ASSERTION AND REASON
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- Peregrine Parker
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1 ASSERTION AND REASON Som qustios (Assrtio Rso typ) r giv low. Ech qustio cotis Sttmt (Assrtio) d Sttmt (Rso). Ech qustio hs choics (A), (B), (C) d (D) out of which ONLY ONE is corrct. So slct th corrct choic : Choics r : (A) Sttmt is Tru, Sttmt is Tru; Sttmt is corrct pltio for Sttmt. (B) Sttmt is Tru, Sttmt is Tru; Sttmt is NOT corrct pltio for Sttmt. (C) Sttmt is Tru, Sttmt is Fls. (D) Sttmt is Fls, Sttmt is Tru. INDEFINITE & DEFINITE INGEGRATION 9. Lt F() idfiit itgrl of cos. Sttmt-: Th fuctio F() stisfis F( + ) = F() rl Sttmt-: cos ( + ) = cos.. Sttmt-: d c ot foud whil d c foud. Sttmt-: is ot diffrtil t =.. Sttmt-: d = t ( ) + C Sttmt-: d = t + C. Sttmt-: If y is fuctio of such tht y( y) = th d log( y) y d Sttmt-: = log ( y) + c y. Sttmt : f() = logsc Sttmt : f() is priodic. Sttmt : 9/ / d l c Sttmt : d l c. Sttmt : t d t ; whr [] = G.I.F. Sttmt : [t ] = for < < t d [t ] = for t <. 6. Sttmt : / d t Sttmt : / / f () d f d d d t cot f () d f ( )d. 7. Sttmt : 8. Sttmt : si d Sttmt : t sc d t c cos d. of 68
2 Sttmt : f () f () d f () c. 9. Sttmt : If f() stisfis th coditios of Roll's thorm i [, ], th f ()d Sttmt : If f() stisfis th coditios of Roll's thorm i [, ], th f ()d. Sttmt : [ si cos ]d, whr [] dots G.I.F. quls 8. Sttmt : If f() = si + cos, th f().. Lt f() cotiuous fuctio such tht Sttmt :. Lt I = f ()d, I f ()d 7 Sttmt : d, N Sttmt I : I. I, I... is icrsig squc. Sttmt II : is icrsig fuctio.. Lt f priodic fuctio of priod. Lt g() = f ()d 7 f (t) dt d h() = g( + ) g(). Sttmt : h is priodic fuctio. Sttmt : g( + ) g() = g(). log d log c Sttmt : f () f () d f () c.. Sttmt :. Sttmt : If I = dt d t / dt I, th I = I. t Sttmt : 6. Sttmt : 8 < mi. [], [ ] d 6 d. Sttmt : If m is th smllst d M is th grtst vlu of fuctio f() i itrvl (, ), th th vlu of th itgrl f ()d is such tht for <, w hv M( ) 7. Sttmt : si d (si cos)+c A Th A is Sttmt : si cos d cos 8. d() Sttmt : is qul to c f () d M( ). 6 of 68
3 / Sttmt : d is / l + + c / 9. Sttmt : is / Sttmt : f ()d f ( ) t d / 6. Sttmt : If f stisfis f( + y) = f() + f(y), y R th Sttmt : If f is odd fuctio th f ()d = f ()d =. Sttmt : If f() is odd fuctio of th f (t)dt is v fuctio of () Sttmt : If grph of y = f() is symmtric out y is th f() is lwys v fuctio.. Sttmt : Ar oudd y y = {}, {} is frctiol prt of =, = d is is. Sttmt : Ar oudd y y = si, =, = is sq. uit.. Sttmt-: lim... r Sttmt-: lim f f ()d r, symols hv thir usul mig.. Sttmt-: If I = t d, th (I + I 6 ) = t. t Sttmt-: If I = t d, th - I - = I, N. d. Sttmt-: If > d c <, th th vlu of th itgrl will of th typ t - c A c, whr A, B, C, r costts. B Sttmt-: If >, c < th + + c c writt s sum of two squrs. 6. Sttmts-: d c Sttmts-: / () (f () f ()d = f() + c 7. Sttmts-: d = log t - ( + /) + c ( ) t d Sttmts-: t c l 8. Sttmts-: c Sttmts-: (f() + f()) d = f() + c. (l ) l 9. Sttmts-: d c Sttmts-: For itgrtio y prts w hv to follow ILATE rul. 6. Sttmts-: A fuctio F() is tidrivtiv of fuctio f() if F () = f() Sttmts-: Th fuctios +,, + r ll tidrivtivs of th fuctio. 6. Sttmts-: d =, < 7 of 68
4 Sttmts-: If f() is fuctio cotiuous vry whr i th itrvl (, ) cpt = c th c f ()d f ()d f ()d c 6. Sttmts-: d Sttmts-: m d M th lst d th mimum vlu of cotiuous fuctio y = f() i [, ] th 6. Sttmts-: d m( ) f ()d M( ) Sttmts-: if f() g() h() i (, ) th 6. Sttmts-: d. f ()d g()d h()d Sttmts-: For y fuctios f() d g(), itgrl o th itrvl (,), th f ()g()d f ()d g ()d 6. Sttmts-: d Sttmts-: If F() is tidrivtiv of cotiuous fuctio (, ) th cos 66. Sttmts-: c itgrtd y sustitutio it si = t. ( si ) Sttmts-: All itgrds r itgrtd y th mthod of sustitutio oly. 67. Sttmt- : si cos d cos = t c Sttmt- : (f() + f ()d = f() + c 68. Sttmts-: ()cos (. )d. si (. ) C Sttmts-: f () '()d, () t quls f (t)dt. 69. Sttmts-: log d log c Sttmts-: 7. Sttmts-: 7. Sttmts-: du uvd u vd vd d d d C ( ) si f ()d F() F() Sttmts-: Sttmts-: 7. Sttmts-: Th vlu of ( )( )d c ot cd 8 f () f '() d f () C f () d f ()d f ( )d 8 of 68
5 Sttmts-: If m f() M [, ] th m( ) f ()d ( )M 7. Sttmts-: / / (si ) d Sttmts-: Ar oudd y y = d y = is / / (si ) (cos ) 9 log 7. Sttmts-: d = log + + c Sttmts-: ( ) 7. Sttmts-: d = t ( ) + c Sttmts-: cos ( ) 76. Sttmt- : f() = Sttmts-: f() = 77. Sttmt- : Sttmts-: Sic 78. Sttmts- : l t dt ( ), th f() = - t t l t dt, th f() + t si d d. t si f f (l ) is odd fuctio. So, tht si d = ( + ) COSt ( t ) c Sttmts-: f () d f ()d f ()d d c si. 79. Sttmts-: Th vlu of th itgrl d logs to [, ] f () d log f () c f () sc d t c f ()d f ()d if f( + ) = f() 9 sq. uits Sttmts-: If m & M r th lowr oud d th uppr ouds of f() ovr [, ] d f is itgrl, th m ( ) f ()d M( ). 8. Sttmts-: [cot ]d = cot, whr [] dots grtst itgr fuctio. Sttmts-: f ()d is dfid oly if f() is cotiuous i (, ) [] fuctio is discotiuous t ll itgrs 8. Sttmts-: d = Sttmts-: f ()d if f() is odd fuctio. 8. Sttmts-: All cotiuous fuctios r itgrl Sttmts-: If fuctio y = f() is cotiuous o itrvl [,] th its dfiit itgrl ovr [, ] ists. 8. Sttmts-: If f() is cotiuous o [, ], d if f ()d, th f() = t lst oc i [, ] Sttmts-: If f is cotiuous o [, ], th t som poit c i [, ] f = 8. Sttmts-: d Sttmts-: c c f ()d f () d f ()d f ()d whr C (A, B) 9 of 68
6 8. Sttmts-: log d Sttmts-: If f is odd fuctio f ()d d k 86. Sttmt- If d th Sttmt- : ( ) = k k d d d 87. Sttmt- : m m! d m { []d Sttmts-: 88. Sttmts-: cos d Sttmts-: f ()d f ()d cos 89. Sttmts-: d Sttmts-: cos cos 9. Sttmts-: [] d( ) Sttmts-: c c d ( )! f ()d f ()d f ()d whr < c <. f ()d f ( )d [ ] [] d d 9. d Sttmts-: Sttmts-: t f ()d f ( )d ANSWER 9. D. B. D. C. A. A. A 6. C 7. D 8. A 9. A. D. D. D. A. A. C 6. A 7. D 8. C 9. A. A. C. C. D. C. A 6. C 7. A 8. A 9. B 6. B 6. A 6. A 6. A 6. A 6. D 66. C 67. C 68. A 69. C 7. A 7. A 7. A 7. B 7. A 7. A 76. D 77. A 78. A 79. D 8. A 8. A 8. B 8. A 8. A 8. A 86. A 87. C 88. A 89. D 9. A 9. A d. cos( )cos( ) () Qu. from Compt. Ems (Idfiit Itgrl) si( ) cos( ) cosc ( ) log c () cosc ( ) log c si( ) cos( ) si( ) cos( ) cosc ( ) log c cosc ( ) log c si( ) cos( ) d. [AISSE 989] () / / [( ) ( ) ( ) ] c () / / [( ) ( ) ( ) ] c / / [( ) ( ) ] c No of ths ( ) cos si. d [EAMCET 99] si cos 7 7 () log( si cos ) () log( si cos ) 7 log( si cos ) No of ths. If (si cos ) d si( c), th th vlu of d c is [Roork 978] 6 of 68
7 () c / d k ( ritrry costt) () c / d / c / d is ritrry costt No of ths. d ( ) () log 8 8 c () log si cos 6. d si cos () si c () si c d 7. ( ) () log( ) log( ) 8. d ( ) p q (t ) () log[ q t q p ( p q q t 9. d () ( ) / c 9 / ( ) 9 / ( ) q (t c / ) c ) ] c [AI CBSE 98] c [IIT 986] log [IIT 979] () () [IIT 98] d. quls [MP PET ] si cos () si c c log log( ) log[ q t No of ths ( 9 log( ) / ) No of ths p ( q (t / ) si c c () t c () t c cot c cot c d. [MP PET 988; BIT Rchi 979] c () c log c c () log c c log c c log c. si d [Roork 977] () [si cos ] c () [si cos ] c [si cos ] c [si cos ] c. d / (9 ) () 9 si. d () c [si ] c () () 9 si c [si si c si c ] c ) ] c c si c No of ths 9 6 of 68
8 . If f( )si cos d log( f( )) c, th f () ( ) () si cos d 6. si cos () t t c () () si t cos [AISSE 986] t c cos t si t c 7. d [MP PET 99] () t c () cot c t c 8. (log ) d [IIT 97, 77] cos No of ths cot si c () (log ) log c () (log ) log c (log ) log c (log ) log c 9. Th vlu of () ( ( ) d will [UPSEAT 999] ) t ( ) ( ) t [ ]. t sc d [IIT 977] () sc sc c 6 sc sc c 9 () () sc sc c 6 No of ths. si d [MP PET 99] () si si. d () c c () ( t / c si si ) t ( si c () si c si c si c si cos si. If,, th cos d si () si c si cos c () si cos cos si. 6 If d A B log(9 ) C, th A, B d C r [IIT 99] 9 () 6 A, B, C log costt () A, B, C log costt 6 c c ) c c 6 of 68
9 A, B, C log costt 6 No of ths. Th vlu of sc d will [UPSEAT 999] () sc t log(sc t ) () sc t log(sc t ) sc t log(sc t ) sc t log(sc t ) 8 6. d ( ) [IIT 98; MP PET 99] () ( ( ) ) c c () ( ) ( ) c c 7. If I si d, th for wht vlu of K, KI (si cos ) costt [MP PET 99] () () 7 8. d Th vlu of will [UPSEAT 999] () log log () log log 9. d [AISSE 98] () / / ( ) ( ) c () / / ( ) ( ) c / / ( ) ( ) c 6 No of ths. cos si cos log d cos si [IIT 99] () () (cos si) (cos si ) (cos si) cos si log cos si cos si log cos si cos si log cos si si log t log sc. d ( si cos ) () si cos si cos si cos si cos () [MNR 989; RPET ] si cos si cos No of ths 6 of 68
10 . If u cos d d v si d, th ( )( u v ) () (). If (log I ) d, th I I () (log ) () (log ) ( ) ( ) ( log ) (log ). / si d [Roork 98] () / cos c () / cos c / si c / si c. If d 9 l( ) 7 l( ) A, th A 6 [MP PET 99] () l( ) costt () l( ) costt Costt No of ths 6. d cos () t t c () t t c t t c No of ths 7. d qul to [MP PET ] () t () t t ( ) No of ths 8. d (si si ) [IIT 98] () log( cos ) log( cos ) log( cos ) 6 () 6 log( cos ) log( cos ) log( cos ) 6 log( cos ) log( cos ) log( cos ) No of ths 9. If d log ( ) ( ) t A, ( )( ) whr A is y ritrry costt, th th vlu of is () / () / /6 /. ( ) d If log C, th th vlus of d r rspctivly ( ) ( ) [Roork ] () /, / (), /, / /, ¾ (Dfiit Itgrl). If I is th grtst of th dfiit itgrls [Krtk CET ] [MP PET 998] 6 of 68
11 I I cos d, I cos d / d, I d, th () I I () I I I I I I. Lt f () fuctio stisfyig f ( ) f( ) with f ( ) d g () th fuctio stisfyig f( ) g( ). Th vlu of itgrl f ( ) g( ) d is qul to () ( 7) () ( ) ( ) No of ths. If m Im (log ) d stisfis th rltio I m k lim, th () k () l m k No of ths. Lt f positiv fuctio. Lt k ) k I k k ) I is [IIT 997 Cclld] f ( d, I f ( d wh k. Th / I () () k /. If f ( t) dt t f( t) dt, th th vlu of f () is () / () / 7 6. () d is qul to [AMU ] () 7. If is y itgr, th cos ( ) d cos () () [IIT 98; RPET 99; UPSEAT ] No of ths d 8. Th vlu of th dfiit itgrl lis i th itrvl [, ]. Th smllst such itrvl is 6 (), () [, ] 7, No of ths 7 9. Lt,,c o-zro rl umrs such tht 8 8 ( cos )( c) d ( cos )( c) d Th th qudrtic qutio c hs () No root i (, ) () At lst o root i (, ) A doul root i (, ) No of ths. If f( ) t dt,, [IIT 98; CEE 99] th [MNR 99] [AIEEE ; DCE ] [IIT 998; AMU ] 6 of 68
12 () f d f r cotious for () f is cotious ut f is ot cotious for f d f r ot cotious t f is cotious t ut f is ot so. Lt g( ) f( t) dt whr f ( t), t [,] d f ( t) for t (, ], th [IIT Scrig ] () g () () g () g () g () cos. Th vlu of d,, is () () f ( ). If ( ), { ( )} f( ) f I g d, d f( ) ))} I I g{ ( d, th th vlu of f ( ) I () (). Lt f : R R d g : R R cotiuous fuctios, th th vlu of th itgrl / [ f ( ) f( )] [ g( ) g( )] d / [IIT 99; DCE ; MP PET ] () () is [IIT Scrig ; AIEEE ] [AIEEE ]. Th umrs P, Q d R for which th fuctio f( ) P Q R stisfis th coditios f ( ), f (log ) d 6. log 9 [ f ( ) R] d r giv y () P, Q, R () P, Q, R P, Q, R P, Q 6, R si 7 7 d si d quls () 7 () 6 7. Lt f ( ) d, f( ) d d f( ) d, th th vlu of ( ) f( ) d [IIT 99] () () 8. Giv tht d, th th vlu of ( )( )( c ) ( )( c)( c ) d is ( )( 9) () 6 () 8 [MP PET ] [Krtk CET 99] 9. If l( m, ) t m ( t) dt, th th prssio for l ( m, ) i trms of l ( m, ) is [IIT Scrig ] () l( m, ) m m () l( m, ) m 66 of 68
13 . l( m, ) m m m l( m, ) lim ()... lim () Zro t. If f( ) d t, t, th f ()... () No of ths [AIEEE ] [IIT Scrig ] 9. For which of th followig vlus of m, th r of th rgio oudd y th curv y d th li y m quls [IIT 999] () (). Ar closd tw th curv y ( ) d li ov -is is [MP PET ] () (). Wht is th r oudd y th curvs y 9 d y 8 is [DCE 999] 9 () () 9 si 6 No of ths. Th r oudd y th curvs y d y is [IIT Scrig ] () () 6. Th volum of sphricl cp of hight h cut off from sphr of rdius is qul to [UPSEAT ] () h ( h) () ( h)( h h) h No of ths 7. If for rl umr y, [ y] is th grtst itgr lss th or qul to y, th th vlu of th itgrl / [ si ] d is () () 8. A If f( ) A si B, f d f ( ) d, th th costts A d B r rspctivly [IIT 99] () d () d 9. If I d, th d () I () I / [IIT 999] 67 of 68
14 I / I. I t d, th lim [ I I ] quls () / (). Th r oudd y th curvs y l, y l, y l d l () sq. uit () 6 sq. uit sq. uit No of ths si. d si, ( N) quls [Kurukshtr CEE 998] () () ( ). If ) d, th ( () () No of ths. si d is [AIEEE ] () () 8 8. ( si ) d cos () / () / is [AIEEE ] y is [AIEEE ] 7 6. O th itrvl,, th grtst vlu of th fuctio f( ) (6 cos t si t) dt / () () Dos ot ist No of ths 7. If I d, I d, I d, I d, th [AIEEE ] () I I () I I I I I I 8. If f( ) f, th d f( ) is qul to () l l () ( l ) No of ths [AIEEE ] [MNR 99; P. CET ; UPSEAT ] [J & K ] 9. If d d d, th th vlu of d will rspctivly [AMU ] (), (),,,. Th si d cosi curvs itrscts ifiitly my tims givig oudd rgios of qul rs. Th r of o of such rgio is [DCE ] () () 68 of 68
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