Time : 1 hr. Test Paper 08 Date 04/01/15 Batch - R Marks : 120

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Lewis Walters
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1 Tim : hr. Tst Papr 8 D 4//5 Bch - R Marks : SINGLE CORRECT CHOICE TYPE [4, ]. If th compl umbr z sisfis th coditio z 3, th th last valu of z is qual to : z (A) 5/3 (B) 8/3 (C) /3 (D) o of ths 5 4. Th itgral, ( cost si t si t cos t) dt has th valu qual to 4 (A) (B) / (C) / (D) 3. A curv is rprstd paramtrically by th quios = t + ad y = t + wh t R ad a >. If th curv touchs th ais of th poit A, th th coordis of th poit A ar (A) (, ) (B) (/, ) (C) (, ) (D) (, ) 4. If z = + iy & = iz th = implis th, i th compl pla : z i (A) z lis o th imagiary ais (C) z lis o th uit circl (B) z lis o th ral ais (D) o 5. Lt ABCD b a ttrahdro such th th dgs AB, AC ad AD ar mutually prpdicular. Lt th ara of triagls ABC, ACD ad ADB b 3, 4 ad 5 sq. uits rspctivly. Th th ara of th triagl BCD, is 5 5 (A) 5 (B) 5 (C) (D) 6. Lt C ad C ar coctric circls of radius ad 8/3 rspctivly havig ctr (3, ) o th argad pla. If th compl umbr z sisfis th iquality, log /3 z 3 > th : z 3 (A) z lis outsid C but isid C (B) z lis isid of both C ad C (C) z lis outsid both of C ad C (D) o of ths 7. Th rgio rprstd by iqualitis Arg Z < 3 ; Z < ; Im(z) > i th Argad diagram is giv by (A) (B) (C) (D) VKR Classs, C Idra Vihar, Kota. Mob. No # #

2 8. Numbr of roots of th fuctio f () = 3 ( ) 3 + si is (A) (B) (C) (D) mor tha 9. A bam of light is st alog th li y =. Which aftr rfractig from th -ais tr th opposit sid by turig through 3º towards th ormal th poit of icidc o th -ais. Th quio of th rfractd ray is (A) ( + 3 ) y = + 3 (B) ( + ) y = + (C) 3 y = 3 (D) No of ths. If a, c, b ar i G.P. th th li a + by + c = : (A) has a fid dirctio (B) always passs through a fid poit (C) forms a traigl with th as whos ara is costat (D) always cuts itrcpts o th as such th thir sum is zro f(b) f(b). Lt f b a ral valud fuctio with drivivs upto ordr two for all R. If = b a for f(a) f(a) ral umbr a & b whr a < b th thr is a umbr c (a, b) for which (A) f(c) = f(c) (B) f(c) = f(c) (C) f(c) = cf(c) (D) No of ths. Writ th corrct squc of Tru & Fals for th followig: (i) At th poit (, ), th tagt li to y = (ii) f() = t t has th grst slop. t dt taks o its maimum valu =,. (iii) Th maimum valu of + is lss tha its miimum valu. (iv) Th maimum valu of / is /, ( > ) (A) FFTT (B) TTFT (C) FTTT (D) o 3. Lt A ad B b mrics ovr th rals. If A B = A + B AB th A B = B A = O if ad oly if :- (A) A is o-sigular (C) (I A) is o-sigular (B) B is o-sigular (D) No of ths 4. Th compl umbr sisfyig th quio 3 = 8i ad lyig i th scod quadrat o th compl pla is (A) 3 + i (B) 3 + i (C) 3 + i (D) 3 + i 5. Th umbr of poits with itgral coordis lyig i th itrior of th quadilral formd by lis + y =, 4 + 5y = ad th coordi as is - (A) 5 (B) 6 (C) 7 (D) o of ths 6. Th tru st of ral valus of for which th poit P with co-ordiat (, ) dos ot li isid th triagl formd by th lis, y = ; + y = & + 3 = is - (A) (, ] (B) [, ) (C) [, ] (D) (, ] [, ) 7. Numbr of imagiary compl umbrs sisfyig th quio, z = (A) (B) (C) (D) 3 VKR Classs, C Idra Vihar, Kota. Mob. No # # z z is

3 8. Two opposit sids of rhombus ar + y = ad + y = 5. If o vrt is (, ) ad th agl th vrt is 45º, a vrt opposit to th giv vrt is. (A) (6 +, ) (B) (6, + ) (C) (6, ) (D) o of ths 9. If f() = sg(si si ) has actly four poits of discotiuity for (, ), N th (A) th miimum valu of is 5 (B) th maimum valu of is 6 (C) thr ar actly two possibl valus of (D) o of ths q d. Lt C 8 r p whr p, q, r N ad d ot b distict, th th valu (p + q + r) quals (A) 64 (B) 6 (C) 6 (D) 6. Lt f () =, dots f ' () = l, g () = l +, dots g ' () = m, h () = d d th th valu of t dt, dots h ' () =, lm, is (A) 3 (B) 3 (C) 3 (D). Cosidr two fuctios f () = si ad g () = f (). Stmt- : Th fuctio h () = f () g() is ot diffrtiabl i [, ] Stmt- : f () is diffrtiabl ad g () is ot diffrtiabl i [, ] (A) Stmt- is tru, stmt- is tru ad stmt- is corrct plaio for stmt-. (B) Stmt- is tru, stmt- is tru ad stmt- is NOT th corrct plaio for stmt-. (C) Stmt- is tru, stmt- is fals. (D) Stmt- is fals, stmt- is tru. 3. I a quadrilral ABCD, AC is th bisctor of th AB AD which is, 3 5 AC 5 AD th cos BA CD is : (A) 4 7 (B) 7 3 = 3 AB = VKR Classs, C Idra Vihar, Kota. Mob. No # 3 # (C) 7 (D) If th vctor 6i 3j 6k is dcomposd ito vctors paralll ad prpdicular to th vctor i j k th th vctors ar : (A) ( î ĵkˆ ) & 7 î ĵ (B) ( î ĵkˆ ) & 8 î ĵ 4kˆ (C) + ( î ĵkˆ ) & 4 î 5 ĵ 8kˆ (D) o 5. Th umbr of poits, whr th fuctio f() = ma ( ta, cos ) is o-diffrtiabl i th itrval (, ), is (A) 4 (B) 6 (C) 3 (D)

4 6. If A ( 4,, 3) ; B (4,, 5) th which o of th followig poits li o th bisctor of th agl btw OA ad OB ('O' is th origi of rfrc) (A) (,, ) (B) (,, 5) (C) (,, ) (D) (,, ) 7. Suppos & ar th poit of maimum ad th poit of miimum rspctivly of th fuctio f() = 3 9 a + a + rspctivly, th for th quality = to b tru th valu of 'a' must b : (A) (B) (C) (D) o i i 8. Th valu of th Lim si i (A) (B) is (C) 9. Four coplaar forcs ar applid a poit O. Each of thm is qual to k, & th agl btw two coscutiv forcs quals 45º. Th th rsultat has th magitud qual to : (D) (A) k (B) k 3 (C) k 4 (D) o 3. If f () = 4 + a 3 +b + c + d b polyomial with ral cofficit ad ral roots. If f ( i ) =, whr i =, th a + b + c + d is qual to (A) (B) (C) (D) ca ot b dtrmid Aswr Sht Studt Nam: Bch : R D : 4//5. A B C D. A B C D 3. A B C D 4. A B C D 5. A B C D 6. A B C D 7. A B C D 8. A B C D 9. A B C D. A B C D. A B C D. A B C D 3. A B C D 4. A B C D 5. A B C D 6. A B C D 7. A B C D 8. A B C D 9. A B C D. A B C D. A B C D. A B C D 3. A B C D 4. A B C D 5. A B C D 6. A B C D 7. A B C D 8. A B C D 9. A B C D 3. A B C D VKR Classs, C Idra Vihar, Kota. Mob. No # 4 #

5 JEE Mais Tst Papr 8 Bch - R D 4//5 /. I = si t cos t dt + / 4 / 5 4 = si t dt si t dt / 4 / zro ths two itgrals cacls Zro] 3. = t + ; y = t + d dy = + a ; = + a dt ; dt ANSWER WITH SOLUTION SOLUTION 5 ( si t cost) (si t cost)dt + dy = d a a dy th poit A, y = ad = for som t = t d 4 si t cost dt a =...() ; also = t + ; = t...(), puttig this valu i () w gt, = t = a ; ow from () a = a = hc A = t + = + = A (, ) As. ] 5. Ara of BCD = BC B D = (bî cĵ) (bî dkˆ ) = bd ĵ bc kˆ dc î = ow 6 = bc ; 8 = cd ; = bd b c + c d + d b = substitutig th valu i () A = b c = 5 As. ] JEE MAINS c d d b...() ANSWER KEY Q A. B A D B A A B C A C B AorC Q A. C A B D C A C C B D C A Q A. A D B B C C VKR Classs, C Idra Vihar, Kota. Mob. No # 5 #

6 3 8. f ' () = 4 ( ) 3 + cos < hc f () is always dcrasig, Also as, f () ad as, f () + hc o positiv ad o giv root Graph is as show. cosidr g() = (f() + f()) From th giv iformio, g(a) = g(b). By Roll's Thorm, thr ists c (a, b) such th g(c) =. Hr g() = (f() f()) g(c) = f(c) = f(c). (i) (F) y = dy = d slop is ot grst. (iii) (T) y = +...() d d y = 3...(3) = (, ) ta = = dy = d...() dy d y = = =, = d y d d >, d < y is ma. if =. y is mi. = (ma). (y) = =, mi.(y) = + = ma. valu < mi. valu ] 3. A B = O if ad oly if (I A) (I B) = I. This implis th I A is o-sigular. Covrsly, if I A is osigular ad lt C b its ivrs ad lt B = I C th C = I B. So (I A) (I B) = I d 7 d. I = ( ) ( ) 7 6 d ( ) = l ( + 7 ) = p + q + r = 6 As. + C lm. l = ; m = 3; = = 3 As. ]. f () = si is diffrtiabl i [, ] g () = si is ot diffrtiabl =. Lt h () = f () g () = si si VKR Classs, C Idra Vihar, Kota. Mob. No # 6 #

7 si( h) si( h) h ' () = Lim h h = h () is diffrtiabl = but g () is ot diffrtiabl = ] 6. OA = 4î 3kˆ ; OB = 4î ĵ 4î 3kˆ â ; 5 4î ĵ bˆ 5 r î 9ĵ4î ĵ 5 r î ĵ 4kˆ 5 r î ĵ kˆ 5 r r 8. T r = si = ] r si r si h si h Lim h h = Sum = put = t S = r r T Lim si r = si d r d = dt si t dt = As. 3. Lt f () = ( )( )( 3 )( 4 ) f ( i ) = 3 4 = = = 3 = 4 = all four roots ar zro f () = 4 a = b = c = d = a + b + c + d = (C) ] VKR Classs, C Idra Vihar, Kota. Mob. No # 7 #

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