Prai paprs A ad B, produd by Edl i 9, wih mark shms Prai Papr A. Fid h valus of for whih 5 osh sih =, givig your aswrs as aural logarihms. (Toal 6 marks) k. A = k, whr k is a ral osa. 9 (a) Fid valus of k for whih A is sigular. () Giv ha A is o-sigular, (b) fid, i rms of k, A. (5) (Toal 9 marks). Th urv wih quaio y = + ah,, has a maimum urig poi A. (a) Fid, i a logarihmi form, h -oordia of A. () (b) Show ha h y-oordia of A is { l( + )}. () (Toal 7 marks) 5. M = 6 9 (a) Fid h igvalus of M. () A rasformaio T : is rprsd by h mari M. Thr is a li hrough h origi for whih vry poi o h li is mappd oo islf udr T. (b) Fid a arsia quaio of his li. () (Toal 7 marks)
5. I = d,. (a) Prov ha, for, I = ( I ). () (b) Fid, i rms of, h a valu of d. (5) (Toal 8 marks) 6. y C O Figur Th urv C, show i Figur, has paramri quaios = l, y =,. (a) Show ha h lgh of C is + l. (7) Th urv is road hrough radias abou h -ais. (b) Fid h a ara of h urvd surfa grad. () (Toal marks)
7. Th pla Π passs hrough h pois P(,, ), Q(,, ) ad R(,, ), whr is a osa. (a) Fid, i rms of, RP RQ. () Giv ha RP RQ = i + dj + k, whr d is a osa, (b) fid h valu of ad show ha d =, () fid a quaio of Π i h form r. = p, whr p is a osa. () () Th poi S has posiio vor i + 5j + k. Th poi S is h imag of S udr rflio i Π. (d) Fid h posiio vor of S. (5) (Toal marks) 8. (a) Show ha h ormal o h ragular hyprbola y =, a h poi P, quaio y = +., has (5) Th ormal o h hyprbola a P ms h hyprbola agai a h poi Q. (b) Fid, i rms of, h oordias of h poi Q. (5) Giv ha h mid-poi of PQ is (X, Y) ad ha ±, () show ha X Y =, () (d) show ha, as varis, h lous of h mid-poi of PQ is giv by h quaio y y + y =. () (Toal marks) TOTAL FOR PAPER: 75 MARKS END
. Th rasformaio R is rprsd by h mari A, whr A =. Fid h igvors of A. Prai papr B (Toal 5 marks). Fid h valus of for whih givig your aswr as aural logarihms. osh + sih = 8, (Toal 6 marks). + + 7 (a + b) +, a >. (a) Fid h valus of a, b ad. (b) Fid h a valu of 5. 5. + + 7 d. () () (Toal 7 marks). (a) Show ha, for = l k, whr k is a posiiv osa, osh = k +. k () Giv ha f() = p ah, whr p is a osa, (b) fid h valu of p for whih f() has a saioary valu a = l, givig your aswr as a a fraio. () (Toal 7 marks)
5. Giv ha y = sih osh, (a) show ha d y d = ( ) sih + sih. () Th igral I is dfid by I = arsih sih d,. (b) Usig h rsul i par (a), or ohrwis, show ha () H fid h valu of I. I = ( )I, () () (Toal 9 marks) 6. Th mari M is giv by M = p, a b whr p, a, b ad ar osas ad a >. Giv ha MM T = ki for som osa k, fid (a) h valu of p, (b) h valu of k, () h valus of a, b ad, (d) d M. () () (6) () (Toal marks)
7. Th hyprbola C has quaio a y =. b (a) Show ha a quaio of h ormal o C a h poi P (a s, b a ) is a si + by = (a + b ) a. (6) Th ormal o C a P us h -ais a h poi A ad S is a fous of C. Giv ha h riiy of C is, ad ha OA = OS, whr O is h origi, (b) drmi h possibl valus of, for <. (8) (Toal marks) 8. Th pla Π passs hrough h pois A (,, ), B (,, ) ad C (,, ). (a) Fid a vor quaio of h li prpdiular o Π whih passs hrough h poi D (,, ). (b) Fid h volum of h rahdro ABCD. () Obai h quaio of Π i h form r. = p. () () () Th prpdiular from D o h pla Π ms Π a h poi E. (d) Fid h oordias of E. () Show ha DE = 5 5. () () (Toal 5 marks) TOTAL FOR PAPER: 75 MARKS END
Furhr Pur Mahmais FP (6669) Prai papr A mark shm Qusio Shm. 5 B 7 M A ( )( 7), 7 M A l or l l7 A (6) (6 marks). (a) d A = k 9k 8 M A Sig o zro ad solvig for k [ 6 ( k ) ] k M k, k 6 A () (b) Cofaors k k 9k 8 k 9 9 k k (B for ah row or olum) B A = k k M Af 9k 8 k d (5) 9 9 k k (9 marks) GCE Furhr Pur Mahmais ad Disio Mahmais mok papr mark shms UA958
Qusio. (a) d y sh d Shm Pu d y osh osh d B M (b) l( ) or 8 l( 7 ) or or 8 7 A l( ) or l(7 ) (or quiv.) A () 8 y l( ) ah(...) (Subsiu for ) M sh ah, ah M y l( ) l () A () (7 marks). (a) Drivig hararisi quaio ( )( 9 ) + = M A + 5 6 = ( + 6)( ) = = 6, = M A () (b) Saig, implyig or showig = assoiad wih poi ivaria li 6 5 9 y y Equaio is 5y = 5y = (ay quiv form) M A () B (7 marks) GCE Furhr Pur Mahmais ad Disio Mahmais mok papr mark shms UA958
GCE Furhr Pur Mahmais ad Disio Mahmais mok papr mark shms UA958 Qusio Shm 5. (a) d I, I I () M A A () (b) d I I I (o orr sam) M I = I I (likig all hr) M A d I I M A (5) (8 marks) 6. (a) d d d d y B B, or M, A l l l d Lgh () M M A (7) (b) Surfa ara = d 8 d M 6 6 ) (8 ) (8 M M A () ( marks)
GCE Furhr Pur Mahmais ad Disio Mahmais mok papr mark shms UA958 Qusio Shm 7. (a) RQ, RP (boh) B RP RQ i j k = ( 5 ) i (6 + 5) j + k M Af () (b) = Af d = 6 5 = () A so () () r. p M Subsiuig poi i pla o giv p, r. 7 M A () (d) Equaio of ormal o pla hrough S : r = 5 B Ms pla whr 7. 5 M Af S has posiio vor 5 = 8 5 M A (5) ( marks)
Qusio 8. (a) d Shm dy y d M A d d d Th ormal o h urv has gradi Th quaio of h ormal is Th quaio may b wri (b) L Q b h poi (q, /q) Th q q ad so B y ( ) M y () A (5) ( q) q ( q ) M A Amp o fid q,.g. ( q )( q ) or quadrai formula M q or A () (d) So Q has oordias, A (5) X ( ), Y ( ) M X Y XY () A () ( ) M X Y XY Y X y y y () A () ( marks) GCE Furhr Pur Mahmais ad Disio Mahmais mok papr mark shms UA958 5
Furhr Pur Mahmais FP (6669) Prai papr B mark shm Qusio Shm. d A = ( ) M 6 8 ( )( ), A :, y, y Eigvor M A :, y, Eigvor y A (5) (5 marks). 5 6 + = = 8 M M A (5 )( ) = A = 5, = l( 5 ), l ap l 5 M A (6) (6 marks). (a) a, b, 6 B B B () (b).5.5 ( ) d 6 ar a 8.5.5 M M A B () (7 marks) GCE Furhr Pur Mahmais ad Disio Mahmais mok papr mark shms UA958
Qusio. (a) osh = (b) f ( ) sh = Shm lk lk (or us = k) M k k k () M A () k p M A For l p, osh 5. (a) Usig produ rul Usig o obai ad osh osh dy d 7 8 B 6 8 p A () 89 89 (7 marks) ( )sih osh sih M sih i drivd prssio M dy ( )sih ( sih ) sih d dy d ( )sih sih () A () (b) = sih arsih osh arsih ( ) sih d arsih + sih d So osh( ar sih) ( ) I I M If sih = h osh = sih = I ( ) I () A () () I ar sih B I I M I I ad us wih prvious rsuls o obai M (arsih ) =.5 (ihr aswr apabl) A () 8 (9 marks) GCE Furhr Pur Mahmais ad Disio Mahmais mok papr mark shms UA958
GCE Furhr Pur Mahmais ad Disio Mahmais mok papr mark shms UA958 Qusio Shm 6. (a) k k k p b a b a p T MM p p M A () (b) 8 k k (f o hir p, if usd) M A () () quaios: a + b = a + = M a ad b i rms of (or quiv.): a = b = (f o hir p) M Af Usig ) 8 ( 8 b a b a b a. M a =, b =, = A,, (6) (d) d M= () ( ) ( ) = 5 M A so () ( marks)
Qusio 7. (a) Shm d d y = a s a, = b s M A d d d y b s b = d a s a a si gradi of ormal is a si b a si y b a = ( a s ) b M A a si + by = (a + b ) a () A so (6) M ( a b ) a a b (b) y = = a si a os B b = a ( ) b = OS = a ad OA = OS 5a M M a + 5a os = = a os M A = 5, A By symmry or (as OA = a b a os a b ) a os = a =, M A (8) ( marks) GCE Furhr Pur Mahmais ad Disio Mahmais mok papr mark shms UA958
Qusio Shm 8. (a) AB 5i + j AC i + j k or BC i j k ABAC i j k 5 = i + 5j + k M A r = i + j + k + ( i + 5j + k) Bf () (b) Volum = AD.( AB AC) AD =i + j + k B 6 = 6 (i + j + k).( i + 5j + k) M = 6 () r.( i + 5j + k) = (i + j).( i + 5j + k) A () M Af = A () (d) [i.( ) + j( + 5) + k( + )].( i + 5j + k) = + 9 + + 5 + + = M Af 5 + = = 5 M E is 68 5 9,, 5 5 5 A () () Disa = 5 i 5j k 5 5 () M A () (5 marks) GCE Furhr Pur Mahmais ad Disio Mahmais mok papr mark shms UA958 5