Cambridge A Level Mathematics 9709 — 2014 May/June Paper 2 · Variant 3

1 (i) Solve the equation x 2 x . [2] + = −13 (ii) Hence solve the equation 3y 2 3y , giving your answer correct to 3 significant figures. + = −13 [2]

Mark scheme: 1 (i) Either Square both sides to obtain linear equation M1 Obtain x = 16530 or 336 or 112 A1 [2] Or Solve linear equation in which, initially, signs of x are different M1 Obtain x + 2 = − x + 13 or equivalent and hence 112 or equivalent A1 [2] (ii) Apply logarithms and use power law M1 Obtain y log 3 = log 112 and hence y = .155 A1 [2]

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2 Solve the equation 3 sin tan 2 for [4] 21 1 = 0Å < 1 < 180Å.

Mark scheme: 2 Use sin 2θ = 2 sin θ cos θ B1 Simplify to obtain form c1sin 2 θ = c 2 or equivalent M1 Find at least one value of θ from equation of form sin θ = k M1 Obtain 35.3° and 144.7° A1 [4]

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8 y M x O The diagram shows the curve y tan x cos2x, for 0 1 = ≤x < 20, and its maximum point M. dy (i) Show that 4 cos2x [5] dx = −sec2x −2. (ii) Hence find the x-coordinate of M, giving your answer correct to 2 decimal places. [4]

Mark scheme: 8 (i) Differentiate using product rule M1 Obtain sec2 x cos 2 x − 2 tan x sin 2 x A1 Use cos 2 x = 2 cos 2 x − 1 or sin 2 x = 2 sin x cos x or both B1 Express derivative in terms of sec x and cos x only M1 Obtain 4 cos 2 x − sec 2 x − 2 with no errors seen (AG) A1 [5] (ii) State 4 cos 4 x − 2 cos 2 x − 1 = 0 B1 Apply quadratic formula to a 3 term quadratic equation in terms of cos 2 x to find the least positive value of cos 2 x M1 1 + 5 Obtain or imply cos2 x = or 0.809… A1 4 Obtain 0.45 A1 [4]

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