PLANCESS RANK ACCELERATOR

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1 PLANCESS RANK ACCELERATOR MATHEMATICS FOR JEE MAIN & ADVANCED Sequeces d Seies 000questios with topic wise execises 000 polems of IIT-JEE & AIEEE exms of lst yes Levels of Execises ctegoized ito JEE Mi & Advced 7 Types of Questios sed o ltest JEE ptte Detiled Solutios of ll questios e ville PlcEssetil Questios ecommeded fo evisio

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3 Sequeces d Seies EXERCISE JEE MAIN/BOARDS Q. I G.P. sum of tems is 6. Fist tem is d commo tio is. Fid. Q. The sum of ifiite geometic pogessio is d the sum of the geometic pogessio mde fom the cues of this ifiite seies i. The fid the seies. Q. Sum of tems of the seies, () () Q. If,, c e i A.P., pove tht () c, c, e lso i A.P. (),, c c e lso i A.P. (c) ( c), (c ), c ( ) e lso i A.P. (d),,c e lso i A.P. c c Q. The sum of thee umes i A.P. is d the sum of thei cues is 88. Fid the umes. Q.6 Fid the sum of the iteges etwee d 00 which e () multiple of () multiple of 7 (c) multiple of d 7 Q.7 The sum of fist tems of two A.P. s e i the tio ( ): ( ). Fid the tio of thei th tems. Q.8 If the p th tem of A.P. is x d q th tem is y, show tht the sum fist (p q) tems is p q x y x y p q Q. If,, c e i H.P. pove tht c,, e i H.P. c c Q. Fid the sum of tems of the seies Q.0 Let,, c, d, e e five el umes such tht,, c e i A.P.;, c, d e i G.P.; c, d, e e i H.P. If d e 8, fid ll possile vlues of, c d d. Q. Fid the sum of fist tems of the seies: Q. Fid the sum of fist tems of the seies: 6. Q. The H.M of two umes is d thei A.M. (A) d G.M. (G) stisfy the eltio A G 7. Fid the umes. Q. Fid the sum of fist 0 tems of the seies: ( ) ( ) ( 6 ).. Q. Fid the sum of fist 0 tems of the seies: 7. Q.6 Fid thee umes,, c etwee d 8 such tht: (i) thei sum is. (ii) the umes,, e cosecutive tems of A.P. d (iii) the umes, c 8 e cosecutive tems of G.P. Q.7 If > 0, > 0 d c > 0, pove tht: ( c) c Q.8 If A, A, G, G, ; d H,H e two A.M. s, G.M. s d H.M s etwee two GG A A umes, the pove tht: H H H H Q. Fid the coefficiet of x d x 8 i the polyomil: (x ) (x ) (x ) (x 00). Q.0 The iteio gles of polygo e i A.P. The smllest gle is 0º d the commo diffeece is º. Fid the ume of sides of the polygo Q. A ume cosists of thee digits i G.P.. The sum of the digits t uits d hudeds plce exceeds twice the digit t tes plce y d the sum of the digits t tes d hudeds plce is two thid of the sum of the digits t tes d uits plce. Fid the ume. Q. tees e plted i stight lie t itevls of metes. To wte them the gdee must ig wte fo ech tee.

4 Sequeces d Seies septely fom well 0 metes fom the fist tee i lie with the tees. How f he will hve to cove i ode to wte ll the tee egiig with the fist if he stts fom the well. Q. Ntul umes hve ee gouped i the followig wy ; (, ) ; (,, 6); (7, 8,, 0) ; Show tht the sum of the umes i the th goup is ( ) Q. I thee seies of GP s, the coespodig umes i G.P. e sutcted d the diffeece of the umes e lso foud to e i G.P. Pove tht the thee sequeces hve the sme commo tio. Q. If,,,. Ae i A.P such tht i 0, show tht S S... Also evlute lim S. Q.6 If ithmetic mes d hmoic mes e iseted etwee d, pove tht 6 A, whee A is y ithmetic me d H H the coespodig hmoic me. Q.7 If x y z d x, y, z e positive umes, show tht ( x) ( y) ( z) 8 xyz. Q.8 Show tht y positive itegl powe (gete th ) of positive itege m, is the sum of m cosecutive odd positive iteges. Fid the fist odd itege fo m ( > )...

5 Sequeces d Seies EXERCISE JEE MAIN Q. If,, c e distict positive el i H.P., the the vlue of the expessio, c c is equl to (A) (B) (C) (D) Q. The sum of ifiity of the seies... is equl to (A) (B) / (C) (D) oe Q. Alog od lies odd ume of stoes plced t itevls of 0 m. These stoes hve to e ssemled oud the middle stoe. A peso c cy oly oe stoe t time. A m cied out the jo sttig with the stoe i the middle, cyig stoes i successio, theey coveig distce of.8 km. The the ume of stoes is (A) (B) (C) (D) Q. If S. () the the vlue of the sum 6.. (00) is (A) S 0 (B) S (C) S (D) S 00 Q. I A.P. with fist tem d the commo diffeece d (, d 0), the tio of the sum of the fist tems to sum of tems succeedig them does ot deped o. The the tio / d the tio, espectively e (A) (C),, (B), (D), Q.6 If x R, the umes ( x x ), / ( x x ) fom A.P. the must lie i the itevl (A) [, ] (B) [, ] (C) [, ] (D) [, ] Q.7 If the sum of the fist tems of ithmeticl pogessio equls tht of the fist tems, the the sum of its fist 0 tems, is (A) equl to 0 (B) equl to (C) equl to (D) o uique Q.8 Lets, s, s.. d t,t,t. e two ithmetic sequece such tht s t 0; s t d vlue of s s t t is 0 (A) 8/ (B) / (C) /8 (D) s t i i i i. The the Q. Let, I A.P. with commo diffeece d d ll whose tems e o-zeo. If ppoches ifiity, the the sum... will ppoch (A) (C) d d (B) d (D) d Q.0 The sum of the fist thee of icesig G.P. is d the sum of thei sques is 8. The the sum of its fist tem is (A) ( ) (B) (C) 6 (D) 6( ) Q. The sum is equl to (A) / (B) / (C) /8 (D) ½ Q. If d (l ) (l ) (l ).. [l (l ) (l ) (l ) ], the is equl to (A) e / (B) e / (C) e / (D) e /.

6 Sequeces d Seies PREVIOUS YEARS QUESTION JEE MAIN Q. If,, c d d p e distict el umes such tht ( c )p ( c cd)p ( c d ) 0, the,, c, d (87) (A) e i H.P. (B) e i G.P. (C) e i H.P. (D) stisfy cd (A) (C) ( )c 6 ( )c (B) (D) ( )c ( )c 6 Q. Sum of the fist tems of the seies 7... is equl to 8 6 (A) (B) (C) (D) Q. If x >, y >, z > e i G.P. the,, e i (8) lx ly lz (A) AP (C) GP (B) HP (D) Noe Q. If,, c, d e positive el ume such tht c d, the M ( ) (c d) stisfies the eltio (000) (A) 0 < M (B) M (C) M (D) M Q. Let the positive umes,, c, d e i A.P. the c, d, cd, cd e (00) (A) ot i AP/GP/HP (B) i AP (C) i GP (D) i HP Q.6 Suppose,, c e i AP d,, c e i G.P. If < < c d c vlue of is (00) (A) (B) (C) (D), the the Q.7 A ifiite GP hs fist tem x d sum, the x elogs to (00) (A) x < 0 (B) 0 < x < 0 (C) 0 < x < 0 (D) x > 0 Q.8 If the sum of fist tems of AP is c, the the sum of sques of these tems is (00).

7 Sequeces d Seies EXERCISE JEE ADVANCED Q. ()The hmoic me of two umes is. The ithmetic me A & the geometic me G stisfy the eltio A G 7. Fid the two umes. () The AM of two umes exceeds thei GM y & HM y 7. Fid the umes. Q. If the 0 th tem of HP is d st tem of the sme HP is 0, the fid the 0 th tem. Q. If six, si x d cosx.six fom icesig geometic sequece, fid the umeicl vlue of cosx. Also fid the commo tio of geometic sequece. Q. If,, c, d, e e umes such tht,, c e i AP;, c, d e i GP & c, d, e e i HP the, (i) Pove tht, c, e e i GP (ii) Pove tht c ( ) / (iii) If & e 8, fid ll possile vlues of, c, d. Q. Let d e two el vlues of fo which the umes,, tke i tht ode fom ithmetic pogessio. If d e two el vlues of fo which the umes,, 6 tke i tht ode fom geometic pogessio, the fid the vlue of ( ). Q.6 Two distict, el ifiite geometic seies ech hve sum of d hve the sme secod tem. The thid tem of oe of the seies is 8. If the secod tem of oth the seies c e witte i the fom m p, whee m, d p e positive iteges, d m is ot divisile y the sque of y pime, fid the vlue of 00m 0p. Q.7 Let S (). Fid [s]. Whee [y] deotes lgest itege less th o equl to y. Q.8 Give tht the cuic x x x 0 ( 0) hs ll thee positive oots. Fid the hmoic me of the oots idepedet of d, hece deduce tht the oot e ll equl. Fid lso the miimum vlue of ( ), if d N. Q. A compute solved sevel polems i successio. The time it took the compute to solve ech successive polem ws the sme ume of times smlle th the time it took to solve the pecedig polem. How my polems wee suggested to the compute if it spet 6. mi to solve ll the polems except fo the fist, 7 mi to solve ll the polems except fo the lst oe,. mi to solve ll the polems except fo the fist two? Q.0 The sequece,,, 8 stisfies the eltio fo,,,. 7 d hs the sum equl to. Evlute k Q. Let d e positive iteges. The vlue of xyz is o k, ccodig s, x, y, z, e i ithmetic pogessio o hmoic pogessio esp.. Fid the vlue of ( ). Q. If the oots of 0x cx x 7 0 e i hmoic pogessio, the fid c d ll the oots. Q. If,, c e i GP & logc, logc, log e i AP, the fid the commo diffeece of the AP. Q. The fist tem of geometic pogessio is equl to, the thid tem is 6, d the ithmetic me of the fist d thid tem to the tem is i the tio :. Fid the positive itegl vlue of. Q. I GP the tio of the sum of the fist eleve tems to the sum of the lst eleve tems is /8 d the tio of the sum of ll the tems without the fist ie to the sum of ll the tems with out the lst ie is. Fid the ume of tems i the GP. Q.6 If sum of fist tems of AP (hvig positive tems) is give y S(T) *( T) whee T is the th tem of seies the T (, N). Fid the vlue of ( )...

8 Sequeces d Seies Q.7 Give thee digit ume whose digits e thee successive tems of G.P. If we sutct 7 fom it, we get ume witte y the sme digits i the evese ode. Now if we sutct fou fom the huded s digit of the iitil ume d leve the othe digits uchged, we get ume whose digits e successive tems of A.P. Fid the ume. Q.8 Fo 0 < <, let S() ( si) cos ( si si ) cos... The fid the vlue of ( ) S. x Q. If t x,t,t x i ode e thee cosecutive tems of G.P., the sum of ll the solutios i [0, ] is k. Fid the vlue of k..6

9 Sequeces d Seies EXERCISE JEE ADVANCED Q. The ithmetic me of the ie umes i the give set {,,, } is digit ume N, ll whose digits e distict. The ume N does ot coti the digit (A) 0 (B) (C) (D) Q. 60 is the tio of two k k k (k k ) eltive pime positive iteges m d. The vlue of (m ) is equl to (A) (B) (C) (D) 7 Q. The sum (A) (C) k k k is equl to k (B) (D) Noe Q. A cicle of dius is iscied i sque. The mid poit of sides of the sque hve ee coected y lie segmet d ew sque esulted. The sides of these sque wee lso coected y segmets so tht ew sque ws otied d so o, the the dius of the cicle iscied i the th sque is (A) (C) (B) (D) Assetio d Reso (A) Sttemet- is tue, sttemet- is tue d sttmet- is coect expltio fo sttmet-. (B) Sttemet- is tue, sttemet- is tue d sttmet- is NOT the coect expltio fo sttmet-. (C) Sttemet- is tue, sttemet - is flse. (D) Sttemet- is flse, sttemet- is tue Q. Sttemet -: If 7 c ( c) d c the 0, c whee,, c e positive el umes. Sttemet-: Fo positive el umes A.M. G.M. Q.6 Sttemet -: If 7 c ( c) d c the 0, whee,, c e c positive el umes. Sttemet-: Fo positive el umes A.M. G.M. Q.7 Let,,... d,,... e ithmetic pogessios such tht, 7 d The, (A) the diffeece etwee successive tems i pogessio is opposite of the diffeece i pogessio. (B) 00, fo y. (C) ( ), ( ), ( ),... e i AP (D) 00 ( ) 0000 Q.8 If si (x y), si x d si (x y) e i H.P. the the vlue of si x.sec (A) (B) / (C) (D) / y Q. The sum of the fist thee tems of the G.P. i which the diffeece etwee the secod d the fist tem is 6 d the diffeece etwee the fouth d the thid tem, is (A) (B) 0. (C) 7 (D) 7 Q.0 If the oots of the equtio. x px qx 0 fom icesig GP, whee p d q e el, the (A) p q 0 (B) p (, ) (C) oe of the oots is uity (D) oe oot is smlle th Q. If the tiplets log, log, log c d (log log ), (log logc), (log c log) e i ithmetic pogessio the (A) 8 ( c) 8( c ) (B),, c e i GP (C),, c e i HP.7

10 Sequeces d Seies (D),, c c e the legths of the sides of tigle. (Assume ll logithmic tems to e defied) Q. x, x e the oots of the equtio x x A 0; x, x e oots of the equtio x x B 0, such tht x, x, x, x fom icesig GP the (A) A (B) B (C) x x (D) x x 0.8

11 Sequeces d Seies PREVIOUS YEARS QUESTION JEE ADVANCED Q. If the fist d the ( )th tem of AP, GP d HP e equl d thei th tems e,, d c espectively, the (88) (A) c (B) c (C) c (D) c 0 Q. Let S, S. Be sques such tht fo ech the legth of side of S equls the legth of digol of S. If the legth of side of S is 0 cm, the fo which of the followig vlues of is the e of S less th sq. cm? () (A) 7 (B) 8 (C) (D) 0 Q. Let Sk, k,,.., 00, deotes the sum of the ifiite geometic seies whose fist tem is k k! vlue of d the commo tio is k. The the 00 (k k )S k is (00) k 00 00! Q. Let,,,.., e el umes stisfyig, 7 > 0 d k k k, fo k,,.,. If , the the vlue of is equl to (00) Q. Le,,,. 00 e ithmetic pogessio with d Sp p i, p 00. Fo y itege with 0, let m. If S S m does ot deped o, the is. (0) Pssge Bsed Polems Pssge I Let A, G, H deote the ithmetic, geometic d hmoic mes, espectively, of two distict positive umes. Fo >, let A d H hve ithmetic, geometic d hmoic mes s A, G, H espectively. Q.6 Which oe of the followig sttemets is coect? i (A) G > G > G >... (B) G < G < G <... (C) G G G... (D) G < G < G <... d G > G > G 6 >... Q.7 Which oe of the followig sttemets is coect? (A) A > A > A >... (B) A < A < A <... (C) A > A > A >... d A < A < A 6 <... (D) A < A < A <... d A > A > A 6 >... Q.8 Which oe of the followig sttemets is coect? (A) H > H > H >... (B) H < H < H <... (C) H > H > H >... d H <H < H 6 <... (D) H < H < H <... d H >H > H 6 >... Pssge II Let V deote the sum of the fist '' tems of ithmetic pogessio (A.P.), whose fist tem is '' d the commo diffeece is ( ). Let T V V d Q T T fo,,... (007) Q. The sum V V... V is (A) (B) (C) ( )( ) ( )( ) ( ) (D) ( ) Q.0 T is lwys (A) odd ume (B) eve ume (C) pime ume (D) composite ume Q. Which oe of the followig is coect sttemet? (A) Q, Q, Q... e i A.P. with commo diffeece. (B) Q, Q, Q... e i A.P. with commo diffeece 6. (C) Q, Q, Q... e i A.P. with commo diffeece. (D) Q Q Q....

12 Sequeces d Seies PLANCESSENTIAL QUESTIONS EXERCISE JEE MAIN/BOARDS Q. Q. Q. Q.7 Q. Q. Q.7 EXERCISE JEE MAIN Q. Q. Q.0 Q. PREVIOUS YEARS QUESTIONS JEE MAIN Q. Q. Q.8 EXERCISE JEE ADVANCED Q.6 Q. Q. Q. Q.7 EXERCISE JEE ADVANCED Q. Q. Q. Q. PREVIOUS YEARS QUESTIONS JEE ADVANCED Q. Q. Q. Q.6 Q.7 Q.8.0

13 Sequeces d Seies ANSWER KEY EXERCISE JEE MAIN/BOARDS Q. 6 Q.... is seies Q. () 8 0 () [0 0] 7 Q. (,, 6) o (6,, ) Q.6 () 66 () 8 (c) Q.7 6: 8 Q.0 c 6,, d ;, c 6, d 8 Q. ( )( )( ) Q. ( ) Q. 6, Q. 60 Q Q.6, 8, c. Q. 00, [(00) 80] Q.0 Q. 6 Q. 70 m Q. ( ) Q.8 m m EXERCISE JEE MAIN Q. B Q. A Q. C Q. D Q. C Q.6 D Q.7 A Q.8 C Q. A Q.0 A Q. C Q. D PREVIOUS YEARS QUESTIONS JEE MAIN Q. B Q. C Q. B Q. A Q. D Q.6 D Q.7 C Q.8 C EXERCISE JEE ADVANCED Q. (A) 6, ; (B) 0, 0 Q. Q. ; Q. (iii), c 6, d o, c 6, d 8 Q. Q.6 8 Q.7 Q.8 8 Q. 8 polems, 7. miutes Q.0 Q. 0 Q. C ; (, /, /) Q. / Q. Q. 8 Q.6 6 Q.7 Q.8 Q. 0 EXERCISE JEE ADVANCED Q. A Q. D Q. B Q. A Q. D Q.6 A Q.7 ABCD Q.8 BC Q. AB PREVIOUS YEARS QUESTIONS JEE ADVANCED Q. ABD Q. BCD Q. Q. 0 Q. o Q.6 C Q.7 A Q.8 B Q. B Q.0 ACD Q. BD Q. ABCD Q.0 D Q. B.

14 Sequeces d Seies SOLUTIONS Sol. Sum of tems is ( ) 6 give, ( ) ( ) Sol. sum of ifiite g.p. is ( ) Seies is,,. ( < ), (), ( ).. () Fist tem of this ifiite seies is d tio is Hece sum of this ifiite seies is Give 8( ) EXERCISE JEE MAIN ( ) ( ) 0 0 ( ) ( ) 0, < Seies is 8 Sol. ) sum upto tems S tems 7[ ] 7 [ ] [ ] [ ( )] 0.( (0.) ) 0. ( (0.) ) ) [ ] 6 [ ] [ ] 0(0 ) 0 0 (0 ) Sol.,, c e i AP ) c, c, e lso i AP,, c e i AP c c c Commo diffeece etwee tem of give AP, c which e equl y equtio (i) Hece give AP c, c, is AP ) c, c, e lso i AP Commo diffeece ( ) c, (c ) c By equtio (i) c ie commo diffeece etwee tems is sme Hece the give seies is i AP C) ( c), (c ), c ( ) Commo diffeece c c c d d c( ) ( ) c c, d S.

15 Sequeces d Seies (c c) ( ) (c c) (c ) [fom eq.(i) d (c ) c (c ) (c c) (c ) d d Hece give seies is AP d) c d d c c, ( ) c ( )( ), c d d c (c ) c c (c )(c ) c ( ) c d ( ) c (c ) c fom eq (i) d d Hece give seies is lso AP Sol. Sum of fist umes i AP is Let,, e the fist umes ( ) ( ) ( ) 6 ± So umes e,,, (,, 6) (fo ) (, ) fo ( ) (6,, ) Sol.6 )sum of iteges etwee & 00 which e multiple of, 6,, 8 66 Hece sum (66) [ l] [ 8] [0] 66 ii) Multiple of 7 7,, 6 8 sum 8 [ l] [7 6] [0] 8 iii)multiple of & 7,, 6 8 Sum [ 8]. 0 Sol.7 sum of fist tems of AP s e i tio Let the AP e, d d AP e, d [ ( )d [ ( )d ] ( ) d ( )d Rtio of th tem well e (i) d d Puttig 7 i equtio (i), we well get desied tio d d Sol.8 give (7) (7) (p )d x (i) (q )d y (ii) sum of fist (p q) tems p q [ (p q )d] (iii) Sutctig (ii) fom (i) (p q)d x y d x y p q (p )(x y) p q x (px py x y) qx py x y x p q p q Puttig vlues of & d i equtio (iii) Spq p q x y yx py (p q )(x y) p q p q x y qx py px qy p q p q (p q) (x y) (x y) p q x y xy (p q) p q p q Sol.,, c e i HP ie c c (i) c S.

16 Sequeces d Seies c c give seies] [whee & c e st & d tems of c ( )( c) c c c c ( c) c c c c c (fom equtio (i) (c) c ( c)( c) c Middle tem of give seies, hece ie give Seies is H.P Sol. 7 Sum of fist tems S S Sutctig (ii) fom (i) S S. ( ) ( ) ( ) c c (i) ( ) s 7 s s s S ( ) ( ) ( ). (ii) 6 6( ) ( ) (. ) Sol.0,, c e i AP c (i), c, d i GP c d (ii) c, d, e e i HP d e 8 ce c e ( ) c fom (i) ( ) ( ) ( ) d fom (ii) ( )8 fom (iii) ( ) ( ) ( 8) , c, 6 d, 8, c, d [, 6, ] d [, 6, 8] Sol. S T ( ) ( ) ( ) S T ( ) ()( )( ) ( ) 6 ( ) ( ) ( ) ( ) ( ) ( 6) Sol. S 6 (fist umes) ( )( )( ) tems 6 tems ( ) S.

17 Sequeces d Seies ( ) ( ) 6 ( ) ( ) ( )( )( ) 6 ( ( ) ( ) ( ) ( )( ) ( )( ) 6 ( )()( ) ( )( ) 6 () [()()( )( )] ( ) ( ) [ ] ( ) Sol. HM of umes is ie (i) AM GM A G [fom (i)] 8 [ 6, ]; [, 6] [ ] Sol. ( ) ( ) (7 6 ) 7 ( 0 ) ( 0 ).( ) (0)(0 ) (. ) 6(. ) (. 6. ) Sol. S...7 T ( ) S T ( ) ( ) ( ) ( ) ( ) 7 7 ( ) S ( )( ) 6 ( ) ( ) (687) 8800 Sol.6,, c (, 8) c (i) (ii) c 8 (iii) c 7 (7 ) 8( ) ( ) ( ) ,, 8 c, umes e (, 8, ) Sol.7 > 0, > 0, c > 0 To pove ( c) c We kow tht AM GM HM AM HM Fo umes,, c AM c HM c S.

18 Sequeces d Seies c c ( c) c Sol.8 Let the ume e,, A, A, e i AP, (, G, G, ) e i GP, (, H, H, ) e i HP A A (i) G G (ii) By popeties of espective seies H H HH H HH H GG HH GG A H A A A H Hece poved Sol. (x ) (x ) (x ) (x 00) Coefficiet of x coefficiet 00 ± coefficiet of x Sol.0 sided polygo d ( )d 0º; d Sum of iteio gle ( ) 80º ( )d (0) ( ) , If 6, the iteio gle will e gete th 80º. Hece the swe is. Sol. Let the ume e c c, c, ( c) c c ( c) c 6 c 7 c c 7 7 ( )(7 ) c ume is 6 Sol. 0 m m m S [0 0 0] 0 0[ 6] m Sol. o of elemets i th goup fist ume i the goup will e S [( ) ( )] [ ] [ ] ( ) Sol. Let the ume i GP e,, & othe umes e,, S.

19 Sequeces d Seies ( ) ( ) ( ) ( ) 0 tio fo thid GP Hece tio of the GP is sme Sol. S ( ) ( )... s... d d( ) S ( d) ( ) lim S lim lim ( d) d d Sol.6, AM, AM AM,, HM, HM HM, Suppose we tke AM & HM 0d d 0 AM d 0, HM, HM 0d d 60 i AP HM HM A 6 H Sol.7 x y z Fo x, y, z. AM HM x y z x y z x y z 6 60 xy yz zx xyz 0 Hece poved (0 ) Hece poved Sol.8 We must pove tht fo some the m d p; M p m m[ m ] [ (m )], fo some odd Let us pove this y iductio P m m[m ] is the equied. m p m p.m m[ m ] m m[m m m] m[m m m m ] m [m (m ) m ] m [A m ] We must pove tht A is odd. A is odd Fo eve m, ms is eve d (m ) is odd A is odd fo odd m, m is odd d (m ) is eve A is odd By iductio hypothesis, M p m poved [ (M )], with odd. Hece S.6

20 Sequeces d Seies Sol. c c c c c c c c c c c c c c c Sol. T T c c ( ) [,, c e i HP] (i) c c c c c c c c c c ( c) c ( c) c [B] S T Sol.... 0m middle stoes stoes [ tems] tems 0 EXERCISE JEE MAIN ( ) 0 Totl o of stoe [C] Sol. S S 6 (00) S S [A] 00 [ 00] 0 [0] 00 Sol., d, ( )d, d, ( ) d S [ ( )d] S [ d ( )d] s s ( )d s d ( )d d ( ) d ( ) ( ) s (s ) d d d s s ( ) [ s s ] ( s) s This is idepedet of ie coefficiet of will e zeo s ; d [c] d Sol.6 x x x x. x Let x t A t t t t t ; x t ( x ) ( x ) t t hece 0 (AM GM)AM GM Sol.7 S S [ 0d] [ 8d] 6 d d s S.7

21 Sequeces d Seies S0 0 [ d] d 0 0 Sol.8 S t S ds (t dt) ds t dt 0 [s ds] [t d ] t 8ds t dt 8ds ds dt dt 6ds 8dt d d s t 8 [c] Sol. The give expessio is equl to (...) d [A] d Sol.0 (i) 8 (ii) squig equtio (i) & the dividig y (ii) ( ) ( ) 8 ( )( ) ( )( ) 0,, GP is, 6, S 8 ( ) ( ) [A] ( )( ) ( ) ( ) ( ) ( ) 8 Sol. l (l ) (l ) l (l ) (l ) l (l) (l)... GP with como tiol l (l) (l)... GP with commo tiol l l l l l 6 l l e / Sol. S.8

22 Sequeces d Seies PREVIOUS YEARS QUESTIONS JEE MAIN Sol. Hee, ( c ) p ( c cd)p ( c d ) 0 ( p p ) ( p cp c ) (c p cdp d ) 0 (p ) (p c) (cp d) 0 (Sice, sum of sques is eve less th zeo ) Ech of the sques is zeo (p ) (p c) (cp d) 0 p c d c,, c e i G.P. Sol. Sum of the tems of the seies 7... upto tems c e witte 8 6 s tems upto... tems 8 Sol. Let the commo tio of the GP e. The, Y x d z x l y lx l dl z l x l LetA l x, D l The,, lx A ly lx l A D lz lx l A D Theefoe,,, e i H.P. lx ly lz Sol. Sice AM GM, the ( ) (c d) ( )(c d) M d Also,( ) (c d) > 0(,, c, d > 0) 0 < M Sol. Sice,, c, d e i A.P. cd cd, cd, cd, d c, cd, c d, cd e i AP e i A.P. cd, cd, d, c e i HP. c, d, cd, cd e i HP. Sol.6 Sice,, c e i AP. Let A D, A, c A D Give, c / (A D) A (A D) / A The umes e Also, D D D D,,, D D D, D e i GP. 6 D So, out of the give vlues, ight choice A D is the Sol.7 We kow tht, the sum of ifiite tem of GP is S,, x S (thus < ) x x o exists oly whe < x i.e., S.

23 Sequeces d Seies o0 < x < 0 0 < x < 0 Sol.8 Let S c S c ( ) c c c T c c( T S S ) T (c c) c c c Sum T c ;( )( ) c c ( ) 6 c c ( )( ) c ( 6c ( ) 6 6 6) c ( ) S.0

24 Sequeces d Seies EXERCISE JEE ADVANCED Sol. () Let umes e, Give HM AM H H 0 0 H 0d GM (GM) (i) AM (ii) A G 7 7fom (i) & (ii) 7 8,, 6 As. (, ) (, 6) () AM GM HM 7 HM G AM 7 (we kow this) (A 7) (A) (A ) 7A 0 A A 7 (i) GM 60 (ii) HM 8 0fom (i) & (ii) Sol. H0 H 0 H 0 0 H d H H 0d 0 d 0 d 0 H Sol. si x, si x, cos x si x e i GP. si x si x cos x si x ( si x cos x) si x cos x si x cos x 6 si x cos x si x cos x cos x si x cos x cos x Si x cos x cos x cos x Cos x cos x Cos x cos cous evet e equl to cos x commo tio cos x si x cos x si x si x six cosx cos x (6 ) / Sol.,, c, d, e e umes c i AP c d i GP c d e i HP c c d i.e si xcos x six S.

25 Sequeces d Seies d ce c e Let e c e d e e e e e e ( ) e c hece,c,e e i GP (ii) ( ) Hece poved (iii) e 8 c ( ) (i) c ± 6 c, d, e (, 6, ); (, 6, 8), d, 8 Sol.,, foms AP (i) ( ) (6 ) 6, (,, ,, Sol.6 GPs ( ) ( ) ( )( ) /8 ( ) /8 ( )( ) 0 If ½ the e 0 If the, >. ( ) 8 m p 00m 0 p Sol.7 S T T T T S T T T T0 T0 [S] Sol.8 x x x 0 HM oots c c c c c HM c c HM(c) c (i) 00 / c c c S.

26 Sequeces d Seies Its give c Equlity of equtio () holds oly if c ie ll the oots e 7 8 is itege mi ( ) s 6 s Sol. Let time tke to solve st polem e s time to solve secod polem will e S S S 7 S. S 7 s (iv) 6. s s 6. (i) s s 6. S S s S 6. s 6. s 7 s s (iii) fom (v) (v) s S s (ii) 6.. S Sol.0 fo 7 ( ) Cotiued k [ 8.] [ ] Sol. xyz o AP/HP Fo, x, y, z, i AP x d y d z d d d [ 7.] ( d) ( d) ( d) (i) Fo, x, y, z, i HP x dh y dh y dh cc to, x, y, z, i dh dh S.

27 Sequeces d Seies xyz d dh dh H Equtio (i) c e witte s (ii) ( ) ( ) ( )( )( ) 6 Equtio (ii) c e witte s ( )( )( ) 6 7 & e iteges ie, 7o 7, ie 0 Sol. 0x cx x 7 0 Let,, y e the oots y c 0 y y y 7 0 Cotiued (iii) (i) 0 (ii) & y e i hmoic pogessio i e y y y y.(iv) Puttig this i equtio (iii) / this i equtio (iv) ( y) y y.6 0 The oots e,, C 0 y C 0 Sol.,, c e i GP c logc, log c, log i AP log c logc log Let c log log (/) log(/) [ log ] log (/) log Let log t [ t] log log t t t (t ) log t t t t t. Sol. 6 ( ) t t t t t t t t ( ) ( 6) ( ) t ( ) ( ) log S.

28 Sequeces d Seies 7 0-7, ve itegl vlue of is. Sol. S S 0 8 S S S S 0 0 (ii) 8 (i) S0 s8 Puttig these vlues i equtio (i) & equtio (ii) Sol.6 S ( T) ( T) S [ ( )d] [ ( ) d] [ ( )d] T ( )d S T T S T T T T S T T T T T & 6 T c c 7 c ( ) c (ew ume) c ( 8) ( 6), So the ume is Sol.8 S() ( si ) cos ( si si ) cos cos cos... si (cos cos...) si 7 cos si cos cos si cos cos cos [ si cos si cos ] S() S ( sicos )( cos ) Sol.7 Let ume e c ( > > c) Sol. t x, t, t x GP e i S.

29 Sequeces d Seies t t x T (A B) ta tb tatb si x si x cos x cos x cosx cos 6 cosx cos 6 t t x Cos x cos t cos 6 6 t cos t 6 t si / cos / cos si cos 6 Cos x O / Solutios e O,,, [ 8 ] 0. 0 K 0 S.6

30 Sequeces d Seies EXERCISE JEE ADVANCED Sol. AM / [. ]/ 678 This does ot coti O Hece Aswe is (A)... Sol. K K (K ) K K K K K K K K K 8 m K K K K K K m M 8 7 [D] Sol. 00 K K K K K (K ) K K (K K)(k k) K K K K K K K k K k Sol. / Cicle iscied i st cicle / i d cicle i d cicle Sol. Sttemets- this is GP with commo tio If ( c) 7 c c Sttemet-AM GM (Tue) We Kow AM GM Fo ume c c (c) / ( c) 7 c Give ( c) 7 c c & ( c) 7 c c So sttemet- is flse Hece [D] 6. S00 eve S00 odd [A] 0 ( 6 00) ( ). [.] [.] Sttemet- is tue Sttemet- is coect & is coect expltio of sttemet- Hece [A] Sol.7, 7, d d 00 S.7

31 Sequeces d Seies d d 0 d db () Hece is coect () ( )d ( )db ( ) (d db) (c) ( ) ( ) ( ) 00, 00, 00 is i AP (d) 00 ( ) 00 (00) 0000 Sol.8 si (x y), si x, si (x y) e i HP Si x si(x y)si(x y) si(x y) si(x y) cosx cosy sixcos y si x cosy cos y cos x Si x (cos y ) cos y Si x cos y cos y/ si x sec y Sol. 6 ( ) ( ) 6 Dividig these () ±, fo 7 sum [A] fo 7 sum 7 7 [A, B] Sol.0 x px qx Roots fom icesig GP Roots e,, p q p (i) q q pp q 0.() [AM GM] p (, ) [B is icoect oe oot () is uity oe oot is & othe is, so if oot is gete th & othe less th [ACD] Sol. log, log, log c, log c e i AP,,,,, 6 6 c c c c c c > c c c > c () c c > c () Hece,, c c fom Log log log c log, log c, log S.8

32 Sequeces d Seies log log log c c ie,, c e i GP[B],, c 8 ( c) 8 8 8c 8 ( c c) 6 ( c c) > so A is icoect As is [B D] Sol. x x A x x xx A x x B x x B x x x x x x A ( ) B ( ) ±, A B x x ( ) x x ( ). 0 [ABCD] S.

33 Sequeces d Seies PREVIOUS YEARS QUESTIONS JEE ADVANCED Sol. Sice, fist d ( )the tems e equl. Let fist tem e x d ( ) th tem y y. whose middle tem is t. Thus i ithmetic pogessio ; t I geometic pogessio : I hmoic pogessio ; t xy xy x y x y c c d > > c (usig AM > GM > HM) Hee, equlity holds (ie, c) oly if ll tems e sme. Hece, optio (), () d (d) e coect. Sol. Let deotes the legth of side of the sque S. We e give legth of digol of S. This show tht,,. Fom GP with commo tio / Theefoe, 00 0 () 00 ( give ) 00 ( 0 give) This is possile fo 8. So (), (c), (d) e the swe. K! Sol. We hve Sk k (k )! Now, (k k ) Sk {(k ) (k ) } (k )! (k )! (k )! 00 k (k 00 00! k )S 00 k! (k Sol. k k k k 00 8! 00! k )S,,., e i AP k... d 0d 0 d 0d 0 d 0d 0 d, Give < [0 0 ] 0 d d d 7 Sol. Give,, m d,,, AP. Sm S is idepedet of. S S [ ( )d] [ ( )d] {(6 d) } idepedet of, (6 d) If d 0, the S S m is idepedet of. Sol.6 Let d e two umes. The, A ;G ;H A H A,G AH, A H H A H Clely, G G G.. S.0

34 Sequeces d Seies Sol.7A is AM of A d H d A > H A > A > H A is AM of A d H d A > H A > A > A : : : A > A > A >. Sol.8 As ove A > H > H, A > H > H H < H < H < Sol. Hee, V [ ( )( )] ( ) V [ ] ( ) ( )( ) ( ) 6 ( ) [( ) ( ) ] ( )( ) Sol.0 V V ( ) [( ) ] () T ( ) ( ) Which is composite ume. Sol. Sice,T T ( ) ( ) Q T T [ ] [] Q 6 S.

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