Cambridge A Level Mathematics 9709 — 2018 May/June Paper 3 · Variant 2

1 Showing all necessary working, solve the equation 3 2x 2x, giving your answers correct to 3 significant figures. −1 = [4] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 1 EITHER: State or imply non-modular equation ( ) ( ) 2 2 2 3 2 1 2 − = x x , or pair of equations ( ) 3 2 1 2 − =± x x M1 2 8 2 18 2 9 0 − + = x x Obtain 2x = 3 2 and 2x = 3 4 or equivalent A1 OR: Obtain 2x = 3 2 by solving an equation B1 Obtain 2x = 3 4 by solving an equation B1 Use correct method for solving an equation of the form 2 = x a , where a > 0 M1 Obtain final answers x = 0.585 and x = – 0.415 only A1 The question requires 3 s.f. Do not ISW if they go on to reject one value 4

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2 sin x 2x sin x 4 (i) Show that [4] −sin 1 2x 1 cosx.  −cos + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ 1 20 2 sin x 2x (ii) Hence, showing all necessary working, find dx, giving your answer in the −sin Ô 1 1 2x 30 −cos form ln k. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 4(i) Use correct double angle formulae and express LHS in terms of cos x and sin x M1 ( ) 2 2sin 2sin cos 1 2cos 1 − − x x x x Obtain a correct expression A1 Complete method to get correct denominator e.g. by factorising to remove a factor of 1 cos − x M1 Obtain the given RHS correctly OR (working R to L): A1 2 sin 1 cos sin sin cos 1 cos 1 cos 1 cos − − × = + − − x x x x x x x x M1A1 2 2sin 2sin cos 2 2cos − = − x x x x Given answer so check working carefully 2sin sin2 1 cos2 − = − x x x M1A1 4 4(ii) State integral of the form ln(1 cos ) + a x M1* If they use the substitution 1 cos = + u x allow M1A1 for ln − u Obtain integral ln(1 cos ) − + x A1 Substitute correct limits in correct order M1(dep)* Obtain answer ( ) 3 ln 2 , or equivalent A1 4

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5 The equation of a curve is x2 x 3y 3. + −y3 = dy x2 2xy (i) Show that . [4] + dx = y2 −x2 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Hence find the exact coordinates of the two points on the curve at which the gradient of the normal is 1. 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Mark scheme: 5(i) State or imply 3 2 d d y y x as derivative of 3 y B1 State or imply 2 d 6 3 d + y xy x x as derivative of 2 3x y OR State or imply ( ) 2 d 2 3 1 3d   + + +     y x x y x x as derivative of ( ) 2 3 + x x y B1 2 2 2 d d 3 6 3 3 0 d d + + − = y y x xy x y x x Equate derivative of the LHS to zero and solve for d d y x M1 Given answer so check working carefully Obtain the given answer A1 4 5(ii) Equate derivative to – 1 and solve for y M1* Use their y = – 2x or equivalent to obtain an equation in x or y M1(dep*) Obtain answer (1, – 2) A1 Obtain answer ( 3 3 , 0) B1 Must be exact e.g. 1 ln3 3 e but ISW if decimals after exact value seen 4

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6 A a 1 rad a B C The diagram shows a triangle ABC in which AB AC a and angle BAC radians. Semicircles are drawn outside the triangle with AB and AC as= diameters.= A circular arc= 1with centre A joins B and C. The area of the shaded segment is equal to the sum of the areas of the semicircles. (i) Show that 1 sin [3] 1 = 20 + 1. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Verify by calculation that lies between 2.2 and 2.4. [2] 1 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Use an iterative formula based on the equation in part (i) to determine correct to 2 decimal places. Give the result of each iteration to 4 decimal places. 1 [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 6(i) Use correct method for finding the area of a segment and area of semicircle and form an equation in θ M1 e.g. 2 2 2 1 1 sin 4 2 2 π θ θ = − a a a State a correct equation in any form A1 Given answer so check working carefully Obtain the given answer correctly A1 3 6(ii) Calculate values of a relevant expression or pair of expressions at 2.2 θ = and θ = 2.4 M1 e.g. ( ) ( ) ( ) f 2.2 2.37... 2.2 f sin f 2.4 2.24... 2.4 2 π θ θ  = >  = +  = <  or ( ) ( ) ( ) f 2.2 0.17... 0 f sin f 2.4 0.15... 0 2 π θ θ θ  = − <  = − −  = + >  Complete the argument correctly with correct calculated values A1 2 Question Answer Marks Guidance 6(iii) Use 1 1 sin 2 θ π θ + = + n n correctly at least once M1 e.g. 2.2 2.3 2.4 2.3793 2.3165 2.2463 2.2614 2.3054 2.3512 2.3417 2.3129 2.2814 2.2881 2.3079 2.3288 2.3244 2.2970 2.3000 2.3185 2.3165 2.3041 2.3054 2.3138 2.3129 2.3072 Obtain final answer 2.31 A1 Show sufficient iterations to 4 d.p. to justify 2.31 to 2 d.p. or show there is a sign change in the interval (2.305, 2.315) A1 3

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7 Throughout this question the use of a calculator is not permitted. The complex numbers i and 2i are denoted by u and v respectively. −3ï3 + ï3 + u (i) Find, in the form x iy, where x and y are real and exact, the complex numbers uv and . [5] v + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) On a sketch of an Argand diagram with origin O, show the points A and B representing the complex numbers u and v respectively. Prove that angle AOB 2 [3] = 30. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 7(i) Substitute in uv, expand the product and use 2i 1 =− M1 Obtain answer uv = 11 5 3i − − A1 EITHER: Substitute in u/v and multiply numerator and denominator by the conjugate of v, or equivalent M1 Obtain numerator 7 7 3i −+ or denominator 7 A1 Obtain final answer 1 3i −+ A1 OR: Substitute in u/v , equate to x + iy and solve for x or for y M1 3 3 3 2 1 2 3 − = −  = +  x y x y Obtain x = – 1 or y = 3 A1 Obtain final answer 1 3 −+ i A1 5 Question Answer Marks Guidance 7(ii) Show the points A and B representing u and v in relatively correct positions B1 Carry out a complete method for finding angle AOB, e.g. calculate arg(u/v) If using ( ) 1 tan 3 θ − = − must refer to ( ) arg u v M1 OR: ( ) 1 2 3 3 3 1 2 tan , tan tan 2 3 3 3 1 9 3 − − − = = ⇒ − = − = − a b a b 2 3 π θ ⇒ = OR: 3 3 3 1 2 9 2 1 cos 14 2 7 28 θ    −       −+ −    = = = 2 3 π θ ⇒ = OR: 28 7 49 1 2 cos 3 2 2 28 7 π θ θ + − = = − ⇒ = Prove the given statement A1 Given answer so check working carefully 3

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8 y M x O 3x The diagram shows the curve y x 1 e−1 and its maximum point M. = + (i) Find the x-coordinate of M. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find the area of the shaded region enclosed by the curve and the axes, giving your answer in terms of e. 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Mark scheme: 8(i) Use correct product or quotient rule M1 ( ) 1 1 1 3 3 3 d 1 e e d − − − = + + x x y x x or ( ) 1 1 3 3 2 3 1 e 1 e d 3 d e − + = x x x x y x Obtain complete correct derivative in any form A1 Equate derivative to zero and solve for x M1 Obtain answer x = 2 with no errors seen A1 4 8(ii) Integrate by parts and reach ( ) 1 1 3 3 1 e e d − − + + ∫ x x a x b x M1* Obtain ( ) 1 1 3 3 3 1 e 3 e d − − − + + ∫ x x x x , or equivalent A1 1 1 1 3 3 3 3 3e d 3e − − − − + − ∫ x x x xe x Complete integration and obtain ( ) 1 1 3 3 3 1 e 9e − − − + − x x x , or equivalent A1 Use correct limits x = – 1 and x = 0 in the correct order, having integrated twice M1(dep*) Obtain answer 1 3 9e 12 − , or equivalent A1 5

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10 Two lines l and m have equations r 2i k s 2i 3j and r i 3j 4k t i 2j k respectively. = −j + + + −k = + + + + + (i) Show that the lines are skew. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ A plane p is parallel to the lines l and m. (ii) Find a vector that is normal to p. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Given that p is equidistant from the lines l and m, find the equation of p. Give your answer in the form ax by cz d. [3] + + = ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 10(i) Equate at least two pairs of components and solve for s or for t M1 4 2 3 5 6 5 13 or 11 or 3 5 7 7 1 6 8 5 3 5 5 − −   = =   = −     − − = = − =       ≠−  − −   −≠ ≠   s s s t t t Obtain correct answer for s or t, e.g. s = – 6, t = – 11 A1 Verify that all three equations are not satisfied and the lines fail to intersect A1 State that the lines are not parallel B1 4 10(ii) EITHER: Use scalar product to obtain a relevant equation in a, b and c, e.g. 2a + 3b – c = 0 B1 Obtain a second equation, e.g. a + 2b +c = 0, and solve for one ratio, e.g. a : b M1 Obtain a : b : c and state correct answer, e.g. 5i – 3j + k, or equivalent A1 OR: Attempt to calculate vector product of relevant vectors, e.g. (2i + 3j – k)×(i + 2j + k) M1 Obtain two correct components A1 Obtain correct answer, e.g. 5i – 3j + k A1 3 Question Answer Marks Guidance 10(iii) EITHER: State position vector or coordinates of the mid-point of a line segment joining points on l and m, e.g. 3 5 2 2 + + i j k B1 OR: Use the result of (ii) to form equations of planes containing l and m B1 Use the result of (ii) and the mid-point to find d M1 Use average of distances to find equation of p. M1 Obtain answer 5x – 3y + z = 7, or equivalent A1 Obtain answer 5x – 3y + z = 7, or equivalent A1 OR: Using the result of part (ii), form an equation in d by equating perpendicular distances to the plane of a point on l and a point on m M1 State a correct equation, e.g. 14 35 −d = 35 −d A1 Solve for d and obtain answer 5x – 3y + z = 7, or equivalent A1 3

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