Cambridge A Level Mathematics 9709 — 2016 May/June Paper 3 · Variant 1

1 (i) Solve the equation 2 x 3 x . [3] −1 = (ii) Hence solve the equation 2 5x 3 5x , giving your answer correct to 3 significant figures. −1 = [2] 1

Mark scheme: 1 (i) EITHER: State or imply non-modular equation (2( x − 1)) 2 = (3 x ) 2 , or pair of linear equations 2( x − 1) = ±3 x B1 Make reasonable solution attempt at a 3-term quadratic, or solve two linear equations M1 Obtain answers x = −and2 x = 52 A1 OR: Obtain answer x = −by2 inspection or by solving a linear equation (B1 Obtain answer x = 52 similarly B2) [3] (ii) Use correct method for solving an equation of the form 5x = a or 5x +1 = a , where a > 0 M1 Obtain answer x =– 0.569 only A1 [2] 2 ∫ 2

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2 2 Find the exact value of dx. [5] Ó 0 xe−2x

Mark scheme: 2 Integrate by parts and reach axe −2 x + b ∫ e −2 x dx M1 Obtain − 12 xe −2 x + 12 ∫ e −2 x dx , or equivalent A1 Complete the integration correctly, obtaining − 12 x e − 2 x − 14 e − 2 x , or equivalent A1 Use limits x = 0 and x = 12 correctly, having integrated twice M1 Obtain answer 14 − 12 e− 1 , or exact equivalent A1 [5]

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3 By expressing the equation cosec 3 sin cot in terms of cos only, solve the equation for 1 = 1 + 1 1 [5] 0Å < 1 < 180Å.

Mark scheme: 3 Correctly restate the equation in terms of sin θ and cos θ B1 Using Pythagoras obtain a horizontal equation in cos θ M1 Reduce the equation to a correct quadratic in cos θ, e.g. 3cos 2 θ − cos θ − 2 = 0 A1 Solve a 3-term quadratic for cos θ M1 Obtain answer θ = 131.8° only A1 [5] [Ignore answers outside the given interval.]

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5 The curve with equation y sin x cos 2x has one stationary point in the interval 0 x 1 Find the = < < 20. x-coordinate of this point, giving your answer correct to 3 significant figures. [6]

Mark scheme: 5 Use product rule M1 Obtain correct derivative in any form, e.g. cos x cos2 x − 2sin x sin 2 x A1 Equate derivative to zero and use double angle formulae M1 Remove factor of cos x and reduce equation to one in a single trig function M1 Obtain 6sin 2 x = 1 , 6cos 2 x = 5 or 5tan 2 x = 1 A1 Solve and obtain x = 0.421 A1 [6] [Alternative: Use double angle formula M1.Use chain rule to differentiate M1. Obtain correct derivative e.g. cos θ − 6sin 2 θ cos θ A1, then as above.]

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6 (i) By sketching a suitable pair of graphs, show that the equation x 5e−x = has one root. [2] (ii) Show that, if a sequence of values given by the iterative formula 1 @25 A xn+1 = 2ln xn converges, then it converges to the root of the equation in part (i). [2] (iii) Use this iterative formula, with initial value x1 1, to calculate the root correct to 2 decimal = places. Give the result of each iteration to 4 decimal places. [3]

Mark scheme: 6 (i) Make recognizable sketch of a relevant graph B1 Sketch the other relevant graph and justify the given statement B1 [2] 1 (ii) State x = ln(25 / x ) B1 2 Rearrange this in the form5e −=x x B1 [2] (iii) Use the iterative formula correctly at least once M1 Obtain final answer 1.43 A1 Show sufficient iterations to 4 d.p. to justify 1.43 to 2 d.p., or show there is a sign change in the interval (1.425, 1.435) A1 [3] 2 dy 2

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7 The equation of a curve is x3 y3 3. −3x2y + = dy x2 (i) Show that −2xy . [4] dx = x2 −y2 (ii) Find the coordinates of the points on the curve where the tangent is parallel to the x-axis. [5]

Mark scheme: 2 dy 27 (i) State or imply 6 xy + 3 x as derivative of 3x y B1 dx 2 dy 3 State 3 y as derivative of y B1 dx dy Equate attempted derivative of the LHS to zero and solve for M1 dx Obtain the given answer A1 [4] (ii) Equate numerator to zero M1* Obtain x = 2y, or equivalent A1 Obtain an equation in x or y DM1* Obtain the point (−2, −1) A1 State the point (0, 1.44) B1 [5] A B C

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