GCE. Mathematics. Mark Scheme for June Advanced Subsidiary GCE 4722 Core Mathematics 2. Oxford Cambridge and RSA Examinations

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1 GCE Mathematics Advanced Subsidiary GCE 7 Core Mathematics Mark Scheme for June 00 Oford Cambridge and RSA Eaminations

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3 7 Mark Scheme June 00 (i) f() = 8 + a a a 6 = 0 a = (ii) f(-) = = -9 * d* ft Attempt f() or equiv, including inspection / long division / coefficient matching Equate attempt at f(), or attempt at remainder, to 0 and attempt to solve Obtain a = Attempt f(-) or equiv, including inspection / long division / coefficient matching Obtain -9 (or a, following their a) (i) area State or imply at least of the correct y-coords, and no others Use correct trapezium rule, any h, to find area between = and = 0 Correct h (soi) for their y-values must be at equal intervals (ii) use more strips / narrower strips Obtain 0.8 (allow 0.7) Any mention of increasing n or decreasing h (i) ( + ½) 0 = Obtain + Attempt at least the third (or fourth) term of the binomial epansion, including coeffs Obtain. Obtain (ii) coeff of = ( ) + (.) + ( ) = 00 ft Attempt at least one relevant term, with or without powers of Obtain correct (unsimplified) terms (not necessarily summed) either coefficients or still with powers of involved Obtain 00 7

4 7 Mark Scheme June 00 (i) u = 6, u =, u = 6 State 6,, 6 (ii) S 0 = 0 / ( ) = 0 Show intention to sum the first 0 terms of a sequence Attempt sum of their AP from (i), with n = 0, a = their u and d = their u u Obtain 0 (iii) w = 6 p + = 6 or 6 + (p ) = 6 p = 7 State or imply w = 6 Attempt to solve u p = k Obtain p = (i) sin 8 sin 6 Attempt use of correct sine rule θ =. o Obtain. o, or better (ii) a 80 ( 6) = 0 o or 6 π / 80 =. 0 π / 80 = 0.87 A.G. π (.) = 0.87 Use conversion factor of π / 80 (ii) b area sector = ½ = 7.9 area triangle = ½ 8 sin 0.87 =. area segment = 7.9. =. 8 Show 0.87 radians convincingly (AG) Attempt area of sector, using (½) r θ Attempt area of triangle using (½) r sin θ Subtract area of triangle from area of sector Obtain.or.

5 7 Mark Scheme June 00 6 a d = ( / + 0) (9 + 8) Attempt integration Obtain / + = 6 / Use limits =, correct order & subtraction b 6 y d y y y c Obtain 6 / or any eact equiv State y Obtain ky Obtain y (condone absence of + c) c 8 d = ( 0 ) ( - ) State or imply Attempt integration of k n = Obtain correct - - (+c) ft Obtain (or -k following their k - ) 7 (i) sin cos sin sin cos cos sin cos cos cos tan Use either sin + cos =, or tan = sin / cos Use other identity to obtain given answer convincingly. (ii) tan = tan tan + tan 6 = 0 (tan )(tan + ) = 0 tan =, tan = - = 6. o, o = 08 o, 88 o State correct equation Attempt to solve three term quadratic in tan Obtain and - as roots of their quadratic Attempt to solve tan = k (at least one root) ft Obtain at least correct roots 6 Obtain all correct roots 8

6 7 Mark Scheme June 00 8 a log w = log 0 (w )log = 0 log 0log w = log w = 7. * * d* Introduce logarithms throughout Use log a b = b log a at least once Obtain (w )log = 0 log or equiv Attempt solution of linear equation y b log y y + = y 9 Obtain 7., or better Use log a log b = log a / b or equiv Use f(y) = as inverse of log f(y) = Attempt to make y the subject of f(y) = Obtain y, or equiv 9 (i) ar = a + d, ar = a + d ar ar = a ar ar + a = 0 r r + = 0 A.G. (ii) f (r) = (r )(r + r ) Attempt to link terms of AP and GP, implicitly or eplicitly. Attempt to eliminate d, implicitly or eplicitly, to show given equation. Show r r + = 0 convincingly Identify (r ) as factor or r = as root r = Hence r = * d* Attempt to find quadratic factor Obtain r + r Attempt to solve quadratic Obtain r = only (iii) a r Equate S to a ( )( ) Obtain a a = 9 / / a = Attempt to find a Obtain a =

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