurrece of eactly of the three evets is 4. (b) probability of occurrece of at least oe of the three evets is 4. (c) probability of occurrece of all the three evets is. 4 (d) probability of occurrece of eactly oe of the three evets is Let a bag cotai 5 balls i which the balls ca have either black colour or white colour. If B is the evet that bag cotais eactly black balls ad its probability is proportioal to, ad E is the evet of gettig a black ball whe a ball is draw radomly from the bag, the : (a) 5 0 P( B ). The prob- ability that + y, give that + y, is : 4 (a) 8 6 (c) (b) 6 (d) 6 5. Let a atural umber 'N' be selected at radom from the set of first hudred atural umbers. The probability 5 that N is ot greater tha 0 is give by : N (a) 0.0 (b) 0.05 (c) 0.5 (d) (b) P( E) B5 5 (c) P E 76 B5 5 (d) P E Let the evets 'A' ad 'B' be mutually eclusive ad ehaustive i ature, the : (a) P( A) P( B) (b) P( A B) 0 (c) P( A B) P( A) P( B) (d) P( A B) P( A) P( B) [ 9 ] Mathematics for JEE-0

44 Probability 0. There are four boes B, B, B ad B 4. Bo B i cotai i cards ad o each card a distict umber is prited, the prited umber varies from to i for bo B i. If a bo is selected radomly, the probability of occ- i urrece of bo B i is give by ad if a card is 0 draw radomly from it the E i represets the evet of occurrece of umber i o the card, the : (a) value of P(E ) is 5 B (b) iverse probability P is E E (c) coditioal probability P is zero B (d) value of P(E ) is 4 Followig questios are assertio ad reasoig type questios. Each of these questios cotais two statemets, Statemet (Assertio) ad Statemet (Reaso). Each of these questios has four alterative aswers, oly oe of them is the correct aswer. Select the correct aswer from the give optios : (a) Both Statemet ad Statemet are true ad Statemet is the correct eplaatio of Statemet. (b) Both Statemet ad Statemet are true but Statemet is ot the correct eplaatio of Statemet. (c) Statemet is true but Statemet is false. (d) Statemet is false but Statemet is true.. Statemet : Let ay two digit umber is raised with power 4K +, where K N, the the probability that uit's place digit of the resultat umber is atural multiple of is / Statemet : If ay two digit umber is raised with power 4K +, K N, the digit at uits place ca be 0,, 4, 5, 6, 9.. Statemet : Let 'A' ad 'B' be two depedet evets ad if { P( A B)} P( B), P( A B) is si, 0 the least value of A P( A B) Statemet : P, B P( B) where P( B) 0. Statemet : I a biomial probability distributio B(, p / 4), if the probability of at least oe success is ot less tha 0.90, the value of '' ca be log Statemet : I the give biomial probability distributio '' is greater tha or equal to log Statemet : Let A ad B be ay two evets of a 4 radom eperimet, where P( A) ad P( B), 5 the the value of P( A B) lies i, 5 Statemet : For ay two evets A ad B, ma P( A), P( B) P( A B) ad P( A B) mi P( A), P( B). 5. Statemet : Let a ellipse of eccetricity 4 5 be iscribed i a circle ad a poit withi the circle be chose radomly, the the probability that the poit lies outside the ellipse is 5 Statemet : The area of a ellipse of eccetricity 'e' is give by a e square uits, where 'a' represets the radius of auiliary circle of the ellipse. [ 40 ] Mathematics for JEE-0

45 Comprehesio passage () ( Questios No. - ) 5. If three balls are picked up at radom from the bag ad all the balls are foud to be of differet colour, the the probability that bag cotaied 4 white balls, is : For a biased coi, let the probability of gettig (a) 7 5 (b) 7 head be ad that of tail be. If A deotes the evet of tossig the coi till the differece of the umber of heads ad tails become '', the aswer the followig questios.. If =, the the probability that the eperimet eds with more umber of heads tha tails, is equal to : (a) 5 (c) 5 9 (b) 4 5 (d) 4 9. If it is give that the eperimet eds with a head for =, the the probability that the eperimet eds i miimum umber of throws, is equal to : (a) 5 (b) 4 9 (c) 8 (d) 5 9. If E is the evet that the last two throws show either two cosecutive heads or tails, the the E value of P is equal to : A (a) 4 (c) 9 5 (b) 9 (d) 0 (c) 4 (d) 0 6. If three balls are picked up at radom ad foud to be oe of each colour, the the probability that bag cotaied equal umber of white ad gree balls is equal to : (a) 4 (c) 7 5 (b) 0 (d) 7 Comprehesio passage () ( Questios No. 7-9 ) A fair die is tossed repeatedly util a si is obtaied. If X deote the umber of tosses required, the aswer the followig questios. 7. The probability that X = equals (a) 5 6 (c) 5 6 (b) 5 6 (d) The probability that X equals (a) 5 6 (b) 5 6 Comprehesio passage () ( Questios No. 4-6 ) Cosider a bag cotaiig si differet balls of three differet colours. If it is kow that the colour of the balls ca be white, gree or red, the aswer the followig questios. 4. The probability that the bag cotais balls of each colour is : (a) 0 (b) 7 (c) 9 (d) 8 (c) 5 6 (d) The coditioal probability that X 6 give X > equals (a) 5 6 (c) 5 6 (b) 5 6 (d) [ 4 ] Mathematics for JEE-0

46 Probability 0. If the papers of 4 studets ca be checked by ay oe of the 7 teachers. If the probability that all the 4 papers are checked by eactly teachers is P, the the value 49P is equal to.... There are two parallel telephoe lies of legth l = 0m which are m apart as showi figure. It is kow that there is a break i each of them, the locatio of the break beig ukow, if the probability that the distace 'R' betwee the breaks is ot larger tha 5m is p, the 5 p is.... A bag cotai black ad white balls, from the bag Joh radomly pick three balls ad the drop balls of red colour ito the bag. If ow Joh radomly pick three balls from the bag ad the probability of gettig all the three balls of differet colour is p, the value of 00 p is.... Let a cubical die has four blak faces, oe face marked with, aother face marked with, if the die is rolled ad the probability of gettig a sum of 6 i throws is p, the value of 4 p is equal to Cosider a cube havig the verte poits A, B, C, D, E, F, G, ad H. If radaoly three corer poits are selected to form a triagle the match the followig colums for the probability of the ature of triagle. Colum (I) (a) Probability that the triagle is scalee (b) Probability that the triagle is right-agled (c) Probability that the triagle is isosceles with eactly Colum (II) two equal sides (s) 4 4. A perso while dialig a telephoe umber forgets the last three digits of the umber but remembers that eactly two of them are same. He dials the umber radomly, if the probability that he dialed the correct umber is P, the value of (080P) is... (p) (q) (r) (d) Probability that the triagle is equailateral (t) 7 [ 4 ] Mathematics for JEE-0

47 6. Five ubiased cubical dice are rolled simultaeously. Let m ad be the smallest ad the largest umber appearig o the upper faces of the dice, the match the probabilitiy give i the colum II correspodig to the evets give i the colum I : Colum (I) Colum (II) (a) m = (b) = 4 (p) (q) (c) m 4 (r) (d) m = ad = 5 (s) [ 4 ] Mathematics for JEE-0

48 Probability. (c). (c). (c) 4. (c) 5. (b) E 6. (b) 7. (b) 8. (b) 9. (b) 0. (d). (a). (c). (d) 4. (b) 5. (c) 6. (c) 7. (d) 8. (d) 9. (c) 0. (b). (a). (c). (c) 4. (b) 5. (a) 6. (b, c, d) 7. (a, b, c, d) 8. (a, b, d) 9. (a, b, c) 0. (a, b, c). (d). (a). (a) 4. (a) 5. (b). (b). (d). (a) 4. (a) 5. (c) E 6. (a) 7. (a) 8. (b) 9. (d) 0. ( 6 ). ( 9 ). ( ). ( 8 ) 4. ( 4 ) 5. (a) t 6. (a) s (b) p (b) s (c) p (c) r (d) r (d) q [ 44 ] Mathematics for JEE-0

49 6. If A ad B are two square matrices of order ad AB = B, BA = A, the A + B = I holds true for the coditio :. Let A a ij ; a si ( ), ij i j the (a) det (A) = si (b) det (A) = 0 (c) det (A) > 0 (d) det (A) < 0 cos si. Let A si cos, the T A A I if the values of ' ' belog to : (a) ; I (b) ( ) ; I 6 (a) A = B = 0 (b) A = B 0 (c) A B 0 7. Let A a ij where ; (d) A ad B are o-zero mi{ i, j} ; i j aij i j ; i j ad [.] represets the greatest iteger fuctio, the det{adj(adj(a))} is equal to : (a) 5 (b) 5 (c) 65 (d) 5 (c) ( ) ; I (d) ; I. Let A a ij, B b ad C c ij be ij three matrices, where det(a) = ad b ij, c ij are the correspodig cofactors of a ij ad b ij respectively, the det(ab T C) is equal to : (a) (c) 4. Let 0 r 0 C r 0 Cr r 00 (b) (d) 0 r C r Cr r A a ij be a matri for which i j aij i ij i j si ( i j) 4, where [.] represets the greatest iteger fuctio, the trace(a) is equal to : (a) 40 (b) 400 (c) 40 (d) Let 'S' be the set of all symmetric matrices for which all the etries are either or, if five of these etries are ad four of them are, the (S) is equal to : ( (S) represets the cardial umber of S ) (a) 0 (b) (c) 0 (d) Total umber of matrices that ca be formed usig all the seve differet oe digit umbers such that o digit is repeated i ay matri, is give by : (a) 7! (b) (7) 7 (c) (7!) (d) 7(7!) 9. Let A a ij ad B b ij be two matrices ad,, {,,}, the which oe of the followig is always true : (a) (b) (c) (d) a. b a. b a. b a. b a. b a. b a. b a. b 0. Let ' ' be the o-real cube root of uity, where 0 0 A 0 0, the A 00 is equal to : 0 0 (a) A (b) A (c) 0 (d) I [ 45 ] Mathematics for JEE-0

50 Matrices. Let cos( / 6) si( / 6) A si( / 6) cos( / 6) ad B ; 0 where T T 00 C ABA, the A C A is equal to : 00 (a) 0 / (b) 00 (c) (d). Let / 00 / k( C ) ad k k Ak 0 k 0. k If (p + q) is equal to : 7 k (a) 00 (b) 508 (c) 04 (d) 40 ( k) C, ad k p 0 Ak 0 q ; the value of 0 0. Let A 0 ad 6A = A + pa + qi, the 0 4 (p + q) is equal to : (a) 0 (b) (c) (d) a b c 4. If matri A b c a, where a, b, c C, abc c a b ad AA T = I, the a b c is equal to : (a) 0 (c) 5. Let (b) a + b + c 8 (d) + a + b + c A a ij represets a matri ad i j j k k i ij jk ki ( ) a ( ) a ( ) a 0 for all i, j, k belogs to {,, }, the 'A' is : (a) symmetric matri. (b) sigular matri. (c) o-sigular matri. (d) orthogoal matri. k 6. Let [0, ),, 6 ad si si cos A si si, the cos si (a) det (A) is idepedet from. (b) det (A) is idepedet from. (c) det ( A), Let (d) det ( A) [, ]. A a ij where, mi{ i, j}, i j aij i j. ; i j 0 If a ij represets the elemet of i th row ad j th colum i matri 'A', the : ([.] represets G.I.F.). (a) det (A) = 0 (b) det (A) = 4 (c) A is symmetric matri (d) Tr(A) = 0 si 8. Let A( ) = i cos i cos, si (a) A( ) is ivertible R. (b) Iverse of A( ) A( ). (c) Iverse of A( ) A( ). (d) A( ) A( ) O. 9. Let (a) P A ( ) ad 0 where i, the A, the 0 (b) Matrices A ad P both are orthogoal matri (c) If A = I + B, the det (B) = 0 (d) det{adj (adj (AP))} = 4 [ 46 ] Mathematics for JEE-0

51 0. Let matri 'A' be sigular matri, ad [0, ]. If A si cos 4si 4 si cos 4si 4, si cos 4si 4 possible values of ' ' ca be : (a) 7 4 (c) 4 (b) 4 (d) 9 4 the Followig questios are assertio ad reasoig type questios. Each of these questios cotais two statemets, Statemet (Assertio) ad Statemet (Reaso). Each of these questios has four alterative aswers, oly oe of them is the correct aswer. Select the correct aswer from the give optios : (a) Both Statemet ad Statemet are true ad Statemet is the correct eplaatio of Statemet. (b) Both Statemet ad Statemet are true but Statemet is ot the correct eplaatio of Statemet. (c) Statemet is true but Statemet is false. (d) Statemet is false but Statemet is true. 0. Statemet : Let A ad A = A I, the A 8 = 55A 56 I Statemet : A 0. Let A ad B be two square matrices of order, ad 'O' represets the ull matri of order. Statemet : If AB = 0, ad A is o-sigular matri, the matri B is ecessarily a sigular matri Statemet : Product of two equal order square matrices ca oly be zero matri if both the matrices are ot o-sigular matrices.. Let A be a matri with real etries, ad satisfy the coditio A = I, where 'I' is uit matri of order. Statemet : If A I ad A I, the det(a) = Statemet : If A I, the Tr(A) 0 4. Statemet : Let A 5 = 0 ad A I for all {,,, 4}, the (I A) = A 4 + A + A + A + I 5 4 Statemet :, where. 5. Let A ad B be square matrices of order, where A a a i j ij ; si ( ). ij T Statemet : If = 7!, the B A B is skew-symmetric matri Statemet : determiat value of skew-symmetric matri of odd order is always zero. [ 47 ] Mathematics for JEE-0

52 Matrices Comprehesio passage () ( Questios No. - ) Let 'S' be the set of all symmetric matrices all of whose etries are either 0 or. If five of these etries are ad four of them are 0, the aswer the followig questios. 5. Let,, R ad matri a 0 0 Q 0 0 c 0 b 0. If 'Q' is orthogoal matri the maimum umber of ordered triplets (,, ) which are possible is give by : (a) (b) 8 (c) (d) 6. The umber of matrices i 'S' is : (a) (b) 6 (c) 9 (d). The umber of matrices A i 'S' for which the system of liear equatio is : (a) less tha 4 (b) at least 4 but less tha 7 (c) at least 7 but less tha 0 (d) at least 0 A y 0 has a uique solutio, z 0. The umber of matrices A i 'S' for which the system of liear equatio A y 0 is icosistet, is z 0 (a) 0 (b) more tha (c) (d) Comprehesio passage () ( Questios No. 4-6 ) For a give square matri 'A', if AA T = A T A = I holds true, the matri is termed as orthogoal matri. If a, b, c R ad matri 'P' is orthogoal, where 0 a a P b b b, the aswer the followig c c c questios : 4. If square matrices of order is formed with the etries 0, a, b ad c, the maimum umber of matrices which ca be formed without repetitio of the etries, is equal to : (a) 840 (b) 4 (c) 56 (d) 9 [ 48 ] 6. If k ; the umber of positive itegral ( abc) solutios for the equatio.. k, is equal to : (a) 8 (b) 0 (c) 6 (d) 7 Let Comprehesio passage () ( Questios No. 7-9 ) 0 0 A 0, ad R, R, R be the row matrices satisfyig the relatios, R A 0 0, R A 0 ad R A. If B is square matri of order with rows R, R, R, the aswer the followig questios. 7. The value of det(b) is equal to : (a) (b) (c) 0 (d) 8. Let C = (A 00.B ) (A 99.B 4 ), the value of det(c) is equal to : (a) 7 (b) 7 (c) 00 (d) Sum of all the elemets of matri B is equal to : (a) 8 (b) 0 (c) 5 (d) 0 0. Let matri A 5 6, the the least positive iteger 'K' for which A K becomes ull matri, is equal to... Mathematics for JEE-0

53 . Let A a ij where A = ad B b ij, If b ij is the cofactor of a ij, ad AB T = C, the sum of diagoal elemets of matri 'C' is equal to.... Let a,, y R, where + y = 0, ad the system of equatios is give by : ay ( a ) ay y y If the system has at least oe solutio, the umber of possible itegral value(s) of 'a' is/are.... Let a,, y R, ad matrices A, B ad C be defied as A a C y, y B ad. If the matri equatios AB = C is havig oly oe solutio, the total umber of possible value(s) of 'a' is/are Let 4 5 A 8 8 6, the match colums (I) ad (II) for the values of ad the rak of matri 'A'. 8 4 Colum (I) Colum (II) (a) If, the rak of matri A is : (p) (b) If, the rak of matri A is : (q) (c) If R {}, the rak of matri A ca be : (r) (d) If 4, the rak of matri A is : (s) 0 5. Match colums (I) ad (II) Colum (I) Colum (II) (a) Let i j A a ij ad B k a ij if (p) 0 ; k A + k B = 0 ; where A 0, the (k + k ) is (b) Maimum value of third order determiat if each of its (q) 4 etries are either or, is (c) If cos cos 0 cos cos cos cos cos 0 cos (r) cos cos cos cos 0 the cos cos cos is equal to : (d) A B where A ad B (s) are determiat of, the (A + B) is equal to [ 49 ] Mathematics for JEE-0

54 Matrices. (b). (c). (c) 4. (c) 5. (b) E 6. (d) 7. (c) 8. (c) 9. (c) 0. (d). (a). (b). (b) 4. (d) 5. (b) 6. (a, c) 7. (b, c, d) 8. (a, c, d) 9. (a, c, d) 0. (a, b, c, d). (d). (a). (c) 4. (b) 5. (d). (a). (b). (b) 4. (d) 5. (b) E 6. (c) 7. (b) 8. (b) 9. (c) 0. ( ). ( 8 ). ( ). ( ) 4. (a) p 5. (a) p (b) q (b) q (c) q, r (c) r (d) r (d) p [ 50 ] Mathematics for JEE-0

55 6. Let f (), g() ad h() be cubic fuctios of ad. If system of equatios : 4 + 5y z = 0, y 4z = 0 ad (K + ) + (K ) y + (K 4) z = 0 have otrivial solutio, the : (a) K = (b) K = 0 (c) K = or 0 (d) K R. Let (, y, z) be poits with iteger co-ordiates satisfyig the system of homogeeous equatio : y z = 0 + z = 0 + y + z = 0. The the umber of such poits for which + y + z 00 are : (a) 6 (b) 5 (c) 0 (d) 7. If [0, ) ad the det (A) lies i the iterval : si A si si ; si (a) [, 4] (b) [, ] (c) [, 4] (d) (, 4) 4. The eistece of uique solutio for the system of equatios, + y + z = p, 5 y + qz = 0 ad + y z = 6 depeds o : (a) 'p' oly. (c) 'p' ad 'q' both. 5. Let (b) 'q' oly. 0 (d) either 'p' or 'q'. f ( ), where [.] ta si [ ] represets the greatest iteger fuctio, the f ( ) d is : (a) cos (b) si + sec (c) + cos si (d) cos + cos f '( ) f "( ) f "'( ) ( ) g '( ) g "( ) g "'( ), the h'( ) h"( ) h"'( ) (a) "( ). (b) graph of ( ) is symmetric about origi. (c) graph of ( ) is symmetric about y-ais. (d) ( ) is polyomial of degree. K K 7. Let P a b, where K N & ( a b) ab 4, K the value of P P P P P P P P 4 (a) 4 (b) 0 (c) 8 (d) is equal to : 8. Let, ad be iteral agles of a triagle, the miimum value of equal to : cos cos cos cos cos cos (a) 0 (b) (c) (d) 9. Let,, ad be the positive real roots of the equatio 4 + p + q + 8 = 0, where p, q, the value of (a) (c) 5 R (b) 9 (d) oe of these is equal to : 0. Let set 'S' cosists of all the determiats of order with etries zero or oe oly ad set 'P' is subset of 'S' cosistig of all the determiats with value. If set 'Q' is subset of 'S' cosistig of all the determiats with value, the : (a) (S) = (P) + (Q) (b) (P) = (Q) (c) (P) = (Q) (d) P Q S is [ 5 ] Mathematics for JEE-0

56 Determiats. Let 'M' be a matri, where det (M) =, the : (a) det( M I) 0. (b) det (M I) is always zero. (c) det (M + I) = 0. (d) det (M + I) is always zero. T MM I ad 7. If a determiat is chose at radom from the set of all determiats of order with elemets zero or oe oly, the the probability that the value of determiat chose is o-egative is equal to : (a) 6 (b) 5 8. Let 4 p q r s t 4 4 be a idetity i, where p, q, r, s ad t are costats, the (q + s) is equal to : (a) 5 (b) 5 (c) 50 (d) 0. Let A, B ad C be the agles of a triagle, where A, B, C, the the value of is equal to : ta A ta B ta C (a) 0 (b) (c) (d) 4. Let a r r i yr j zr k, where r =,, be three vectors ad a r r, a. a a. a a. a, the value of y z y z y z is equal to : (a) 4 (b) 6 5. Let (c) 6 (d) 6 f ( ) ; the f () is idepedet of : (a) ad (b) ad (c) ad (d), ad 6. If a homogeous system of equatios is represeted by: a by cz 0, b cy az 0 ad c ay bz 0, ad ifiite ordered triplets (, y, z) are possible without ay liear costrait, the (a) a b c 0 ad a b c ab bc ca (b) a b c 0 ad a b c ab bc ca (c) a b c 0 ad a b c ab bc ca (d) a b c 0 ad a b c ab bc ca (c) 8 (d) 6 8. Let,, be o-zero real umbers, the system of equatios i, y ad z, y z ad (a) o solutio. (b) uique solutio. (c) ifiitely may solutios. (d) fiitely may solutios. y z, y z has : 9. The umber of values of 'K' for which the system of equatios (K + ) + (K + )y + K + = 0 ad (5K + ) + (7K + )y + 4K + = 0 is cosistet ad idetermiate is give by : (a) 0 (b) (c) (d) ifiite 0. If the system of equatios ; + y = 0, 6 + ky 4 =0 ad 6 + y 0 = 0 is cosistet, the (a) k = (b) k = (c) k = or (d) k. Let a, b, c be o-zero real umbers ad fuctio f () is give by is divisible by : a ab ac ab b bc ac bc c (a) 4 (b) 6 (c) a b c (d) + a + b + c, the f (). System of equatios : + y + z = 6, y z 7, + y + z = has : (a) Ifiitely may solutios if 4, 6. (b) No solutio if 5, 7. (c) Uique solutio if 5, 7. (d) No solutio if, 5. [ 5 ] Mathematics for JEE-0

57 . Cosider the system of liear equatios i, y, z : 7y 7z 0 (si ) y z 0 (cos ) 4y z 0 If the system has o-trivial solutios, the agle ' ' ca be : (a) 5 6 (b) 7 6 K K 6. Statemet : Let K cos i si for all 9 9 K W, the value of determiat zero is (c) 4 (d) 7 6 Statemet : 9 K K 0 4. Let determiat 'D' is havig all the elemets as either or. If the product of all the elemets of ay row or ay colum of 'D' is egative, the it is represeted by 'D N '. If the order of 'D' is, the : (a) miimum value of D N is. (b) miimum value of D N is 4. (c) total umber of D N is 6. (d) total umber of D N is. 5. Let f () be real valued polyomial fuctio, ad f '( ) f ( ), the (a) (b) f ( ) d 0 f ( ) d 0 (c) y f ( ) is odd fuctio (d) y = f () is symmetrical about lie = 0 7. Let f (), g() ad h() be the polyomial fuctios of degree, 4 ad 5 respectively, where f '( ) g '( ) h'( ) ( ) f '( ) g '( ) h'( ) ad R. f "( ) g "( ) h"( ) Statemet : ( ) is divisible by ( ) Statemet : ( ) '( ) 0 8. Let S {,,,..., } be the set of determiats that ca be formed with the distict ozero real umbers a, a, a,... a 9, where repeatitio of elemets is ot permissible, the Statemet : i 0 Statemet : i 9 i i Followig questios are assertio ad reasoig type questios. Each of these questios cotais two statemets, Statemet (Assertio) ad Statemet (Reaso). Each of these questios has four alterative aswers, oly oe of them is the correct aswer. Select the correct aswer from the give optios : (a) Both Statemet ad Statemet are true ad Statemet is the correct eplaatio of Statemet. (b) Both Statemet ad Statemet are true but Statemet is ot the correct eplaatio of Statemet. (c) Statemet is true but Statemet is false. (d) Statemet is false but Statemet is true. 9. Let f ( ) si si 4 si 6 cos cos 4 cos 6 Statemet : If 0,, the umber of solutios of the equatio f () = 0 are five Statemet : si 0 is havig five solutios if R. [ 5 ] Mathematics for JEE-0

58 Determiats 0. Let 'A r ' represets the umber of positive itegral solutios of y z r, where r N {, }, ad A A A r r r A A A r r r A A A r r r 4. Statemet : Value of 0 Statemet : I a determiat if ay two rows or ay two colums are idetical, the determiat value is zero. [ 54 ] Mathematics for JEE-0

59 4. Matri additio for B + B + B B 00 is equal to : Comprehesio passage () ( Questios No. - ) (a) 00 B (b) 99 B (c) 99 I (d) 98 I Let y f ( ) be quadratic fuctio, ad 4a 4a f ( ) a a 4c 4c f () c c 4b 4b f () b b. If a, b ad c are distict real umbers, ad maimum value of f () occurs at poit 'V', the aswer the followig questios.. Let i j A a ij, a ij f i j a 0 for all i j, the det (A) is equal to : ij (a) 0 (b) (c) (d) ad. Let 'A' is the poit of itersectio of y f ( ) with -ais ad poit B(, f ( )) is such that chord AB subteds a right agle at 'V', the area (i square uits) eclosed by f () with chord AB is : (a) 50 (c) 75 4 (b) 5 (d) 0. Let g( ) (, ), the total umber f ( ) of poits of discotiuity for y g( ) i, are give by : ( [.] represets G.I.F. ) (a) 4 (b) 6 (c) 8 (d) Comprehesio passage () ( Questios No. 4-6 ) Cosider the matrices, 4 A 4, 4 C, C 0 ad C. 4 4 Let matri 'B ' of order is formed with the colum vectors of the matrices C, C ad C, ad B adj( B ), N, the aswer the followig questios : Let M = AB A B A B... A B, the det (M) is equal to : 00 (a) 00 (b) 00 (c) 0 (d) For a variable matri X. the matri equatio AX = C will have : (a) Uique solutio (b) No solutio (c) Fiitely may solutios (d) Ifiitely may solutios 7. a a b a c If a, b, c I ad ab b b c, ac bc c the total umber of possible triplets of (a, b, c) is/are Let / 0 cos si si f ( ) si si cos, si cos 0 ad I f ( ) f '( ) d, the the least iteger just greater tha 'I' is equal to Let U / cos d, cos the value of 0 U U U U U U U U U is equal to Cosider the system of equatios : (si ) y (cos ) z 0 (cos ) y (si ) z 0 (si ) y (cos ) z 0 If ad are real umbers, ad the system of equatios has o-trivial solutios, the umber of itegral values of which are possible for differet values of are... [ 55 ] Mathematics for JEE-0

60 Determiats. Let f () be polyomial fuctio havig local miima at = 5 ad f (0) = f () =. If for all a a a b R, f '( ) b b a b a b a b followig colum (I) ad II. Colum (I) where 'a' ad 'b' are some costats, the match the Colum (II) (a) Value of (a + b) (p) (b) Value of f (5) (q) 0 (c) Number of solutios for 4 f ( ) (r) (d) f ( ) lim f ( ). Cosider the system of equatios : K y z Ky z K y Kz K (s) Match colum (I) ad (II) for the values of 'K' ad the ature of solutio for the system of equatios. Colum (I) Colum (II) (a) If K, the system of equatios have (p) Uique solutio. (b) If K, the system of equatios may have (q) Ifiitely may solutios. (c) If K R {, }, the system of equatios have (r) No solutio. (d) If K {, }, the system of equatios may have (s) Fiitely may solutios. [ 56 ] Mathematics for JEE-0

61 . (d). (d). (a) 4. (b) 5. (d) E 6. (c) 7. (c) 8. (c) 9. (d) 0. (c). (b). (b). (c) 4. (b) 5. (d) 6. (d) 7. (d) 8. (d) 9. (c) 0. (d). (a, d). (a, b, d). (a, b, c) 4. (b, c) 5. (a, c, d) 6. (b) 7. (a) 8. (b) 9. (b) 0. (d). (a). (b). (b) 4. (b) 5. (c) E 6. (b) 7. ( ) 8. ( 4 ) 9. ( 0 ) 0. ( ). (a) r. (a) q (b) s (b) p, r (c) p (c) p (d) q (d) q, r [ 57 ] Mathematics for JEE-0

62 . If log 7 m, the log 8, 49 is equal to : (a) (m + ) (b) m (c) m. If l l a lb (d) m + a b, where a, b R the relatio betwee a ad b is : (a) a = b (c) a = b (b) (d) b a b a 4 log log 6 log 9. The value of (8) (7) () is : 4. (a) 49 (b) 65 (c) 6 (d) log 5log log (a) 0 (b) (c) log (d) log 5. If 4 is equal to : A log log log 56 log, the A is : (a) (b) (c) 5 (d) 7 6. If log ( bc), y log ( ac), z log ( ab), the a b c which oe of the followig is equal to? (a) y z (b) y z (c) y z (d) ( ) ( y) ( z) 7. If a log4, b log6 4, c log486, the value of ( + abc) is : (a) ab (b) ac (c) bc (d) 0 y z 8. If a b, b c, c a, the value of (yz) is : (a) 0 (b) (c) (d) 89 o log(ta( r )) is equal to : r (a) (b) (c) (d) 0 is equal to : log a r r ( ) (a) loga ( ). (c) log a 4. If log log ( ) (b) log a (d) oe of these 7 5 ( 5) 0, the is equal to : (a) (b) (c) 4 (d) ( ) 0. The value of (0.05) log is : (a) 8 (b) 8 (c) 0 (d) 0. If log 7 a (a) a a (c) 4 a a, the log 6 6 is : a (b) a 4 a (d) 4 a [ 58 ] Mathematics for JEE-0

63 Logarithm 4. If 00!, the equal to : (a) (b) 0 (c) (d)... is log log log The umber of solutio(s) of log ( 5) 6 is/are : (a) (b) 0 (c) (d) 6. If logcos si, the the values of si lies i the iterval : (a) 5, (c) 0, (b) 0, (d) 5 5 5, 4 7. log (si ) 0, [0, 4 ], the umber of values of which are itegral multiples of 4, is : (a) 4 (b) (c) (d) 0 8. Set of real values of satisfyig the iequatio log is : ( 6 ) 0.5 (a) (, ] (b), 4 (c) [4, ) (d) oe of these. If. If, the set of '' cotais : 4 (a) (, 0) (b) (, ) (c) (, ) 5 log 4 of is : (d) oe of these 0, the ehaustive set of values (a) [0, 4] (b) (0, 4] {} (c) (0, 4). The value of 96 (d) oe of these log 4 log9 is : log log (a) (b) 0 (c) (d) 7 4. If log, log( 5) ad log are i A.P., the is equal to : (a) (b) (c) 4 (d) 8 5. If log log log log 6, the is : (a) 0 (b) 9 (c) (d) log 9. Set of real for which ( ) ( 5) is : 6. If 5 log log ( ) 4 4, the has :, (a) (, ) (4, ) (b) (4, ) (c) (, 4) (d) [, 4) (4, ) 0. If log0., the belogs to : 5 (a), (0, ) 5 (b), (c) (, ) (0, ) (d) oe of these (a) oe positive itegral value. (b) oe irratioal value. (c) two positive ratioal values. (d) o real value. 7. If 9 satisfy the equatio 8a l( 5 a ) l( a ) l, a (a) value of 'a' is (b) value of 'a' is 9 5 (c) = 5 is other solutio (d) = is other solutio the [ 59 ] Mathematics for JEE-0

64 8. Let l p, the the correct statemets are : l 0 (a) p is a ratioal umber (b) p is a irratioal umber (c) p lies i, (d) p lies i, 4 9. Let set 'S' cotai the values of for which the equatio the : log 0 log0 is satisfied, (a) total umber of elemets i 'S' are 4 (b) set 'S' cotais oly oe fractioal umber (c) set 'S' cotais oly oe irratioal umber (d) total umber of elemets i 'S' are 0. If set 'S' cotais all the real values of for which log ( ) is true, the set 'S' cotai : (a) log 5, log 7 (b) log 4, log 8 (c), (d) (, 0). Let (, ) ad y (, 6), where y = 6. If ad y satisfy the relatio value of ( y) is equal to... 8 log y log y, the 6 loga. log0a. loga 5. If a R {}, ( a) ad 5. If log0 0 log00 log4 () (9), where 0, the value of M is equal to r log 5 r equal to... si, 4 the value of () M is 4. If l a lb lc, ( y z) ( z ) ( y) y yz z z z y y the value of ( a).( b).( c) is equal to Total umber of itegral solutio(s) of the equatio log ( ) log 5 log 6 is/are... [ 60 ] Mathematics for JEE-0

65 Logarithm. (b). (a). (d) 4. (c) 5. (c) 6. (b) 7. (c) 8. (b) 9. (d) 0. (a). (c). (a). (c) 4. (c) 5. (d) 6. (b) 7. (a) 8. (b) 9. (b) 0. (a). (d). (d). (a) 4. (b) 5. (b) 6. (a, b, c) 7. (a, c) 8. (b, c) 9. (a, b) 0. (a, b, d). ( 6 ). ( 5 ). ( 5 ) 4. ( ) 5. ( ) [ 6 ] Mathematics for JEE-0

66 6. The values of 'a' ad 'b' for which equatio. Which oe of the followig fuctios is a odd fuctio? (a) 4 f ( ) loge ( ) ( )( ) (b) f ( ) loge ( )( ) (c) f ( ), where f () + f (y) = f ( + y). f ( y) for all (d), y R e f ( ) e. Domai of fuctio f ( ) log() ( ) is : (a) (, ) (c) (0, ) (b), (d),0 (0, ). If [.] represets the greatest iteger fuctio, the b a has four distict solutios, are : e (a) a (, ) ; b 0 (b) a (, ) ; b R (c) a (, ) ; b R (d) a (, ) ; b 0 7. Let f : R R ad g : R R be fuctios defied as 8. If 0 ; Q ; Q f ( ) ad g( ), the compositio f () g() is ; Q 0 ; Q : (a) oe-oe oto (c) may-oe oto (b) oe-oe ito (d) may-oe ito log 0 ad f ( ), the domai of f () is : (a) [, ) (, ) (b) [, ) (, ] (c) (, ) (, 5) (d) oe of these 9. The umber of solutios of equatio 6 cos = 0 i [0, ] are : (a) 6 (b) 4, (c) (d) 99 r 4 00 is equal to : r0 (a) 0 (b) 70 (c) 75 (d) Let f () = si a + cos a ad g () = si + cos have equal fudametal period, the 'a' is : (a) (b) (c) (d) 4 5. Let f () + f ( ) = R ad g() = f (), the g() is symmetrical about : (a) the lie (c) the lie = (b) the poit (, 0) (d) the poit, 0 0. If f : (, 6) (, 5) is a fuctio defied as. If f ( ), where [.] represets the greatest i- teger fuctio, the f () is give by : (a) + (b) + (c) + is : 5 (a), (c), R, the rage of (d) oe of these 4 ( )( ) f ( ) (b) [6, ) 5 (d), [ 6 ] Mathematics for JEE-0

67 Fuctios. If 59 f ( ) f 0 0 R {}, the f (7) is equal to : (a) 7 (b) 5 (c) 4 (d). Let f : [, ] R be a odd fuctio defied as f () = + ta +, the belogs to : 8. Area eclosed by iequality y y 4 is : (a) sq. uits (b) 5 sq. uits (c) 4 sq. uits (d) 8 sq. uits 9. Number of solutios of equatio are : (a) (b) 4 (c) 5 (d) e (a) (5, ) (b) (7, ) (c) R (d) R + 4. If f ( ) si.cos[ ] cos.si[ ], where [.] represets the greatest iteger fuctio, the fudametal period of f () is : (a) (b) (c) 6 (d) 6 5. Let f : R [0, / ) be defied as f () = ta ( + + a) the set of values of 'a' for which f () is oto, is : (a), 4 (c), 8 (b) [4, ) (d) oe of these 6. If f : R R be a fuctio satisfyig f ( + ) + f ( + 7) = R the fudametal period of f () is : (a) (b) 4 (c) 8 (d) 6 0. The umber of itegral values of 'm' for which fuc- tio f ( ) ( m ) ( m 5) is ivertible, are : (a) 4 (b) 0 (c) 6 (d) 8 log log. If a a, where a {}, the value of is : log R a log (a) a (b) log log a (c) a (d). If is havig all four real roots, the ehaustive set for ' ' belogs to : (a) [, 67] (b) [, 6] (c) [0, 75] (d) [, 8]. Domai of fuctio f () = si ( 5 + 5) is : (a) [, ] [4, 5] (b) [, ] [, 4] (c) [, ] [4, 5] (d) [, ] [, 5] 4. Let f ( ) cos cos cos.cos, the f is equal to : 8 7. Iterval of satisfyig the iequality 5 6 is give by : (a) 4 (c) 4 5 (b) 5 4 (d) (a) 0,, 4 (c), (4, 5] (b) 0,, 5 5 (d) 0,, 4 5. Domai of f ( ) 0[ ] [ ], where [.] is greatest iteger fuctio, is : (a) [, 8) (b) [, 7] (c) (, 7] (d) (, 8) [ 6 ] Mathematics for JEE-0

68 6. Let f () = (si + si ) si, the R, f ( ) is : (a) positive (b) o-positive (c) egative (d) o-egative 7. Number of solut io( s) of the equatio e = 0 is/are : (a) 0 (b) (c) (d) 4 8. If f : R R is defied by rage of f () is : f ( ) (a) [, ] (b) (, ] (c) [, ), the (d),, 9. Number of itegral values of which satisfy the iequality 4 ( ) ( 4) 4 5 ( ) (6 ) (a) ifiite (b) 8 0 (c) 9 (d) 0 are : 0. Let f () = + (a b) + ( a b) cuts the -ais at two distict poits for all values of b, where a,br, the the iterval of 'a' is : (a) [, ) (b) (, ) (c) (, ) (d) (, ] 4. Let f ( ), the f () is : e (a) eve fuctio (b) odd fuctio (c) either eve or odd fuctio (d) both eve ad odd fuctio 5. Let 6. If [ ] f ( ), where [.] is greatest iteger [ ] fuctio, the rage of f () is : (a) 0, (c) 0, (b) [0, ) (d) [0, ] ( K) f ( ) ; K 0 ( K) K of the followig statemets is true :, the which oe (a) f () + f ( ) = (b) f () + f ( ) = (c) f () + f ( + ) = (d) f () = f ( ) 7. Let f () = ad g() = [], where [.] represets the greatest iteger fuctio, the the iequality g( f ()) f (g()) is valid, if (a) (, 0) I (b) I (c) (, 0) (d) R 8. Let f () = si a ad g() = si b, where a < 0, b < 0. If umber of roots of f () = 0 is greater tha umber of roots of g() = 0, the :. If ( l ) l, the belogs to : ( l ) (a) (0, e) (b) (, e) (c) (, e) (d) (0, e). Let g() = + [] ad f () = sg (), where [.] is greatest iteger fuctio, the for all R f (g()) is :. (a) f () (c) [g()] (b) g () (d) ; if Q Let f ( ) ; if Q (a) 0 (c) (b) f (), the f ( f ( f ())) is : (d) (a) a < b (b) a > b (c) ab (d) a + b = Let f ( ),, the f 00 (009), where f ( f ( f ())) is represeted by f (), is : (a) 00 (b) 009 (c) 40 (d) oe of these 40. If f () + 6 = f () + 4 +, the f () is ecessarily o-egative i : (a) [, ] (b) (, ) (, ) (c) 6, 6 (d) oe of these [ 64 ] Mathematics for JEE-0

69 Fuctios 4. Let f : R R be a fuctio defied as f () = + k cos. If f () is ivertible fuctio, the possible values of 'k' may lie i the iterval : (a) (, ) (b) (, 5) (c) (, ) (d) ( e, ) 4. Let f () be real valued fuctio ad 4. Let 44. Let f ( + y) = f () f (a y) + f (y) f (a ) for all, y R. If for some real 'a', f (0) = 0, the : (a) f () is eve fuctio. (b) f () is periodic fuctio. (c) f () = R. (d) f " () is both eve ad odd fuctio. f ( ). f f ( ) f R {0}, the fuctio f () may be : (a) f ( ) (b) f ( ) ta (c) f ( ) (d) f ( ) 4 l 0 ; f ( ) ;, where N ad [ ] represets the greatest iteger just less tha or equal to, the which of the followig statemet(s) are true : (a) f () is odd fuctio. (b) f () is ot periodic. (c) sg ( f ()) = R. (d) f () is eve fuctio. 45. Let f : R R be a fuctio defied as f () = + +, the : (a) f () is surjective fuctio. (b) umber of itegral solutios of the equatio f ( ) 4 0 are si. 46. Let (c) umber of real solutios of the equatio f ( ) 4 0 are ifiitely may. (d) umber of real solutios of the equatio f ( ) 4si 0 are more tha eight. ( ) ; f ( ) ad ; 4 g( ) [, ]. If h() = g( f ()), the : (a) Rage of h() is [, ]. (b) Domai of h() is [0, ]. (c) Domai of h() is [, ]. (d) Number of solutios of the equatio h() sg( + + 8) = 0 are two. 47. Let A R : [5si ] [cos ] 6 0,, where [.] represets the greatest it eger fuctio. If f () = si cos A, the : (a) value of f () is less tha ta. (b) value of f () is less tha cos( ). (c) value of f () is more tha 4. 5 (d) value of f () is more tha Let N ad [.] represets the greatest iteger fuctio, where be defied as f :[0, ], f ( ) r si, r r the : (a) f () is oe-oe fuctio. (b) f () is oto fuctio. (c) f () is ito fuctio. (d) f () is may-oe fuctio. 49. Let,, be o-zero real umbers ad f :[0, ] [0, ] be a fuctio defied as f ( ) the :. If f () is bijective fuctio, (a) value of is 0. (b) value of is. (c) is root of 0. (d) oe of the possible values of ' ' ca be /. [ 65 ] Mathematics for JEE-0

70 50. Cosider the fuctio f () = 4 8k + 4 (6 k) + 4 for all R. If the graph of fuctio f () is cove dowwards, the possible values of 'k' ca be : (a) cos (cos ) (b) cot (cot e) (c) ta (d) ta Statemet : graph of y = + si ad y = cos itersect each other at three distict poits i (0, ). 5. If [] represets the greatest iteger fuctio ad f ( ) si cos, the Statemet : Rage of f () is, Followig questios are assertio ad reasoig type questios. Each of these questios cotais two statemets, Statemet (Assertio) ad Statemet (Reaso). Each of these questios has four alterative aswers, oly oe of them is the correct aswer. Select the correct aswer from the give optios : (a) Both Statemet ad Statemet are true ad Statemet is the correct eplaatio of Statemet. (b) Both Statemet ad Statemet are true but Statemet is ot the correct eplaatio of Statemet. (c) Statemet is true but Statemet is false. (d) Statemet is false but Statemet is true. 5. Let f : R R ad g : R R be two bijective fuctios ad both the fuctios are mirror images of oe aother about the lie y = 0. Statemet : If h : R R be a fuctio defied as h() = f () + g(), the h() is may oe oto fuctio Statemet : h() = h( ) = Statemet : If (0, ), the the equatio ta sec cos is havig three distict solutios Statemet : si + cos = for all [, ]. 54. Cosider the fuctio ( ) log f ( ) si ( ) 4cos ( ), the Statemet : Rage of f () is [ 5, 5] 9 ad Statemet : If R, the value of ( asi bcos ) lies i a b, a b. 55. Let fuctio f : N N be defied as f ( ) sg(cos ), the Statemet : f ( ) is bijective i ature Statemet : sg(cos ), [ 66 ] Mathematics for JEE-0

71 Fuctios Comprehesio passage () ( Questios No. - ) Let A {(, y) : ma{ y, y } 0 ;, y R} ad B {(, y) : ma{ y, y } 0 ;, y R}. O the basis of give set of ordered pairs (, y) i the -dimesioal plae, aswer the followig questios.. Area of the regio which cotai all the ordered pairs (, y) that belogs to the set of A B is equal to : (a) 00 square uits. (b) 800 square uits. (c) 400 square uits. (d) 600 square uits.. Let the ordered pair (, y) be termed as itegral poit if both ad y belog to the set of itegers, the total umber of itegral poits which belog to the set of A B are : (a) 600 (b) 000 (c) 660 (d) 860. Number of ordered pairs (, y) which satisfy the coditio y 0 ad belog to set 'A', where 0 { } represets the fractioal part of, are : (a) 00 (b) 40 (c) fiitely may (d) ifiitely may. Comprehesio passage () ( Questios No. 4-6 ) Let f : A B be bijective fuctio ad its iverse eists, where the iverse fuctio of f () is give by g : B A. If the fuctios y f ( ) ad y g( ) are represeted graphically by the cotiuous curves C ad C respectively, the aswer the followig questios. 4. If the poits (4, ) ad (, 4) lie o the curve 'C ' the miimum umber(s) of solutios of the equatio f () g () = 0 is/are : (a) (b) (c) 6 (d) 5. Let f ( ) (cos ) d, where f (0) = 0, the which oe of the followig statemets is true : (a) C ad C meet oly at poit (0, 0). (b) C ad C meet at ifiitely may poits o the lie y = 0. (c) C ad C meet at fiitely may poits o the lie y + = 0. (d) All the poits of itersectio of C ad C lie o the lie y + = Let p A ad q B, where p q 0. If poit (p, q) lies o C but ot o C, the : (a) C ad C ca't meet o the lie y = 0. (b) C ad C do't meet each other. (c) either C ad C do't meet each other or they meet o the lie y = 0. (d) C ad C meet o the lie y = 0. Comprehesio passage () ( Questios No. 7-9 ) Let f : N N be a fuctio defied by f () = D, where D k represets the largest atural umber which ca be obtaied by rearragig the digits of atural umber k. For eample : f (7) = 7, f (568) = 865, f (89) = etc. O the basis of give defiitio of f (), aswer the followig questios. 7. Fuctio f () is : (a) oe-oe ad ito. (b) may-oe ad ito. (c) oe-oe ad oto. (d) may-oe ad oto. 8. If atural umber ' 0 ' divides f ( ) for every N, the maimum possible value of ' 0 ' is equal to : (a) (b) 4 (c) 9 (d) 9. Let f ( ) 9985, where N, the maimum umber of possible distict values of ' ' are : (a) more tha 00. (b) less tha 50. (c) more tha 55. (d) less tha 0. [ 67 ] Mathematics for JEE-0

72 Comprehesio passage (4) ( Questios No. 0- ) Let f : R R be a fuctio defied as f () = , ad ma{ f ( t) ; 4 t } ; 4 0 g( ) mi{ f ( t) ; 0 t } ; 0 f ( ) 6 ; O the basis of give defiitios of f () ad g(), aswer the followig questios. 0. Total umber of locatio(s) at which the graph of y g( ) breaks i [ 4, ) is/are :. Let 5 f ( ) si.cos( ), where I, ad the period of f () is, the total umber of possible values of '' is equal to Total umber of itegral values of i for which the equatio, 4 4 si (si ) si (si ) is satisfied, are... (a) (b) (c) 0 (d) 4. If the equatio f ( ) 0 is havig eactly three distict real roots, the total umber of possible itegral values of ' ' are : (a) 0 (b) (c) 40 (d) 4. If the equatio g( ) 0 is havig ifiitely may real solutios, the umber of possible itegral values of ' ' is/are : (a) 0 (b) (c) (d) 5. Let N, ad f : N N be a fuctio defied by f ( ) ( r)!. If P () ad Q () are polyomials i r such that f ( + ) = P () f ( + ) + Q () f () for all N, the value of P (0) + Q (6) is equal to Let the equatio (P + ) ( ) (P ) ( + + ) = 0 is havig two distict ad real roots ad f ( ), where f f ( ) f f P, the value of ' ' is Let f () ad g() be eve ad odd fuctios respec- tively, where f ( ) f g( ), the value of f (4) is equal to Let f ( ) R, ad [] represets the greatest iteger fuctio of, the match the coditios/epressios i colum (I) with statemet(s) i colum (II). Colum (I) Colum (II) (a) If (, ), the f () satisfies (p) 0 [ f ( )] (b) If [, ], the f () satisfies (q) [ f ( )] 0 (c) If [ 4, ], the f () satisfies (r) [ f ( )] 0 (d) If [, ), the f () satisfies (s) [ f ( )] [ 68 ] Mathematics for JEE-0

73 Fuctios 9. Match the fuctios i colum (I) with their correspodig rage i colum (II). Colum (I) (a) f ( ) cos(si ) si(cos ) for all, (b) f ( ) cos(cos (si )) for all [0, ] (c) f ( ) cos(cos ) all, (d) f ( ) cos(si ) for all 0, 8 Colum (II) (p) [cos, ] (q) [cos, cos(cos)] (r) [cos(cos), cos] (s) [cos, si] (t) si, cos 0. Match the followig colums (I) ad (II) Colum (I) Colum (II) ta cot (a) Domai of f () = cos cotai(s) (p) 4 (b) Domai of f ( ) log si ( ) / (c) Rage of cotai(s) (q) f ( ) ta cotai(s) (r) 8 (d) If [ ] represets the greatest iteger fuctio of, (s) 8 ad f ( ) [cos ] [si ], the domai of f () (t) 4 cotai(s) [ 69 ] Mathematics for JEE-0

74 . (a). (a). (c) 4. (d) 5. (d) E 6. (c) 7. (a) 8. (b) 9. (c) 0. (a). (b). (c). (a) 4. (b) 5. (d) 6. (c) 7. (d) 8. (a) 9. (b) 0. (c). (c). (d). (b) 4. (a) 5. (a) 6. (d) 7. (a) 8. (b) 9. (c) 0. (b). (a). (c). (b) 4. (a) 5. (c) 6. (b) 7. (d) 8. (b) 9. (b) 40. (a) 4. (a, c) 4. (a, b, c, d) 4. (a, b, c, d) 44. (a, d) 45. (a,b, c, d) 46. (a, d) 47. (a, d) 48. (c, d) 49. (b, c, d) 50. (a, b, d) 5. (d) 5. (d) 5. (d) 54. (a) 55. (b). (d). (c). (d) 4. (b) 5. (d) E 6. (c) 7. (b) 8. (c) 9. (c) 0. (c). (d). (c). ( 8 ) 4. ( 7 ) 5. ( 5 ) 6. ( ) 7. ( 0 ) 8. (a) p, q, s 9. (a) s 0. (a) r (b) p, q, s (b) q (b) p, t (c) p, q, s (c) p (c) r, s, t (d) p, q, r (d) p (d) q, r, t [ 70 ] Mathematics for JEE-0

75 .. t. e t dt 0 lim 0 cos( ) (a) (c) 4 is equal to : (b) (d) ( ) lim si.si...si (a) 4 4/ (b) e / (c) e /8 (d) e / is equal to :. Let f () be differetiable ad f () = ad f '() = 4, 4. the f ( ) lim f () is equal to : (a) (b) e (c) 0 (d) e ( 4... ) lim 8 (a) 0 (b) 5 (c) 6 (d) 4 is : l(cos ) lim si 0 is : (a) (b) (c) lim(cos ) 0 cot( ) is : (d) e (a) (b) e (c) e (d) / 8. The value of lim e is equal to : (a) (c) 0 (b) e (d) oe of these 9. Let f () be real fuctio ad g() is bouded fuctio for all (a) f () R f ( ). e g( ), the lim e (b) g() (c) 0 (d) 0. If the graph of fuctio y = f () is havig a uique taget of fiite slope at locatio (a, 0), the log e ( 6 f ( )) lim is equal to : a f ( ) (a) 0 (b) is : 5. If f ( ) is differetiable ad f (0) 0, such that ( ) ( ) ( ) f y f y y f y, the f ( ) lim is equal to : (a) (b) 0 (c) (d). Let (c) (d) / ( a cos ) b si lim, the (a + b) is : 0 (a) (b) (c) 4 (d) [ 7 ] Mathematics for JEE-0

76 Limits lim... is equal to : (a) 6 (b) 6 (c) 64 (d) oe of these. If ormal to curve y = f () at = 0 is y + = 0, the lim 0 ( ) 5 (4 ) 4 (7 f f f ) is : 7. I which of the followig case(s), the limit does't eist? (a) lim 0 sec (c) lim 4 ta (b) lim(si ) 0 (d) lim( l 0 ) 8. Let f () be differetiable fuctio for all R ad (a) (c) 4. Let f :[, ] R ad 5. (b) (d) where 0 lim cos, f (0) 0, f '(0) lim f, the value of lim ( ) cos is equal to : (a) (c) (b) 0 (d) si( ( si )) lim 0 ta is equal to : (a) (b) (c) (d) f (). If the : (a) 9. Let 7 f () 6 f ( ) f ( ) lim for every > 0, (b) f ( ) has local miima at / () (c) f ( ) is strictly icreasig for all (d) f "( ) 0 R f ( ) lim cot, k 0 the k (a) f ( ) is icreasig fuctio for all R. (b) f () is differetiable for all R {0}. (c) [ f ( )] d 0, where [.] represets greatest iteger fuctio. (d) f ( ) is odd fuctio If lim( ) lim, the : (a) (b) 0 (c) 4 (d) 6. Let m, I ad f ( ) log e ( ) m cos ( ) for all (0, ).If g( ) e R ad lim f ( ) g '( ), the : (a) m + = 5 (b) m + = 4 (c) m = (d) m = 0 Followig questios are assertio ad reasoig type questios. Each of these questios cotais two statemets, Statemet (Assertio) ad Statemet (Reaso). Each of these questios has four alterative aswers, oly oe of them is the correct aswer. Select the correct aswer from the give optios : [ 7 ] Mathematics for JEE-0

77 (a) Both Statemet ad Statemet are true ad Statemet is the correct eplaatio of Statemet. (b) Both Statemet ad Statemet are true but Statemet is ot the correct eplaatio of Statemet. (c) Statemet is true but Statemet is false. (d) Statemet is false but Statemet is true.. Statemet : Let L lim 4 7, the limitig value 'L' approaches to positive ifiity Statemet : The form of idetermiacy i 'L' is form. a. Statemet : Let a ad a a. Let all N, the lim ( a ) is equal to, for Statemet : Sequece {a } for all N is covergig i ature. S r., the ( r )! r0 r Statemet : lim S Statemet : lim 0 ( )! 4. Statemet : Let L lim... value of limit 'L' is equal to Statemet : r lim f f ( ) d r 0 5. Statemet : Let L value of si (L) =, the lim (si) (cos), the si cos.si Statemet : lim 0 ta si [ 7 ] Mathematics for JEE-0

78 Limits 4. If area of triagles PAB ad PCD are 'A ' ad 'A '. Comprehesio passage () ( Questios No. - ) Let f () ad g() be cotiuous fuctios for all R ad f (0) = g (0) = 0. If g( ).si lim 0 f ( cos ) f ( ) ad lim, the aswer the followig 0 questios. lim. g is equal to : (a) (c) 4 (b) (d) 4 g(cos ). lim is equal to : 0 4. (a) (c) lim si( ) 0 (a) 6 (b) (d) f ( ) g( ) is equal to : (b) 4 respectively, the lim A is equal to : 0 A (a) (b) 4 (c) (d) 6 5. If area of triagle PAB is 'A ' ad area eclosed by arc AB with the chord AB is 'A ', the lim A is 0 A equal to : (a) / (b) 5/ (c) (d) 6. If area of triagle PCD is 'A ' ad area eclosed by arc AB with the chord AB is 'A ', the lim A is 0 A equal to : (a) 8 (b) 5 4 (c) (d) 6 f ( ) 7. Let f ( ) si(si ) si ad L lim 0. If limitig value 'L' is o-zero ad fiite, the value of '' must be equal to... si 8. Let L lim 0 a e bl( ) c e. If the value of L is /, the (b + a c) is equal to... (c) 6 (d) 4 Comprehesio passage () ( Questios No. 4-6 ) Let poits 'A' ad 'B' lies o t he circle C : + y = 0, where AOB, 'O' beig the origi. If tagets draw at 'A' ad 'B' to 'C ' meet at 'P', ad the taget to 'C ' draw at the mid-poit of arc AB meet the lies PA ad PB at 'C' ad 'D' respectively, the aswer the followig questios. 9. Let lim ad S r r r... 4 S to... r r r is equal to 'L', the value of L is equal 0. Let p() be a polyomial of degree 4 havig the poits of etremum at = ad =, where p( ) lim. The value of p() is [ 74 ] Mathematics for JEE-0

79 . Let [] represets the greatest iteger which is just less tha or equal to, the match the followig colums (I) ad (II). Colum (I) Colum (II) (a) si ta lim 0 (p) (b) si lim si 0 (c) lim [ ] [ ] 0 (q) 0 (r) (d) lim 0 4. Let L 4 a b 4 c d (s) 4 (t) limit does't eist lim, the match the colums (I) ad (II). Colum (I) Colum (II) (a) If L = 4, the value of (c a) is (p) (b) If L =, the value of 'c' is (q) (c) If L = 6, b, the value of (a + b) ca be (r) R (s) 4 (d) If L =, d, the value of (c + d) ca be (t) 0 R [ 75 ] Mathematics for JEE-0

80 Limits. (b). (a). (b) 4. (a) 5. (c) E 6. (a) 7. (d) 8. (a) 9. (a) 0. (c). (c). (d). (b) 4. (c) 5. (a) 6. (a, b, d) 7. (a, b) 8. (a, b, c, d) 9. (b, d) 0. (b, d). (d). (b). (a) 4. (b) 5. (b). (c). (a). (c) 4. (b) 5. (a) E 6. (a) 7. ( 6 ) 8. ( 6 ) 9. ( 8 ) 0. ( 0 ). (a) r. (a) r (b) s (b) p (c) t (c) r, s (d) p (d) p, q, t [ 76 ] Mathematics for JEE-0

81 (c) 5 (d) 5. Let f () = mi {, 4 5, }, the total umber of poits of o-differetiability is/are : (a) 4 (b) (c) (d). Total umber of locatios of o-differetiability for the fuctio f () = + cos + ta 4 i the iterval (, ) is/are : (a) (b) (c) (d) 4. If fuctio f : R R satisfy the coditio f ( y) f ( y) cos si y f ( y) f ( y) si cos y ad f '(0), the : (a) f " () f () = 0 (b) 4 f " () + f ' () = 0 (c) 4 f "() + f () = 0 (d) 4 f ' () + f " () = 0 4. The umber of poits of o-differetiability of f () = ma{si, cos, 0} i (0, ), where N, are give by : (a) 4 (c) 6 (b) (d) f ( ) ; 0 7. Let f () = + ad g( ), f ( ) ; 0 the : (a) g() is cotiuous R (b) g() is cotiuous R (c) g() is discotiuous R (d) g() is cotiuous R 8. Let 9. If a ; f ( ) b ; fuctio for all R, the ( a b) is : (a) 0 (b) 8 (c) 5 (d) 5 be a differetiable f ( ) a b has eactly three poits of o-differetiability, the (a) b R, a 0 (b) a > 0, b = 0 (c) b = 0, a R (d) a < 0, b = 0 0. If f () = [ 5], [.] is greatest iteger fuctio, the total umber of poits i (, ) where f () is ot cotiuous is/are : (a) 0 (b) (c) (d) 5 5. Let f ( ) e the f () is o-differetiable for belogs to : (a) {0, } (b) {0, } (c) {, l } (d) {0, l } 6. Let f () = , the f ( h) f () lim, is equal to : h0 h h (a) (b) 5. cosec (cos si ) ; 0 Let f ( ) a ; 0 / / / e e e ; 0 / / ae be cotiuous at locatio = 0, the value of (a + b) is : (a) e (b) e e e (c) e e (d) e e be [ 77 ] Mathematics for JEE-0