Version.0 General Certificate of Eucation (A-level) January 0 Mathematics MFP (Specification 660) Further Pure Mark Scheme
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Mark Scheme General Certificate of Eucation (A-level) Mathematics Further Pure January 0 Key to mark scheme abbreviations M mark is for metho m or M mark is epenent on one or more M marks an is for metho A mark is epenent on M or m marks an is for accuracy B mark is inepenent of M or m marks an is for metho an accuracy E mark is for explanation or ft or F follow through from previous incorrect result CAO correct answer only CSO correct solution only AWFW anything which falls within AWRT anything which rouns to ACF any correct form AG answer given SC special case OE or equivalent A, or (or 0) accuracy marks x EE euct x marks for each error NMS no metho shown PI possibly implie SCA substantially correct approach c caniate sf significant figure(s) p ecimal place(s) No Metho Shown Where the question specifically requires a particular metho to be use, we must usually see evience of use of this metho for any marks to be aware. Where the answer can be reasonably obtaine without showing working an it is very unlikely that the correct answer can be obtaine by using an incorrect metho, we must awar full marks. However, the obvious penalty to caniates showing no working is that incorrect answers, however close, earn no marks. Where a question asks the caniate to state or write own a result, no metho nee be shown for full marks. Where the permitte calculator has functions which reasonably allow the solution of the question irectly, the correct answer without working earns full marks, unless it is given to less than the egree of accuracy accepte in the mark scheme, when it gains no marks. Otherwise we require evience of a correct metho for any marks to be aware.
Mark Scheme General Certificate of Eucation (A-level) Mathematics Further Pure January 0 MFP k ( ) 0. + (0.5) k 0. f (.,.5) k 0. (.+.5) 0.5... PI accept p or better y(.) y() + [ + ] k k + 0.5.0. m Dep on previous two Ms an numerical values for k s y(.).5 5 Must be.5 Total 5 (a) pcos x qsin x+ 5 psin x+ 5qcos x cos x Differentiation an subst. into DE p+ 5q ; 5 p q 0 m Equating coeffs. 5 p ; q OE Nee both (b) Aux. eqn. m + 5 0 PI. Or solving y (x)+5y0 as far as y ( ycf 5 ) Ae x OE 5x 5 c s CF + c s PI with exactly one ( ygs ) Ae + sinx+ cosx BF arbitrary constant OE Total 6 (a) r + rcosθ r + x B rcosθ x state or use r x x + y ( x) r x + y use y x 5 Must be in the form y f(x) but accept ACF for f(x). (b) Equation of line: rcosθ x Use of rcosθ x x OE y y ± ; [Pts, ± ] Distance between pts (0.75, ) an (0.75, ) is Altn: 5 ( elimination of either r or θ) At pts of intersection, r an cosθ OE () 5 (For A conone slight prem approx.) Distance PQ r sinθ () Or use of cosine rule or Pythag. 5 5 () Must be from exact values. Total 9
Mark Scheme General Certificate of Eucation (A-level) Mathematics Further Pure January 0 MFP(cont) x x IF is e Awar even if negative sign missing ln( x) ( + c) ln( ) ( + ) e OE Conone missing c e x c (k)x F Ft earlier sign error y x x y xe ( x y) x e x x LHS as /(y IF) PI x y x e x x x (e ) x e x e x Integration by parts in correct irn x x x y xe e (+c) ACF When x, y e so m Bounary conition use e e e + c to fin c after integration. 5 e c x x 5 y x e x e x e 9 Must be in the form y f(x) Total 9 5
Mark Scheme General Certificate of Eucation (A-level) Mathematics Further Pure January 0 MFP(cont) 5(a) x+ 8 x 5 (x+ )(x+ ) (x+ )(x+ ) B Accept C 5 (b) 0 ( )( ) x x x+ x+ x+ x+ [ ln(x ) ln(x ) ] + + (+c) OE I lim a a 0 (x+ )(x+ ) replace by a an lim a (OE) [ a a ] a lim lim a ln( + ) ln( + ) (ln 5 ln 5) ln a + a + 6 ln ln 9 lim a + ln a + a m,m 6 CSO Total 7 Limiting process shown. Depenent on the previous 6
Mark Scheme General Certificate of Eucation (A-level) Mathematics Further Pure January 0 MFP(cont) sin θ cos θ θ Use of r θ 6 Area ( ) cos θ sin θ θ 0 B ( ) ( ) B r cosθsin θ or better Correct limits 6sin θ cos θ θ 0 sin k sin cos ( ) θ θ θ (k>0) 8sin θ (-sin θ) sinθ 0 m Substitution or another vali metho to 8sin θ 8sin θ 5 0 5 8 8 6 0 5 5 F integrate sin 7 CSO AG θcos θ Correct integration of p sin θ cos θ Alternatives for the last four marks cosθ 0 cos θ cos θ θ () Area ( ) ( cos θ cos θ) θ (cos θsinθ sin θcos θ) 5 Area ( 0) + ( ) [ 0 (0)] 6 5 5 (m) (F) () cosθsin θ λcosθ + μ cos θcosθ ( λ, μ 0) Integration by parts twice or use of cos θ cosθ ( cos5θ + cosθ) Correct integration of p cos θcosθ [eg p sin 5θ + sin θ 0 6 ] CSO AG 6 { + } 0 6 5 Total 7 7
Mark Scheme General Certificate of Eucation (A-level) Mathematics Further Pure January 0 MFP(cont) 7(a)(i) B Accept coeffs unsimplifie, even! cos x + sin x + x x x 6 for 6. (ii) 9 ln(+ ) x x () x + () x x x + 9x B Accept coeffs unsimplifie (b)(i) tan y e x, y tanx sec x e x Chain rule ACF eg ysec x y tanx tanx sec xtan x e sec x e x + m Prouct rule OE ACF tanx sec x e ( tan x+ sec x) (tan tan x + + x ) y ( + tan x) 5 AG Completion; CSO any vali x metho. (ii) y ( + tan x) sec x + ( + tan x) When x 0, CSO (iii) y(0) ; y (0) ; y (0) ; y (0) ; y(x) y(0) + x y (0) + x y (0) +! x y (0) (c) tan e x + x + x + x tan x lim e (cosx + sin x) x 0 xln( + x) x x x x lim + x+ + x+ + 6 x 0 9 x x x +... lim x + x +.. x 0 9 lim + x +.. x x... x 0 9 x... CSO AG m Total Using series expns. Diviing numerator an enominator by x to get constant terms. OE following a slip. 8
Mark Scheme General Certificate of Eucation (A-level) Mathematics Further Pure January 0 MFP(cont) 8(a) y y Chain rule e t y y y y x t CSO AG (b) y x ; x y x y OE x y + x m Prouct rule (ep on previous M) y x + x x OE y x x + y lnx becomes y x x + y lnx x y t + y lne (using (a) y + y t m 5 CSO AG (c) Auxl eqn m m + 0 PI (m ) 0, m PI CF: ( y ) ( )e t C At+ B Ft wrong value of m provie equal roots an arb. constants in CF. Conone x for t here PI Try ( yp ) at+ b If extras, coeffs. must be shown to be 0. a+ at+ b t a b Correct PI. Conone x for t here GS {y} (At +B)e t +0.5(t + ) BF 6 Ft on c s CF + PI, provie PI is non-zero an CF has two arbitrary constants an RHS is fn of t only () y (Alnx + B)x +0.5(lnx + ) y.5 when x B F Ft one earlier slip y (x) (A lnx + B ) x + Ax + 0.5 x m Prouct rule y () 0.5 A F Ft one earlier slip y ( ln x) x + (ln x+ ) 5 ACF Total 8 TOTAL 75 9