WBJEEM MATHEMATICS. Q.No. μ β γ δ 56 C A C B

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1 WBJEEM - MATHEMATICS Q.No. μ β γ δ C A C B B A C C A B C A B B D B 5 A C A C 6 A A C C 7 B A B D 8 C B B C 9 A C A A C C A B B A C A B D A C D A A B C B A A 5 C A C B 6 A C D C 7 B A C A 8 A A A A 9 A C B B A D A A D A A C D A C A C B D D A A B A 5 B C C A 6 C B B A 7 A A A A 8 A C A C 9 B A D A C C C D B C C C C B A C B B D A B A A 5 C A B C 6 A C C D 7 D A A 8 A B A C 9 C C A B D C A D A D B C C A A A A D C A A A A 5 A B C D B C A C 7 A C A B 8 C D A B 9 B C D B 5 B C C C 5 A C D C 5 A D C C 5 A A C B 5 A A B A 55 C A C A 56 C A C B 57 D B B A 58 B A B A 59 C C A C 6 C D A D 6 D B C A 6 A A A C 6 D B A A 6 B C A C 65 B C D A 66 A D D C 67 A D C C 68 C D A D 69 A A A D 7 D A B A 7 C C B D 7 C A C A 7 A A D B 7 C A A B 75 A C C A 76 A,D A,B,D C,D A,B 77 A,B A,B A,B A,B 78 A,B C,D A,B,D A,D 79 C,D A,B A,D A,B,D 8 A,B,D A,D A,B C,D

2 WBJEEM - (Aswrs & Hits) Cod-μ CATEGORY - I Q. to Q.6 carry o mark ach, for which oly o optio is corrct. Ay wrog aswr will lad to dductio of / mark.. y Lt th quatio of a llips b 5, ad passig through th foci of th llips is 9 (B) 7 (C) (D) 5 Hits : a, b 5 P (, ), S(a, ) Radius PS, S ( 9, ) PS +. Th th radius of th circl with tr ( ). If y + is paralll to a tagt to th parabola y, th its distac from th ormal paralll to th giv li is 9 (B) 7 7 As : (B) Hits : m slop of li ; a y m am am (Norm l) (C) 7 y 6.. Distac 9 7. I a ΔABC, taa ad tab ar th roots of pq( + ) r. Th ΔABC is a right agld triagl (B) a acut agld triagl (C) a obtus agld triagl (D) a quilatral triagl Hits : pq r + pq taa tab so ta(a+b) is udfid C π/. Lt th umbr of lmts of th sts A ad B b p ad q rspctivly. Th th umbr of rlatios from th st A to th st B is p+q (B) pq (C) p + q (D) pq As : (B) Hits : p (B) q ; (A B) pq ANSWERS & HINTS for WBJEEM - SUB : MATHEMATICS (D) 7

3 WBJEEM - (Aswrs & Hits) 5. Th fuctio π ta π[ ] f(), whr [] dots th gratst itgr, is + [] cotiuous for all valus of (B) discotiuous at (C) ot diffrtiabl for som valus of (D) discotiuous at Hits : f() R 6. Lt z, z b two fid compl umbrs i th Argad pla ad z b a arbitrary poit satisfyig z z + z z z z. Th th locus of z will b a llips (B) a straight li joiig z ad z (C) a parabola (D) a bisctor of th li sgmt joiig z ad z Hits : Possibility of llips P(z), S (z ), S (z ) PS + PS a S S a So it is a llips + 7. Lt S C + C+ C C. Th S quals + As : (B) + + Hits : P + + r+ (B) + + r Cr ; S r + C. r, r (C) S C. r π (D) 8. Out of 7 cosoats ad vowls, th umbr of words (ot cssarily maigful) that ca b mad, ach cosistig of cosoats ad vowls, is 8 (B) 5 (C) 5 (D) 5 Hits : 7 C C 5! 9. Th rmaidr obtaid wh! +! +! ! is dividd by is 9 (B) 8 (C) 7 (D) 6 Hits : divids!, 5! tc. Rmaidr

4 WBJEEM - (Aswrs & Hits) Lt S dot th sum of th ifiit sris Th!!! 5! S < 8 (B) S > (C) 8 < S < (D) S 8 Hits : th trm of, 8,,, 65,... ( ) r r.(r ) + r! 7 5. For vry ral umbr, lt f() Th th quatio f() has!!!! o ral solutio (B) actly o ral solutio (C) actly two ral solutios (D) iifiit umbr of ral solutios As : (B) Hits : is a solutio r ( r ) r! r. Th cofficit of i th ifiit sris pasio of, for <, is ( )( ) /6 (B) 5/8 (C) /8 (D) 5/6 As : (B) Hits : Ep Cofficit 5/8 ( ) ( ) If α, β ar th roots of th quadratic quatio + p + q, th th valus of α + β ad α + α β + β ar rspctivly pq p ad p p q + q (B) p(q p ) ad (p q)(p + q) (C) pq ad p q (D) pq p ad (p q) (p q) As : (D) Hits : α + β + α β α + β (α+β) αβ (α + β) (α + β ) α β p + pq (p q) q. A fair si-facd di is rolld tims. Th probability that ach fac turs up twic is qual to! 6!6!6 (B) 6 6! (C) 6 6! (D) 66 As : ( C) Hits : C C... C. 6

5 WBJEEM - (Aswrs & Hits) 5. Lt f() b a diffrtiabl fuctio i [, 7]. If f() ad f () 5 for all i (, 7), th th maimum possibl valu of f() at 7 is 7 (B) 5 (C) 8 (D) θ 5 Hits : ta 5 5 θ 5 7 So f(7) Th valu of π π π ta + ta + cot is cot 5 π Hits : Add, subtract cot 5 π (B) cot π 5 (C) cot π 5 (D) cot π 5 7. Lt b th st of all ral umbrs ad f : b giv by f() +. Th th st f ([, 6]) is 5 5,, (B) 5 5, As : (B) Hits : f () 6 > if >, < if < f() f(α) 6. So α± 5 (C), (D) 5 5, 5 [Not : f() is is ivrtibl ithr f r > or for < so th right aswr should b ithr, or 5, 8. Th ara of th rgio boudd by th curvs y ad y is / (B) / (C) / (D) Hits : ( ) 9. Th poit o th parabola y 6 which is arst to th li + y + 5 has coordiats (9, ) (B) (, 8) (C) (, 6) (D) ( 9, ) Hits : Normal at P(am, am) has slop m. a 6, m. Th quatio of th commo tagt with positiv slop to th parabola y 8 ad th hyprbola y is y 6 + (B) y 6 (C) y + (D) y Hits : y a, a

6 WBJEEM - (Aswrs & Hits) y m + a m ; m >, a.m m, m 6. Lt p,q b ral umbrs. If α is th root of +p +5q, β is a root of +9p +5q ad <α<β, th th quatio +6p +q has a root γ that always satisfis γ α/+β (B) β < γ (C) γ α/+β (D) α < γ < β As : (D) Hits : Lt, f() +6p +q f(α) α +6p α+q (α +p α+5q ) + (p α+5q ) + p α+5q > Agai, f(β) β +6p β+q (β +9p β+5q ) (p β+5q ) (p β + 5q ) < So, thr is o root γ such that, α<γ<β. Th valu of th sum ( C ) + ( C ) + ( C ) ( C ) is ( C ) (B) C (C) C + (D) C As : (D) Hits : (+) C + C + C C, (+) C + C +C C So, cofficit of i [(+) (+) ] i.. (+) is ( C +C C ), which is C So, C C +C + C C, ( C ) + ( C ) + ( C ) ( C ) C C C. Ram is visitig a frid. Ram kows that his frid has childr ad of thm is a boy. Assumig that a child is qually likly to b a boy or a girl, th th probability that th othr child is a girl, is / (B) / (C) / (D) 7/ Hits : Evt that at last o of thm is a boy A, Evt that othr is girl B, So, probability rquird P(B/A) ( A) P( A) PB ( ) ( ) PB A, Now, total cass ar ( BG, BB GG) P A (As, B A {BG} ad A {BG,BB}) ( ) ( ) / cos π/ si π/ si. Lt b a itgr, A ( π/) cos( π/) ad Ι is th idtity matri of ordr. Th A Ι ad A Ι (B) A m Ι for ay positiv itgr m (C) A is ot ivrtibl (D) A m for a positiv itgr m Hits : A cos θ si θ siθ cos θ, A cosθ si θ si θ cos θ, So, hr, A cosπ si π si π cos π,, ad A Ι

7 WBJEEM - (Aswrs & Hits) 5. Lt Ι dot th idtity matri ad P b a matri obtaid by rarragig th colums of Ι. Th Thr ar si distict choics for P ad dt(p) (B) Thr ar si distict choics for P ad dt(p) ± (C) Thr ar mor tha o choics for P ad som of thm ar ot ivrtibl (D) Thr ar mor tha o choics for P ad P Ι i ach choic As : (B) Hits : Ι, diffrt colums ca b arragd i,! i.. 6 ways, I ach cas, if thr ar v umbr of itrchags of colums, dtrmiat rmais ad for odd umbr of itrchags, dtrmiat taks th gativ valu i..! 6. Th sum of th sris π si 7 is π si 8 +si π 6 +si π 5 (C) (B) π si 6 +si π +si π +si π 6 π si 6 +si π +si π +si π 6 +si π 7 (D) si π 8 +s π 6! Hits : π si 7,! π! π si5! π! π si + si si π si 6 +si π +si π +si π 6 6 π s 7 + si k π, whr k, so si k π k 6 7. Lt α,β b th roots of ad S α +β, for all itgrs. Th for vry itgr S +S S + (B) S S S (C) S S + (D) S +S S + Hits : α+β, S + S, (α +α )+ β +β ), α (α+) + β (β+), ow sic α α & β β α.α + β.β α + +β + S + 8. I a ΔABC, a,b,c ar th sid of th triagl opposi to th agls A,B,C rspctivly. Th th valu of a si(b C) + b si(c A) + c si(a B) is quqal to (B) (C) (D) 9. I th Argad pla, th distict roots of +z+z +z (z is a compl umbr) rprst vrtics of a squar (B) a quilatral triagl (C) a rhombus (D) a rctagl As : (B) Hits : +z+z +z, (+z) (+z ), z,, ω, ω, whr ω is a cub root of uity, so, distict roots ar : (,),,,,. Distac btw ach of thm is. So, thy form a quilatral triagl. Th umbr of digits i (giv log.) is 6 (B) (C) 9 (D) 9 Hits : log log. 9.6, so, 9 digits

8 WBJEEM - (Aswrs & Hits). If y cos, th it satisfis th diffrtial quatio ( ) c, whr c is qual to (B) (C) (D) As : (D) Hits : y cos y (cos ), cos, cos, +, (- ). Th itgratig factor of th diffrtial quatio (+ ) y + ta is ta (B) + (C) ta (D) log (+ ) Hits : + y ta, I.F ta. Th solutio of th quatio log log 7 ( ) is (B) 7 (C) 9 (D) 9 Hits : log log 7 ( ), log 7 ( ), , ,, 9. If α,β ar th roots of a +b+c (a ) a d α + h, β + h ar th roots of p +q+r (p ) th th ratio of th squars of thir discrimiats is a :p (B) a:p (C) a :p (D) a:p D Hits : D a (α β), D P (α β) ; D a p [ Not : Corrct aswr is 5. Lt f() +5+. If w writ f() as f() a(+)( ) +b( )( )+c( )(+) for ral umbrs a,b,c th thr ar ifiit umbr of choics for a,b,c (B) oly o choic for a but ifiit umbr of choics for b ad c (C) actly o choic for ach of a,b,c (D) mor tha o but fiit umbr of choics for a,b,c a p for D Hits : f() (a+b+c) + ( a b) a+b c, a+b+c, a b 5, a+b c, a, b /, c 9 6. Lt f() + /. th th umbr of ral valus of for which th thr uqual trms f(), f(), f() ar i H.P. is (B) (C) (D) D ]

9 WBJEEM - (Aswrs & Hits) Hits : f() f()f(), f(), f(), f(), f(), f() ar i H.P., So, f() f() + f(),,, at, trms ar qual so oly solutio is 7. Th fuctio f() +b+c, whr b ad c ral costats, dscribs o-to-o mappig (B) oto mappig (C) ot o-to-o but oto mappig (D) ithr o-to-o or oto mappig As : (D) Hits : Upward parabola f() has a miimum valu. So, it is ot oto, also symmtric about its ais which is a straight li paralll to Y-ais, so it is ot o-to-o 8. Suppos that th quatio f() +b+c has two distict ral roots α ad β. Th agl btw th tagt to α+β α+β th curv y f() at th poit,f ad th positiv dirctio of th -ais is (B) (C) 6 (D) 9 Hits : f() +b+c rprsts upward parabola which cuts -ais at α ad β. As th graph is symmtric, so, α+β α+β tagt at,f paralll to -ais. Hc, M α β N ( α+ β, f ( α+ β )) y ϕ y 9. Th solutio of th diffrtial quatio y + y is (whr c is a costat) ϕ y y y ϕ c (B) ϕ c (C) ϕ y c (D) ϕ c Hits : Lt, y ν, y ν ν + d ν d substitutig, v (ν + φν ν ( ) ) ν + φ ( ν ) ( ) νφ ν, dv, φν ( ) [Lt, φ(v ) z, φ (v )v dv dz], dz z, z / + k, z c, y φ c. Lt f() b a diffrtiabl fuctio ad f () 5. Th ( ) f( ) f lim (B) 5 (C) (D) As : (D) quals

10 WBJEEM - (Aswrs & Hits) Hits : lim ( ) f( ) f,( form, so usig L Hospital s rul), lim ( ) f, f () 5. Th valu of cos(t )dt lim is si (B) (C) (D) log Hits : cos lim si + cos cos lim si + cos +. Th rag of th fuctio y si, Hits : y si π 6 (B) [, ] π 6 is (C), (D) [, ) y ma si π/, y mi. Thr is a group of 65 prsos who lik ithr s gig or dacig or paitig. I this group lik sigig, lik dacig ad 55 lik paitig. If 6 prsos lik bot sigig ad dacig, lik both sigig ad paitig ad lik all thr activitis, th th umbr of prs s who lik oly dacig ad paitig is (B) (C) (D) Hits : (S P D) 65 (S) (D) (P) 55 (S D) 6 (S P) (S D P) (S P D) (S) + (D) + (P) (S D) (D P) (P S) + (S D P) (P D) + (P D) (P D) (P D S). Th curv y (cos + y) / satisfis th diffrtial quatio (y ) + + cos (B) y + cos (C) (y ) + cos (D) (y ) + cos

11 WBJEEM - (Aswrs & Hits) Hits : y (cos + y) / y cos + y y si + +.y. cos +, (y ) + + cos 5. Suppos that z, z, z ar thr vrtics of a quilatral triagl i th Argad pla. Lt α ( + i ) ad β b a o-zro compl umbr. Th poits αz + β, αz + β, αz + β will b Th vrtics of a quilatral triagl (B) Th vrtics of a isoscls triagl (C) Colliar (D) Th vrtics of a scal triagl Hits : + + ( α z +β) ( α z +β) ( α z +β) ( α z +β) ( α z +β) ( α z +β) + + α(z z ) α(z z ) α(z z ) + + α (z z ) (z z ) (z z ) Hc, αz + β, αz + β, αz + β ar vrtics of quilat ral triagl. a si si 6. If lim ists ad is qual to, th th valu of a is ta (B) (C) (D) As : (B) Hits : 8 a( ) ( ) lim +... a (a ) lim +... a a +, 7. If f() th, > f() is 7/ (B) 5/ (C) / (D) 7/ Hits : f() + f() ( + ) + ( ) + + 5/ + 7/

12 WBJEEM - (Aswrs & Hits) 8. Th valu of z + z + z i is miimum wh z quals i (B) 5 + i (C) Hits : z + z + z i + y + ( ) + y + + (y ) + y 6 y + i + (D) i [ + y.y. ] + i z Th umbr of solutio(s) of th quatio + is/ar (B) (C) (D) As : (B) Hits : + Squarig, / Which dos ot satisfis th quat o. Hc, o solutio 5. Th valus of λ for which th curv (7 + 5) + (7y + ) λ ( + y ) rprsts a parabola is ± 6 5 As : (B) (B) ± 7 5 Hits : 9 [( + 5/7) + (y + /7) ] 5λ + y 5 (C) ± 5 (D) ± 5 5λ 9 λ 9 5 λ ±7/5

13 WBJEEM - (Aswrs & Hits) 5. If si + cosc π, th th valu of is 5 (B) (C) (D) Hits : si π cosc sc cos 5 si si 5 5. Th straight lis + y, 5 + y ad + 5y form a isoscls triagl (B) a quilatral triagl (C) a scal tri gl (D) a right agld triagl Hits : Thir poit of itrsctio ar (, ) (, ) ad (/, /) which is th vrtics of isocls triagl. 5. If I ( α ), th α lis i th itrval (, ) (B) (, ) (C) (, ) (D) (, ) Hits : I ( α ) > ( α) should b somwhr positiv ad somwhr gativ so α (, ) Hc, a (, ) 5. If th cofficit of 8 i a + b is qual to th cofficit of 8 i a b, th a ad b will satisfy th rlatio ab + (B) ab (C) a b (D) a + b Hits : a + b Co-fficit of 8 i a + b

14 WBJEEM - (Aswrs & Hits) C 6 a 7. 6 b () Co-fficit 8 i a b C 7 a 6 7 b C 7 a 6. 7 b () Sic, C 6 a 7 /b 6 C 7 a 6 /b 7 a b ab Th fuctio f() a si + b is diffrtiabl at wh a + b (B) a b (C) a + b (D) a b Hits : f() a si + b f() a si + b a si + b < f () acos + b acos b < at a + b a b a + b 56. If a, b ad c ar positiv umbrs i a G.P., th th roots of th quadratic quatio (log a) (log b) + (log c) ar log c log c ad log a (B) ad log a Hits : b ac log a log b + log c ( ) ( ) log a log b + log c Sic, satisfis th quatio Thrfor is o root ad othr root say β (C) ad log a c (D) ad log c a.β log c log a log a c β log c log a log c a

15 WBJEEM - (Aswrs & Hits) si, 57. Lt R b th st of all ral umbrs ad f: [, ] R b dfid by f(), Th, f satisfis th coditios of Roll s thorm o [, ] (B) f satisfis th coditios of Lagrag s Ma Valu Thorm o [, ] (C) f satisfis th coditios of Roll s thorm o [, ] (D) f satisfis th coditios of Lagrag s Ma Valu Thorm o [, ] As : (D) Hits : f() is odiffrtiabl at 58. Lt z b a fid poit o th circl of radius ctrd at th origi i th Argad pla ad z ±. Cosidr a quilatral triagl iscribd i th circl with z, z, z as th vrtics tak i th coutr clockwis dirctio. Th z z z is qual to z (B) z (C) z (D) z As : (B) π Hits : Lt z r iα, z r i( α+ ), z r z z z r π α+α+ +α+ r i(α + π) r iα (r iα ) π i( ) π i( α+ ) z 59. Suppos that f() is a diffrtiabl fuctio s ch tha f () is cotiuous, f () ad f () dos ot ist. Lt g() f (). Th g () dos ot ist (B) g () (C) g () (D) g () Hits : g().f () g( + h) g() g () lim h h (+ h) f( + h) f() g () lim h h g () + lim f ( + h) h f () z z z

16 WBJEEM - (Aswrs & Hits) 6. Lt [] dot th gratst itgr lss tha or qual to for ay ral umbr. Th lim is qual to (B) (C) (D) Hits : lim < + < + lim lim < CATEGORY - II Q.6 to Q.75 carry two marks ach, for which oly o optio is corrct. Ay wrog aswr will lad to dductio of / mark 6. W dfi a biary rlatio ~ o th st of all ral mat ics as A ~ B if ad oly if thr ist ivrtibl matrics P ad Q such that B PAQ. Th biary rlatio ~ is Nithr rfliv or symmtric (B) Rfliv ad symmtric but ot trasitiv (C) Symmtric ad trasitiv but ot rfl iv (D) A quivalc rlatio As : (D) Hits : For Rfliv, A.I IA, A IAI so rfliv. For Symmtric, B PAQ, BQ PA, P BQ A or A (P ) B. (Q ),so symmtric. For Trasitiv, B PAQ, C PBQ P.PAQ.Q (PP)A(QQ), so trasitiv 6. Th miimum valu of si + co is / (B) / + (C) (D) si cos + si + cos Hits :, si cos +., si cos + 6. For ay two ral umbrs θ ad ϕ, w dfi θrϕ if ad oly if sc θ ta ϕ. Th rlatio R is Rfliv but ot trasitiv (B) Symmtric but ot rfliv (C) Both rfliv ad symmtric but ot trasitiv (D) A quivalc rlatio As : (D) Hits: For rfliv, θ φ so sc θ ta θ, Hc Rfliv For symmtric, sc θ ta φ so, ( + ta θ) (sc φ ) so, sc φ ta θ. Hc symmtric For Trasitiv, lt sc θ ta φ ad sc φ ta γ so, + ta φ ta γ or, sc θ ta γ. Hc Trasitiv

17 WBJEEM - (Aswrs & Hits) 6. A particl startig from a poit A ad movig with a positiv costat acclratio alog a straight li rachs aothr poit B i tim T. Suppos that th iitial vlocity of th particl is u > ad P is th midpoit of th li AB. If th vlocity of th particl at poit P is v ad if th vlocity at tim T is v, th v v (B) v > v (C) v < v (D) v v As : (B) t tt/ tt A (v ) P(v ) B (u>) Hits : Sic th particl is movig with a positiv costat acclratio hc it s vlocity should icras. So th tim tak to travl AP is mor that th timtak for PB. So th istat T is bfor P. Hc v > v sic vlocity icrass from A to B. 65. Lt t dot th th trm of th ifiit sris 9...! +! +! +! + 5! +. Th lim t is (B) (C) (D) As : (B) Hits : t + 6 6, lim sic domiator is vry larg compard to umrator 66. Lt α, β dot th cub roots of uity othr tha ad α β. Lt s ( ) α β. Th th valu of s is Eithr ω or ω (B) Eithr ω or ω (C) Eithr ω or ω (D) Eithr ω or ω Hits : a ω, β ω α ω β ω ω + ω ω + ω...+ ω, S ( ).( ω) ( ) ( ) α ω α ω, β ω ω, S (ω β ω ) (ω ) +(ω )...+(ω ) ω ω ω ( ) ( ω) ω ω ω 67. Th quatio of hyprbola whos coordiats of th foci ar (±8, ) ad th lgth of latus rctum is uits, is y 8 (B) y 8 (C) y 8 (D) y 8 b Hits : a 8, a, y a a + b or, 6 a + a so a, b 8, so y Applyig Lagrag s Ma Valu Thorm for a suitabl fuctio f() i [, h], w hav f(h) f() + hf (θh), < θ <. Th for f() cos, th valu of lim h + θ is (B) (C) (D)

18 WBJEEM - (Aswrs & Hits) Hits : For f() cos, cos h + h ( si(θh)), cosh siθ h, h cosh si h θ h cosh h si si h lim θ lim lim h h h h h 69. Lt X {z + iy : z } for all itgrs. Th is X A siglto st (B) Not a fiit st (C) A mpty st (D) A fiit st with mor tha o lmts Hits : X ( z ) (,) X ( z ), X ( z ), Th rquird rgios ar shadd for,, so clarly will b oly th poit circl origi. So a siglto st π/cos 7. Suppos M N, + π+ As : (D) (B) π/ ( + ) X si cos. Th th valu of (M N) quals π π π π si cos N si Hits : + ( ) + ( + ) + π cos cost Rplacig t, dt M ( + ) ( t+. So M N ) π+ π (C) π π + cos π+ + ( ) (D) π+ 7. π π 6π cos + cos + cos is qual to zro (B) lis btw ad (C) is a gativ umbr (D) lis btw ad 6 As : (c)

19 WBJEEM - (Aswrs & Hits) π si π π 6π Hits : cos cos cos 7 π + + cos si π 7 7. Clarly it is a gativ o. 7. A studt aswrs a multipl choic qustio with 5 altrativs, of which actly o is corrct. Th probability that h kows th corrct aswr is p, < p <. If h dos ot kow th corrct aswr, h radomly ticks o aswr. Giv that h has aswrd th qustio corrctly, th probability that h did ot tick th aswr radomly, is p p + (B) 5p p + (C) 5p p + As : (c) Hits : K H kows th aswrs, NK H radomly ticks th aswrs, C H is corrct C PK ( ) P K K P C C C PK ( ) P + PNK ( ) P K NK P P + ( P) 5 5P P+ (D) p p + 7. A pokr had cosists of 5 cards draw at radom from a wll-shuffld pack of 5 cards. Th th probability that a pokr had cosists of a pair ad a tripl of qual fac valus (for ampl svs ad kigs or acs ad qus, tc.) is 6 65 Hits : (B) 5 C5 65 C C C C (C) (D) Lt f( ) ma{ +, [ ]}, whr [] d ts th gratst itgr. Th th valu of ( ) (B) 5/ (C) / (D) As : (c) Hits : Rquird ara Th solutio of th diffrtial quatio K f is y + udr th coditio y wh is log NK C y y log + (B) y log + (C) y log log + (D) y log log log + Hits : Itgratig factor y ( log ) + log log log ( log ) log y log + c + c,c

20 WBJEEM - (Aswrs & Hits) CATEGORY - III Q. 76 Q. 8 carry two marks ach, for which o or mor tha o optios may b corrct. Markig of corrct optios will lad to a maimum mark of two o pro rata basis. Thr will b o gativ markig for ths qustios. Howvr, ay markig of wrog optio will lad to award of zro mark agaist th rspctiv qustio irrspctiv of th umbr of corrct optios markd. 76. Lt f( ) t dt, >, Th, f() is cotiuous at (B) f() is ot cotiuous at (C) f() is diffrtiabl at (D) f() is ot diffrtiabl at As : (A,D) Hits : t dt ( t) dt + ( t ) dt +, f( ) f ( ) Clarly f() is cotiuous but ot diffrtiabl at. +, >, >,, 77. Th agl of itrsctio btw th curvs y si + cos ad + y, whr [] dots th gratst itgr, is ta (B) ta ( ) (C) ta (D) ta ( ) / As : (A,B) Hits : si + cos + si So, s + cos. y si + cos. (,) y y, So, agl is ithr ta ( ) or ta (). (,) y 78. If u() ad v() ar two idp dt solutios of th diffrtial quatio b cy, + th additioal solutio(s) of th giv diffrtial qua io is (ar) y 5 u() + 8 v() (B) y c {u() v()} + c v(), c ad c ar arbitrary costats (C) y c u() v() + c u()/v(), c ad c ar arbitrary costats (D) y u() v() As : (A,B) Hits : Ay liar combiatio of u() ad v() will also b a solutio. 79. For two vts A ad B, lt P.7 ad P(B).6. Th cssarily fals statmts(s) is/ar P( A B).5 (B) P( A B).5 (C) P( A B).65 (D) ( ) As : (C,D) P A B.8 Hits : P( A B) P( A B) ow PA ( ) PA ( B),.7. P( A B), ( ). P A B.6

21 WBJEEM - (Aswrs & Hits) 8. If th circl + y + g + fy + c cuts th thr circls + y 5, + y 8 6y + ad + y + y at th trmitis of thir diamtrs, th C 5 (B) fg 7/5 (C) g + f c + (D) f g As : (A,B,D) Hits : Commo chords of th circl will pass through th ctrs. c 5, 8g + 6f 5, g f 7 so, g,f 5

PURE MATHEMATICS A-LEVEL PAPER 1

-AL P MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION PURE MATHEMATICS A-LEVEL PAPER 8 am am ( hours) This papr must b aswrd i Eglish This papr cosists of Sctio A ad Sctio