Cambridge A Level Mathematics 9709 — 2017 May/June Paper 2 · Variant 3

1 Solve the equation x a 2x , giving x in terms of the positive constant a. [3] + = −5a ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 1 2 2 2 5 x a x a + = − or pair of linear equations B1 SR B1 for 6 x a Attempt solution of quadratic equation or of pair of linear equations M1 Allow M1 if 4 3 and 6 seen Obtain, as final answers, 6a and 4 3 a A1 Total: 3

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2 Use logarithms to solve the equation 52x, giving your answer correct to 3 significant figures. 3x+4 = [4] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 2 Apply logarithms to both sides and apply power law *M1 Obtain ( ) 4 log3 2 log5 x x + = or equivalent A1 Solve linear equation for x DM1 dep *M Obtain 2.07 A1 Allow greater accuracy Total: 4 May be implied by part graph in first quadrant

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3 (i) By sketching a suitable pair of graphs, show that the equation x3 11 = −2x has exactly one real root. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Use the iterative formula xn+1 = 3 11 −2xn to find the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 3(i) Draw sketch of 3 y x = *B1 Draw straight line with negative gradient crossing positive y-axis and indicate one intersection DB1 dep *B Total: 2 3(ii) Use iterative formula correctly at least once M1 Obtain final answer 1.926 A1 Show sufficient iterations to justify 4 sf or show sign change in interval ( ) 1.9255,1.9265 A1 Total: 3

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e4x 4 Find the equation of the tangent to the curve y at the point on the curve for which x 0. 2x 3 = = Give your answer in the form ax by c 0 where a,+b and c are integers. [5] + + = ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 4 Use quotient rule (or product rule) to find first derivative M1 Obtain ( ) 4 4 2 8 e 10e 2 3 x x x x + + or equivalent A1 Substitute 0 x = to obtain gradient 10 9 A1 Form equation of tangent through ( ) 1 3 0, with numerical gradient M1 Obtain 10 9 3 0 x y − + = A1 Total: 5

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7 (a) Find 2 cos cos 1 [4] Ó 1 −3 1 + d1. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ @ A 4 1 (b) (i) Find dx. [2] 2x Ô 2x 1 + + ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ 4 @ A 4 1 (ii) Hence find dx, giving your answer in the form ln k. [3] 2x 1 2x + Ô 1 + ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................

Mark scheme: 7(a) 2 2cos cos 3 d θ θ θ − − ∫ B1 Attempt use of identity to obtain integrand involving cos2θ and cosθ M1 Integrate to obtain form 1 2 3 sin2 sin k k k θ θ θ + + for non-zero constants M1 Obtain 1 2 sin2 sin 2 c θ θ θ − − + A1 Total: 4 Question Answer Marks Guidance 7(b)(i) Integrate to obtain form ( ) ( ) 1 2 ln 2 1 ln k x k x + + or ( ) ( ) 1 2 ln 2 1 ln 2 k x k x + + M1 Obtain ( ) 1 2 2ln 2 1 ln x x + + or ( ) ( ) 1 2 2ln 2 1 ln 2 x x + + A1 Total: 2 7(b)(ii) Use relevant logarithm power law for expression obtained from application of limits M1 Use relevant logarithm addition / subtraction laws M1 Obtain ln18 A1 Total: 3

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8 y 1, 4 3, 3 x O The diagram shows the curve with parametric equations x 2 2t, y 2 sin3t 3 cos3t 1 = −cos = + + for 0 The end-points of the curve are 1, 4 and 3, 3 . ≤t ≤120. dy 3 (i) Show that sin t cos t. [5] dx 2 4 = −9 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find the coordinates of the minimum point, giving each coordinate correct to 3 significant figures. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Find the exact gradient of the normal to the curve at the point for which x 2. [3] = ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 8(i) Obtain d d 2sin2 x t t = B1 Obtain d 2 2 d 6sin cos 9cos sin y t t t t t = − B1 Use d d d / d d d y y x x t t = for their first derivatives M1 Use identity sin 2 2sin cos t t t = B1 Simplify to obtain 3 9 2 4 sin cos t t − with necessary detail present A1 Total: 5 Question Answer Marks Guidance 8(ii) Equate d d y x to zero and obtain tant k = M1 Obtain 3 2 tant = or equivalent A1 Substitute value of t to obtain coordinates ( ) 2.38, 2.66 A1 Total: 3 8(iii) Identify 1 4 t π = B1 Substitute to obtain exact value for gradient of the normal M1 Obtain gradient 4 3 2 , 8 3 2 or similarly simplified exact equivalent A1 Total: 3

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