Cambridge A Level Mathematics 9709 — 2020 Oct/Nov Paper 3 · Variant 2

3 2 (a) Expand 1 6x in ascending powers of x, up to and including the term in x3, simplifying the 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 2 x or 3 x term M1 For the given expression State correct first two terms 1 + 2x A1 Obtain the next two terms 2 3 40 4 3 x x − + A1 + A1 One mark for each correct term. ISW Accept 1 3 13 The question asks for simplified coefficients, so candidates should cancel fractions. 4 2(b) State answer 1 6 x < B1 OE. Strict inequality 1

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3 The variables x and y satisfy the relation 2y = 31−2x. (a) By taking logarithms, show that the graph of y against x is a straight line. State the exact value of the gradient of this line. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the exact x-coordinate of the point of intersection of this line with the line y 3x. Give your = ln a answer in the form where a and b are integers. [2] ln b, ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 3(a) State or imply log2 log3 2 log3 y x = − B1 Accept ( ) ln 2 1 2 ln3 y x = − State that the graph of y against x has an equation which is linear in x and y, or is of the form ay = bx + c B1 Correct equation. Need a clear statement/comparison with matching linear form. Clear indication that the gradient is – 2ln3 ln2 B1 Must be exact. Any equivalent e.g. 3 2 2log log k k − , 2 1 log 9 3 3(b) Substitute y = 3x in an equation involving logarithms and solve for x M1 Obtain answer x = ln3 ln72 A1 Allow M1A1 for the correct answer following decimals 2

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5 y P x O The diagram shows the curve with parametric equations x tan y = 1, = cos21, for 1 −12π < 1 < 2π. (a) Show that the gradient of the curve at the point with parameter is sin [3] 1 −2 1 cos31. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ The gradient of the curve has its maximum value at the point P. (b) Find the exact value of the x-coordinate of P. 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Mark scheme: 5(a) State d d x θ = 2 sec θ or d d y θ = 2sin cos θ θ − B1 CWO, AEF. Use d d y x = d d y θ ÷ d d x θ M1 Obtain 3 d 2sin cos d y x θ θ = − from correct working A1 AG Alternative method for question 5(a) Convert to Cartesian form and differentiate M1 1 2 1 y x = + ( ) 2 2 d 2 d 1 y x x x − = + A1 OE Obtain 3 d 2sin cos d y x θ θ = − from correct working A1 AG 3 Question Answer Marks Guidance 5(b) Use correct product rule to obtain ( ) 3 d 2cos sin d θ θ θ ± M1 Condone incorrect naming of the derivative For work done in correct context Obtain correct derivative in any form A1 e.g. ( ) 4 2 2 2cos 6sin cos θ θ θ ± − + Equate derivative to zero and obtain an equation in one trig ratio A1 e.g. 2 3 tan 1 θ = , or 2 4 sin 1 θ = or 2 4 cos 3 θ = Obtain answer x = – 1 3 A1 Or 3 3 − Alternative method for question 5(b) Use correct quotient rule to obtain 2 2 d d y x M1 Obtain correct derivative in any form A1 ( ) ( ) ( ) 2 2 2 4 2 2 1 2 2 2 1 1 x x x x x − + + × × + + Equate derivative to zero and obtain an equation in x2 A1 e.g. 6x2 = 2 Obtain answer x = – 1 3 A1 4

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6 The complex number u is defined by 7 i u + = 1 −i. (a) Express u in the form x iy, where x and y are real. 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(b) Show on a sketch of an Argand diagram the points A, B and C representing u, 7 i and 1 + −i respectively. [2] (c) By considering the arguments of 7 i and 1 show that + −i, 4 1 1 tan−1 tan−1 [3] 3 = 7 + 4π. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 6(a) Multiply numerator and denominator by 1 + i, or equivalent M1 Must multiply out Obtain numerator 6 + 8i or denominator 2 A1 Obtain final answer u = 3 + 4i A1 Alternative method for question 6(a) Multiply out ( )( ) 1 7 i x iy i − + = + and compare real and imaginary parts M1 Obtain 7 or 1 x y y x + = − = A1 Obtain final answer u = 3 + 4i A1 3 6(b) Show the point A representing u in a relatively correct position B1 FT The FT is on xy ≠ 0. Show the other two points B and C in relatively correct positions: approximately equal distance above / below real axis B1 Take the position of A as a guide to ‘scale’ if axes not marked 2 4 2 2 5 C B A Question Answer Marks Guidance 6(c) State or imply arg(1 – i) = 1 4π − B1 ArgC Substitute exact arguments in arg(7 + i) – arg(1 – i) = arg u M1 Must see a statement about the relationship between the Args e.g. Arg Arg Arg A B C = − or equivalent exact method Obtain 1 1 4 1 1 tan tan π 3 7 4 − −     = +         correctly A1 Obtain given answer correctly from their ( ) 3 4 u k i = + 3

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7 The variables x and t satisfy the differential equation dx e3t cos22x, dt = for t It is given that x 0 when t 0. ≥0. = = (a) Solve the differential equation and obtain an expression for x in terms of t. 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(b) State what happens to the value of x when t tends to infinity. [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 7(a) Correct separation of variables B1 2 3 sec 2 d e d t x x t − =   Needs correct structure Obtain term 3 1e 3 t − − B1 Obtain term of the form tan 2 k x M1 From correct working Obtain term 1 tan2 2 x A1 Use x = 0, t = 0 to evaluate a constant, or as limits in a solution containing terms of the form tan 2 a x and 3 e t b −, where ab ≠ 0 M1 Obtain correct solution in any form A1 e.g. 3 1 1 1 tan2 e 2 3 3 t x − = − + Obtain final answer ( ) 1 3 1 2 tan 1 e 2 3 t x − −   = −     A1 7 7(b) State that x approaches 1 1 2 tan 2 3 −      B1 FT Correct value. Accept 0.294 x → The FT is dependent on letting 3 e 0 t − → in a solution containing 3 e t −. 1

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8 With respect to the origin O, the position vectors of the points A, B, C and D are given by ` a ` a ` a ` a −−¿OA 21 , −−¿OB 4 , −−¿OC 11 and −−¿OD 32 . = = −1 = = 5 1 2 3 (a) Show that AB 2CD. 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(b) Find the angle between the directions of −−¿AB and −−¿CD. 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(c) Show that the line through A and B does not intersect the line through C and D. 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Mark scheme: 8(a) Obtain 2 2 2 and 1 4 1 AB CD       = − =       −      B1 Or equivalent seen or implied Use the correct process for calculating the modulus of both vectors to obtain AB and CD M1 24, 6 AB CD = = Using exact values, verify that 2 AB CD = A1 Obtain given statement from correct work Allow from BA = 2DC, OE 3 8(b) Use the correct process to calculate the scalar product of the relevant vectors (their and AB CD   ) M1 2 2 2 4 2 and 1 or 2 and 2 4 1 4 2             − −             − −       Divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result M1 Obtain answer 99.6° (or 1.74 radians) or better A1 Do not ISW if go on to subtract from 180° (99.594…, 1.738...) Accept 260.4° 3 Question Answer Marks Guidance 8(c) State correct vector equations for AB and CD in any form, e.g. ( ) ( ) 2 2 1 2 1 2 and 1 1 5 4 2 1 λ μ           = + − = +           −      r r B1ft Follow their and AB CD   Alternative: ( ) ( ) 4 2 3 2 1 2 and 2 1 1 4 3 1 λ μ             = − + − = +             −       r r Equate at least two pairs of components of their lines and solve for λ or for µ M1 Obtain correct pair of values from correct equations A1 Alternatives when taking A or B as point on line A λ µ B λ µ ij 1 6 − 1 3 17 7 3 3 ≠ ij 7 6 − 2 3 − 17 7 3 3 ≠ ik 1 2 1 0 2 ≠ ik 1 2 − 0 0 2 ≠ jk 3 2 -3 5 5 ≠− jk 1 2 -4 5 5 ≠− Verify that all three equations are not satisfied and that the lines do not intersect A1 CWO with conclusion e.g. 17 7 3 3 ≠ or 17 7 3 3 = is inconsistent or equivalent 4

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10 y R 3 x O a 2π M The diagram shows the curve y x cos x, for 0 and its minimum point M, where x a. = ≤x ≤32π, = The shaded region between the curve and the x-axis is denoted by R. 1 (a) Show that a satisfies the equation tan a [3] = 2a. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ @ A 1 , with initial value (b) The sequence of values given by the iterative formula an+1 = π + tan−1 2an x1 3, converges to a. = Use this formula to determine a correct to 2 decimal places. Give the result of each iteration to 4 decimal places. 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(c) Find the volume of the solid obtained when the region R is rotated completely about the x-axis. Give your answer in terms of [6] π. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 10(a) Use correct product rule M1 Obtain correct derivative in any form A1 e.g. d 1 cos sin d 2 y x x x x x = − . Accept in a or in x Equate derivative to zero and obtain 1 tan 2 a a = A1 Obtain given answer from correct working. The question says ‘show that ..’ so there should be an intermediate step e.g. cos 2 sin x x x = . Allow 1 tan 2 x x = 3 10(b) Use the iterative process correctly at least once (get one value and go on to use it in a second use of the formula) M1 Must be working in radians Degrees gives 1, 12.6039, 5.4133, ... M0 Obtain final answer 3.29 A1 Clear conclusion Show sufficient iterations to at least 4 d.p.to justify 3.29, or show there is a sign change in the interval (3.285, 3.295) A1 3, 3.3067, 3.2917, 3.2923 Allow more than 4d.p. Condone truncation. 3 Question Answer Marks Guidance 10(c) State or imply the indefinite integral for the volume is ( ) 2 π cos d x x x  B1 [If π omitted, or 2π or 1 π 2 used, give B0 and follow through. 4/6 available] Use correct cos 2A formula, commence integration by parts and reach ( sin2 ) sin2 d x ax b x ax b x x + ±  + *M1 Alternative: 2 1 sin 2 sin 2 d 4 4 4 x x x x x + − Obtain 1 1 1 1 ( sin2 ) sin2 d 2 4 2 4 x x x x x x + − + , or equivalent A1 Complete integration and obtain 2 1 1 1 sin2 cos2 4 4 8 x x x x + + A1 OE Substitute limits x = 0 and x = 1 π 2 , having integrated twice DM1 2 π π 1 1 0 0 0 2 8 4 4   + − − − −     Obtain answer ( ) 2 1 π π 4 16 − , or exact equivalent A1 CAO 6

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