Cambridge A Level Mathematics 9709 — 2017 May/June Paper 3 · Variant 1

3 It is given that x ln 1 y, where 0 y 1. = −y −ln < < e−x (i) Show that y . [2] = 1 e−x + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ 1 @ A 2e (ii) Hence show that y dx ln . [4] Ó 0 = e 1 + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: Question Answer Marks 3(i) 1 − y B1 Remove logarithms correctly and obtain ex = y e − x B1 Obtain the given answer y = − x following full working 1 + e Total: 2 3(ii) State integral k ln(1 + e − x ) where k = ± 1 *M1 State correct integral − ln(1 + e − x ) A1 Use limits correctly DM1  2e  A1 Obtain the given answer ln   following full working  e + 1  Total: 4

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4 The parametric equations of a curve are x ln cos y = 1, = 31 −tan 1, where 0 1 ≤1 < 20. dy (i) Express in terms of tan [5] dx 1. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find the exact y-coordinate of the point on the curve at which the gradient of the normal is equal to 1. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 4(i)  d x sin θ  M1 Use chain rule to differentiate x  = −   dθ cos θ  dy 2 B1 State = 3 − sec θ dθ dy dy dx M1 Use = ÷ dx dθ dθ d y 3 − sec 2 θ A1 Obtain correct in any form e.g. d x − tan θ dy tan 2 θ − 2 A1 Obtain = , or equivalent dx tan θ Total: 5 4(ii) Equate gradient to −1 and obtain an equation in tanθ M1 Solve a 3 term quadratic ( tan 2 θ + tan θ − 2 = 0 ) in tanθ M1 π 3π A1 Obtain θ = and y = − 1 only 4 4 Total: 3

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6 The plane with equation 2x 2y 5 is denoted by m. Relative to the origin O, the points A and B + −z = have coordinates 3, 4, 0 and 0, 2 respectively. −1, (i) Show that the plane m bisects AB at right angles. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ A second plane p is parallel to m and nearer to O. The perpendicular distance between the planes is 1. (ii) Find the equation of p, giving your answer in the form ax by cz d. [3] + + = ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 6(i) State or obtain coordinates (1, 2, 1) for the mid-point of AB B1 Verify that the midpoint lies on m B1 State or imply a correct normal vector to the plane, e.g. 2i + 2 j − k B1 State or imply a direction vector for the segment AB, e.g. − 4 i − 4 j + 2k B1 Confirm that m is perpendicular to AB B1 Total: 5 6(ii) State or imply that the perpendicular distance of m from the origin is 53 , or B1 unsimplified equivalent State or imply that n has an equation of the form 2 x + 2 y − z = k B1 Obtain answer 2 x + 2 y − z = 2 B1 Total: 3

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7 Throughout this question the use of a calculator is not permitted. The complex numbers u and w are defined by u 7i and w 3 4i. = −1 + = + (i) Showing all your working, find in the form x iy, where x and y are real, the complex numbers + u u and [4] −2w w. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ In an Argand diagram with origin O, the points A, B and C represent the complex numbers u, w and u respectively. −2w (ii) Prove that angle AOB 1 [2] = 40. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) State fully the geometrical relation between the line segments OB and CA. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 7(i) State that u – 2w = – 7 – i B1 EITHER: u (M1 Multiply numerator and denominator of by 3 – 4i, or equivalent w Simplify the numerator to 25 + 25i or denominator to 25 A1 Obtain final answer 1 + i A1) OR: Obtain two equations in x and y and solve for x or for y (M1 Obtain x = 1 or y = 1 A1 Obtain final answer 1 + i A1) Total: 4 7(ii) u M1 Find the argument of w Obtain the given answer A1 Total: 2 7(iii) State that OB and CA are parallel B1 State that CA = 2OB, or equivalent B1 Total: 2

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10 y M x O 140 The diagram shows the curve y sin x cos22x for 0 and its maximum point M. = ≤x ≤140 (i) Using the substitution u cosx, find by integration the exact area of the shaded region bounded = by the curve and the x-axis. [6] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find the x-coordinate of M. Give your answer correct to 2 decimal places. [6] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 10(i) State or imply d u = − sin x d x B1 Using correct double angle formula, express the integral in terms of u and du M1 Obtain integrand ± (2u 2 − 1) 2 A1 1 A1 Change limits and obtain correct integral (2u 2 − 1) 2 d u with no errors seen ∫ 1 2 Substitute limits in an integral of the form au 5 + bu 3 + cu M1 Obtain answer 151 (7 − 4 2) , or exact simplified equivalent A1 Total: 6 10(ii) Use product rule and chain rule at least once M1 Obtain correct derivative in any form A1 Equate derivative to zero and use trig formulae to obtain an equation in M1 cos x and sin x Use correct methods to obtain an equation in cos x or sin x only M1 Obtain 10cos 2 x = 9 or 10sin 2 x = 1 , or equivalent A1 Obtain answer 0.32 A1 Total: 6

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