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1 R Pen Towe Rod No Conttos Ae Bistupu Jmshedpu 8 Tel (67)89 IIT JEE themtis Ppe II PART III ATHEATICS SECTION I (Totl ks : ) (Single Coet Answe Type) This setion ontins 8 multiple hoie questions. Eh question hs fou hoies (A) (B) (C) nd (D) out of whih ONLY ONE is oet.. Let f () nd g() sin fo ll R. Then the set of ll stisfying (f o g o g o f)() (g o g o f) () whee (f o g) () f (g ()) is (A) ± n n {...} (B) ± n n {...} (C) ( / ) n n {......} (D) n n {......}. (A) (f o g o g o f)() (g o g o f) () [sin sin ] sin sin sin sin o ± n n {...}.. Let ( y) e ny point on the pol y. Let P e the point tht divides the line segment fom ( ) to ( y) in the tio :. Then the lous of P is (A) y (B) y (C) y (D) y. (C) y Let P e (h k) h k k h Lous of P is y. y. Let P(6 ) e point on the hypeol. If the noml t the point P intesets the is t (9 ) then the eentiity of the hypeol is (A) ( / ) (B) ( / ) (C) (D). (B) Noml to hypeol t P(6 ) 6 y 6 / / y 6 6 ( / ) 9 / / e ( / ) / e ( / ).. A vlue of fo whih the equtions hve one oot in ommon is (A) (B) i (C) i (D) IIT JEE ( Ap ) Questions & Solutions Ppe II www. penlsses.om

2 . (B) α e the ommon oot α α α α α α i.. Let e ue oot of unity nd S e the set of ll non singul mties of the fom whee eh of nd is eithe o. Then the nume of distint mties in the set S is (A) (B) 6 (C) (D) 8. (A) ( ) Clely o o o n tke ny vlue o ut s it mkes the deteminnt zeo. lso o No of suh mties. 6. The ile pssing though the point ( ) nd touhing the y is t ( ) lso psses though the point (A) ( / ) (B) ( / ) (C) ( / / ) (D) ( ) 6. (D) Let C e (h ) Eqution of ile: ( h) (y ) h Q ile psses though ( ) h / Eqution of ile will e ( ( / )) (y ) / Only ( ) stisfies. / 7. If Lim [ ln( )] sin θ > nd θ ( ] then the vlue of θ is (A) ± / (B) ± / (C) ± / 6 (D) ± / 7. (D) sin θ sin θ sin θ ± sin θ sin θ ± os θ ( sin θ ) Fo to e el os θ θ ± /. IIT JEE ( Ap ) Questions & Solutions Ppe II www. penlsses.om

3 8. Let f : [ ] [ ) e ontinuous funtion suh tht f () f ( ) fo ll [ ]. Let R f ( ) d nd R e the e of the egion ounded y y f () nd the is. Then (A) R R (B) R R (C) R R (D) R R 8. (C) R ( ) f ( ) d R R R R. SECTION II (Totl ks : 6) ultiple Coet Choie Type This setion ontins multiple hoie questions. Eh question hs fou hoies (A) (B) (C) nd (D) out of whih ONE OR ORE my e oet. 9. Let L e noml to the pol y. If L psses though the point (9 6) then L is given y (A) y (B) y (C) y (D) y 9. (A)(B)(D) Eqution of noml y m m m i.e y m m m Pssing though (9 6) m 7m 6 m Eqution of nomls e y y y.. Let f : ( ) R e defined y f ( ) whee is onstnt suh tht < <. Then (A) f is not invetile on ( ) (B) f f on ( ) nd f ( ) f ( ) (C) f f on ( ) nd f ( ) (D) f is diffeentile on ( ) f ( ) ( ). (C)(D) f ( ) < ( ) s ( ) ( ) Also f () is ontinuous fo ll ( ) s domin of definition is R { / } whe / > Hene f () is stitly deesing so invetile. f ( ) y y f ( ) f ( ) y f ( ) nd f ' () Sine f ' () eist fo ll ( ) f is diffeentile fo ll ( ). IIT JEE ( Ap ) Questions & Solutions Ppe II www. penlsses.om

4 . Let E nd F e two independent events. The poility tht etly one of them ous is / nd the poility of none of them ouing is /. If P(T) denotes the poility of ouene of the event T then (A) P(E) / P(F) / (B) P(E) / P(F) / (C) P(E) / P(F) / (D) P(E) / P(F) /. (A)(D) P ( E F ) P ( E F ) / P ( E F ) / Let P(E) P(F) y P ( E ). P ( F ) / ( ) ( y) / Also P ( E ). P ( F ) P ( E ). P ( F ) / ( y) ( ) y / Solving / o / nd y / o /.. If f ( ) os < then < ln > (A) f () is ontinuous t / (B) f () is not diffeentile t (C) f () is diffeentile t (D) f () is diffeentile /. (ABCD) f ( ) os < < ln > Continuity t / L.H.L R.H.L f ( / ) f ( ) sin < < / > Clely f () is not diffeentile t f () is diffeentile t nd f () is diffeentile /. IIT JEE ( Ap ) Questions & Solutions Ppe II www. penlsses.om

5 IIT JEE ( Ap ) Questions & Solutions Ppe II www. penlsses.om Setion III (Totl ks : ) (Intege Answe Type) This setion ontins 6 questions. The nswe to eh of the questions is single digit intege nging fom to 9. The ule oesponding to the oet nswe is to e dkened in the ORS.. Let i e nd y z e non zeo omple numes suh tht z y Then the vlue of z y is.. y z y y z z ( ) ( ) ( ) ( ) ( ) ( ) ( ) z y. Let y'() y () g '() g() g'() y () R whee f '() denotes d f d ) ( nd g() is given non onstnt diffeentile funtion on R with g() g (). Then the vlue of y () is.. y ' () y () g' () g () g ' () Line diffeentil eqution with integting fto e g() y (). e g() g(). g' (). e g() d y (). e g() e g() (g() ) Sine y () nd g() y () (g() ) e g() y () (g() ) e g().. Let e mti stisfying nd Then the sum of the digonl enties of is

6 IIT JEE ( Ap ) Questions & Solutions Ppe II 6 www. penlsses.om ; Sum of digonl enties The stight line y divides the iul egion y 6 into two pts. If 8 S * then the nume of point(s) in S lying inside the smlle pt is 6.. y meets o odinte es t &. lies outside the ile hene uled out. Fo y : y > Fo y : y > Fo 8 y : y < & e the only two points. 7. Let k j i nd j i k i e thee given vetos. If is veto suh tht. nd v then the vlue of. is ( ) λ ( λ) i ( λ) j k If i y j z k then. z z Also λ y λ z λ λ i 6 j k. 6 9.

7 8. The nume of distint el oots of is 8.. Let us ssume tht ll fou oots e el nd distint. Hene f ' () must hve distint el oots nd f '' () must hve distint el oots ut tht is not tue s f '' () ( ) with D <. Hene f () n t hve ll fou oots el. As f () f () 9 nd f ( ) f () must hve two distint oots one in ( ) nd the othe one in ( ). SECTION II ( Totl ks : 6) (ultiple Coet Answes Type) This setion ontins questions. Eh question hs fou sttements (A B C nd D) given in Column I nd five sttements (pq s nd t) in Column II. Any given sttement in Column I n hve oet mthing with ONE nd ORE sttement(s) given in Column II. Fo emple if fo given question sttement B mthes with the sttments given in q nd then fo the ptiul question ginst sttement B dken the ules oesponding to q nd in the ORS. 9. th the sttements given in Column I with the vlues given in Column II. Column I Column II (A) If j k j k nd k fom tingle then the intenl ngle of the tingle etween nd e (B) If ( f ( ) ) d then the vlue of (q) f is 6 (C) The vlue of (p) se ( ) d ln is () 6 (D) The mimum vluye of z z is given y Ag z fo () (t) IIT JEE ( Ap ) Questions & Solutions Ppe II 7 www. penlsses.om

8 9. (A) (q) (B) (p) (C) (s) (D) (t) Fo (A): Sine side lengths e hene ngle etween nd is Fo (B): Fo (C): os θ Fo (D): z e iθ / θ /... f ( ) d ( ) / 6 ((ln se tn )) [ln ln 7 / 6 ln ( os θ ) i sin θ z ( os θ ) sin θ f () f ( / 6) / 6. ln ] sin θ g f ( θ ) whih is mimum when θ /. z os θ 6. th the sttements given in Column I with the intevls/ union of intevls given in Column II Column I (A) The set Column II iz Re : z is omple nume z z ± (p) ( ) ( ) z is (B) The domin of the funtion 8 ( ) f () sin ( ) is (q) ( ) ( ) tn θ (C) If f ( θ ) tn θ tn θ then the set () [ ) tn θ f ( θ ) : θ < is (D) If f ( ) ( ) then f () is (s) ( ] [ ) inesing in (t) ( ] [ ) IIT JEE ( Ap ) Questions & Solutions Ppe II 8 www. penlsses.om

9 iθ iz ie i (os θ i sin θ ) 6. (A) (s) Let k Re Re iθ Re z e sin θ i sin θ os θ i (os θ i sin θ ) Re i sin θ (os θ i sin θ ) sin θ It is defined only when k ( ] [ ) 8.( ) (B) (t) Sine 8. 9 Put y 8 y 9 y 8 y 9 y ( y 9 )( y ) ( y )( y ) ( y 9 ) ( Q y nd y e lwys ( ) ve) ( y ) y < nd y 9 ( ) [ ) 8 y ( y 9 )( y ) nd 9 y ( y )( y ) y nd y > ( y ) ( y ) ( ] ( ) The ommon solution is ( ] ( ). tn θ (C) () Sine f ( θ ) tn θ tn θ se θ f (θ) [ ) tn θ (D) () Let f() / ( ) f '() ( / ) / ( ) / Fo inesing ` f ' () [( / ) ]. ( ). IIT JEE ( Ap ) Questions & Solutions Ppe II 9 www. penlsses.om

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