Cambridge A Level Mathematics 9709 — 2015 Oct/Nov Paper 2 · Variant 2

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

Mark scheme: 1 (i) Either Square both sides to obtain three-term quadratic equation M1 Solve three-term quadratic equation to obtain two values M1 Obtain –1 and 73 A1 Or Obtain 73 from graphical method, inspection or linear equation B1 Obtain –1 similarly B2 [3] (ii) Use logarithmic method to solve an equation of the form 5 y = k where k > 0 M1 Obtain 0.526 and no others A1 [2]

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2 The sequence of values given by the iterative formula 4 2 xn+1 = + x2n 2xn 4, + + with initial value x1 2, converges to = !. (i) Determine the value of correct to 3 decimal places, giving the result of each iteration to 5 decimal places. ! [3] (ii) State an equation satisfied by and hence find the exact value of [2] ! !.

Mark scheme: 2 (i) Use the iterative formula correctly at least once M1 Obtain final answer 2.289 A1 Show sufficient iterations to justify accuracy to 3 d.p. or show sign change in interval (2.2885, 2.2895) A1 [3] 4 (ii) State x = 2 + 2 or equivalent B1 x + 2 x + 4 Obtain 3 12 B1 [2]

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5 Find the x-coordinates of the stationary points of the following curves: [3] (i) y = 4xe−3x; 4x2 (ii) y x 1. [5] = +

Mark scheme: 5 (i) Use product rule to obtain form k1e −3 x + k 2 xe −3 x M1 Obtain correct 4 e − 3 x − 12 x e − 3 x A1 Obtain x = 13 or 0.333 or better and no other A1 [3] (ii) Use quotient rule or equivalent M1* Obtain correct numerator 8 x ( x + )1 − 4 x 2 or equivalent A1 Equate numerator to zero and solve to find at least one value M1 dep Obtain x = − 2 A1 Obtain x = 0 A1 [5] dx

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1 @ A 30 1 1 97 (i) Show that the exact value of cos2x dx is [6] cos2x 60 + 8ï3. Ô 0 + (ii) y x O 130 1 The diagram shows the curve y cos x for 0 The shaded region is bounded cosx = 1 + ≤x ≤130. by the curve and the lines x 0, x and y 0. Find the exact volume of the solid obtained when the shaded region is rotated= completely= 30 about= the x-axis. [4]

Mark scheme: 7 (i) Express cos 2 x in form k1 + k 2 cos 2 x M1 Obtain correct 12 + 12 cos 2 x A1 Rewrite second term as sec 2 x B1 Integrate to obtain at least terms k 3 sin 2 x and k 4 tan x M1 Obtain 12 x + 14 sin 2 x + tan x A1 Confirm given result 16 π + 89 3 A1 [6] 1 (ii) State volume is π ∫ (cos x + cos x 2) (π maybe implied by later appearance) B1 1 1 + 2) dx B1 Expand to obtain π ∫ (cos 2 x + cos + 2) dx or ∫ (cos 2 x + cos 2 x 2 x Integrate integrand involving three terms (in part using part (i) or otherwise i.e. k 3 sin 2 x + k 4 tan x + k 5 x ) M1 Obtain 56 π 2 + 89 3π or exact equivalent A1 [4]

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