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1 6 [Q. Booklet Numer] KOLKATA WB- B-J J E E - 9 MATHEMATICS QUESTIONS & ANSWERS. If C is the reflecto of A (, ) i -is d B is the reflectio of C i y-is, the AB is As : Hits : A (,); C (, ) ; B (, ) y A (, ) AB ( ( )) ( ( )) O. The vlue of cos cos º º 7 si 7 is B (, ) C(, ) As : 8 6 Hits : cosºcos 7 si 7 si 7 cos 7.(cosº ) siº (cosº) (siº cosº ) si 8. The vlue of itegrl d is As : Hits : I d, + v d dv - I v dv v v v dv dv

2 . The lie y t itersects the ellipse y i rel poits if 9 t t t t As : Hits : y ; 9 y t t 9 9 t 9( t ) t ( t )( t ) t ( t ) t 9( t. Geerl solutio of si + cos mi, 6 IR ) is. ( ) As : ( ) Hits : si cos mi, 6 IR ( ) ( ) + 6 ( ) + mi( 6) IR mi, 6 mi{, } IR si + cos si cos si si, ( ). ( ) 6. If A d B squre mtrices of the sme order d AB I, the A is equl to B B As : Hits : AB I, A.AB.A I B A A B B B []

3 7. The co-ordites of the focus of the prol descried prmetriclly y t +, y t + re (7, ) (, ) (, ) ( 7, ) As : Hits : t + ; y t +, or, (y ) ( ) y y 8. For y two sets A d B, A (A B) equls B A B A B A C B C As : Hits : A (A B) A (A B c ) A (A B c ) c A (A c c B) (A A ) (A B) A B 9. If, 6, A º, the o trigle is possile oe trigle is possile two trigle re possile either o trigle or two trigles re possile As : Hits : ; 6;A sia sib sib sia 6 sib siº. No trigle is possile sice sib >. A Mppig from IN to IN is defied s follows : f : IN IN f() ( + ), IN (IN is the set of turl umers). The f is ot oe-to-oe f is oto f is oth oe-to-oe d oto f is oe-to-oe ut ot oto As : Hits : f : IN IN ; f() ( + ) ( + ) ( + ) ( ) ( + + ) oe-to-oe There does ot eist IN such tht ( + ) Hece f is ot oto []

4 . I trigle ABC if si A si B, the the trigle is c equilterl isosceles right gled otuse gled As : Hits : sia sib c c siasib sia sib c c si C sic C 9º sic d. si cos equls l t c l t c l t c l t c where c is ritrry costt As : Hits : d si cos d si cos d si cosec d log t c 6 l t c 6. The vlue of cos cos cos cos 7 6 As : is Hits : cos cos cos cos []

5 . If P si cos the As : P P P P 6 6 Hits : P si cos si si si 6 si si 6 6 P. A positive cute gle is divided ito two prts whose tgets re d. The the gle is As : 6 Hits : Agle t t t. t / 6 t / 6 () / 6. If f() f( ) the f()d is equl to f()d f()d f()d f()d As : Hits : f() f( ), I f()d ( )f( )d ( )f()d I f()d I f()d I f()d []

6 d 7. The vlue of ( )( 9) is 6 As : 8 Hits : d / ( )( 9) sec d (puttig t) (t )(t 9) / {(9 t ) ( t )}sec d (t )(t 9) / sec d t / sec d 9 t t / t / t t / si 8. If I d d I os d, the, / c I I I < I I > I I I + / As : / / y cos Hits : I si d ; I cos d y si P / I,, cos > si / cos d si d O / / I > I i.e. I < I 9. The secod order derivtive of si t with respect to cos t t t is As : Hits : y si t ; cos t dy si t cost; dt dy d dy dt d dt si tcost cos tsit d cos t sit dt sit cost tt [6]

7 d y d d d dy d d d d dt dt tt tt. d ( sec t) cos t sit cos t sit d d y t/.. The smllest vlue of cos + is 7 7 As : Hits : cos +, cos cos cos + + cos + 7. The geerl solutio of the differetil equtio dy d y y e e is e y e e + c e y e e + c e y e + e + c e y e + e + c where c is ritrry costt As : Hits : e y y dy (e e )d Itegrte y e e e c, e e e c. Product of y r cosecutive turl umers is lwys divisile y r! (r + )! (r + )! (r + )! As : Hits : ( + ) ( + )... ( + r) ( r)!! ( r)! r r! r! C! r! dy. The itegrtig fctor of the differetil equtio log y log is give y d e log log (log ) As : Hits : If e dy. y d log log d e log(log ) e log / d log [7]

8 . If + y the y y (y) y y ( y) y y ( y) yy (y) As : Hits : + yy + yy yy ( y). If c, c, c,..., c deote the co-efficiets i the epsio of ( + ) the the vlue of c + c + c c is. ( + ) ( + ) ( + ) As. Hits : ( ) c c c... c ( ) Put () c c c c... c... c c 6. A polygo hs digols. The umer of its sides is As : Hits : C ( ) ( ) 88 ( ) 8 7. If e the roots of ( ) +, the the vlue of As : Hits : + + ( + ) ( + ) + 8. The gle etwee the lies joiig the foci of ellipse to oe prticulr etremity of the mior is is 9º. The eccetricity of the ellipse is 8 [8]

9 As : Hits : t e e e e S (-e, ) (, ) (, ) O S(e, ) e e e e 9. The order of the differetil equtio d y d dy d As :. The sum of ll rel roots of the equtio + 7 As : Hits : Put y y + y (y ) (y + ) y y (Not possile) ± ±, Sum. If f ( ) d d { f ( )} d 7 the the vlue of f() d As : Hits : f ( ) d ( ) f ( ) d7 f ( ) d f ( ) d f ( ) d f ( ) d is f ( ) d ( ) [9]

10 . For ech N, is divisile y where N is set of turl umers As : Hits : (8) ( + 7) + C 7 + C C 7 7[ C + C ]. The Rolle s theorem is pplicle i the itervl for the fuctio f() f() f() + f() As : Hits : f() d f() f( ) for f() ut t, f() is ot differetile hece is the correct optio. f () f ( ). The distce covered y prticle i t secods is give y + 8t t. After secod velocity will e uit/secod uits/secod uits/secod 7 uits/secod As : d Hits : v 8 8t dt t, v 8 8. If the co-efficiets of d i the epsio of ( + ) 9 e sme, the the vlue of is 7 As : Hits : ( + ) 9 9 C C 8 () + 9 C 7 () + 9 C 6 () 9 C 7 9 C The vlue of is log log As : Hits : log + log log [] 7. If log c, y log c, z log c, the the vlue of will e y z + y + z + c + c c As : Hits : + log + log c log c log log z c c, Similrly log c y c, As. log (c) c

11 8. Usig iomil theorem, the vlue of (.999) correct to deciml plces is As : Hits : C C (. ) C (.) C (.). + (.) (.) If the rte of icrese of the rdius of circle is cm/.sec., the the rte of icrese of its re, whe the rdius is cm, will e As : dr Hits : A r dt da dr r () dt dt. The qudrtic equtio whose roots re three times the roots of + + c is + + c + + c c + + c As : Hits : c 9. c + + c. Agle etwee y d y t the origi is t t As : His : Agle etwee es (sice co-ordite es re the tgets for the give curve).. I trigle ABC,, d si A, the B is equl to º 6º 9º º As : Hits : sia si B.si A B sib. []

12 []. e is equl to As : e e e e (e ) His : I [] e d e ( e ) ( e ) Period of fuctio is. The coefficiet of, where is y positive iteger, i the epsio of ( ) ½ is As : + + Hits : s s s( ) s ( ) f(), f() ( ) The circles + y + 6 d + y itersect t two distict poits if < < < 8 > 8 As. Hits : C (, ) r 6 C (, ) r r & r < C C < r + r < < < + < < < < < 8 < < 8 []

13 si 6. d is equl to log (si ) + c (si ) c where c is ritrry costt As : Hits : I tdt log c si (cos ) + c si t t c d dt (si ) c 7. The umer of poits o the lie + y which re uit distce prt from the lie + y is Ifiity As : Hits : + y y PQ 8. Simplest form of cos is As : sec sec cosec Hits :.cos cos. cos sec cos cos 9. If y t si si, the the vlue of dy t d 6 is As : []

14 Hits : y t cos cos t si cos t t dy d. If three positive rel umers,, c re i A.P. d c the miimum possile vlue of is As : Hits : ( - d) ( + d) ( d ) + d (). If cos cos, whe ( < < ), the the vlues of re :,cos cos, cos As : Hits : cos cos (cos ) cos cos cos ( cos )(cos ) cos cos cos cos. For y comple umer z, the miimum vlue of z + z is As : Hits : z (z ) z + z []

15 . For the two circles + y 6 d + y y there is / re oe pir of commo tgets oly oe commo tget three commo tgets o commo tget As : Hits : C (, ) r C (, ) r C C r r C C < r r. If C is poit o the lie segmet joiig A (, ) d B (, ) such tht AC BC, the the coordite of C is, As : Hits :, (, 7) (7, ) A (, ) C B (, ) C, C,. If,, c re rel, the oth the roots of the equtio ( ) ( c) + ( c) ( ) + ( ) ( ) re lwys positive egtive rel imgiry As : Hits : ( c) c c D ( c).( c c) ( c c c) [( ) ( c) ( c ) ] [( ) ( c) ( c ) ] 6. The sum of the ifiite series is!! 6! e e e As : e Hits : T...( ) []

16 (...)... e ep e e e 7. The poit (, ) is the verte of squre d oe of its digols is 7 y + 8. The equtio of the other digol is 7 y + 7y + 7y + 7y As : C Hits : + 7y k...() + k k + 7y A B (, ) 8. The domi of defiitio of the fuctio f ( ) log e ( ) is As : Hits : < + log e ( ) log e ( ) e e e e m m 9. For wht vlue of m, m m e e e is the rithmetic me of d? Noe As : Hits : m m m m m Stisfy. [6]

17 si( e ) 6. The vlue of the limit lim log is e As : e Hits : si( ) Lt log( h) h e h Put h h si( e ) ( e ) Lt. h h ( e ) log( h) h si( e ) ( e ) h Lt.. h h ( e ) h log( h).. h h 6. Let f ( ) the the vlue of Lt f ( ) is does ot eist As : Hits : Becuse o left hd side of fuctio is ot defied. 6. f() + is cotiuous for (, ) (, ) {} oly > o vlue of As : Hits : ; f ( ) ; y y y O 6. t cos t cos is equl to As : Hits : Let cos, the cos [7]

18 t cos t cos t t t t cos 6. If i d is positive iteger, the i + i + + i + + i + is euql to i i As : Hits : i ( i i i ) i ( i i) d 6. ( ) equls l c l c l c l c where c is ritrry costt. As : d d d l l C l C ( ) Hits : d 66. If,, c re i G.P. ( >, >, c > ), the for y rel umer (with >, ), log, log, log c re i G..P. A.P. H.P. G..P. ut ot i H.P. As : Hits :,, c re i G.P. log,log,log c re i A.P. log, log, log c rei H.P. log,log,log c rei H.P. 67. A lie through the poit A (, ) which mkes gle of º with the positive directio of -is is rotted out A i clockwise directio through gle º. The the equtio of the stright lie i the ew positio is y y y y As : Hits : Equtio of lie i ew positio : y tº ( ) y y [8]

19 y ( ) y ( ) y 68. The equtio si cos hs oly oe solutio two solutios ifiitely my solutios o solutio As : Hits : si cos si. Therefore 6 si cos cot hve solutio 69. The slope t y poit of curve y f () is give y dy d it psses through (, ). The equtio of the curve is d y + y y + y + As : Hits : dy d dy d y C Curve psses through (, ). Hece + C C y + 7. The modulus of i i i is uit As : uit uit uit Hits : i i i i( i) i i 7i ( 7i)( i) i ( i) ( i) ( i) (9 ) i i 7 i i Modulus 7. The equtio of the tget to the coic y 8 + y + t (, ) is y + As : Hits : Equtio of tget t (, y ) is yy ( + ) + (y + y ) + ; y Equtio of tget is y ( ) ( y ) or 8 + uit [9]

20 or + or or or 7. A d B re two idepedet evets such tht P(AB').8 d P.. The P is As : Hits : Let P P(AB') P + P(B') P(AB'). + ( ).( ) or.8 +. or.7.8 or.7. or 7 7. The totl umer of tgets through the poit (, ) tht c e drw to the ellipses + y d + 9y is As : Hits : (, ) lies outside the ellipse y d o the ellipse 9y. Therefore there will e tgets for the first ellipse d oe tget for the secod ellipse. 7. The vlue of lim... is As : log zero Hits : lim... lim d lim t r r r r 7. A prticle is movig i stright lie. At time t, the distce etwee the prticle from its strtig poit is give y t 6t + t. Its ccelertio will e zero t t uit time t uit time t uit time t uit time As : Hits : t 6t t d t t dt d Accelertio dt Accelertio 6t t d 6t dt []

21 76. Three umers re chose t rdom from to. The proility tht they re cosecutive is As : Hits : Totl umer of cses ; C Totl umer of fvourle cses 8 Required proility The co-ordites of the foot of the perpediculr from (, ) upo the lie + y re (, ) (, ) (, ) (, ) As : Hits : Let P e the foot of the perpediculr. P lies o lie perpediculr to + y. Equtio of the lie o which P lies is of the form : y + k (, ) But this lie psses through (, ). P k Hece, co-ordites of P my e otied y solvig + y d y, y Hece, P (, ) O (, ) 78. If A is squre mtri the, A + A T is symmetric AA T is skew - symmetric A T + A is skew-symmetric A T A is skew symmetric As : Hits : (A + A T ) T A T + (A T ) T A T + A A + A T 79. The equtio of the chord of the circle + y whose mid poit is (, ) is y y As : Hits : O(, ) Equtio : Chord with mid-poit (, ) 8. If A A + I, the the iverse of the mtri A is A I I A A + I A As : Hits : A A + I A A I A.A A.A A A I A A I A []

22 MATHEMATICS SECTION-II. A tri movig with costt ccelertio tkes t secods to pss certi fied poit d the frot d ck ed of the tri pss the fied poit with velocities u d v respectively. Show tht the legth of the tri is (u + v)t. A. v u + t v u t v u + S. Show tht v u ( v u)( v u) t( v u) u v S S t si si si 9 (t 7 t) cos cos9 cos 7 A. T si cos si. cos cos.cos.cos si( ). cos.cos T (t t) T (t9 t) T (t 7 t9) T + T + T (t 7 t ). If si t, y si t, prove tht d y dy y d d A. y si ( si ) dy d cos(si ). dy d cos(si ) []

23 ( ( dy ) d dy ) d.cos [ y Agi differetite (si ] ) [ si dy d y dy ( ).. ( ) 8y d d d Divide y dy d dy d (si d y dy ( ) y d d. Show tht, for positive iteger, the coefficiet of k ( K ) i the epsio of + ( + ) + ( + ) ( + ) is + C k. )] A. ( ) S ( ) ( ) Coefficiet of k i ( ) Coefficiet of k+ i ( + ) + + C k+ + C k. If m, e itegers, the fid the vlue of (cosm si) d A. I (cos m si si.cos m) d cos m. d si. d si.cos m. d cos m. d si. d (Odd...) ( cosm) d ( cos) d m (si m) (si ) + ( ) ( ) m []

24 6. Fid the gle suteded y the doule ordite of legth of the prol y t its verte. A. y,, [ put y ] A (, ), B(, ) Slope OA Slope of OB As. 7. If f is differetile t, fid the vlue of (O, C) A B f() f() Lt. A. Lt f ( ) f ( ), form y LH f ( ) Lt f ( ) f() f () 8. Fid the vlues of for which the epressio ( ) + + is lwys positve. A. ( ) + + > D < ( ) ( + ) < < + < ( 9) ( ) < < < 9 9. Fid the sum of the first terms of the series A. S (.) (.) (.)... 9 (.... terms) []

25 (.)[ (.) ] 9 9 [ (.)] 9 9 (.) [ (.) ] 9 (.9) 8 (.) 8. The equtio to the pirs of opposite sides of prllelogrm re + 6 d y 6y +. Fid the equtios of its digols. A....(i)... (ii) y... (iii) y... (iv) A (, ), B (, ), C (, ), D(, ) Equtio of AC y, y 8 y, y 7 Equtio of BD y y, + y + y []

Important Facts You Need To Know/Review:

Important Facts You Need To Know/Review: Importt Fcts You Need To Kow/Review: Clculus: If fuctio is cotiuous o itervl I, the its grph is coected o I If f is cotiuous, d lim g Emple: lim eists, the lim lim f g f g d lim cos cos lim 3 si lim, t