PURE MATHEMATICS A-LEVEL PAPER 1

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Geraldine Caldwell
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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 B Aswr ALL qustios i Sctio A ad ay FOUR qustios i Sctio B You ar providd with o AL(E) aswr book ad four AL(D) aswr books Sctio A : Writ your aswrs i th AL(E) aswr book Sctio B : Us a sparat AL(D) aswr book for ach qustio ad put th qustio umbr o th frot covr of ach aswr book Th four AL(D) aswr books should b tid togthr with th gr tag providd Th AL(E) aswr book ad th four AL(D) aswr books must b hadd i sparatly at th d of th amiatio FORMULAS FOR REFERENCE si( A B) si A cos B cos Asi B cos( A B) cos A cos B si Asi B ta A ta B ta( A B) ta A ta B A B A B si A si B si cos A B A B si A si B cos si A B A B cos A cos B cos cos A B A B cos A cos B si si si A cos B si( A B) si( A B) cos A cos B cos( A B) cos( A B) si Asi B cos( A B) cos( A B) 5 Ulss othrwis spcifid, all workig must b clarly show Hog Kog Eamiatios Authority All Rights Rsrvd -AL-P MATH -AL MATH All Rights Rsrvd

2 SECTION A ( marks) Aswr ALL qustios i this sctio Writ your aswrs i th AL(E) aswr book A squc { a } is dfid by a, a ad a a a for,,, Prov by mathmatical iductio that a ( ) ( ) for,,, Lt i = (,, ), j = (,, ), k = (,, ) ad a = i, b = i + j, c = j + k (a) Prov that a is ot prpdicular to b c Fid all uit vctors which ar prpdicular to both a ad b c (c) If [, ] is th agl btw a ad b c, prov that (a) Eprss z z i th form of z c r, whr c ad r ar costats Shad th rgio rprstd by pla z z C : i th Argad z (a) Writ dow th matri A rprstig th rotatio i th Cartsia pla aticlockwis about th origi by 5 Writ dow th matri B rprstig th largmt i th Cartsia pla with scal factor (c) Lt X ad V BAX, whr A ad B ar th matrics y dfid i (a) ad If V, prss y i trms of 5 (a) Lt f ( ) ad g( ) b polyomials Prov that a o-zro polyomial u( ) is a commo factor of f ( ) ad g( ) if ad oly if u( ) is a commo factor of f ( ) g( ) ad g( ) Lt f ( ) 6 5 ad g( ) 8 7 Usig (a) or othrwis, fid th HCF of f ( ) ad g( ) (7 marks) 6 For k,,, lt zk cos k i si k b compl umbrs, whr (a) Evaluat z z z Prov that cos k ( z k ) ad cos k ( zk ) zk z k Hc or othrwis, prov that cos cos cos cos cos cos -AL MATH - All Rights Rsrvd Go o to th t pag -AL MATH - All Rights Rsrvd

3 7 (a) Lt m ad k b positiv itgrs ad k m m ( m ) ( m k ) ( m ) m( m k ) Prov that k k m ( m ) Prov that th abov iquality dos ot hold wh k Lt m b a positiv itgr Usig (a) or othrwis, prov that ( ) m m ( ) m m for k SECTION B Aswr ay FOUR qustios i this sctio Each qustio carris 5 marks Us a sparat AL(D) aswr book for ach qustio 8 (a) Cosidr th systm of liar quatios i, y, z a y z (S) : y z b, whr a, b R y az b (i) Show that (S) has a uiqu solutio if ad oly if a Solv (S) i this cas (ii) For ach of th followig cass, dtrmi th valu(s) of b for which (S) is cosistt, ad solv (S) for such valu(s) of b () a, () a (9 marks) Cosidr th systm of liar quatios i, y, z (T) : a 5 y y y y z z az z Fid all th valus of a for which (T) is cosistt Solv (T) for ach of ths valus of a a, whr a R -AL MATH 5 All Rights Rsrvd Go o to th t pag -AL MATH 6 5 All Rights Rsrvd

4 9 Vctors u, v ad w i u v v w w u R ar said to b orthogoal if ad oly if (a) (i) Show that if u, v ad w ar o-zro orthogoal vctors, th u, v ad w ar liarly idpdt (ii) Giv a coutr ampl to show that th covrs of th statmt i (i) is ot tru Lt u ( u, u, u ), v ( v, v, v ) ad w ( w, w, w ) b thr o-zro orthogoal vctors i R (i) By computig th product u u u u v w v v v u v w, w w w u v w show that u u u v v v w w w (ii) Lt p = ( p, p, p ) b a vctor i R Show that p is a liar combiatio of u, v ad w (c) Lt = (,, ), y = (,, ), z = ( q = (,, ) b vctors i R,, ) ad (i) Show that, y ad z ar orthogoal (ii) Eprss q as a liar combiatio of, y ad z ( marks) (a) Lt a, a,, a b ral umbrs ad b, b,, b b o-zro ral umbrs By cosidrig ( a i i b i ), or othrwis, prov Schwarz s iquality a i ibi a i i b i i, ad that th a a a quality holds if ad oly if b b b (i) i i Prov that ar ral umbrs i i, whr,,, (ii) Prov that i i i i i i i i, whr,,, ar ral umbrs ad,,, ar positiv umbrs Fid a cssary ad sufficit coditio for th quality to hold (iii) Usig (ii) or othrwis, prov that y y y y y y, whr t t t t t t y, y,, y ar ral umbrs, ot all zro, ad t (9 marks) -AL MATH 7 6 All Rights Rsrvd Go o to th t pag -AL MATH 8 7 All Rights Rsrvd

5 (a) Lt f ( ) p, whr p R (i) Show that th quatio f ( ) has at last o ral root (a) Lt A b a matri such that A A A I, whr I is th idtity matri (ii) Usig diffrtiatio or othrwis, show that if p, th th quatio f ( ) has o ad oly o ral root (iii) If p, fid th rag of valus of p for ach of th followig cass: () th quatio f ( ) has actly o ral root, () th quatio f ( ) has actly two distict ral roots, () th quatio f ( ) has thr distict ral roots (9 marks) Lt g( ) a, whr a R (i) Prov that th quatio g( ) has at most two ral roots (ii) Prov that th quatio g( ) has two distict ral roots if ad oly if a < (i) Prov that A has a ivrs, ad fid (ii) Prov that A I (iii) Prov that ( A ) ( A ) A I (iv) Lt (i) (ii) (iii) Fid a ivrtibl matri B such that B B B I X Usig (a)(i) or othrwis, fid X A i trms of A Lt b a positiv itgr Fid X Fid two matrics Y ad Z, othr tha X, such that Y Y Y I, Z Z Z I (9 marks) -AL MATH 9 8 All Rights Rsrvd Go o to th t pag -AL MATH 9 All Rights Rsrvd

6 Lt b a squc of ral umbrs such that ad for,, (a) (i) Show that for, ( ) ( ) (ii) Show that th squc,, 5, strictly icrasig (i) For ay positiv itgr, show that is strictly dcrasig ad that th squc,, (ii) Show that th squcs,, ad,, p, 5, 6 covrg to th sam limit (c) By cosidrig ( or othrwis, fid lim i trms of ad ) is, 6 -AL P MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION PURE MATHEMATICS A-LEVEL PAPER pm pm ( hours) This papr must b aswrd i Eglish [You may us th fact, without proof, that from (ii), lim ists] ( marks) This papr cosists of Sctio A ad Sctio B Aswr ALL qustios i Sctio A ad ay FOUR qustios i Sctio B END OF PAPER You ar providd with o AL(E) aswr book ad four AL(D) aswr books Sctio A : Writ your aswrs i th AL(E) aswr book Sctio B : Us a sparat AL(D) aswr book for ach qustio ad put th qustio umbr o th frot covr of ach aswr book Th four AL(D) aswr books should b tid togthr with th gr tag providd Th AL(E) aswr book ad th four AL(D) aswr books must b hadd i sparatly at th d of th amiatio 5 Ulss othrwis spcifid, all workig must b clarly show Hog Kog Eamiatios Authority All Rights Rsrvd -AL MATH -- All Rights Rsrvd -AL-P MATH

7 FORMULAS FOR REFERENCE si( A B) si A cos B cos Asi B cos( A B) cos A cos B si Asi B ta A ta B ta( A B) ta A ta B SECTION A ( marks) Aswr ALL qustios i this sctio Writ your aswrs i th AL(E) aswr book Fid th idfiit itgral d ( ) ( ) Hc valuat th impropr itgral d ( ) ( ) A B A B si A si B si cos A B A B si A si B cos si A B A B cos A cos B cos cos A B A B cos A cos B si si si A cos B si( A B) si( A B) cos A cos B cos( A B) cos( A B) si Asi B cos( A B) cos( A B) a if, Lt f( ) b si if b a If f is cotiuous at, show that Furthrmor, if f is diffrtiabl at, fid th valus of a ad b Lt f : R R b a cotiuous fuctio satisfyig th followig coditios: f ( ) (i) lim ; (ii) f ( y) f( ) f ( y) for all ad y (a) Prov that f ( ) ists ad f ( ) f ( ) for vry R By cosidrig th drivativ of f (), show that f ( ) -AL P MATH - All Rights Rsrvd -AL P MATH -- All Rights Rsrvd Go o to th t pag

8 Th quatio of a straight li i R is t y t, t R z t Lt A ad B b two poits o with OA OB r, whr O is th origi (a) Eprss th distac btw A ad B i trms of r If OAB is a quilatral triagl, fid th valu of r 5 (a) If th fuctio g: R R is both v ad odd, show that g( ) for all R For ay fuctio f : R R, dfi F( ) [f ( ) f ( )] ad G( ) [f ( ) f ( )] (i) Show that F is a v fuctio ad G is a odd fuctio (ii) If f ( ) M( ) N( ) for all R, whr M is v ad N is odd, show that M( ) F( ) ad N( ) G( ) for all R 6 (a) Fid Lt Fid lim si si t f ( t ) t lim if t, if t f ( t)dt 7 Lt b th curv with polar quatio r cos, (a) Fid th polar coordiats of all th poits o that ar farthst from th pol O Sktch th curv (c) Fid th ara closd by (7 marks) -AL P MATH -- All Rights Rsrvd -AL P MATH --5 All Rights Rsrvd Go o to th t pag

9 SECTION B Aswr ay FOUR qustios i this sctio Each qustio carris 5 marks Us a sparat AL(D) aswr book for ach qustio 9 (a) Fid si d ( marks) 8 8 Lt f ( ) ( ) (a) Fid f ( ) ad f ( ) ( marks) Dtrmi th rag of valus of for ach of th followig cass: (i) f ( ), (ii) f ( ), (iii) f ( ), (iv) f ( ) ( marks) (c) Fid th rlativ trm poit(s) ad poit(s) of iflio of f ( ) ( marks) (d) Fid th asymptot(s) of th graph of f ( ) () Sktch th graph of f ( ) (f) Lt g ( ) f ( ) ( ) (i) Is g ( ) diffrtiabl at? Why? ( mark) ( marks) Lt f : R [, ) b a priodic fuctio with priod T bkt kt (i) Prov that ( )d akt positiv itgr k T (ii) Lt I f ( )d b f f ( ) d for ay T Prov that I I T for ay positiv itgr (iii) If is a positiv umbr ad is a positiv itgr such that T ( ) T, prov that T a ( ) T I f ( )d I T T Hc fid th impropr itgral f ( )d i trms of I ad T (9 marks) (c) Usig th rsults of (a) ad (iii), valuat si d ( marks) (ii) Sktch th graph of g ( ) -AL P MATH All Rights Rsrvd -AL P MATH All Rights Rsrvd Go o to th t pag

10 Lt f ad g b cotiuous fuctios dfid o [, ] such that f is dcrasig ad g( ) for all [, ] For [,], dfi G ( ) g( t)dt ad () G( ) f( t )dt f ( t)g( t)dt (a) (i) Prov that G ( ) Hc prov that () for all (ii) (, ) Evaluat () ad hc prov that G() f ( t )g( t)dt f ( t)dt Lt H ( ) [ g( t)]dt for all [,] (7 marks) Cosidr th parabolas C : y ( ) ad C : y Lt P( p, p) b a poit o C Th two tagts draw from P to C touch C at th poits S ( s, s ) ad T ( t, t ) (a) Fid th quatios of PS ad PT ad hc show that s t p, st p ( marks) Q ( q, q ) is a poit o th arc ST of C Prov that th ara of SQT is a maimum if ad oly if q p (c) Lt Q b th poit i whr th ara of SQT is a maimum If th straight li PQ cuts th chord ST at M, fid th quatio of th locus of M as P movs alog C (i) Prov that G() H() (ii) Usig (a)(ii), prov that f ( )dt G() t f ( t)g( t)dt (c) Usig th rsults of (a)(ii) ad (ii), prov that f ( )dt f ( t)dt t f ( t) t dt, whr is a positiv itgr Hc show that lim f ( t) t dt = ( marks) -AL P MATH All Rights Rsrvd -AL P MATH All Rights Rsrvd Go o to th t pag

11 (a) Lt g () b a fuctio cotiuous o [a, b], diffrtiabl i (a, b), with g ( ) dcrasig o (a, b) ad g (a) = g = Usig Ma Valu Thorm, show that thr ists c (a, b) such that g is icrasig o (a, c) ad dcrasig o (c, b) Hc show that g( ) for all [ a, b] Lt f b a twic diffrtiabl fuctio ad f ( ) o a op itrval I Suppos a, b, I with a b By cosidrig th fuctio g( ) ( b )f ( a) ( a)f ( b) ( b a)f ( ) or othrwis, show that b a f ( ) f ( a) f ( b) b a b a Hc, or othrwis, prov that f ( ) f ( ) f ( ) for all, I, whr, with (c) Lt ad b positiv umbrs (i) If with, prov that, (ii) If, ar positiv umbrs, prov that (a) (i) Lt I ( ) = ta u du, whr is a o-gativ itgr ad ta Show that I ( ) I ( ) for all (ii) Usig th substitutio t ta u, or othrwis, show that t dt ( ) t for ay positiv itgr ( ) ta (i) Lt ad b a positiv itgr Prov that t d ( )( ) t t (ii) (iii) Usig (a) or othrwis, show that ( ) ( ) p p p Suppos that ta Evaluat ta ad ta, ad 5 show that ta ta 5 9 Hc prov that p ( ) p p p p 5 9 ( ) 5 9 ( marks) END OF PAPER -AL P MATH -- 9 All Rights Rsrvd -AL P MATH - All Rights Rsrvd

1985 AP Calculus BC: Section I

1985 AP Calculus BC: Section I 985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b