Cambridge A Level Mathematics 9709 — 2016 May/June Paper 2 · Variant 1

2 Solve the equation 5 tan 4 cot for [5] 21 = 1 0Å < 1 < 180Å.

Mark scheme: 2 Use cot θ = 1 ÷ tan θ B1 Form equation involving tanθ only and with no denominators involving θ M1 Obtain tan 2 θ = 72 A1 Obtain 28.1 A1 Obtain 151.9 A1 [5] Allow other valid methods 2

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3 Given that 3ex 14, find the possible values of ex and hence solve the equation 3ex 14 correct to 3 significant+ 8e−x =figures. + 8e−x = [6]

Mark scheme: 3 Rearrange to 3e 2 x − 14e x + 8 = 0 or equivalent involving substitution B1 Solve quadratic equation in ex to find two values of ex *M1 Obtain 23 and 4 A1 Use natural logarithms to solve equation of form ex = k where k > 0 dep on DM1 Allow M mark if left in exact form M1 Obtain − 0.405 A1 Obtain 1.39 A1 [6] 2

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3x2 6 The equation of a curve is y At the point on the curve with positive x-coordinate p, the = x2 4. gradient of the curve is 1 + 2. _P48p Q (i) Show that p . [5] −16 = p2 8 + (ii) Show by calculation that 2 p 3. [2] < < (iii) Use an iterative formula based on the equation in part (i) to find the value of p correct to 4 significant figures. Give the result of each iteration to 6 significant figures. [3]

Mark scheme: 6 (i) Use quotient rule or equivalent *M1 6 x ( x 2 + 4) − 6 x 3 Obtain 2 2 or equivalent A1 ( x + 4) Equate first derivative to 12 and remove algebraic denominators dep on *M1 DM1 Obtain 48 p = p 4 + 8 p 2 + 16 or 48 x = x 4 + 8 x 2 + 16 or equivalent A1 48 p − 16 Confirm given result p = 2 A1 [5] p + 8 48 p − 16 (ii) Consider sign of p − 2 at 2 and 3 or equivalent M1 p + 8 Complete argument correctly with appropriate calculations A1 [2] (iii) Carry out iteration process correctly at least once M1 Obtain final answer 2.728 A1 Show sufficient iterations to justify accuracy to 4 sf or show sign change in interval (2.7275, 2.7285) B1 [3] 2 2

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