National Quali cations

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1 Ntiol Quli ctios AH07 X77/77/ Mthmtics FRIDAY, 5 MAY 9:00 AM :00 NOON Totl mrks 00 Attmpt ALL qustios. You my us clcultor. Full crdit will b giv oly to solutios which coti pproprit workig. Stt th uits for your swr whr pproprit. Aswrs obtid by rdigs from scl drwigs will ot rciv y crdit. Writ your swrs clrly i th swr booklt providd. I th swr booklt, you must clrly idtify th qustio umbr you r ttmptig. Us blu or blck ik. Bfor lvig th mitio room you must giv your swr booklt to th Ivigiltor; if you do ot, you my los ll th mrks for this ppr. *X7777* A/HTP

2 FORMULAE LIST Stdrd drivtivs Stdrd itgrls f ( ) f ( ) f ( ) f ( ) d si cos ( ) sc t( ) + c si + c t + + t + c t sc l + c cot cosc + c sc sc t cosc l cosc cot Summtios S d (Arithmtic sris) = + ( ) (Gomtric sris) S = ( r ) r ( + ) ( + )( + ) 3 ( + ) r =, r =, r = 6 r= r= r= Biomil thorm r r + b = b r= 0 r whr ( )! = C = r r r!( r)! Mcluri psio 3 iv f ( 0) f ( 0) f ( 0) f( ) = f( 0) + f ( 0) ! 3!! Pg 0

3 FORMULAE LIST (cotiud) D Moivr s thorm Vctor product [ (cos si )] r θ + i θ = r ( cos θ + isi θ ) i j k b= bsi θ ˆ = 3 = i j + k b b b b b b b b b Mtri trsformtio Ati-clockwis rottio through gl, θ, bout th origi, cosθ si θ si θ cosθ [Tur ovr Pg 03

4 Totl mrks 00 Attmpt ALL qustios MARKS. Writ dow th biomil psio of y 5y 3 d simplify your swr.. Eprss ( + )( ) 6+ 0 i prtil frctios. 3. O suitbl domi, fuctio is dfid by f ( ) Fid f ( ), simplifyig your swr. =. 3. Th fifth trm of rithmtic squc is -6 d th twlfth trm is -3. () Dtrmi th vlus of th first trm d th commo diffrc. (b) Obti lgbriclly th vlu of for which S = () (i) Us Gussi limitio o th systm of qutios blow to giv prssio for z i trms of λ. + y z =3 y+ 3z = 3+ y+ λz = 8 (ii) For wht vlu of λ is this systm of qutios icosistt? (b) Dtrmi th solutio of this systm wh λ = Us th substitutio u = 5 to fid th ct vlu of d. 6 Pg 0

5 MARKS 7. Mtrics P d Q r dfid by P = 5 d 3 Q =, whr y,. y () Giv th dtrmit of P is, obti: (i) Th vlu of. (ii) P. (iii) P Q, whr Q is th trspos of Q. 5 (b) Th mtri R is dfid by R=, whr z. z 6 Dtrmi th vlu of z such tht R is sigulr. 8. Us th Euclid lgorithm to fid itgrs d b such tht b = 9. dy d = + giv tht wh = 0, y =. 9. Solv ( y ) Eprss y i trms of S is dfid by r r + r= 3. () Fid prssio for S, fully fctorisig your swr. (b) Hc fid prssio for r + r 3 whr p > 5. p r= 0 [Tur ovr Pg 05

6 3 +. Giv y =, us logrithmic diffrtitio to fid dy d. Writ your swr i trms of. MARKS 5. I th digrm blow prt of th grph of y = f ( ) hs b omittd. Th poit (, ) lis o th grph d th li y = 3 is symptot. y (, ) 0 3 Giv tht f ( ) is odd fuctio: () Copy d complt th digrm, icludig y symptots d y poits you kow to b o th grph. (b) g( ) = f ( ). O sprt digrm, sktch g( ). Iclud kow symptots d poits. (c) Stt th rg of vlus of f ( ) giv tht ( ) f 0 =. 3. Lt b itgr. Usig proof by cotrpositiv, show tht if is v, th is v. Pg 06

7 . Fid th prticulr solutio of th diffrtil qutio MARKS d y 6 d dy + 9 y = 8 si + 9 cos d giv tht y = 7 d dy d = wh = () A bm of light psss through th poits B(7, 8, ) d T(-3, -, 6). Obti prmtric qutios of th li rprstig th bm of light. (b) A sht of mtl is rprstd by pl cotiig th poits P(,, 9), Q(,, 7) d R(-3, 7, ). Fid th Crtsi qutio of th pl. (c) Th bm of light psss through hol i th mtl t poit H. Fid th coordits of H O suitbl domi, curv is dfid by th qutio + 9y = 36. A sctio of th curv i th first qudrt, illustrtd i th digrm blow, is rottd 360 bout th y-is. y 0 Clcult th ct vlu of th volum grtd. 5 [Tur ovr for t qustio Pg 07

8 7. Th compl umbr z = + i is root of th polyomil qutio 3 z 6z + 6z z+ q = 0, whr q. () Stt scod root of th qutio. (b) Fid th vlu of q d th rmiig roots. 3 (c) Show th solutios to z 6z + 6z z+ q = 0 o Argd digrm. MARKS 6 8. Th positio of prticl t tim t is giv by th prmtric qutios = t cos t, y = t si t, t 0. () Fid prssio for th isttous spd of th prticl. 5 Th digrm blow shows th pth tht th prticl tks. y A 0 (b) Clcult th isttous spd of th prticl t poit A. [END OF QUESTION PAPER] Pg 08

National Quali cations

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