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

cotx x 1 Prove the identity cos 2x. [3] −tan cotx tan x  + ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: Question Answer Marks 1 Express the LHS in terms of either cos x and sin x or in terms of tan x B1 Use Pythagoras M1 Obtain the given answer A1 Total: 3

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3 Using the substitution u ex, solve the equation 3ex 4. Give your answer correct to 3 significant figures. = 4e−x = + [4] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 3 Rearrange as 3u 2 + 4u − 4 = 0 , or 3e 2 x + 4e x − 4 = 0 , or equivalent B1 Solve a 3-term quadratic for ex or for u M1 Obtain e x = 23 or u = 23 A1 Obtain answer x = –0.405 and no other A1 Total: 4

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1 20 14 Find the exact value of sin [4] Ó 0 1 21 d1. ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 4 1 1 *M1 cos 2 θ dθ Integrate by parts and reach aθcos 2 θ+ b ∫ Complete integration and obtain indefinite integral −2θcos 12 θ+ 4sin 12 θ A1 Substitute limits correctly, having integrated twice DM1 Obtain final answer ( 4 − π) / 2 , or exact equivalent A1 Total: 4

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5 A curve has equation y 2 ln 1 3 cos2x for 0 3 = + ≤x ≤120. dy (i) Express in terms of tan x. [4] dx ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Hence find the x-coordinate of the point on the curve where the gradient is Give your answer correct to 3 significant figures. −1. 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Mark scheme: 5(i) Use the chain rule M1 Obtain correct derivative in any form A1 Use correct trigonometry to express derivative in terms of tan x M1 d y 4tan x A1 Obtain = − , or equivalent d x 4 + tan 2 x Total: 4 5(ii) Equate derivative to –1 and solve a 3–term quadratic for tan x M1 Obtain answer x=1.11 and no other in the given interval A1 Total: 2

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6 The equation cot x 1 has one root in the interval 0 x denoted by = −x < < 0, !. (i) Show by calculation that is greater than 2.5. [2] ! ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Show that, if a sequence of values in the interval 0 x given by the iterative formula @ A 1 < < 0 converges, then it converges to [2] tan−1 1 xn+1 = 0 + !. −xn ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Use this iterative formula to determine correct to 3 decimal places. Give the result of each iteration to 5 decimal places. ! 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Mark scheme: 6(i) Calculate the value of a relevant expression or expressions at x = 2.5 and at another M1 relevant value, e.g. x = 3 Complete the argument correctly with correct calculated values A1 Total: 2 6(ii) State a suitable equation, e.g. x = π + tan − 1 (1/ (1 − x ) ) without suffices B1 Rearrange this as cot x = 1 − x , or commence working vice versa B1 Total: 2 6(iii) Use the iterative formula correctly at least once M1 Obtain final answer 2.576 only A1 Show sufficient iterations to 5 d.p. to justify 2.576 to 3 d.p., or show there is a sign A1 change in the interval (2.5755, 2.5765) Total: 3

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7 y M x O 12x e The diagram shows a sketch of the curve y for x 0, and its minimum point M. x = > (i) Find the x-coordinate of M. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Use the trapezium rule with two intervals to estimate the value of 3 12x e dx, x Ô1 giving your answer correct to 2 decimal places. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) The estimate found in part (ii) is denoted by E. Explain, without further calculation, whether another estimate found using the trapezium rule with four intervals would be greater than E or less than E. [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 7(i) Use correct quotient rule or product rule M1 Obtain correct derivative in any form A1 Equate derivative to zero and solve for x M1 Obtain x = 2 A1 Total: 4 7(ii) State or imply ordinates 1.6487…, 1.3591…, 1.4938… B1 Use correct formula, or equivalent, with h = 1 and three ordinates M1 Obtain answer 2.93 only A1 Total: 3 7(iii) Explain why the estimate would be less than E B1 Total: 1

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10 The points A and B have position vectors given by OA i 2k and OB 3i j k. The line l −−→ −−→ has equation r 2i j mk i , where m is=a constant.−2j + = + + = + + + - −2j −4k (i) Given that the line l intersects the line passing through A and B, find the value of m. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find the equation of the plane which is parallel to i and contains the points A and B. Give your answer in the form ax by cz d. −2j −4k [5] + + = ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 10(i) Carry out a correct method for finding a vector equation for AB M1 Obtain r = i − 2 j + 2k + λ( 2i + 3 j − k ) , or equivalent A1 Equate two pairs of components of general points on AB and l and solve for λ or for M1 µ Obtain correct answer for λor µ, e.g. λ= 75 or µ= 73 A1 Obtain m = 3 A1 Total: 5 10(ii) EITHER: (B1 Use scalar product to obtain an equation in a, b and c, e.g. a − 2b − 4 c = 0 Form a second relevant equation, e.g. 2 a + 3b − c = 0 and solve for one ratio, e.g. a M1 : b Obtain final answer a : b : c = 14 : – 7 : 7 A1 Use coordinates of a relevant point and values of a, b and c and find d M1 Obtain answer 14 x − 7 y + 7 z = 42 , or equivalent A1) OR 1: Attempt to calculate the vector product of relevant vectors, e.g. (M1 ( i − 2 j − 4k ) × ( 2i + 3 j − k ) Obtain two correct components A1 Obtain correct answer, e.g. 14 i − 7 j + 7 k A1 Substitute coordinates of a relevant point in 14 x − 7 y + 7 z = d , or equivalent, and M1 find d Obtain answer 14 x − 7 y + 7 z = 42 , or equivalent A1) OR 2: Using a relevant point and relevant vectors, form a 2–parameter equation for the (M1 plane State a correct equation, e.g. r = i − 2 j + 2k + s ( i − 2 j − 4k ) + t ( 2i + 3 j − k ) A1 State 3 correct equations in x, y, z, s and t A1 Eliminate s and t M1 Obtain answer 2x – y + z = 6, or equivalent A1) OR 3: (M1 Using a relevant point and relevant vectors, form a determinant equation for the plane x − 1 y + 2 z − 1 A1 State a correct equation, e.g. 1 − 2 − 4 = 0 2 3 − 1 Attempt to expand the determinant M1 Obtain or imply two correct cofactors A1 Obtain answer 14 x − 7 y + 7 z = 42 , or equivalent A1) Total: 5

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