Cambridge A Level Mathematics 9709 — 2018 Oct/Nov Paper 3 · Variant 1

1 Find the set of values of x satisfying the inequality 2 2x x 3a , where a is a positive constant. −a < + [4] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 1 EITHER: State or imply non-modular inequality ( ) ( ) 2 2 2 2 2 3 x a x a − < + , or corresponding quadratic equation, or pair of linear equations 2(2x – a) = ± (x + 3a) B1 Make reasonable attempt at solving a 3-term quadratic, or solve two linear equations for x M1 Obtain critical values 5 3 x a = and 1 5 x a = − A1 State final answer 1 5 5 3 a x a − < < A1 OR: Obtain critical value 5 3 x a = from a graphical method, or by inspection, or by solving a linear equation or an inequality B1 Obtain critical value 1 5 x a = − similarly B2 State final answer 1 5 5 3 a x a − < < [Do not condone ⩽ for < in the final answer.] B1 4

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2ex 2 Showing all necessary working, solve the equation + e−x 4, giving your answer correct to ex = −e−x 2 decimal places. [4] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 2 Rearrange the equation in the form 2e x a b = or e e x x a b − = M1 Obtain correct equation in either form with a = 2 and b = 5 A1 Use correct method to solve for x M1 Obtain answer x = 0.46 A1 4

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3 (i) By sketching a suitable pair of graphs, show that the equation x3 3 has exactly one real = −x root. [2] (ii) Show that if a sequence of real values given by the iterative formula 2x3 3 n + xn+1 = 3x2 1 n + converges, then it converges to the root of the equation in part (i). 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(iii) Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places. 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Mark scheme: 3 (i) Sketch a relevant graph, e.g. 3 y x = B1 Sketch a second relevant graph, e.g. y = 3 – x, and justify the given statement B1 Consideration of behaviour for x < 0 is needed for the second B1 2 3(ii) State or imply the equation ( ) ( ) 3 2 2 3 / 3 1 x x x = + + B1 Rearrange this in the form 3 3 x x = − , or commence work vice versa B1 2 Question Answer Marks Guidance 3(iii) Use the iterative formula correctly at least once M1 Obtain final answer 1.213 A1 Show sufficient iterations to 5 d.p. or more to justify 1.213 to 3 d.p., or show there is a sign change in the interval (1.2125, 1.2135) A1 3

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4 The parametric equations of a curve are x 2 sin sin y 2 cos cos = 1 + 21, = 1 + 21, where 0 < 1 < 0. dy (i) Obtain an expression for in terms of [3] dx 1. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Hence find the exact coordinates of the point on the curve at which the tangent is parallel to the y-axis. 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Mark scheme: 4(i) Obtain d d x θ 2cos 2cos2 θ θ = + or d d y θ = 2sin 2sin 2 θ θ − − B1 Use dy/dx = dy/dθ ÷ dx/dθ M1 Obtain correct d d y x in any form, e.g. – 2sin 2sin2 2cos 2cos2 θ θ θ θ + + A1 3 4(ii) Equate denominator to zero and use any correct double angle formula M1* Obtain correct 3-term quadratic in cos θ in any form A1 Solve for θ depM1* Obtain x = 3 3 / 2 and y = 1 2 , or exact equivalents A1 4

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7 y M R x O 120 The diagram shows the curve y 5 sin2x cos3x for 0 and its maximum point M. The shaded = ≤x ≤120, region R is bounded by the curve and the x-axis. (i) Find the x-coordinate of M, giving your answer correct to 3 decimal places. 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(ii) Using the substitution u sin x and showing all necessary working, find the exact area of R. [4] = ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 7(i) Use product rule M1* Obtain correct derivative in any form A1 Equate derivative to zero and obtain an equation in a single trig function depM1* Obtain a correct equation, e.g. 2 3 tan 2 x = A1 Obtain answer x = 0.685 A1 5 Question Answer Marks Guidance 7(ii) Use the given substitution and reach ( ) 2 4 d a u u u ∫ − M1 Obtain correct integral with a = 5 and limits 0 and 1 A1 Use correct limits in an integral of the form 3 5 1 1 3 5 a u u   −     M1 Obtain answer 2 3 A1 4

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2 3i8 (a) Showing all necessary working, express the complex number + in the form rei1, where r 0 1 > −2i and Give the values of r and correct to 3 significant figures. [5] −0 < 1 ≤0. 1 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) On an Argand diagram sketch the locus of points representing complex numbers z satisfying the equation z 2i 1. Find the least value of z for points on this locus, giving your answer −3 + = in an exact form. 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Mark scheme: 8(i) EITHER: Multiply numerator and denominator by 1 + 2i, or equivalent, or equate to x + iy, obtain two equations in x and y and solve for x or for y M1 Obtain quotient 4 7 5 5 − + i, or equivalent A1 Use correct method to find either r or θ M1 Obtain r = 1.61 A1 Obtain θ = 2.09 A1 OR: Find modulus or argument of 2 + 3i or of 1 – 2i B1 Use correct method to find r M1 Obtain r = 1.61 A1 Use correct method to find θ M1 Obtain θ = 2.09 A1 5 8(ii) Show a circle with centre 3 – 2i B1 Show a circle with radius 1 B1ft Centre not at the origin Carry out a correct method for finding the least value of z M1 Obtain answer 13 – 1 A1 4

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10 The planes m and n have equations 3x y 10 and x 2z 5 respectively. The line l has + −2z = −2y + = equation r 4i 2j k i j 2k . = + + + , + + (i) Show that l is parallel to m. 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(ii) Calculate the acute angle between the planes m and n. 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(iii) A point P lies on the line l. The perpendicular distance of P from the plane n is equal to 2. Find the position vectors of the two possible positions of P. 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Mark scheme: 10(i) EITHER: Expand scalar product of a normal to m and a direction vector of l M1 Verify scalar product is zero A1 Verify that one point of l does not lie in the plane A1 OR: Substitute coordinates of a general point of l in the equation of the plane m M1 Obtain correct equation in λ in any form A1 Verify that the equation is not satisfied for any value of λ A1 3 10(ii) Use correct method to evaluate a scalar product of normal vectors to m and n 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 74.5° or 1.30 radians A1 3 10(iii) EITHER: Using the components of a general point P of l form an equation in λ by equating the perpendicular distance from n to 2 M1 OR: Take a point Q on l, e.g. (5, 3, 3) and form an equation in λ by equating the length of the projection of QP onto a normal to plane n to 2 M1 Obtain a correct modular or non-modular equation in any form A1 Solve for λ and obtain a position vector for P, e.g. 7i + 5j + 7j from λ = 3 A1 Obtain position vector of the second point, e.g. 3i + j – k from λ = – 1 A1 4

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