Cambridge A Level Mathematics 9709 — 2018 May/June Paper 3 · Variant 1

e3x 3 A curve has equation y Find the x-coordinates of the stationary points of the curve in the 1 = tan 2x. interval 0 x Give your answers correct to 3 decimal places. [6] < < 0. ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 3 Use quotient or product rule M1 Obtain correct derivative in any form A1 1 M1* Equate derivative to zero and obtain a quadratic in tan x or an equation of the 2 form a sin x = b Solve for x M1(dep*) Obtain answer 0.340 A1 Obtain second answer 2.802 and no other in the given interval A1 6

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4 The polynomial x4 2x3 ax b, where a and b are constants, is divisible by x2 1. Find the values of a and b. + + + −x + [5] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 4 EITHER: Commence division by x 2 − x + 1 and reach a partial quotient M1 of the form x 2 + kx Obtain quotient x 2 + 3 x + 2 A1 Either Set remainder identically equal to zero and solve for a or for b, or M1 multiply given divisor and found quotient and obtain a or b Obtain a = 1 A1 Obtain b = 2 A1 OR: Assume an unknown factor x 2 + Bx + C and obtain an equation M1 in B and/or C Obtain B = 3 and A = 2 A1 Either Use equations to obtain a or b or multiply given divisor and M1 found factor to obtain a or b Obtain a = 1 A1 Obtain b = 2 A1 5

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4 x 5 Let I dx. 1 = Ô 1 4 −x 1 30 (i) Using the substitution x show that I 2 [4] 1 = cos21, = Ó cos21 d1. 60 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Hence find the exact value of I. 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Mark scheme: 5(i) State or imply dx = −2cosθsinθdθ, or equivalent B1 Substitute for x and dx, and use Pythagoras M1 Obtain integrand ± 2cos 2θ A1 Justify change of limits and obtain given answer correctly A1 4 5(ii) Obtain indefinite integral of the form aθ+ b sin 2θ M1* 1 A1 Obtain θ+ sin2θ 2 Use correct limits correctly M1(dep*) 1 A1 Obtain answer 6π with no errors seen 4

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6 In a certain chemical reaction the amount, x grams, of a substance is decreasing. The differential equation relating x and t, the time in seconds since the reaction started, is dx t, dt = −kx where k is a positive constant. It is given that x 100 at the start of the reaction. = (i) Solve the differential equation, obtaining a relation between x, t and k. 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(ii) Given that t 25 when x 80, find the value of t when x 40. [3] = = = ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 6(i) Separate variables correctly and integrate at least one side B1 Obtain term ln x B1 2 B1 Obtain term − kt t , or equivalent 3 Evaluate a constant, or use limits x = 100 and t = 0, in a solution containing M1 terms a ln x and bt t 2 A1 Obtain correct solution in any form, e.g. 1n x = − kt t + 1n 100 3 5 6(ii) Substitute x = 80 and t = 25 to form equation in k M1 Substitute x = 40 and eliminate k M1 Obtain answer t = 64.1 A1 3

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10 The point P has position vector 3i k. The line l has equation r 4i 2j 5k i 2j 3k . −2j + = + + + - + + (i) Find the length of the perpendicular from P to l, giving your answer correct to 3 significant figures. 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(ii) Find the equation of the plane containing l and P, giving your answer in the form ax by cz d. + + =[5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: JJJG JJJG 10(a) EITHER: Find PQ (or QP ) for a general point Q on l, e.g. B1 (1 + µ ) i + ( 4 + 2µ) j + ( 4 + 3µ) k JJJG Calculate the scalar product of PQ and a direction vector for l M1 and equate to zero 3 A1 Solve and obtain correct solution e.g. µ=− 2 Carry out method to calculate PQ M1 Obtain answer 1.22 A1 JJJG JJJG OR1: Find PQ (or QP ) for a general point Q on l B1 Use a correct method to express PQ 2 (or PQ) in terms of µ M1 Obtain a correct expression in any form A1 Carry out a complete method for finding its minimum M1 Obtain answer 1.22 A1 JJJG JJJG OR2: Calling (4, 2, 5) A, state PA (or AP ) in component form, e.g. i + 4j + 4k B1 JJJG JJJG Use a scalar product to find the projection of PA (or AP ) on l M1 Obtain correct answer 21/ 14 , or equivalent A1 Use Pythagoras to find the perpendicular M1 Obtain answer 1.22 A1 JJJG JJJG OR3: State PA (or AP ) in component form B1 JJJG Calculate vector product of PA and a direction vector for l M1 Obtain correct answer, e.g. 4i + j – 2k A1 Divide modulus of the product by that of the direction vector M1 Obtain answer 1.22 A1 5 10(ii) EITHER: Use scalar product to obtain a relevant equation in a, b and c, B1 e.g. a + 2b + 3c = 0 JJJG Obtain a second relevant equation, e.g. using PA a + 4b + 4c = 0, M1 and solve for one ratio Obtain a : b : c = 4 : 1 : – 2, or equivalent A1 Substitute a relevant point and values of a, b, c in general equation M1 and find d Obtain correct answer, 4x + y – 2z = 8, or equivalent A1 OR1: Attempt to calculate vector product of relevant vectors, e.g. M1 (i + 4j + 4k) × (i + 2j + 3k) Obtain two correct components A1 Obtain correct answer, e.g. 4i + j – 2k A1 Substitute a relevant point and find d M1 Obtain correct answer, 4x + y – 2z = 8, or equivalent A1 OR2: Using a relevant point and relevant vectors form a 2-parameter M1 equation for the plane State a correct equation, e.g. A1 r = 4i + 2j + 5k + λ(i + 4j + 4k) + µ(i + 2j + 3k) State three correct equations in x, y, z, λand µ A1 Eliminate λand µ M1 Obtain correct answer 4x + y – 2z = 8, or equivalent A1 5

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