4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16.

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Griffin Lindsey
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1 (n )0 7 (M) 0(n ) 00 n (A) S ((7) 0(0)) (M) (7 00) 8897 (A). (5a b) (5a)... (M) (a b ) 5 5 a b (M)(A) So th cofficint is 75 (A) (C) [] S (7 7) (M) () 8897 (A) (C) [] 5. x.55 0 (x )log 0 log 0.55 (M) log0.55 x log (A) 0 x (A) x (A) (C) []. 8 Rquird trm is (x) 5 ( ) 5 (A)(A)(A) Thrfor th cofficint of x 5 is (A) (C) []. a 5 a d 0 (may b implid) 5 d (M) (A) T 5 5 (A). S 5 5 { } (M)(A)(A) 50 or or.7 ( sf) (A) (C) S 5 85 a, a d d 0 d 7.5 S 5 5 ( (7.5)) 5 ( 0) (A) (M) (A) (M) log a 5 7. (a) log 5 log x y a (M) (A) (C) [] S 5 85 (A) (C) [] log a 0 log a log a 5 or log a log a 0 (M) log a log a 5 x y (A) (C) [] u 8. S r (M)(A)

2 5 (A) 5 (A) (C) [] 000( ) (M) 0 000(.075)(.075 ).075 (M) 508 (narst dollar) (A) (C) [] 9. (a b) Cofficint of a 5 b 7 is 5 7 (M)(A) 79 (A) (C) 0. (a) Plan A: 000, 080, 0... Plan B: 000, 000(.0), 000(.0) nd month: 00, rd month:.0 (A)(A) []. P log 0 P log 0 QR QR (M) P log 0 (log 0 P log 0 (QR )) QR (M) (og 0 P log 0 Q log 0 R) (M) (x y z) x y z or (x y z) (A) [] For Plan A, T a d 000 (80) (M) 880 (A) For Plan B, T 000(.0) (M) 898 (to th narst dollar) (A). (a) a 000, a n 000 (n ) (M) n Sh runs 0 km on th 7th day. (A) (c) (i) For Plan A, S [000 (80)] (M) (880) 780 (to th narst dollar) (A) 7 S 7 ( ) (M) Sh has run a total of 0.5 km (A) [] 000(.0 ) (ii) For Plan B, S (M) (to th narst dollar) (A). (a) (narst dollar) (A) (C) [0]. Th constant trm will b th trm indpndnt of th variabl x. (R) x x 9x x x x x x (M) 9 8 x 8x x x (A) 7 (A) []

3 5. (x y) (x) (x) (y) (x) (y) (x)(y) (y) (A) 8x x y x y 9xy y (A)(A)(A) (C) [] 8. (a) ( ) () ( ) ( ) (M) (A) (C). (a) u 7, d.5 (M) u u (n )d 7 ( ).5 07 (A) (C) S 0 n [u (n )d] ( ) (M) (A) (C) [] 7. METHOD 0 [(7) (0 ).5] (M) 0() (A) (C) log 9 8 log 9 log9 9 log9 x (M) (A) x 9 (M) x 7 (A) (C) METHOD, ) log 8 log 9 log9 log * ' 9 ( (M) log 9 7 (A) x 7 (A) (C) [] [] (a) r.5 (A) is th th yar. (M) u 0(.5) (M) 0759 (Accpt 070 or 0800.) (A) (c) (.5) n 5000 (.5) n 0 (M) 5000 log (n )log.5 0 (M) 5000 log 0 n 8.9 (A) log.5 n th yar 999 Using a gdc with u 0, u k uk, u 9 00, u 0 50 (A) (M) 999 (G) 5

4 .5 (d) S 0 (M) (Accpt 90 or 000.) (A) Statmnt (a) Is th statmnt tru for all ral numbrs x? (Ys/No) If not tru, xampl A No x l (log 0 0. ) (a) (A) (C) B No x 0 (cos 0 ) (A) (C) C Ys N/A () Narly vryon would hav bought a portabl tlphon so thr would b fwr popl lft wanting to buy on. (R) Sals would saturat. (R) 0. (a) u u l d or d (M) ( ) d (M) (A) (C) [] Nots: (a) Award (A) for ach corrct answr. Award (A) marks for statmnts A and B only if NO in column (a). Award (A) for a corrct countr xampl to statmnt A, (A) for a corrct countr xampl to statmnt B (ignor othr incorrct xampls). Spcial Cas for statmnt C: Award (A) if candidats writ NO, and giv a valid rason (g 5 π arctan ). u n u l (n ) or 998 (n l) (M) 998 n (A) 00 (A) (C). (a) Ashly AP... to 5 trms (M) 5 S 5 [() ()] (M) 5 90 hours (A). (a) 0 (A) (C) Billi GP, (.), (.) (M) (x ) [for corrct xponnts] x 9 x or 8 x x x constant 8 (M)(A) (A) (A) (C) (i) In wk, (.) (A).5 hours (AG) ( ) 5 [. ] (ii) S 5 (M). 8 hours ( sf) (A). log 7 (x(x 0.)) l (M)(A) x 0.x 7 (M) x 5. or x 5 (G) x 5. (A) (C) Not: Award (C5) for giving both roots.. 7 8

5 (c) (.) n > 50 (M) (.) n 50 > (A) 50 (n ) ln. > ln (ii) st nd 8 nd rd 8 (M) 8 Gomtric progrssion, r 8 (A) 50 ln n > ln. n >.97 n > 5.97 Wk (A) (A) (c) (i) u u r (M) ( , sf) (A) (.) n > 50 (M) By trial and rror (.) 5., (.) (A) n l 5 (A) n (Wk ) (A) [] u (ii) S r (M) (A) [0] 5. Trm involving x 5 is () ( x) (A)(A)(A) 5 0 (A) Thrfor, trm 0x (A) Th cofficint is 0 (A) (C). (a) (i) PQ AP AQ (M) ( ) cm (A)(AG) 7. Arithmtic squnc d (may b implid) (M)(A) n 50 (A) S ( 750) or S ( 9 ) (M) 5 5 (A) (C) 8. Slcting on trm (may b implid) (M) 7 5 (x ) 5 (A)(A)(A) 800x 0 (A)(A) (C) Not: Award C5 for 800. (ii) Ara of PQRS ( )( ) 8 cm (A) (i) Sid of third squar ( ) ( ) cm Ara of third squar cm (A) 9 0

6 9. x f Σf (ax) (ax) (ax) (M)(M)(M)... a x 8a x a x (A)(A)(A) (C) Nots: Award C if brackts omittd, lading to ax 8ax ax. Award C if corrct xprssion with brackts as in first lin of markschm is givn as final answr (a) m (A) (C). Arithmtic squnc (M) a 00 d 0 (A) (a) Distanc in final wk (M) 70 m (A) (C) Q 5 (A) (C) (c) Q 8 (A) IQR 8 5 (M) (accpt 5 8 or [5, 8]) (C) 5 Total distanc [ ] (M) 5080 m (A) (C) Not: Pnaliz onc for absnc of units i award A0 th first tim units ar omittd, A th nxt tim. 0. (a) log 5 x log 5 x (M) y (A) (C). (a) (i) Ara B, ara C (A)(A) log 5 x log5 x (M) y (A) (C) (ii) (Ratio is th sam.) (M)(R) (c) log log 5 x 5 log5 5 (M) y (A) (C) (iii) Common ratio (A) 5

7 5 (i) Total ara (S ) ( 0.5) (0., sf) (A) (ii) 8 ' ' Rquird ara S 8 (M) 0.8 (7...) (A) 0.8 ( sf) (A) Not: Accpt rsult of adding togthr ight aras corrctly. METHOD x 0, y 0 p, r z 0 (A)(A)(A) p log 0 x 0 log 0 r y z q 0 0 (A) r p q log 0 r 0 p q (A) (C)(C)(C) (c) Sum to infinity (A) (A) [] 5. (a) (i) Nithr (ii) (iii) Gomtric sris Arithmtic sris (iv) Nithr (C) Not: Award (A) for gomtric corrct, (A) for arithmtic corrct and (A) for both nithr. Ths may b implid by blanks only if GP and AP corrct.. METHOD log 0 x log 0 x log 0 y log 0 y z z (A)(A)(A) log 0 y log 0 y (A) log 0 z log z (A) log 0 x log 0 x log y log z y z p q r (A) (C)(C)(C) (Sris (ii) is a GP with a sum to infinity) Common ratio (A) a S r (M) (A) (C) Not: Do not allow ft from an incorrct sris.

8 ( ax ) accpt (A)(A)(A) (A) 0 7 a 70 (M) a 7 a (A) (C) 0 Not: Award (A)(A)(A0) for 7 ax. If this lads to th answr a 7, do not award th final (A). METHOD b a c x 0, y 0, z 0 (A)(A)(A) b a x y 0 0 log0 log 0 c z 0 ' ' (A) b a c b log0 0 a c ' ' (A) 9. (a) (i) 00, 800 (A) 7. 8 () ( x ) ' Trm is 5 8. METHOD log x 8 Accpt 5 (M)(A)(A)(A) 8 8x (A) (C) log x (A) log y log y (A) log z log z (A) log x log y log z (A)(A) a b c (A) (C) (ii) Total salary 0 ( ) (A) 8000 (A) (N) (i) 0700, 9 (A)(A) (ii) 0 th yar salary (.07) (A) or 800 or 885 (A) (N) 5

9 (c) EITHER n A 000 ( ) 00 (A) Schm A S ( n ) n 0 000(.07 ) Schm B SB (A).07 Solving SB > SA (accpt SB SA, giving n. ) (may b implid) (M) Minimum valu of n is 7 yars. (A) (N) Using trial and rror (M) (c) (i) 5000(.0) n > 0000 or (.0) n > A (ii) Attmpting to solv th inquality «log (.0) > log (M) n >.5... (A) yars A Not: Candidats ar likly to us TABLE or LIST on a GDC to find n. A good way of communicating this is suggstd blow. Lt y.0 x (M) Whn x, y.958, whn x, y.08 (A) x i yars A Arturo Bill yars yars (A)(A) Not: Award (A) for both valus for yars, and (A) for both valus for 7 yars.. (a) u S 7 (A) (C) u S u 8 7 (A) d 7 (M) Thrfor, minimum numbr of yars is 7. (A) (N) [] (A) (C) (c) u u ( n ) d 7 () (M) 0. (a) Rcognizing an AP (M) u 5 d n 0 (A) substituting into u 0 5 (0 ) M 5 (that is, 5 sats in th 0th row) A u 9 (A) (C) Substituting into S 0 0 ((5) (0 )) (or into 0 (5 5)) M 80 (that is, 80 sats in total) A. (a) trms (A) (C) 5 0, ( ) 8, x ' ( ) (A)(A)(A) fourth trm is 80x (A). (a) 5000(.0) n A for xtracting th cofficint A 80 (A) (C5) Valu 5000(.0) 5 ( ) 790 to sf (Accpt 78, or 78.5) A 7 8

10 . METHOD 9, 7 (A)(A) x x xprssing as a powr of, ( ) ( ) (M) x x (A) x x (A) 7x x (A) (C) 7 METHOD ( x) xlog9 log 7 (M)(A)(A) x log 7 x log9 ' (A) x x (A) 7x x (A) (C) 7 Nots: Candidats may us a graphical mthod. Award (M)(A)(A) for a sktch, (A) for showing th point of x 5. (a) logx log ( x 5) log x 5 ' intrsction, (A) for 0.85., and (A) for 7. (A) x A (A) (C) x 5 Not: If candidats hav an incorrct or no answr to part (a) award (A)(A0) x if log sn in part. x 5 EITHER x log x 5 '. METHOD x x 5 0 ( ) x x 5 x 5 (M)(A)(A) 5 x (A) (C) x log0 x 5 ' (M)(A) log x log0 log0 x 5 ' (A) x 7.5 (A) (C) Using binomial xpansion ( 7 ) ( 7 ) ( 7 ) ( 7 ) (M) (A) (A) ( 7 ) 90 7 (so p 90, q ) (A)(A)(C)(C) METHOD For multiplying ( 7 ) ( 7 ) ( 9 7 7)( 7 ) (M) (A) ( 7 ) ( ) (A) (so p 90, q ) (A)(A)(C)(C) 7. For using u u r 8 (M) 8 8r (A) 9 0

11 r r ± (A)(A) (c) For attmpting to us infinit sum formula for a GP (M) S 5 A N u S, r 5 S 5, ( 0.8) (A)(A)(C)(C) (a) (i) r A N (ii) u 5 ( ) (A) 95 (accpt 900) A N 8. (a) (i) log c 5 log c log c 5 (A) p q A N (i),, 8 A N (ii) r A N (ii) log c 5 log c 5 (A) METHOD q A N d M d A N METHOD For changing bas g log log 0 0,log d 0 log 0 d d A N M (c) Stting up quation (or a sktch) M x x 8 (or corrct sktch with rlvant information) x x x x x x x 5 x 5 or x 5 A (A) x 5 A N Nots: If trial and rror is usd, work must b documntd with svral trials shown. Award full marks for a corrct answr with this approach. If th work is not documntd, award N for a corrct answr. 9. (a) For taking an appropriat ratio of conscutiv trms (M) r A N For attmpting to us th formula for th n th trm of a GP (M) u 5.9 A N (d) (i) r (ii) For attmpting to us infinit sum formula for a GP (M) 8 S S A N Not: Award M0A0 if candidats us a valu of r whr r >, or r <. A N []

12 5. (a) (i) S 0 A N (ii) u, d (A) ln a b ln a ln b (A)(A) Attmpting to us formula for S n M S A N ln a b p q A N (i) M 0 A (ii) For writing M as M M or M M or 0 0 M M A M 0 AG N N0 5. Not: Throughout this qustion, th first and last trms ar intrchangabl. (a) For rcognizing th arithmtic squnc (M) u, n 0, u 0 0 (u, n 0, d ) (A) Evidnc of using sum of an AP M S A ( ) S 0 (or ( 9 )) S 0 0 AG N0 (c) (i) M 8 0 (ii) T A (M) AA N N Lt thr b n cans in bottom row Evidnc of using S n 0 (M) n n n ( ) g 0, ( ( n ) ) 0, ( n ( n )( ) ) 0 n n n 80 0 A n 80 or n 8 (A) n 80 A N (d) T (M) AA N 5. (a) ln a b ln a ln b (A)(A) ln a b p q A N

13 ( n ) (c) (i) Evidnc of using S (ii) METHOD n S n n (M) n n S 0 AG N0 Substituting S 00 ( g n n ) n 00 0, 00 EITHER n., n 5. n Any valid rason which includs rfrnc to intgr bing ndd, R and pointing out that intgr not possibl hr. R N g n must b a (positiv) intgr, this quation dos not hav intgr solutions. A A A (c) (i) (ii) 9 u 0 5 (M) ,.8 0 5, 5 n 785 ' u n 5 A N 5 For attmpting to us infinit sum formula for a GP 5 5 A (M) 5 A N S.5 (. to s f ) N Discriminant 80 A Valid rason which includs rfrnc to intgr bing ndd, and pointing out that intgr not possibl hr. R N g this discriminant is not a prfct squar, thrfor no intgr solution as ndd. R 55. (a) log a 0 log a (5 ) (M) log a 5 log a p q A N METHOD Trial and rror S 080, S 5 5 AA Any valid rason which includs rfrnc to intgr bing ndd, R and pointing out that intgr not possibl hr. R N [] log a 8 log a (M) log a q A N (c) log a.5 log a 5 (M) 5. (a) (0.) A N 5 log a 5 log a p q A N 5. (a) 7 trms A N 5

14 A valid approach (M) ' x Corrct trm chosn ( ) ( ) ' x Calculating 0, ( ) 7 A (A)(A) Trm is 50x A N 57. Idntifying th rquird trm (sn anywhr) M ' 0 g (A) y,, (A) a 80 A N 58. (a) For taking thr ratios of conscutiv trms (M) ( ) A hnc gomtric AG N0 (i) Evidnc of using th sum of an AP M 0 g ( 0 ) 0 n n 0 A N (ii) METHOD 00 Corrct calculation for n 00 n g ( 99 ), 550 Evidnc of subtraction g n (A) (M) n 50 A N METHOD Rcognising that first trm is, th numbr of trms is 80 (A)(A) g ( 00), ( 79 ) 00 n n 50 A N (i) r (A) u n 8 n A N (ii) For a valid attmpt to solv 8 n 088 (M) g trial and rror, logs n A N 59. (a),, 9 A N 7 8

15 9 0. (a) For finding scond, third and fourth trms corrctly (A)(A)(A) Scond trm, third trm, fourth trm For finding first and last trms, and adding thm to thir thr trms (A) 0 N (A) Adding givs 0 accpt A N. (a) d (A) vidnc of substitution into u n a (n ) d (M) g u 0 00 u 0 0 A N 0 corrct approach (M) g 5 (n ) corrct simplification (A) g 50 (n ), 50 n, 5 n n 5 A N. (a) vidnc of dividing two trms (M) g , r 0. A N vidnc of substituting into th formula for th 0 th trm (M) g u 0 000( 0.) 9 u 0 0. (accpt th xact valu 0.088) A N (c) vidnc of substituting into th formula for th infinit sum (M) S g S 875 A N. vidnc of using binomial xpansion (M) g slcting corrct trm, b a b a b a vidnc of calculating th factors, in any ordr AAA g 5, ( ) ,, ' ' x 0x (accpt 00x to sf) A N [5]

Note If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.

Note If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark. . (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold