AIEEE CBSE ENG A function f from the set of natural numbers to integers defined by

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1 AIEEE CBSE ENG. A futio f from the set of turl umers to itegers defied y, whe is odd f (), whe is eve is (A) oe oe ut ot oto (B) oto ut ot oe oe (C) oe oe d oto oth (D) either oe oe or oto. Let z d z e two roots of the equtio z + z +, z eig omple. Further, ssume tht the origi, z d z form equilterl trigle, the (A) (B) (C) (D). If z d re two o zero omple umers suh tht z, d Arg (z) Arg () π, the z is equl to (A) (B) (C) i (D) i + i. If, the i (A), where is y positive iteger (B), where is y positive iteger (C) +, where is y positive iteger (D) +, where is y positive iteger 5. If d vetors (,, ) (,, ) d (,, ) re o oplr, the the produt equls (A) (B) (C) (D) 6. If the system of lier equtios + y + z + y + z + y + z hs o zero solutio, the,, (A) re i A. P. (B) re i G.P. (C) re i H.P. (D) stisfy If the sum of the roots of the qudrti equtio + + is equl to the sum of the squres of their reiprols, the, d re i (A) rithmeti progressio (B) geometri progressio (C) hrmoi progressio (D) rithmeti geometri progressio 8. The umer of rel solutios of the equtio + is (A) (B) (C) (D)

2 9. The vlue of for whih oe root of the qudrti equtio ( 5 + ) + ( ) + is twie s lrge s the other, is (A) (B) (C) (D) I. If A d A α β, the β α (A) α +, β (B) α +, β (C) α +, β (D) α, β +. A studet is to swer out of questios i emitio suh tht he must hoose t lest from the first five questios. The umer of hoies ville to him is (A) (B) 96 (C) 8 (D) 6. The umer of wys i whih 6 me d 5 wome die t roud tle if o two wome re to sit together is give y (A) 6! 5! (B) (C) 5!! (D) 7! 5!. If,, re the ue roots of uity, the is equl to (A) (B) (C) (D). If C r deotes the umer of omitios of thigs tke r t time, the the epressio C r+ + C r + C r equls (A) + C r (B) + C r+ (C) + C r (D) + C r+ 5. The umer of itegrl terms i the epsio of ( 8 + 5) is (A) (B) (C) (D) 5 6. If is positive, the first egtive term i the epsio of ( + ) 7/5 is (A) 7 th term (B) 5 th term (C) 8 th term (D) 6 th term 7. The sum of the series + upto is equl to (A) log e (B) log (C) log e (D) log e e 8. Let f () e polyomil futio of seod degree. If f () f ( ) d,, re i A. P., the f (), f () d f () re i (A) A.P. (B) G.P. (C) H. P. (D) rithmeti geometri progressio 56

3 9. If,, d y, y, y re oth i G.P. with the sme ommo rtio, the the poits (, y ) (, y ) d (, y ) (A) lie o stright lie (B) lie o ellipse (C) lie o irle (D) re verties of trigle. The sum of the rdii of isried d irumsried irles for sided regulr polygo of side, is π π (A) ot (B) ot π π (C) ot (D) ot. If i trigle ABC os C + os A, the the sides, d (A) re i A.P. (B) re i G.P. (C) re i H.P. (D) stisfy +. I trigle ABC, medis AD d BE re drw. If AD, DAB 6 π d ABE π, the the re of the ABC is (A) 8 (C) 6 (B) 6 (D). The trigoometri equtio si si, hs solutio for (A) < < (B) ll rel vlues of (C) < (D). The upper th portio of vertil pole suteds gle t t poit i the horizotl 5 ple through its foot d t diste m from the foot. A possile height of the vertil pole is (A) m (B) m (C) 6 m (D) 8 m 5. The rel umer whe dded to its iverse gives the miimum vlue of the sum t equl to (A) (B) (C) (D) 6. If f : R R stisfies f ( + y) f () + f (y), for ll, y R d f () 7, the f (r) is 7 (A) (C) 7 ( + ) 7 ( + ) (B) 7 ( + ) (D) r

4 7. If f () f () f () f () ( ) f (), the the vlue of f () !!!! (A) (B) (C) (D) is 8. Domi of defiitio of the futio f () + log ( ), is (A) (, ) (B) (, ) (, ) (C) (, ) (, ) (D) (, ) (, ) (, ) 9. t lim π / + t (A) 8 [ si] [ π ] is (B) (C) (D). log( + ) log( ) If lim k, the vlue of k is (A) (B) (C) (D). Let f () g () k d their th derivtives f (), g () eist d re ot equl for some. f()g() f() g()f() + g() Further if lim g() f(), the the vlue of k is (A) (B) (C) (D). The futio f () log ( + + ), is (A) eve futio (C) periodi futio (B) odd futio (D) either eve or odd futio +. If f () e, the f () is, (A) otiuous s well s differetile for ll (B) otiuous for ll ut ot differetile t (C) either differetile or otiuous t (D) disotiuous everywhere. If the futio f () 9 + +, where >, ttis its mimum d miimum t p d q respetively suh tht p q, the equls (A) (B) (C) (D)

5 t 5. If f (y) e y, g (y) y; y > d F (t) f (t y) g (y) dy, the (A) F (t) e t ( + t) (B) F (t) e t ( + t) (C) F (t) t e t (D) F (t) t e t 6. If f ( + ) f (), the f () d is equl to (A) f( )d (C) + f()d (B) + + f()d (D) f( + )d se t dt 7. The vlue of lim si is (A) (B) (C) (D) 8. The vlue of the itegrl I (A) + (C) + + ( ) d is (B) (D) lim lim is 5 5 (A) (B) zero (C) (D) 5. d e Let F () d si, >. If e d F (k) F (), the oe of the possile vlues of k, is (A) 5 (B) 6 (C) 6 (D) 6. The re of the regio ouded y the urves y d y is (A) sq uits (B) sq uits (C) sq uits (D) 6 sq uits. Let f () e futio stisfyig f () f () with f () d g () e futio tht stisfies f () + g (). The the vlue of the itegrl f () g () d, is

6 e 5 (A) e (B) e + e (C) e (D) e + e e +. The degree d order of the differetil equtio of the fmily of ll prols whose is is is, re respetively (A), (B), (C), (D), 5. The solutio of the differetil equtio ( + y ) + ( t y (A) ( ) k e (B) e t y (C) e t y + k (D) e t y dy e ), is d t y t y + k t y e + k 5. If the equtio of the lous of poit equidistt from the poits (, ) d (, ) is ( ) + ( ) y +, the the vlue of is (A) ( + ) (B) + + (C) ( + ) (D) + 6. Lous of etroid of the trigle whose verties re ( os t, si t), ( si t, os t) d (, ), where t is prmeter, is (A) ( ) + (y) (B) ( ) + (y) + (C) ( + ) + (y) + (D) ( + ) + (y) 7. If the pir of stright lies py y d qy y e suh tht eh pir isets the gle etwee the other pir, the (A) p q (B) p q (C) pq (D) pq 8. squre of side lies ove the is d hs oe verte t the origi. The side pssig through the origi mkes gle α ( < α < π ) with the positive diretio of is. The equtio of its digol ot pssig through the origi is (A) y (os α si α) (si α os α) (B) y (os α + si α) + (si α os α) (C) y (os α + si α) + (si α + os α) (D) y (os α + si α) + (os α si α) 9. If the two irles ( ) + (y ) r d + y 8 + y + 8 iterset i two distit poits, the (A) < r < 8 (B) r < (C) r (D) r > 5. The lies y 5 d y 7 re dimeters of irle hvig re s 5 sq uits. The the equtio of the irle is (A) + y + y 6 (B) + y + y 7 (C) + y + y 7 (D) + y + y 6 5. The orml t the poit (t, t ) o prol meets the prol gi i the poit (t, t ), the

7 (A) t t t (D) t t t (B) t t + t (D) t t + t 5. y y The foi of the ellipse + d the hyperol oiide. The the vlue of is (A) (B) 5 (C) 7 (D) 9 5. A tetrhedro hs verties t O (,, ), A (,, ), B (,, ) d C (,, ). The the gle etwee the fes OAB d ABC will e (A) os 9 (B) os 7 5 (C) (D) 9 5. The rdius of the irle i whih the sphere + y + z + y z 9 is ut y the ple + y + z + 7 is (A) (B) (C) (D) 55. y z y z 5 The lies d k k re oplr if (A) k or (B) k or (C) k or (D) k or 56. The two lies y +, z y + d d y +, z y + d will e perpediulr, if d oly if (A) (B) + + (C) ( + ) ( + ) + ( + ) (D) The shortest diste from the ple + y + z 7 to the sphere + y + z + y 6z 55 is (A) 6 (B) (C) (D) Two systems of retgulr es hve the sme origi. If ple uts them t distes,, d,, from the origi, the (A) (C) r r r r r r r r r r r r r r r r,, re vetors, suh tht + +,,,, the + + is equl to (A) (B) 7 (C) 7 (D) 6. If u, v r r r r r r r u + v w) (u v) (v w ) equls r r r (A) (B) u v w

8 r r r r r r (C) u w v (D) u v w 6. Cosider poits A, B, C d D with positio vetors 7 î ĵ + 7kˆ, î 6ĵ + kˆ, î ĵ + kˆ d 5 î ĵ + 5kˆ respetively. The ABCD is (A) squre (B) rhomus (C) retgle (D) prllelogrm ut ot rhomus 6. The vetors AB î + kˆ, d AC 5î ĵ + kˆ re the sides of trigle ABC. The legth of the medi through A is (A) 8 (B) 7 (C) (D) A prtile ted o y ostt fores î + ĵ kˆ d î + ĵ kˆ is displed from the poit î + ĵ + kˆ to the poit 5 î + ĵ + kˆ. The totl work doe y the fores is (A) uits (B) uits (C) uits (D) 5 uits r r 6. Let u î + ĵ, v î ĵ d the w r ˆ is equl to r w î + ĵ + kˆ (A) (B) (C) (D) r. If ˆ is uit vetor suh tht u ˆ r d v ˆ 65. The medi of set of 9 distit oservtios is.5. If eh of the lrgest oservtios of the set is iresed y, the the medi of the ew set (A) is iresed y (B) is deresed y (C) is two times the origil medi (D) remis the sme s tht of the origil set 66. I eperimet with 5 oservtios o, the followig results were ville: 8, 7 Oe oservtio tht ws ws foud to e wrog d ws repled y the orret vlue. The the orreted vrie is (A) 78. (B) (C) 77. (D) Five horses re i re. Mr. A selets two of the horses t rdom d ets o them. The proility tht Mr. A seleted the wiig horse is (A) (B) 5 5, (C) 5 (D) Evets A, B, C re mutully elusive evets suh tht P (A), P (B) P (C). The set of possile vlues of re i the itervl (A), (B), (C), (D) [, ] d

9 69. The me d vrie of rdom vrile hvig iomil distriutio re d respetively, the P (X ) is (A) (B) 6 (C) 8 (D) 7. The resultt of fores P r d Q r is R r. If Q r is douled the R r is douled. If the diretio of Q r is reversed, the R r is gi douled. The P : Q : R is (A) : : (B) : : (C) : : (D) : : 7. Let R d R respetively e the mimum rges up d dow ilied ple d R e the mimum rge o the horizotl ple. The R, R, R re i (A) rithmeti geometri progressio (B) A.P. (C) G.P. (D) H.P. 7. A ouple is of momet G r d the fore formig the ouple is P r. If P r is tured through right gle, the momet of the ouple thus formed is H r. If isted, the fores P r re tured through gle α, the the momet of ouple eomes (A) G r si α H r os α (B) H r os α + G r si α (C) G r os α H r si α (D) H r si α G r os α 7. Two prtiles strt simulteously from the sme poit d move log two stright lies, oe with uiform veloity u r d the other from rest with uiform elertio r f. Let α e the gle etwee their diretios of motio. The reltive veloity of the seod prtile with respet to the first is lest fter time usiα f osα (A) (B) f u uosα (C) u si α (D) f 7. Two stoes re projeted from the top of liff h meters high, with the sme speed u so s to hit the groud t the sme spot. If oe of the stoes is projeted horizotlly d the other is projeted t gle θ to the horizotl the t θ equls u u (A) (B) g gh h (C) h g u (D) u gh 75. A ody trvels distes s i t seods. It strts from rest d eds t rest. I the first prt of the jourey, it moves with ostt elertio f d i the seod prt with ostt retrdtio r. The vlue of t is give y s (A) s + (B) f r + f r (C) s(f + r) (D) s + f r

10 Solutios. Clerly oth oe oe d oto Beuse if is odd, vlues re set of ll o egtive itegers d if is eve, vlues re set of ll egtive itegers. Hee, (C) is the orret swer.. z + z z z (z + z ) z z. Hee, (C) is the orret swer ( + ). Hee, (B) is the orret swer. + i (+ i) i i + i i i. Hee, (A) is the orret swer. 6. Coeffiiet determit +. Hee, (C) is the orret swer 8. + ( ) ( ) ±, ±. Hee, (B) is the orret swer 7. Let α, β e the roots α + β + α β α + β α + β αβ ( α + β) ( + )

11 ,, re i H.P. Hee, (C) is the orret swer. A A + + α +, β. Hee, (B) is the orret swer. 9. β α α 5 + α ( ) ( 5 + ) Hee, (A) is the orret swer. Clerly 5! 6! (A) is the orret swer. Numer of hoies 5 C 8 C C 5 8 C Hee, (B) is the orret swer Sie, + +, if is ot multiple of Therefore, the roots re idetil. Hee, (A) is the orret swer. 7. C r+ + C r + C r + C r + C r+ + + C r + C r+. Hee, (B) is the orret swer + + +

12 log log e log. e Hee, (D) is the orret swer. 5. Geerl term 56 C r ( ) 56 r [(5) /8 ] r From itegrl terms, or should e 8k k to. Hee, (B) is the orret swer. 8. f () + + f () + + f ( ) lso + f () + f () f () f () AP. Hee, (A) is the orret swer. 9. Result (A) is orret swer.. (B) + osc + os A Hee, (A) is the orret swer 6. f () 7 f ( + ) f () + f () f () 7 oly f () 7 r 7 5. (B) f (r) 7 ( ) ( + ). π si π. π si π ()

13 . Hee, (D) is the orret swer 7. LHS! ( ) ( )( ) + +!! C + C. Hee, (C) is the orret swer. + lim +. Hee, (C) is the orret swer. 8. ± > ( + ) ( ) >. Hee (D) is the orret swer. 9. lim π /. π t ( si) π ( π ) Hee, (C) is the orret swer.. f ( ) f () Hee, (B) is the orret swer.. si (θ + α) si. Hee, (B) is the orret swer θ t (/) / /. f () t p, q 6p + 8p + 6q + 8q + f () < t p d f () > t q.. Applyig L. Hospitl s Rule f()g () g()f () lim g () f ()

14 k(g () ff ()) (g () f ()) k. Hee, (A) is the orret swer. 6. f () d ( + ) f ( + ) d. Hee, (B) is the orret swer.. f () f ( h) f ( + h) LHD RHD. Hee, (B) is the orret swer. 7. t( ) lim si t( ) lim si. Hee (C) is the orret swer. 8. ( ) d ( ) + ( ). + + Hee, (C) is the orret swer. t 5. F (t) f (t y) f (y) dy t f (y) f (t y) dy t y e (t y) dy t ( + t). Hee, (B) is the orret swer.. Clerly f () > for q < for p or p q. Hee, (C) is the orret swer. si e. F ()

15 e 6 e si si d F (k) F () d F (k) F () 6 F () d F (k) F () F (6) F () F (k) F () k 6. Hee, (D) is the orret swer.. Clerly re sq uits (,) (,) (,) 5. Let p (, y) ( ) + (y ) ( ) + (y ) ( ) + ( ) y + ( + ). Hee, (A) is the orret swer. os t + sit + sit os t + 6., y + y +. 9 Hee, (B) is the orret swer.. Equtio y 9 h) yy yy yy y. Hee (B) is the orret swer.. f () [ f ()] d solvig this y puttig f () f (). Hee, (B) is the orret swer. 5. Itersetio of dimeter is the poit (, ) πs 5 s 9 ( ) + (y + ) 9 Hee, (C) is the orret swer. 7. (D) 9. d ( + y dy ) si y (e )

16 d + dy α + y e su y + y y e 5. (C) e 6 7. Hee, (C) is the orret swer p pq q, p 8 p ( ) 8 C 8. Hee, (A) is the orret swer. 9. ( ) + (y ) r ( ) + (y + ) ( ) + (y + ). 67. Selet out of 5 5. Hee, (D) is the orret swer Hee, (C) is the orret swer.

17 z π. Arg z z i or + i.

[Q. Booklet Number]

[Q. Booklet Number] 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 (,