GCE Mathematics (MEI) Advanced GCE 4753 Methods for Advanced Mathematics (C3) Mark Scheme for June 00 Oxford Cambridge and RSA Examinations
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4753 Mark Scheme June 00 Section A /6 /6 cosd x sin 0 3 0 = sin 0 3 = /3 cao fg( x) x gf( x) x k sin, k > 0, k 3 k = ()/3 0.33 or better soi from correctly-shaped graphs (i.e. without intercepts) or for u = cos u d u condone 90 in limit 3 or for sin u 3 so: sin : B0, sin : M0B0, 3sin : M0B0, /3 sin : M0 but must indicate which is which bod gf if negative x values are missing - [4] graph of x only graph of x V shape with (, 0) and (0, ) labelled V shape with (0, ) labelled (0, ) 3(i) / y(3 x ) dy / dx (3 x ). 6x = ( + ) / / chain rule ½ u / o.e., but must be 3 can isw here (ii) / y x(3 x ) dy / dx x..(3 x ) 6x * / ft [4] product rule ft their dy/dx from (i) common denominator or factoring (+ ) / www must show this step for
4753 Mark Scheme June 00 4 p = 00/x = 00 x dp/dx = 00x = 00/x dp/dt = dp/dx dx/dt dx/dt = 0 When x = 50, dp/dx =( 00/50 ) dp/dt = 0 0.04 = 0.4 dep cao [6] attempt to differentiate 00x o.e. o.e. soi soi substituting x = 50 into their dp/dx dep nd o.e. e.g. decreasing at 0.4 condone poor notation if chain rule correct or x = 50 + 0 t P = 00/x = 00/(50 + 0 t) dp/dt = 00(50 + 0 t) 0 = 000/(50 + 0t) When t = 0, dp/dt = 000/50 = 0.4 5 y 3 = xy x 3y dy/dx = x dy/dx + y x 3y dy/dx x dy/dx = y x (3y x) dy/dx = y x dy/dx = (y x)/(3y x) * 3y dy/dx x dy/dx + y x collecting terms in dy/dx only must show x dy/dx + y on one side TP when dy/dx = 0 y x = 0 y = x (x) 3 = x.x x 8x 3 = x x = /8 *(or 0) [7] or x = /8 and dy/dx = 0 y = ¼ or (/4) 3 = (/8)(/4) (/8) or verifying e.g. /64 = /64 or x = /8 y 3 = (/8)y /64 verifying that y = ¼ is a solution (must show evidence*) dy/dx = (¼ (/8))/( ) = 0 *just stating that y = ¼ is M0 E0 6 f(x) = + sin = y x y x = + sin 3y sin 3y = (x )/ 3y = arcsin [(x )/] x y arcsin so x 3 f ( x) arcsin 3 Range of f is to 3 x 3 [6] attempt to invert must be y = or f (x) = or (x )/ must be x, not y or f(x) at least one step attempted, or reasonable attempt at flow chart inversion (or any other variable provided same used on each side) condone < s for allow unsupported correct answers; to 3 is A0 7 (A) True, (B) True, (C) False Counterexample, e.g. + ( ) = 0 B,,0
4753 Mark Scheme June 00 8(i) When x =, y = 3 ln + = 0 [] (ii) dy 3 x dx x At R, dy 3 0 x dx x 3 + x x = 0 (3 x)( + x) = 0 x =.5, (or ) y = 3 ln.5 +.5.5 = 0.466 (3 s.f.) d y 3 dx x When x =.5, d y/dx (= 0/3) < 0 max cao cao ft [9] d/dx (ln x) = /x re-arranging into a quadratic = 0 factorising or formula or completing square substituting their x ft their dy/dx on equivalent work www don t need to calculate 0/3 SC for x =.5 unsupported, SC3 if verified but condone rounding errors on 0.466 (iii) Let u = ln x, du/dx = /x dv/dx =, v = x ln xdx xln x x. dx x x ln x. dx = x ln x x + c parts condone no c allow correct result to be quoted (SC3).05 (3ln ) A xxx dx.05 3 lnx x x 3 =.5057 +.833.. = 0.33 ( s.f.) ft dep cao [7] correct integral and limits (soi) 3 3their' x ln xx' x x 3 substituting correct limits dep st 3
4753 Mark Scheme June 00 9(i) (0, ½ ) allow y = ½, but not (x =) ½ or ( ½,0) [] nor P = / x x x x (ii) dy (e )e e.e x dx ( e ) x e x ( e ) When x = 0, dy/dx = e 0 /(+e 0 ) = ½ x (iii) A = e d x 0 x e ln( e x ) 0 or let u = + e x, du/dx = e x e A = / du = ln u u ln( e ) ln e ln * e x x x x (iv) e e e e g( x) x x e e g( x) x x e e Rotational symmetry of order about O x x x x x x (v)(a) e e e e e e gx ( )..( ) x x x x e e e e x e.( ) x x e e x x x e.e e f( x) x x x x e(ee ) e (B) Translation 0 / (C) Rotational symmetry [of order ]about P ft [4] [5] [6] Quotient or product rule correct expression condone missing bracket cao mark final answer follow through their derivative correct integral and limits (soi) k ln( + e x ) k = ½ or v = e x, dv/dx = e x o.e. [½ ln u] or [½ ln (v + )] substituting correct limits www substituting x for x in g(x) completion www taking out ve must be clear must have rotational about O, order (oe) combining fractions (correctly) translation in y direction up ½ unit dep translation used o.e. condone omission of 80/order product rule: dy x x x x x e.e ( )(e ) e ( e ) dx x e from (udv vdu)/v SC x ( e ) condone no dx allow missing dx s or incompatible limits, but penalise missing brackets not g(x) g(x). Condone use of f for g. allow shift, move in correct direction for. 0 alone is SC. / 4
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