Cambridge A Level Mathematics 9709 — 2020 May/June Paper 3 · Variant 1

2 (a) Expand 2 in ascending powers of x, up to and including the term in x2, simplifying the −3x −2 coefficients. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) State the set of values of x for which the expansion is valid. [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 2(a) State a correct unsimplified version of the x or x2 term of the expansion of (2 – 3x)–2 or 2 3 1 2 −   −     x State correct first term 1 4 B1 Obtain the next two terms 2 3 27 4 16 + x x A1 + A1 4 2(b) State answer 2 3 < x , or equivalent B1 1

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4 The curve with equation y e2x sin x 3 cos x has a stationary point in the interval 0 = + ≤x ≤π. (a) Find the x-coordinate of this point, giving your answer correct to 2 decimal places. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Determine whether the stationary point is a maximum or a minimum. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 4(a) Use product rule M1 Obtain derivative in any correct form e.g. ( ) ( ) 2 2 2e sin 3cos e cos 3sin x x x x x x + + − A1 Equate derivative to zero and obtain an equation in one trigonometric ratio M1 Obtain x = 1.43 only A1 4 4(b) Use a correct method to determine the nature of the stationary point e.g. 2.84 2.88 1.42, 0.06e 0 1.44, 0.07e 0 x y x y ′ = = > ′ = = − < M1 Show that it is a maximum point A1 2

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5 (a) Find the quotient and remainder when 2x3 6x 3 is divided by x2 3. [3] −x2 + + + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ 3 2x3 6x 3 (b) Using your answer to part (a), find the exact value of dx. [5] −x2 + + x2 3 Ô1 + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 5(a) M1 Obtain quotient 2x – 1 A1 Obtain remainder 6 A1 3 5(b) Obtain terms x2 – x (FT on quotient of the form 2x + k) B1FT Obtain term of the form 1 tan 3 −      x a M1 Obtain term 1 6 tan 3 3 −      x (FT on a constant remainder) A1FT Use x = 1 and x = 3 as limits in a solution containing a term of the form ( ) 1 tan− a bx M1 Obtain final answer 1 6 3 π + , or exact equivalent A1 5

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y 8 A certain curve is such that its gradient at a point x, y is proportional to The curve passes x x. through the points with coordinates 1, 1 and 4, e . (a) By setting up and solving a differential equation, find the equation of the curve, expressing y in terms of x. 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(b) Describe what happens to y as x tends to infinity. [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 8(a) State d d = y y k x x x , or equivalent B1 Separate variables correctly and attempt integration of at least one side M1 Obtain term ln y, or equivalent A1 Obtain term 1 2 −k x , or equivalent A1 Use given coordinates to find k or a constant of integration c in a solution containing terms of the form a ln y and b x , where ab ≠ 0 M1 Obtain k = 1 and c = 2 A1 + A1 Obtain final answer y = exp 2 2   − +     x , or equivalent A1 8 Question Answer Marks 8(b) State that y approaches eଶ (FT their c in part (a) of the correct form) B1FT 1 ( )   ( )  

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9 With respect to the origin O, the vertices of a triangle ABC have position vectors OA 2i 5k, OB 3i 2j 3k and OC i j k. −−¿ −−¿ −−¿ = + = + + = + + (a) Using a scalar product, show that angle ABC is a right angle. 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(b) Show that triangle ABC is isosceles. 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(c) Find the exact length of the perpendicular from O to the line through B and C. 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Mark scheme: 9(a) State or AB BA and or BC CB in vector form B1 Calculate their scalar product M1 Show product is zero and confirm angle ABC is a right angle A1 3 9(b) Use correct method to calculate the lengths of AB and BC M1 Show that AB = BC and the triangle is isosceles A1 2 9(c) State a correct equation for the line through B and C, e.g. r = i + j + k + λ (2i + j + 2k) or ( ) 3 2 3 2 2 μ = + + + − −− r i j k i j k B1 Taking a general point of BC to be P, form an equation in λ by either equating the scalar product of  OP and  BC to zero, or applying Pythagoras to triangle OBP (or OCP), or setting the derivative of  OP to zero M1 Solve and obtain λ = – 5 9 A1 Obtain answer 1 2 3 , or equivalent A1

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3i 10 (a) The complex number u is defined by u a 2i, where a is real. = + (i) Express u in the Cartesian form x iy, where x and y are in terms of a. 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(ii) Find the exact value of a for which arg u* 1 [3] = 3π. ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ (b) (i) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities z z and z [4] −2i ≤ −1 −i −2 −i ≤2. (ii) Calculate the least value of arg z for points in this region. 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Mark scheme: 10(a)(i) M1 Use i2 = –1 at least once A1 Obtain answer 2 2 6 3 4 4 + + + ai a a A1 3 10(a)(ii) Either state that arg u = 1 3π − or express u* in terms of a (FT on u) B1 Use correct method to form an equation in a M1 Obtain answer a = 2 3 − A1 3 Question Answer Marks 10(b)(i) Show the perpendicular bisector of points representing 2i and 1 + i B1 Show the point representing 2 + i B1 Show a circle with radius 2 and centre 2 + i (FT on the position of the point for 2 + i) B1FT Shade the correct region B1 4 10(b)(ii) State or imply the critical point 2 + 3i B1 Obtain answer 56.3° or 0.983 radians B1 2

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