Cambridge A Level Mathematics 9709 — 2015 Oct/Nov Paper 2 · Variant 3

35 3 1 Find the exact value of dx, giving the answer in the form ln k. [5] Ô 2x 5 −1 +

Mark scheme: 1 Integrate to obtain k ln( 2 x + 5) M1 Obtain correct 32 ln( 2 x + 5) A1 Apply limits and use logarithm law for ln a − ln b M1 Use logarithm power law M1 Obtain ln 125 A1 [5] 2 2

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2 (i) Solve the equation 2x 3 x 8 . [3] + = + (ii) Hence, using 3 2y 8 . Give the answer correct to logarithms, solve the equation 2y+1 + = + 3 significant figures. [2]

Mark scheme: 2 (i) Either State or imply non-modulus equation ( 2 x + 3) 2 = ( x + 8) 2 or corresponding pair of linear equations B1 Solve 3-term quadratic equation or 2 linear equations M1 Obtain x = − 113 and x = 5 A1 Or Obtain x = 5 from graphical method, inspection, equation, … B1 Obtain x = − 113 similarly B2 [3] (ii) Use logarithms to solve equation of form 2 y = k where k > 0 M1 Obtain 2.32 A1 [2] dx t t

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sin 2x 7 The equation of a curve is y cosx 1. = + dy 2 cos2x cosx (i) Show that . [7] + −1 dx cosx 1 = + (ii) Find the x-coordinate of each stationary point of the curve in the interval x Give each answer correct to 3 significant figures. −0 < < 0. [3]

Mark scheme: 7 (i) Use quotient rule or equivalent to find first derivative M1 2 cos 2 x (cos x + )1 + sin 2 x sin x Obtain or equivalent A1 (cos x + 2)1 Use at least one of cos 2 x = 2 cos 2 x − 1 and 2 x = 2 sin x cos x B1 Express first derivative in terms of cos x only M1 2 cos 3 x + 4 cos 2 x − 2 Obtain 2 or equivalent A1 (cos x + )1 Factorise numerator or divide numerator by (cos x + )1 or equivalent M1 2 (cos 2 x + cos x − )1 Confirm given answer correctly A1 [7] cos x + 1 (ii) Use quadratic formula or equivalent to find value of cos x M1 Obtain x-coordinate 0.905 A1 Obtain x-coordinate –0.905 and no others in range A1 [3]

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