Cambridge A Level Mathematics 9709 — 2015 May/June Paper 2 · Variant 2

1 (i) Use logarithms to solve the equation 2x = 205, giving the answer correct to 3 significant figures. [2] (ii) Hence determine the number of integers n satisfying 20−5 < 2n < 205.

Mark scheme: 1 (i) Introduce logarithms and use power law M1 Obtain x = 21.6 A1 [2] (ii) Obtain or imply –21.6 or –21 as lower value B1 State 43 B1 [2]

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4 y M x O The diagram shows the curve y = ex + 4e−2x and its minimum point M. (i) Show that the x-coordinate of M is ln 2. [3] (ii) The region shaded in the diagram is enclosed by the curve and the lines x = 0, x = ln 2 and y = 0. Use integration to show that the area of the shaded region is 2.5 [4]

Mark scheme: 4 (i) Differentiate to obtain e x − 8e −2 x B1 Use correct process to solve equation of form a e x + b e −2 x = 0 M1 Confirm given answer ln 2 correctly A1 [3] (ii) Integrate to obtain expression of form p e x + q e − 2 x M1 Obtain correct e x − 2e −2 x A1 Apply both limits correctly M1 depM 5 Confirm given answer A1 [4] 2 4

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6 y P x O The diagram shows part of the curve with equation y = 4 sin2x + 8 sin x + 3 and its point of intersection P with the x-axis. (i) Find the exact x-coordinate of P. [3] (ii) Show that the equation of the curve can be written y = 5 + 8 sin x −2 cos 2x, and use integration to find the exact area of the shaded region enclosed by the curve and the axes. [6]

Mark scheme: 6 (i) Solve three-term quadratic equation for sin x M1 Obtain at least sin x = − 12 and no errors seen A1 Obtain x = 76 π A1 [3] (ii) State sin 2 x = 12 − 12 cos2 x B1 Obtain given 5 + 8sin x − 2cos2 x with necessary detail seen B1 Integrate to obtain expression of form ax + b cos x + c sin 2 x M1 Obtain correct 5 x − 8cos x − sin 2 x A1 Apply limits 0 and their x-value correctly M1 depM Obtain 356 π + 72 3 + 8 or exact equivalent A1 [6] 4 dy

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