Cambridge A Level Mathematics 9709 — 2017 Oct/Nov Paper 2 · Variant 3

3 It is given that the variable x is such that 1.32x 80 and 3x 3x . < −1 > −10 Find the set of possible values of x, giving your answer in the form a x b where the constants a and b are correct to 3 significant figures. < < [7] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 3 Take logarithms of both sides and apply power M1 Condone incorrect inequality law signs until final answer. The first 6 marks are for obtaining the correct critical values. ln80 A1 Obtain 2 x < or equivalent using log10 ln1.3 Obtain x = 8.35... A1 State or imply non-modulus inequality B1 (3 x − 1) 2 > (3 x − 10) 2 or corresponding equation or linear equation 3 x −=1 − (3 x − 10) Attempt solution of inequality or equation M1 (obtaining 3 terms when squaring each bracket or solving linear equation with signs of 3x different) Obtain x = 116 or x = 1.83... A1 Conclude 1.83 < x < 8.35 A1 7

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4 4 (a) Find [4] + sin21 Ô 1 d1. −sin21 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ a 2 (b) Given that dx ln 16, find the value of the positive constant a. [4] 3x 1 = Ô0 + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 4(a) Obtain integrand of form a sec 2 θ+ b M1 Obtain correct 5sec 2 θ− 1 A1 Integrate to obtain form a tanθ+ bθ M1 Obtain 5tanθ− θ+ c A1 4 4(b) Obtain integral of form k ln(3 x + 1) *M1 Apply limits and obtain 23 ln(3a + 1) = ln16 A1 Obtain equation with no presence of ln DM1 Obtain 21 A1 4

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6 The parametric equations of a curve are x 2e2t 4et, y 5te2t. = + = dy (i) Find in terms of t and hence find the coordinates of the stationary point, giving each coordinate dx correct to 2 decimal places. 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(ii) Find the gradient of the normal to the curve at the point where the curve crosses the x-axis. 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Mark scheme: 6(i) Obtain ddxt = 4e 2 t + 4e t B1 Use product rule to find ddyt M1 dy 5e 2 t + 10te 2 t A1 Obtain = or equivalent dx 4e 2 t + 4e t ae 2 t + bte 2 t M1 Equate first derivative of the form ce 2 t + de t to zero and solve to find t Obtain t = − 12 from completely correct work A1 Obtain (3.16, − 0.92) A1 6 6(ii) Identify t = 0 B1 Substitute t = 0 in expression for first derivative M1 and find negative reciprocal Obtain − 85 or equivalent A1 3

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7 y P Q x O The diagram shows the curve y x2 3x 1 5 cos 12x. = + + + The curve crosses the y-axis at the point P and the gradient of the curve at P is m. The point Q on the curve has x-coordinate q and the gradient of the curve at Q is −m. (i) Find the value of m and hence show that q satisfies the equation x a sin 2x1 b, = + where the values of the constants a and b are to be determined. 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(ii) Show by calculation that q [2] −4.5 < < −4.0. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Use an iterative formula based on the equation in part (i) to find the value of q correct to 3 significant figures. Give the result of each iteration to 5 significant figures. 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Mark scheme: 7(i) Differentiate to obtain form k1 x + k 2 + k 3 sin 12 x *M1 Obtain correct 2 x + 3 − 52 sin 12 x and deduce or A1 imply gradient at P is 3 Equate first derivative to their − 3 and rearrange DM1 Obtain x = 54 sin 12 x − 3 A1 4 7(ii) Consider sign of their 2 x + 6 − 52 sin 12 x at − 4.5 M1 and − 4.0 or equivalent Complete argument correctly for correct A1 expression with appropriate calculations 2 7(iii) Use iteration formula correctly at least once M1 Obtain final answer − 4.11 A1 Show sufficient iterations to justify accuracy to A1 3 sf or show sign change in interval ( −4.115, − 4.105) 3

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