Cambridge A Level Mathematics 9709 — 2018 May/June Paper 3 · Variant 3

2 Showing all necessary working, solve the equation 52x 5x 5. Give your answer correct to 3 decimal places. = + [5] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 2 State or imply u 2 = u + 5 , or equivalent in 5x B1 Solve for u, or 5x M1 1 A1 Obtain root (1 + 21 ), or decimal in [2.79, 2.80] 2 Use correct method for finding x from a positive root M1 Obtain answer x = 0.638 and no other answer A1 Total: 5

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1 603 Showing all necessary working, find the value of x cos 3x dx, giving your answer in terms of Ó 0 0. [5] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 3 Integrate by parts and reach ax sin3 x + b ∫ sin3 xdx M1* 1 `1 A1 Obtain x sin3 x − ∫ sin3 x d x , or equivalent 3 3 1 1 A1 Complete the integration and obtain x sin3 x + cos3 x , or equivalent 3 9 Substitute limits correctly having integrated twice and obtained ax sin 3x + b cos 3x M1(dep*) 1 A1 Obtain answer (π − 2 ) OE 18 Total: 5

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ln x 4 The curve with equation y has a stationary point at x p. 3 x = = + 3 (i) Show that p satisfies the equation ln x 1 . [3] x = + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) By sketching suitable graphs, show that the equation in part (i) has only one root. [2] 3 x (iii) It is given that the equation in part (i) can be written in the form x . Use an iterative + ln x = formula based on this rearrangement to determine the value of p correct to 2 decimal places. Give the result of each iteration to 4 decimal places. 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Mark scheme: 4(i) Use the quotient or product rule M1 Obtain correct derivative in any form A1 Equate derivative to zero and obtain the given equation A1 Total: 3 4(ii) Sketch a relevant graph, e.g. y = ln x B1 3 B1 Sketch a second relevant graph, e.g. y = 1 + , and justify the given statement x Total: 2 4(iii) 3 + x M1 Use iterative formula nx +1 = correctly at least once ln x n Obtain final answer 4.97 A1 Show sufficient iterations to 4 d.p.to justify 4.97 to 2 d.p. or show there is a sign A1 change in the interval (4.965, 4.975) Total: 3

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5 (i) By first expanding cos2x sin2x 3, or otherwise, show that + cos6x sin6x 1 sin22x. [4] 4 + = −3 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Hence solve the equation cos6x sin6x 23, + = for x [4] 0Å < < 180Å. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 5(i) Attempt cubic expansion and equate to 1 M1 Obtain a correct equation A1 Use Pythagoras and double angle formula in the expansion M1 Obtain the given result correctly A1 Total: 4 5(ii) Use the identity and carry out a method for finding a root M1 Obtain answer 20.9° A1 Obtain a second answer, e.g. 69.1° A1FT Obtain the remaining answers, e.g. 110.9° and 159.1°, and no others in the given A1FT interval Total: 4

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1 6 (i) Express in partial fractions. [2] 4 −y2 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) The variables x and y satisfy the differential equation xdy 4 dx = −y2, and y 1 when x 1. Solve the differential equation, obtaining an expression for y in terms of x. = = [6] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ 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Mark scheme: 6(i) Carry out relevant method to find A and B such that M1 1 A B ≡ + 4 − y 2 2 + y 2 − y 1 A1 Obtain A = B = 4 Total: 2 6(ii) Separate variables correctly and integrate at least one side to obtain one of the terms M1 a ln x, b ln (2 + y) or c ln (2 – y) Obtain term ln x B1 1 1 A1FT Integrate and obtain terms ln ( 2 + y ) − ln ( 2 − y ) 4 4 Use x = 1 and y = 1 to evaluate a constant, or as limits, in a solution containing at M1 least two terms of the form a ln x, b ln (2 + y) and c ln (2 – y) Obtain a correct solution in any form, e.g. A1 1 1 1 ln x = ln ( 2 + y ) − ln ( 2 − y ) − ln3 4 4 4 4 A1 2 ( 3 x − 1) Rearrange as , or equivalent ( 3 x 4 + 1) Total: 6

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8 The equation of a curve is 2x3 2a3, where a is a non-zero constant. −y3 −3xy2 = dy 2x2 (i) Show that [4] −y2 dx = y2 2xy. + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find the coordinates of the two points on the curve at which the tangent is parallel to the y-axis. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 8(i) 2 d y B1 State or imply 3y as derivative of 3y d x 2 d y 2 B1 State or imply 3 y + 6 xy as derivative of 3xy d x d y M1 Equate derivative of LHS to zero and solve for d x Obtain the given answer A1 Total: 4 8(ii) Equate denominator to zero and solve for y M1* Obtain y = 0 and x = a A1 Obtain y = αx and substitute in curve equation to find x or M1(dep*) to find y Obtain x = – a A1 Obtain y = 2a A1 Total: 5

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9 (a) Find the complex number z satisfying the equation 3z 1 5i, −iz* = + where z* denotes the complex conjugate of z. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z which satisfy both the inequalities z and Im z where Im z denotes the imaginary part of z. Calculate the greatest value of arg≤3z for points ≥2,in this region. Give your answer in radians correct to 2 decimal places. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 9(a) Substitute and obtain a correct equation in x and y B1 Use 2i =−1 and equate real and imaginary parts M1 Obtain two correct equations in x and y, e.g. A1 3x – y = 1 and 3y – x = 5 Solve and obtain answer z = 1 + 2(i) A1 Total: 4 9(b) Show a circle with radius 3 B1 Show the line y = 2 extending in both quadrants B1 Shade the correct region B1 Carry out a complete method for finding the greatest value of arg z M1 Obtain answer 2.41 A1 Total: 5

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10 The points A and B have position vectors 2i j 3k and 4i j k respectively. The line l has equation r 4i 6j i 2j . + + + + = + + - + −2k (i) Show that l does not intersect the line passing through A and B. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ The point P, with parameter t, lies on l and is such that angle PAB is equal to 120Å. (ii) Show that 3t2 8t 4 0. Hence find the position vector of P. [6] + + = ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 10(i) Carry out a correct method for finding a vector equation for AB M1 Obtain r = 2i + j +3k +λ(2i – 2k), or equivalent A1 Equate pair(s) of components AB and l and solve for λor µ M1(dep*) Obtain correct answer for λor µ A1 Verify that all three component equations are not satisfied A1 Total: 5 10(ii) State or imply a direction vector for AP has components B1 (2 + t, 5 + 2t, – 3 – 2t) JJJG JJJG State or imply that cos 120° equals the scalar product of AP and AB divided by the M1 product of their moduli Carry out the correct processes for finding the scalar product and the product of the M1 moduli in terms of t, and obtain an equation in terms of t Obtain the given equation correctly A1 Solve the quadratic and use a root to find a position vector for P M1 2 A1 Obtain position vector 2i + 2j + 4k from t = – 2, having rejected the root t = − 3 Total: 6

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