SOLVED EXAMPLES. Ex.1 If f(x) = , then. is equal to- Ex.5. f(x) equals - (A) 2 (B) 1/2 (C) 0 (D) 1 (A) 1 (B) 2. (D) Does not exist = [2(1 h)+1]= 3
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1 SOLVED EXAMPLES E. If f() E.,,, th f() f() h h LHL RHL, so / / Lim f() quls - (D) Dos ot ist [( h)+] [(+h) + ] f(). LHL E. RHL h h h / h / h / h / h / h / h As.[C] (D) Dos ot ist LHL RHL, so giv it dos ot ist. As.[D] E. quls - / / / (D) ( / ( / ) ) quls - Limit As.[B] / As.[B] E.5 Limit E.6 E.7 is qul to- / (D) ( )( ) ( )( ) ( ) (D) ( ) (D.L.Hospitl rul), form LHL h h RHL ( h ) h ( h ) h h h E.8 If f() ( h ) h ( h ) h As.[A] (D) Dos ot ist As.[A] LHL RHL, so it dos ot ist.as.[d], th f() (D) Dos ot ist IIT JEE PREPRETION LIMIT 5
2 LHL E.9 RHL h h h h h h h h h () LHL RHL dos ot ist. As.[D] Limit E. / (D) ( ) ( ) log( ). As.[C] quls - / / (D) E. E. t si / / / (D) Th giv it is i th form, thrfor pplyig L 'Hospitl's rul, w gt Limit sc cos si log( ) / / As.[C] (D) Dos ot ist It is i / form, so usig Hospitl rul, w hv Limit cos si ( ) (/ form) / As.[C]...! E. Th vlu of Limit / As.[C] si / / / (D) / si. si...!...!...!... is - / As.[C] E. IIT JEE PREPRETION LIMIT 5 E.5 si quls - si (D) Dos ot ist ( fiit umbr btw d )/ As.[B] Limit t / / (D) / /... / [, so glctig highr powrs of ] / / / As.[B]
3 E.6 If f() f() E.7 If G () th si cos, th { (cos f() quls - { (si / )} / )}. As.[C] G ( ) 5, G () quls - / /5 Hr G() Giv it E.8 If f(9) 9 d f'(9), th E.9 qul to - form (By L Hospitl rul) As.[D] f ( ) 9 (D) 9 Giv it is i / form, so usig Hospitl rul, w gt Limit f (9 ). f (9 ) 9 9..f ( ) f ( ) As.[C] (D) is Limit E. E. / / [( / ). /. /. / ( / ) ]. / /. As.[B] Th vlu of (log ) is - () /. (log ) (log ) (log ) (log ). 6 (log ). form form (log ) 6 log form 6 (log ) 6 si Limit log ( / ) º form. As.[A] (D) /8 8 si( / 8 ) ( / 8 ) cos( / 8 ) As.[D] IIT JEE PREPRETION LIMIT 5
4 E. If f() E., /,, th Hr d f() f() f() f() quls - (D) Dos ot ist ( ) f() f() dos ot ist. ( ) quls - log log As.[D] / log (D) Giv Limit ( ). ( ) log. log As.[B] E. If,b,c,d r positiv rl umbrs, th E.5 d/b b (c+d)/(+b) c d b c d c/ (D) ( form) b c d b c d / b (c + d) d/b As.[A] t / quls - (D) Lt y E.6 t (cot ) / log y ( cos ( ( cot log cot ) cos ( ) ) ) / form ( form) form y.as.[b] quls - () () log () log () No of ths Th giv it E.7 log. si. cos / si( / ) log. As.[C] Th vlu of cot is - (D) / Giv Limit si( / ) cos( / ) si( / ) cos( / ) si( / ) si( / ) cos( / ) si( / ) cos( cos( / ) / ) cos( / ) ( / ) si( / ) form ( / ) si( / ) As.[A] IIT JEE PREPRETION LIMIT 5
5 LEVEL- Qustio bsd o Q. If f() Eistc of it,,,, th Q. If f() Q. Q. Q.5 f(),,,, f() (D) Dos ot ist f() (D) si quls si si th- f() f() (D) Dos ot ist (D) Dos ot ist Q.6 Lt f() ( ) [/], whr [ ] rprst grtst itgr fuctio th f() is - (D) Dos ot ist Q.8 If f () Q.9,,, th, f() - (D) dos ot ist / [] quls - / (D) / Q. Which of th followig its ists- [] Q. ( Q. If f() si / ) / /, f() / (D) All th bov ( N) / / (D) Dos ot ist, th (D) Dos ot ist Q. If f is odd fuctio d f() f() ists th Q. If [] grtst itgr, th qul to - ± ( ) [] is Q.7 Which of th followig its dos ot ist- Qustio bsd o (D) { + } { } Q ( ) / / (D) IIT JEE PREPRETION LIMIT 5
6 Q.6 Th vlu of 7 5 / 7/ /5 (D) Q.5 / / Q.7 Th vlu of Q.8 Q.9 ( ) ( + ) ( ) ( ( ) ( 5 ) (5 ) 6 ) / /5 (D) / si 5 5 /5 (D) Q. Th vlu of... 5 / Q.6 Q.7 ( )! ( )! ( )!... / (D) Q.8 Th vlu of Q /... is - / / (D) Q. Q (D) ( + 5 ) / quls 5 Qustio bsd o Fctoristio mthod Q. Th vlu of / si cos (D) Q. si cos Q. Th vlu of 8 is udfid Q.... is qul to- / (D) / Q. 7 5 / / / (D) / IIT JEE PREPRETION LIMIT 55
7 Q. / / / / / (D) / Qustio bsd o Rtiolistio mthod Q. / (D) Dos ot ist Q. Q. cos si 8 (D) Q.5 Q.6 Q.7 Q.8 / / ( ) ( ) (D) / si ( ) / (D) / Q.9 Th vlu of b Q. Th vlu of b (D) b b b ( > b) is - / ( ) is Qustio bsd o Epsio mthod Q. Q. Q.5 Q.6 Q.7 Q.8 si si (D) cos si cos / (D) si cos log ( ) / / si t / (D) / cos si (D). cos log log IIT JEE PREPRETION LIMIT 56
8 Q.9 ( t ) / / (D) Q.57 ( si ) / ( si ) / / (D) / Qustio bsd o L Hospitl rul Q.58 h / ( h ) h / Q.5 Q.5 Q.5 Q.5 log / (D) m m m/ m m (D) / m t log si Q.5 Lt f() [ / ] f ( ) f () 8 m log /9 / Q.55 Th vlu of, th th vlu of (D), th Q.59 / /... Q.6 Th vlu of / (D) Q.6 Th vlu of Q.6 h / (D) / ( ) [ t (/) sc ] h (8 h ) h / / / 6/ (D) /8 / si Q.6 Th vlu of cos (D) / (D) Q.6 Th vlu of / sc log / / (D) / Q.56 Th vlu of (6 5 ) / / 5 ( ) /5 5/6 /8 Q.65 Th vlu of / cos log (t ) IIT JEE PREPRETION LIMIT 57
9 Q.66 log / (D) / Q.75 Th vlu of cot p cot q q/p (D) p/q Q.67 Th vlu of si( cos h ) log( h h ) si log h + log si cos cos + log si (D) cos log + si Q.76 Q.77 8 t / 7 si cos (D) Dos ot ist (D) / Q.68 / t si is qul to- (D) Q.69 If f(), f ' (), g(), g'(), th Q.7 g ( ) f ( ) g ( ) f ( ) 5 (D) 5 h ( h ) si ( h ) h si cos + si (cos + si ) (cos + si ) Q.7 Th vlu of Q.7 ( ) si ( ) (D) (D) / Q.7 Th vlu of (D) 5 Qustio bsd o Som stdrd it Q.78 cos cos / Q.79 Th vlu of y (y ) cosc (y ) (D) / Q.8 Th vlu of Q.8 Q.8 Q.8 Q.8 [log (+) log ] (D) ( ) / / (D) / si si (D) (D) Q.7 Th vlu of log ( k cos ) is - k (D) k Q.85 si (D) / IIT JEE PREPRETION LIMIT 58
10 Q.86 If Q.87 t k si 5, th th vlu of k 5 (D) 5 ( / ) (D) Q.88 Th vlu of Q.89 Q.9 If Q.9 b log b d d si (b/ ) is ( >) - cos d log b is qul to- / / (D) k si cos 8 8 k, th vlu of / / / (D) /8 8 cos cos cos /6 / 8 Qustio bsd o,, Forms Q.9 Q.9 Q.9 log ( ) / (D) 9 cos (D) / [ + t] cot quls - (+ ) / (D) / Q.95 Q.96 / / (D) / / (sc ) cot / Q.97 Th vlu of (cosc ) /log is - (D) / Q.98 Th vlu of Q.99 If f() Q. Q. / (t ) t (D), th- f() 6 f() (D) f() f() (D) (D) Q. Th vlu of / is - Q. Th vlu of ( + ) / is - (D) IIT JEE PREPRETION LIMIT 59
11 LEVEL- Q. If f() g() si,,,, 5,, Z othrwis, th d g[f ()] (D) 5 Q. If [] dots th grtst itgr, th Q. quls - {[ ] + [ ] + [ ] +. + [ ] / / /6 (D)..... ( ) / (D) Q.8 If f() Q.9 si( [ ]) [ ] for [ ] for [ ] whr [] dots th grtst itgr, f() quls - si /8 / / (D) Q. If g() is polyomil stisfyig g() g(y) g() + g(y) + g (y) for ll rl d y d g() 5, th 8 8 g() is - Q. Th vlu of (cos ) / / (D) Q. 5 t Q.5 {log () log ( + )... log (k )}, k k N is - dos ot ist Q.6 Th vlu of k log t Q. Q. ( ) 6 (D) dos ot ist 8 ( ) 5 ( ) (D) Q.7 h si h cos h 6 6 h ( cosh si h ) / / (D) / is qul to Q. /5 (D) Dos ot ist IIT JEE PREPRETION LIMIT 6
12 Q.5 Th vlu of sc sc sc sc / (D) Q. si log ( ) / / / (D) / Q.6 If > d g is boudd fuctio Q.7 Q.8 f ( ) g() ( g ( ) is - (si ) ) f() / (D) /... / / Q.9 Th vlu of Q. Q. 5 6 is qul to- cos (si ) cos ( ) qul to- ( ) (D) (D) Dos ot ist... ( ) (D) is Q. Th vlu of t t t t t Q.5 Th vlu of Q.6 Q.7 Q.8 Q.9 Q. 5 (D) si (si ) m (m < ) is qul to- /m cos ( ) (D) m/ / / / (D) /8 log 5 (log 5 5 ) quls - / / / cot cos ( ) is - ( >, > ) is qul to- Q. si ( / ) (D) 6 6 IIT JEE PREPRETION LIMIT 6
13 Q. If si si is o-zro dfiit, th must b - Q.5 t t (D) Q. cos (si ) / t (si ) (D) Q.6 si si Q. If f '' (), th th vlu of f ( ) f ( ) f ( ) (D) Q. ( + ( []) ) whr [] rprst grtst itgr fuctio. (D) IIT JEE PREPRETION LIMIT 6
14 Q. If [] dots th grtst itgr lss th or qul to, th [ ] [ ] [ ]... [ ] quls - / / (D) Q. Th vlu of ( t ) / / / (D) / Q. If { } rprst frctiol prt of th [ ] { } { } { } is qul to whr [ ] rprst G.I.F. / LEVEL- Q.7 Lt mi m { + + } R d b cos th vlu of r.. Q.8 Th vlu of Q.9. r r b is - si si si -... / (!) (!) /! (D) (!) Q. Lt f (), th - f (), for > f () for < f () is ot dfid for y vlu of (D) f () for Q.5 If f () h (), g () ( ) [f() + g() +h()] th Q.6 If A i 7 i i < < <.... Th m is qul to ( ) m is qul to ( ) m+ is qul to ( ) m (D) dos ot ist (D) d, i,,..., d if m (A A..A ), Q. Q. ( ) / / ( ) / /... /... ( ) [(mi m (y y + )) si / ] whr [ ] rprst grtst itgr fuctio is si si Q. If f() is th itgrl of th fid f () - / /, Q. If f () is cotiuous fuctio from f : R R d ttis oly irrtiol vlu s th r r f ( r ) r f ( r ) f ( r ) IIT JEE PREPRETION LIMIT
15 Q. Th vlu of cos( ) ( ) / Q.5 Giv rl vlud fuctio f such tht Q.6 f() t { }, ( [ ] ), { } cot{ }, is - whr [ ] rprst G.I.F. d { } rprst frctiol prt of f() t f() cot f ( ) (D) All of th bov si [cos ] cos [cos ] is - / dos ot ist Q.7 Th vlu of Q.8 If si 99 si whr [ ] rprst grtst itgr fuctio ( b ) th, b, b, b Sttmt typ Qustios All qustios r Assrtio & Rso typ qustios. Ech of ths qustios cotis two sttmts: Sttmt- (Assrtio) d Sttmt- (Rso). Aswr ths qustios from th followig four optio. Sttmt-I d Sttmt-II r tru Sttmt-II is th corrct pltio of Sttmt-I Sttmt-I Sttmt-II r tru but Sttmt-II is ot th corrct pltio of Sttmt-I. Sttmt-I is tru but Sttmt-II is fls (D) Sttmt-I is fls but Sttmt-II is tru. Q.9 Sttmt I : [] / / (whr [ ] rprst grtst itgr fuctio) dos ot ist. Sttmt-II : / / dos ot ist. Q. Sttmt-I : Th grph of th fuctio y f() hs uiqu tgt t th poit (, ) through which th grph psss th log ( 6 (f ( )) f ( ) Sttmt-II : Sic th grph psss through (, ). Thrfor f(), wh f() giv it is zro by zro form. So tht it c b vlut by usig L Hospitl s rul. Q. Sttmt-I : wh <, log ( ) cos Sttmt-II : For < <, s,. Q. Sttmt -I : Sttmt -II : Q. Sttmt -I : Sttmt -II : si log( + ) y si y y cos it is qul to right hd it. Q. Sttmt -I : Vlu of Sttmt -II: If f() d ist's. f() ists if th lft hd / ( + f()) g() is g() (si) t is. f ( ) g ( ), IIT JEE PREPRETION LIMIT
16 Pssg Bsd Qustios Pssg :- Lt m, r o zro itgrs d t m si itgr. O th bsis of bov iformtio, swr th followig qustios- Q.5 Which of th followig sttmt is tru m is should b v but is odd both m & should b odd m is odd d is v (D) both m & r v itgrs Q.6 Th vlu of it i trms of m & is m 6 m Q.7 Is m & r rltd s m m m Q.8 Th vlu of it for m is Q.9 If 6 t (m ) si 6 for m, th vlu of it is ot itgr th Colum Mtchig Qustios Q. Q. Mtch th try i Colum with th try i Colum. f() is lss th qul to, whr Colum-I f ( ) f ( ) f ( ) (P) si Colum-II (Q) (R) (D) ( + si ) cosc (S) f() f() f(), whr f() is s i colum-i Colum-I t[ [5 / ] si t[ t t t whr [] is th grtst itgr fuctio f() (D) f() cos log( ) si cos Colum-II ] ] [5 / ] (P) / 8 (Q) 5 (R) (S) / IIT JEE PREPRETION LIMIT
17 SECTION A Q. If f (), f (), th LEVEL- (Qustio skd i prvious AIEEE d IIT-JEE) f ( ) [AIEEE ] (D) Q. Th vlu of Q. Q. Q.5 If ( cos ) si si / / 6/5 (D) 5/6 5 5 [AIEEE ] [AIEEE ] (D) log [ ], N, (whr [] dots [ ] grtst itgr lss th or qul to ) [AIEEE-] hs vlu hs vlu hs vlu (D) dos ot ist log ( ) log ( ) k, th vlu of k is - [AIEEE ] (D) Q.6 Lt f() g() k d thir th drivtivs f (), g () ist d r ot qul for som. Q.7 Furthr if f ( )g ( ) f ( ) g ( ) f ( ) g ( )f ( ) g ( ) th th vlu of k [AIEEE ] (D) / t t [ si [ ] ] [AIEEE ] Q.8 If b, th th vlus of d b, r- [AIEEE ] R, b R, b R R, b (D) d b Q.9 Lt d b th distict roots of Q. Q. Q. + b + c, th cos ( ( ) b c ) ( ) ( ) (D) cos{ ( )} [AIEEE-8] ( ) dos ot ist quls quls (D) quls / cot SECTION-B cos ( ) ( ) ( ) 5 [AIEEE-] [IIT-99] 5 (D) [IIT-99] (D) 8 6 (D) 8 IIT JEE PREPRETION LIMIT
18 Q. ( cos ) (D) No [IIT -99] Q. t t ( cos ) (D) [IIT-999] Q. Q.5 for [IIT-99] o vlu of is y whol umbr oly (D) oly t / [IIT-99] (D) Q. For R, Q. 5 (D) 5 si ( cos ) [IIT Scr. ] quls - [IIT Scr. ] / (D) Q.6 Q.7 t / (D) 5 / [IIT- 99] [IIT- 996] (D) Q.8 Th vlu of Q.9 h log ( h ) h log( h ) [IIT-997] cos ( ) [IIT-998 similr to IIT- 99] dos ot ist bcus LHL RHL Q. Th vlu of Itgr ; for which (cos ) (cos ) is fiit o zro umbr- [IIT Scr. ] (D) Q. Lt f : R R such tht f() d f () 6. Q.5 If th f ( ) f () / quls - / (D) (si vlu of ) [( ) t ] [IIT Scr. ] th th [IIT Scr.] ists d it quls dos ot ist bcus (D) ists d it quls + (D) IIT JEE PREPRETION LIMIT 5
19 Q.6 If f() is diffrtibl fuctio d f () 6, f (), f (c) rprsts th diffrtitio of f() t c, th my ist f ( h h ) f ( ) h f ( h h ) f ( ) will ot ist is qul to (D) is qul to [IIT Scr.] Q.7 Lt f() b strictly icrsig d diffrtibl, th f ( ) f ( ) f ( ) f ( ) (D) [IIT Scr.] Q.9 Lt L, >. If L is fiit, th [IIT- 9] L 6 (D) L Q. If Lim l ( b ) bsi, b > d (, ], th th vlu of is - [IIT- ] (D) 6 Q.8 si / (si ), for > - [IIT-6] (D) IIT JEE PREPRETION LIMIT 6
20 ANSWER KEY LEVEL- Q.No As. A C D D B B A A C D D D A D A A A D C B Q.No As. A B B C A A B C A B B B B D D D B D D A Q.No As. B B D B B C B B B D C A B D B B C B D A Q.No As. D C D B C C D D B A B B A D C A C A D A Q.No As. D C B A C D B C A D C D B C B C D B D D Q.No. As. C B D LEVEL- Q.No As. B B C C B A B B A B A C C B C B C B B D Q.No As. A C B B D A C B B B A B B D C B LEVEL- Q.No As. A C D A,B C D C A B C B A B D D B B C B A Q.No As. A D D A D B C A A. P,R,S ; P,S ; P,Q,R,S ; (D) P. Q ; R ; S ; (D) P LEVEL- SECTION-A Q.No As. A A A A D B D B A A, SECTION-B Q.No As. B C D B B C A B A A C B Q.No As. C C C C B D A,C D IIT JEE PREPRETION LIMIT 7
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