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1 Qestion Answer Marks (i) a = ½ B allow = ½ y y d y ( ). d ( ) 6 ( ) () dy * d y ( ) dy/d = 0 when = 0 ( ) = 0, = 0 or ¾ y = (¾) /½ = 7/, y = 0.95 (sf) [] B [9] y dy/d Gi Qotient (or prodct) rle consistent with their derivatives; (v d + dv)/v M0 correct RHS epression condone missing bracket penalise omission of bracket in QR at this stage if in addition = 0 giving = ½, A0 mst se = ¾ ; if (0, 0) given as an additional TP, then A0 can infer from answer in range 0.9 to 0.95 inclsive

2 Qestion Answer Marks (iii) = d = d Gi ( ) d d / / / / d ( )d * area 5 d when =, =, when =.5, = 8 ( ) if missing brackets, withhold / ½ d condone missing d here, bt withhold correct integral and limits may be inferred from a change of limits andp their attempt to integrate (their) ¼ ( / + / ) =, 8 (or sbstitting back to s and sing and.5) 8 / ( / 5 5/ / )d = or B 5/ / [8] 5 o.e. e.g. [ 5/ /(5/) + / /(/)] o.e. correct epression (may be inferred from a correct final answer) cao, mst be eact; mark final answer

3 (i) When =, y = /( ) = So P is (, ) which lies on y = [] sbstitting = (both s) y = and completion ( = is enogh) or = / ( ) = (by solving or verifying)...( ) d y / d / / ( ) ( ) * / ( ) When =, dy/d = ½ / = ½ This gradient wold be if crve were symmetrical abot y = cao [7] Qotient or prodct rle PR: ½( ) / + ( ) / correct epression top and bottom by ( ) o.e. e.g. taking ot factor of ( ) / sbstitting = or an eqivalent valid argment If correct formla stated, allow one error; otherwise QR mst be on correct and v, with nmerator consistent with their derivatives and denominator correct initially allow ft on correct eqivalent algebra from their incorrect epression

4 (iii) = d/d = d = d When =, = when =, = 9 9 d / d 9 ( / / 9 / )d / = (8 + ) (/ + ) 5 * Area nder y = is ½ ( + ) 8 = 56 Area = (area nder y = ) (area nder crve) so reqired area B B cao B cao [9] or d/d = / (d ) splitting their fraction (correctly) and / / = / (or ) / / (o.e) sbstitting correct limits o.e. (e.g ) soi from working 0.7 or better No credit for integrating initial integral by parts. Condone d =.Condone missing d s in sbseqent working. or integration by parts: / (+) / d (mst be flly correct condone missing bracket by parts: [ / (+) / /] F(9) F() () or F() F() () dep sbstittion and integration attempted mst be trapezim area: is M0

5 Qestion Answer Marks Gidance (i) (A) (0, 6) and (, BB Condone P and Q incorrectly labelled (or (B), 5) and (0, ) BB nlabelled) (iii) (iv) f'() ( ). ( ). ( ) f() = 0 ( + ) ( + ) = 0 + = 0 ( )( + ) = 0 = or = When =, y = /( ) = 6 so other TP is (, 6) f( ) = ( ) = * b b ( ) d ln a a ( ( b b ln b) a alna) Area is f()d 0 So taking a = and b = area = ( + ln ) ( ½ + ln ) = ln ½ [] dep BBcao [6] [] B cao [5] Qotient or prodct rle consistent with their derivatives, condone missing brackets correct epression their derivative = 0 obtaining correct qadratic eqation (soi) dep st bt withhold if denominator also set to zero mst be from correct work (bt see note re qadratic) sbstitting for both s in f ln F(b) F(a) condone missing brackets oe (mark final answer) mst be simplified with ln = 0 PR: ( +)( )(+) + (+) If formla stated correctly, allow one sbstittion error. condone missing brackets if sbseqent working implies they are intended Some candidates get + +, then realise this shold be +, and correct back, bt not for every occrrence. Treat this sympathetically. Mst be spported, bt cold be verified by sbstittion into correct derivative allow slip for F mst show evidence of integration of at least one term or f() = + + /(+) A = f( )d ln( ) 0 0 = ½ + ln = ln ½

6 (i) or. ln. d y = d ln = ln = d y = ln+ ( ) d = ln + B [] B [] qotient rle with = ln and v = d/d (ln ) = / soi correct epression (o.e.) o.e. cao, mark final answer, bt mst have divided top and bottom by prodct rle with = and v = ln d/d (ln ) = / soi correct epression o.e. cao, mark final answer, mst simplify the.(/) term. Consistent with their derivatives. dv ± vd in the qotient rle is M0 Condone ln. = ln for this (provided ln. is shown) e. ln, or vice-versa ln ln d let = ln, d/d = / dv/d = /, v = = ln. d + = ln d + = ln + = (ln + ) +c * [] Integration by parts with = ln, d/d = /, dv/d = /, v = mst be correct, condone + c condone missing c mst have c shown in final answer Mst be correct at this stage. Need to see /