National Quali cations

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1 PRINT COPY OF BRAILLE Ntiol Quli ctios AH08 X747/77/ Mthmtics THURSDAY, MAY INSTRUCTIONS TO CANDIDATES Cdidts should tr thir surm, form(s), dt of birth, Scottish cdidt umbr d th m d Lvl of th subjct t th top of thir first swr sht. 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 o your swr sht. Qustios mrkd with strisk diffr i som rspcts from thos i th pritd ppr. Mrks r show i squr brckts t th d of ch qustio or prt qustio. A OW i th mrgi idicts w qustio. A sprt formul sht is providd. PCBr

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 r ( + ) ( + )( + ) ( + ) r =, r =, r = 6 4 r= r= r= Biomil thorm r r + b = b r= 0 r whr ( )! = C = r r r!( r)! Mcluri psio iv 4 f ( 0) f ( 0) f ( 0) f( ) = f( 0) + f ( 0) !! 4! 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 = = 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 pg 0

4 Totl mrks 00 Attmpt ALL qustios. () Giv f( ) = si, fid f ( ). [ mrks] 5 (b) Diffrtit y =. [ mrks] 7 + (c) For ycos + y = 6, us implicit diffrtitio to fid dy. [4 mrks] d. Us prtil frctios to fid 7 d. [4 mrks] 5. () Writ dow d simplify th grl trm i th biomil psio of 5 +. [ mrks] (b) Hc, or othrwis, fid th trm idpdt of. [ mrks] 9 4. Giv tht z = + i d z = p6 i, p, fid: () zz ; [ mrks] (b) th vlu of p such tht z z is rl umbr. [ mrk] 5. Us th Euclid lgorithm to fid itgrs d b such tht 06+ 9b= 7. [4 mrks] pg 04

5 6. O suitbl domi, curv is dfid prmtriclly by = t + Fid th qutio of th tgt to th curv whr t =. [5 mrks] d y = l ( t+ ). 7. Mtrics C d D r giv by: C = 0 0 d D= k+ 0, whr k. () Obti C D whr C is th trspos of C. [ mrks] (b) (i) Fid d simplify prssio for th dtrmit of D. [ mrks] (ii) Stt th vlu of k such tht D dos ot ist. [ mrk] 8. Usig th substitutio u = siθ, or othrwis, vlut π 4 si θcosθdθ. [4 mrks] π 6 9. Prov dirctly tht: () th sum of y thr coscutiv itgrs is divisibl by ; [ mrks] (b) y odd itgr c b prssd s th sum of two coscutiv itgrs. [ mrk] pg 05

6 0. Giv z = + iy, sktch th locus i th compl pl giv by z = z + i. [ mrks]. () Obti th mtri, A, ssocitd with ticlockwis rottio of π origi. [ mrk] rdis bout th (b) Fid th mtri, B, ssocitd with rflctio i th -is. [ mrk] (c) Hc obti th mtri, P, ssocitd with ticlockwis rottio of π rdis bout th origi followd by rflctio i th -is, prssig your swr usig ct vlus. [ mrks] (d) Epli why mtri P is ot ssocitd with rottio bout th origi. [ mrk]. Prov by iductio tht, for ll positiv itgrs, = ( ). [5 mrks] r r= *. Rfr to th digrm for Qustio. A gir hs dsigd liftig dvic. Th hdl turs scrw which shorts th horizotl lgth d icrss th vrticl hight. Th dvic is modlld by rhombus, with ch sid 5 cm, s show i th digrm. Th horizotl lgth is cm, d th vrticl hight is h cm s show. () Show tht h = 500. [ mrk] (b) Th horizotl lgth dcrss t rt of 0 cm pr scod s th hdl is turd. Fid th rt of chg of th vrticl hight wh = 0. [5 mrks] pg 06

7 4. A gomtric squc hs first trm 80 d commo rtio. () For this squc, clcult: (i) th 7 th trm; [ mrks] (ii) th sum to ifiity of th ssocitd gomtric sris. [ mrks] Th first trm of this gomtric squc is qul to th first trm of rithmtic squc. Th sum of th first fiv trms of this rithmtic squc is 40. (b) (i) Fid th commo diffrc of this squc. [ mrks] (ii) Writ dow d simplify prssio for th th trm. [ mrk] Lt S rprst th sum of th first trms of this rithmtic squc. (c) Fid th vlus of for which S = 44. [ mrks] 5. () Us itgrtio by prts to fid si d. [ mrks] (b) Hc fid th prticulr solutio of dy y = si, 0 d giv tht =π wh y = 0. Eprss your swr i th form y = f( ). [7 mrks] pg 07

8 6. Pls π, π d π hv qutios: π : π : y+ z =4 5y z = π : 7 + y + z = whr. () Us Gussi limitio to fid th vlu of such tht th itrsctio of th pls π, π d π is li. [4 mrks] (b) Fid th qutio of th li of itrsctio of th pls wh tks this vlu. [ mrks] Th pl π 4 hs qutio 9+ 5y+ 6z = 0. (c) Fid th cut gl btw π d π 4. [ mrks] (d) Dscrib th gomtricl rltioship btw π d π 4. Justify your swr. [ mrk] 7. () Giv f ( ) =, obti th Mcluri psio for f ( ) th trm i. [ mrks] (b) O suitbl domi, lt g( ) = t. (i) Show tht th third drivtiv of g( ) is giv by 4 g ( ) = sc + t sc 4. [ mrks] up to, d icludig, (ii) Hc obti th Mcluri psio for g( ) up to d icludig th trm i. [ mrks] (c) Hc, or othrwis, obti th Mcluri psio for icludig, th trm i. [ mrks] t up to, d (d) Writ dow th first thr o-zro trms i th Mcluri psio for t + sc. [ mrk] [END OF QUESTION PAPER] pg 08

9 X747/77/ Mthmtics AH 08 Q 5 cm h X747/77/ Mthmtics AH 08 Ppr Q