Cambridge A Level Mathematics 9709 — 2020 May/June Paper 3 · Variant 2

1 Find the quotient and remainder when 6x4 x3 5x is divided by 2x2 1. [3] + −x2 + −6 −x + ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 1 Obtain quotient 3x2 + 2x –1 A1 Obtain remainder 2x – 5 A1 4

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3 Find the exact value of 4 3 x 2 ln x dx. [5] Ó1 ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 3 Commence integration and reach 5 5 2 2 1 ln . d +  ax x b x x x Obtain 5 5 2 2 2 2 1 ln . d 5 5 − x x x x x A1 Complete the integration and obtain 5 5 2 2 2 4 ln 5 25 − x x x , or equivalent A1 Use limits correctly, having integrated twice e.g 2 4 2 4 32ln 4 32 0 5 25 5 25 × − × − × +       DM1 Obtain answer 128 124 ln2 5 25 − , or exact equivalent A1 5

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4 A curve has equation y cos x sin 2x. = Find the x-coordinate of the stationary point in the interval 0 x 1 giving your answer correct to < < 2π, 3 significant figures. [6] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 4 Use correct product rule M1 Obtain correct derivative in any form, e.g. –sin x sin 2x + 2cos x cos 2x A1 Use double angle formula to express derivative in terms of sin x and cos x M1 Equate derivative to zero and obtain an equation in one trig function M1 Obtain 3 sin 2x = 1, or 3 cos 2x = 2 or 2 tan 2x = 1 A1 Solve and obtain x = 0.615 A1 6

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5 (a) Express 2 cos x 5 sin x in the form R cos x , where R 0 and Give the − + ! > 0Å < ! < 90Å. exact value of R and the value of correct to 3 decimal places. 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Mark scheme: 5(a) State 7 = R Use trig formulae to find α M1 Obtain α = 57.688° A1 3 5(b) Evaluate 1 cos 1 7   −    to at least 3 d.p. (67.792°) (FT is on their R) B1 FT Use correct method to find a value of ߠ in the interval M1 Obtain answer, e.g. 5.1° A1 Obtain second answer, e.g.117.3°, only A1 4

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6 y M x O 1 x The diagram shows the curve y , for x and its maximum point M. = 1 3x4 ≥0, + (a) Find the x-coordinate of M, giving your answer correct to 3 decimal places. 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(b) Using the substitution u 3x2, find by integration the exact area of the shaded region bounded = by the curve, the x-axis and the line x 1. [5] = ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 6(a) Use quotient or product rule M1 Obtain correct derivative in any form e.g. ( ) ( ) 4 3 2 4 1 3 12 1 3 x x x x + − × + A1 Equate derivative to zero and solve for x M1 Obtain answer 0.577 A1 4 Question Answer Marks 6(b) State or imply d 2 3 d = u x x, or equivalent B1 Substitute for x and dx M1 Obtain integrand ( ) 2 1 2 3 1+ u , or equivalent A1 State integral of the form ܽtanିଵݑ and use limits u = 0 and u = 3 (or x = 0 and x = 1) correctly M1 Obtain answer 3 π 18 , or exact equivalent A1 5

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9 y 1 x O p 12π k The diagram shows the curves y cos x and y where k is a constant, for 0 The = = 1 x, ≤x ≤12π. + curves touch at the point where x p. = 1 (a) Show that p satisfies the equation tan p [5] = 1 p. + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ @ A 1 (b) Use the iterative formula pn+1 = tan−1 1 pn + to determine the value of p correct to 3 decimal places. Give the result of each iteration to 5 decimal places. 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(c) Hence find the value of k correct to 2 decimal places. 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Mark scheme: 9(a) State cos 1 = + k p p B1 Differentiate both equations and equate derivatives at x = p M1 Obtain a correct equation in any form, e.g. ( ) 2 sin 1 − = − + k p p A1 Eliminate k M1 Obtain the given answer showing sufficient working A1 5 9(b) Use the iterative formula correctly at least once M1 Obtain final answer p = 0.568 A1 Show sufficient iterations to justify 0.568 to 3 d.p., or show there is a sign change in the interval (0.5675, 0.5685) A1 3 9(c) Use a correct method to find k M1 Obtain answer k = 1.32 A1 2

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10 With respect to the origin O, the points A and B have position vectors given by −−¿OA 6i 2j and = + −−¿OB 2i 2j 3k. The midpoint of OA is M. The point N lying on AB, between A and B, is such = + + that AN 2NB. = (a) Find a vector equation for the line through M and N. 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The line through M and N intersects the line through O and B at the point P. (b) Find the position vector of P. 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(c) Calculate angle OPM, giving your answer in degrees. 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Mark scheme: 10(a) State that the position vector of M is 3i + j B1 Use a correct method to find the position vector of N M1 Obtain answer 10 2 2 3 + + i j k A1 Use a correct method to form an equation for MN M1 Obtain correct answer in any form, e.g. 1 3 2 3 λ   = + + + +     r i j i j k A1 5 10(b) State or imply r = µ(2i + 2j + 3k) as equation for OB B1 Equate sufficient components of MN and OB and solve for λ or for µ M1 Obtain λ = 3 or µ = 2 and position vector 4i + 4j + 6k for P A1 3 10(c) Carry out correct process for evaluating the scalar product of direction vectors for OP and MP, or equivalent M1 Using the correct process for the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result M1 Obtain answer 21.6° A1 3

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