Qn Suggested Solution Marking Scheme 1 y. G1 Shape with at least 2 [2]

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1 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme y G Shpe with t lest [] fetures correct y = f'( ) G ll fetures correct SR: The mimum poit could be i the first or secod qudrt. -itercept (,). -itercept ( c,). H.A y = m (bove -is) 4. y-itercept bove H.A y = m [4] Let required equtio be y = + b + c+ d, bcd,,,. Substitute (,) ito equtio, d = b+.5c=.. +. b +.c = b +.6c =. Usig GC, 5 5 =, b =, c = Required equtio is y = + (*) 7 Substitute =.45 ito (*), y =.5 the mufcturer's clim is ccurte. B d = [SOI] M Formulte system of equtios with t lest equtio correct A Correct equtio B Correct coclusio with substitutio of =.45 ito (*) (SOI) () [] Q O l p ( q) is the shortest distce (or perpediculr distce) % % % from Q to l. B Correct Geometricl Iterprettio Pge of 8

2 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme OR p ( q) is the re of the prllelogrm with djcet % % % vectors p d q % % %. (b) [4] 4(i) [7] By cosie rule, b = + b b cosθ % % % % %% ( b ).( b ) =. + b. b b cosθ % % % % %% %% %% bb. b. b. +. =. + bb. bcosθ %% %% %% %% %% %% % %. b= bcosθ %% % % b. = bcos θ (show) %% % % = 4 Sice >, + > > > 4 + < 4 + < (show) 4 + > > 4 4 B Writig rd side of Δ s b % % M Apply cosie rule to obti b = + b b cosθ % % % % %% M Write b % % s ( b ).( b ) d epd % % % % AG Obti b. = bcosθ d show %% % % tht b. = bcosθ %% % % AG Mke use of >, compre + > to show 4 + < 4 (ii) M Plce ll terms o oe side d tke commo deomitor (Algebric method) Pge of 8

3 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme + M Attempt to fctorise + by completig the squre or + usig formul Method : (Algebric method) ( ) ( ) + ( ) ( 4+ ) 4 ( ) ( + 4 ) M Fid ll the criticl vlues d ttempt to usig umber lie determie rge of ( )( + ) A Correct swer Sice is positive costt, 4 + < (from (i)) (Grphicl method) Ad + + > > 4 M Attempt to sketch grph to locte prts of the grph below -is + 4 or < M Attempt to fid - itercepts of grph Method : (Grphicl method) y O A Correct swer Pge of 8

4 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme + Let = + = ± 9 4( ) = = ± Sice, + 4 or < (i) [8] 5(ii) Replce i with, we obti ( ) ( ) ( ) Hece + 4 or < or + 4 < f( r) f( r ) = cos[ α + ( r+ ) β ] cos[ α + ( r ) β )] = si( α + rβ)si β (By fctor formul) [f ( r) f ( r )] = [ si( α + rβ)si β] r= r= = si βsi( α + rβ) r= M Replce with A Correct swer M Apply fctor formul A Correct swer M Able to use prt (i) Pge 4 of 8

5 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme siα + si( α + β ) + si( α + β ) si( α + β ) = si( α + rβ) r= r= si( α + rβ) = [f ( ) f ( )] r r r= si β f() f() f() f() + = +... si β + f ( ) f ( ) + f ( ) f ( )] M Attempt to use MOD, must show the ccelltio d t lest rows i frot d t the ed = [f ( ) f ( )] si β M Apply fctor formul = {cos[ α + ( + ) β ] cos( α β )} si β + = [ si( α + β)si( ) β] si β + si( α + β)si( ) β = () si β + Where p = d q =. 5(iii) Sub β = π Differetite () wrt α : cosα + cos( α + π) + cos( α + π) cos( α + π) π + cosα + si π = π si π + = cosα + si π A Obti d b correctly M Sub β = π d ttempt to differetite () wrt α A Show ll steps clerly with the correct swer Pge 5 of 8

6 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme Altertive Solutio: M Use of cosα + cos( α + π) + cos( α + π) cos( α + π) π cosθ = si θ = cos( α + rβ ), β π 6(i) [4] 6(ii) [5] r= π = si α r β r= π π + si α si π = π si = si π π α si π + = cos π α + si π + d = cost dy = si t dy si t t t = = d cost Equtio of tget T is t p y cos p = ( si p) t p si p y = + cos p + cos p cos p + si p t p = + cos p t p = + sec p y+ t p = ( + sec p) Whe y =, t p = + sec p + secp = t p Whe =, y = + sec p A Show ll steps clerly with the correct swer M Either d A Correct d y d or d y correct i terms of t M Form equtio of tget AG Correct equtio of T M Substitute =, y = to fid A d B. Also ccept = cot p + cosec p or = +. t p si p Pge 6 of 8

7 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme Method : M Apply mid-poit formul + secp + secp Coordites of M =, t p. +, + y M = + sec p A Obti d y coordites y = sec p = y of M. + sec p y = = t p t p Usig + t p= sec p, y + = (y ) 4y y = y + = 4y 4y+ = y( ) y = Method : sec p Coordites of M = cot p+ cosec p, +. + sec p y = sec p = y cos p + = cot p+ cosec p= si p Usig si p+ cos p=, M Form crtesi equtio of locus of M usig + t p= sec p A Correct crtesi equtio of locus of M. M Epress +, y + M y = A Obti d y coordites of M. M Form crtesi equtio of locus of M usig si p+ cos p= Pge 7 of 8

8 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme cos p + A Correct crtesi equtio + = of locus of M. y + y + = (y ) ( y) + (y ) (y ) = 4 y + = (4y 4y+ ) y = y y = y( ) y = 7 [] (i) By Pl A, re pited o the th dy = 7 ( ).5 B Correct formul (ii) = (7) ( ).5 4 Totl re pited o dys = [ ] Method : M Form iequlity or use GC tble ivolvig the sum of AP formul SR: If studets form equtio isted of iequlity, wrd out of mrks. Are < > 4 A Correct swer He will fiish pitig his room o the 8th dy. Method : Totl re pited o dys = M Form iequlity or use GC tble ivolvig the sum of AP formul Pge 8 of 8

Lest = 8. He will fiish pitig his room o the 8th dy. (iii) For 5, re pited o the th dy = d pitig stops.

9 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme [ (7) ( ).5] 4 SR: If studets form equtio 8 ( ) 6 isted of iequlity, wrd out of mrks A Correct swer But > < 5 So 8 4. Lest = 8. He will fiish pitig his room o the 8th dy. (iii) For 5, re pited o the th dy = d pitig stops. Therefore Pl A cot be pplied to rbitrrily lrge wll. B Correct reso 4 By GC or usig [ (7) (4 ).5 ] = 5.5, lrgest re tht Pl A c be pplied to is 5.5 m. B Correct lrgest re. Pge 9 of 8

10 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme (iv).8 M Apply correct GP formul for S.8 4 M Form iequlity S..7 4 d ttempt to solve.8.8 lgebriclly..8. l.8 l. A Correct umber of dys. 7. He eeds 8 dys to fiish pitig t lest 7% of his room. (v) If he chooses Pl B, totl re pited fter ifiite 7 umber of dys = = < B Correct coclusio with. He cot fiish reso with correct formul for pitig his room. Therefore he should choose Pl A. sum to ifiity used d the swer is < 4 8[] + (i) y = ( + k) dy ( + k) ( + )( + k) = M Attempt to fid d y d ( + k) d usig quotiet/product rule + k k 4 k = ( + k) + 4+ k = ( + k) dy = + + k = d For o sttiory poits, Let 4 Discrimit < ( k )( ) 6 4 < k > + Whe k =, y = =, ( + ) No sttiory poits whe k = M Use Discrimit < to obti lier iequlity ivolvig k A correct swer k k Pge of 8

11 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme 8(ii) y G Shpe (with turig poits) G t lest 4 out of 6 fetures G All correct. -itercept(,). mi( 4.8,.858). m(.88,.9) 4. V.A = 5. V.A = (SOI) 6. H.A y = (SOI) 8(iii) m =,.858 or.9 ( to s.f ) B All vlues correct 8(iv) = = + ( )( ) = + + = ( ) M ttempt to fid the epressio of the secod curve by rerrgig the terms Ad sketch the correct curve y= Sketch the grph of y= i (ii) y AG coclude with referece to digrm Pge of 8

12 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme Sice the grph y= itersect the grph + y = t two distict poits, there re rel roots to ( ) the equtio =. 9[] () 4iz w= 9i () * (4+ i) w = z+ i -----() Let w= + iy 4i[(4+i)( i y) i] ( + i y) = 9i 4i[(4 +i 4iy+ y i] i y) = 9i 6i 8+ 6y+ 8yi+ iy = 9i Compre rel d imgiry prts 6y 9 = () 6y+ 8y y = y = (4) From () 6y + 5 = 9 From (4) y + + 7y = y y = 8 9y = 9 y = 6( ) + 5 = = 9 w = i z = (4 + i)( + i) i =+i M Either let z = + iy,, y or w= + ib, b d ttempt to solve the simulteous equtios M Use of w* = iyd ttempt to compre rel d imgiry prts to form two simulteous equtio A Either Obti w = i or z =+i A All the swers correct 9(bi) u u = = * * v v 5 6 u = u * rg rg rg v* () v π i π π i i 6 v = 6e e = 6e π rg v* = 6 From () B Correct vlue of r i.e u 5 = * v 6 π B Correct rg v* = 6 M Apply correct properties rg u rg v* Pge of 8

13 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme u rg * v 7π π = 6 4π = A Correct swer for Arg u 4π π v * = π + = 9 (bii) π i u 5 = e v * 6 u v * 5e = 6e 7 π i π i 6 7 π π + i 6 5 = e 6 To be purely imgiry 7π + π (k + ) π =, k k+ 4 = 7 k =, = 4 k = 9, = 6 k = 6, = 8 u v * M Apply properties of rg u v * usig previous prt d (k + ) π equte it to either or (k ) π OR to compre it with π π 5π ±, ±, ±... or to equte rel prt to zero d solve for A All three swers for re correct by observtio [4] ()(i) + d d c 5 = ( ) ( ) = ( ) ( ) M Idetify correct u d v ' d use the correct formul for itegrtio by prts A Correct swer u = u' = v' = v= ( ) Pge of 8

14 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme ()(ii) y + = + + = + 9 = ( ) 9 = = B Attempt to solve for = V = π + + d c [ ] = π + + d = 9π + + d M Correct V c ( ) 9π + or = 9π + ( ) ( ) d 5 = 9π + 4 π ( ) ( ) 5 = 9π π = 9 + π uits 5 ( ) V = 9π + Vc 78 = 9 π( + ) (9 + ) π 5 774π = uits 5 M Fid the required volume usig V = 9π ( + ) Vc A Obti Ect volume π Pge 4 of 8

15 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme (c)(i) y O Are of st rectgle = l + Are of d rectgle = l + Are of rd rectgle = l + Μ Are of ( th ) rectgle = l + A l l l l = L + + = l L = l L ( + )( + )( + ) K ( l ) = M Attempt to fid re of rectgles with the re of the first d the lst rectgle correct AG Apply lws of logrithms to show the epressio give (c)(ii) u = l( + ) u' = + v' = v = ( ) lim A = l + d M Use = ( + ) lim A l d d pply itegrtio by prts correctly Pge 5 of 8

Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme = [ l( ) ] d M Rewrite s d + + [ ] = l( + ) [ l( + )] = l [ l ] = l uits d itegrte correctly A Correct swer i ect form (i) 4 Floor

16 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme = [ l( ) ] d M Rewrite s d + + [ ] = l( + ) [ l( + )] = l [ l ] = l uits d itegrte correctly A Correct swer i ect form (i) 4 Floor re y = 4 y = y Are A= (4 ) + (.y ) + M Epress A i terms of d y 4 4 = = = (ii) / da 64 8 = = d / = 8 64 = = + 6 =.4 (iii) M Substitute 4 y = AG Able to simplify to give epressio M Apply Chi Rule to get ( f() ) / k. M Equte d A d = d simplify to give epressio A Correct vlue of i y form e.g. = =. By GC, = 4.64 = 4 to s.f. B Correct vlue of to s.f. Pge 6 of 8

17 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme (iv) Method : da ve +ve d \ / Method : / d A = + 4 d / > whe = 4.64 Hece A is miimum whe = 4 to s.f. 64 Miimum Cost = 8(4.64) = $47.4 (erest cets) = $4 (to s.f.) / da From (ii), = 8 + d Whe =, da = da d d / = 8 + =.577 =. uit /mi ( s.f.) B Apply st or d derivtive test [oly wrd mrk if vlue of is correct] B Correct miimum cost M Attempt Chi rule to fid da or pply implicit d differetitio to obti equtio ivolvig d A d d M Attempt to substitute = d d = (i) the cocetrtio of A d B t time t re ( )d ( b ) mol/dm respectively A Correct swer to s.f. B both swers correct (ii) (iii) d + = k ( )( b ), k d M vlue for is, Q = d fter = there is o more cocetrtio of substce A for rectio to cotiue. B Correct swer B Correct M vlue for is B stte correct reso Pge 7 of 8

18 Mrkig Scheme for HCI 8 Prelim Pper Q Suggested Solutio Mrkig Scheme (iv) d M correct method of = k ( ) itegrtio ( ) d ( ) d= kt kt C ( ) = + = () kt + C Whe t =, = c = = kt + = kt + A correct epressio () ccept Aswer without costt M Use of iitil vlue to fid costt A Correct prticulr solutio = kt + S t A correct grph with symptote d iitil vlue lbelled (v) d (vi) d = 4 S S B B oe mrk for ech curve showig clerly reltive positio Do ot Accept grphs with S d S overlppig ech other t the til eds t (vii) From the grph, α < β. d d Q for S > for S d both curves strt from the origi. B correct α < β B correct reso epli usig the cocetrtio of the Pge 8 of 8

National Quali cations AHEXEMPLAR PAPER ONLY

National Quali cations AHEXEMPLAR PAPER ONLY Ntiol Quli ctios AHEXEMPLAR PAPER ONLY EP/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite