Cambridge A Level Mathematics 9709 — 2021 Oct/Nov Paper 2 · Variant 3

1 Find the exact value of 4e2x 2e x dx. [4] −1 ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 1 Integrate to obtain 2 B1 Integrate to obtain 2e−x B1 Apply limits correctly to integral of the form 2 1 2 e e− + x x k k M1 1 4 ≠ k . Condone one error. Obtain 4 2e 2e − A1 or exact equivalent. 4

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2 (a) Sketch, on the same diagram, the graphs of y 3x and y x 3 . [2] (b) Find the coordinates of the point where the two graphs intersect. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (c) Deduce the solution of the inequality 3x x . [1] < −3 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 2(a) Draw V-shaped graph with vertex on positive x-axis B1 Must be straight lines. Draw straight line through origin with positive gradient greater than gradient of first graph, together with a V shaped graph B1 Must have the first B1. 2 2(b) Solve linear equation with signs of 3x and x different or solve non-modulus equation 2 2 (3 ) ( 3) = − x x M1 Obtain 3 4 = x A1 Obtain 9 4 = y A1 And no other point. 3 2(c) State 3 4 < x B1 FT Following their (single) x-coordinate from part (b). 1

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3 y 2.58, 9.00 1.03, 6.36 ln x O The variables x and y satisfy the equation ay kx, where a and k are constants. The graph of y against = ln x is a straight line passing through the points 1.03, 6.36 and 2.58, 9.00 , as shown in the diagram. Find the values of a and k, giving each value correct to 2 significant figures. [5] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 3 State or imply equation is ln ln ln = + y a k x Equate gradient of line to 1 ln a M1 Or eliminate lnk from simultaneous equations. Obtain 1 2.64 ln 1.55 = a or equivalent and hence 1.8 = a A1 AWRT Substitute appropriate values to find lnk M1 Obtain ln 2.7... = k and hence 15 = k A1 AWRT Alternative method for question 3 6.36 1.03 e = a k and 9 2.58 e = a k B1 For both. Elimination of k to obtain an equation in a only ( ) 2.64 1.55 e = a M1 Must have previous B1. Use of a correct method to obtain a M1 Allow for 0.59 e = a . 1.8 = a A1 15 = k A1 5

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4 The curve with equation y xe2x 5e x has a minimum point M. (a) Show that the x-coordinate of M satisfies the equation x 1 ln 5 ln 1 2x . [5] = 3 −13 + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Use an iterative formula, based on the equation in part (a), to find the x-coordinate of M correct to 3 significant figures. Use an initial value of 0.35 and give the result of each iteration to 5 significant figures. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 4(a) Attempt use of product rule to differentiate 2 e x x M1 Obtain 2 2 e 2 e 5e− + − x x x x A1 Equate first derivative to zero and multiply by e x to obtain an equation involving 3e x M1 Obtain 3e (1 2 ) 5 + = x x or equivalent A1 Confirm given result 1 1 3 3 ln5 ln(1 2 ) = − + x x with sufficient detail A1 AG 5 4(b) Use iteration process correctly at least once M1 Need 0.35 and 2 correct values. Obtain final answer 0.357 A1 Answer required to exactly 3sf. Allow recovery. Show sufficient iterations to 5sf to justify answer or show sign change in interval [0.3565, 0.3575] A1 3

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5 y x O B A The diagram shows the curve with parametric equations 2t x ln 2t 3 , y −3 = + = 2t 3. + The curve crosses the y-axis at the point A and the x-axis at the point B. dy 6 (a) Show that [4] dx = 2t 3. + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the gradient of the curve at A. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (c) Find the gradient of the curve at B. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 5(a) Obtain d 2 d 2 3 = + x t t Use quotient rule, or equivalent, to find d d y t M1 Obtain 2 d 2(2 3) 2(2 3) d (2 3) + − − = + y t t t t A1 OE Divide to confirm d 6 d 2 3 = + y x t A1 AG 4 5(b) Attempt to find value of t corresponding to 0 = x M1 Obtain 1 = − t and hence gradient is 6 A1 2 5(c) Attempt to find value of t corresponding to 0 = y M1 Obtain 3 2 = t and hence gradient is 1 A1 2

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6 The polynomials f x and g x are defined by f x 4x3 ax2 8x 15 and g x x2 bx 18, = + + + = + + where a and b are constants. (a) Given that x 3 is a factor of f x , find the value of a. [2] + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Given that the remainder is 40 when g x is divided by x , find the value of b. [2] −2 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (c) When a and b have these values, factorise f x x completely. [3] −g ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (d) Hence solve the equation f cosec cosec 0 for 0 [3] −g = < < 2π. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 6(a) Substitute = − , equate to zero and attempt solution for a M1 Allow attempt at synthetic division, must be a complete method. Allow one sign error carried through. Allow attempt at algebraic long division, must be complete with the remainder equated to zero. Obtain 13 = a A1 2 6(b) Substitute 2 = x , equate to 40 and attempt solution for b M1 Obtain 9 = b A1 2 6(c) Identify 3 + x as factor of f ( ) g( ) − x x B1 May be implied by synthetic division. If working backwards from solutions from a calculator then B0 M0. Attempt, by division or equivalent, to find quadratic factor M1 ( )( )( ) 3 2 1 2 1 + + − k x x x where 1 ≠ k gets B1 M1. Obtain ( 3)(2 1)(2 1) + − + x x x A1 3 6(d) Attempt correct process to find at least 1 value from cosecθ = k where 1 < − k M1 Allow for o 199.5 or o 19.5 − . Obtain 3.48or 5.94 A1 Obtain a second correct solution A1 And no others within the range. 3

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7 (a) By first expanding cos 2 , show that cos 3 4 cos3 3 cos . [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the exact value of 2 cos3 5 cos 5 . [2] 18π −32 18π ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (c) Find 12 cos3x cos3 3x dx. [4] −4 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ Additional Page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown. ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................ ........................................................................................................................................................................

Mark scheme: 7(a) θ θ θ θ − B1 Attempt correct relevant identities to express in terms of cosθ only M1 M0 if moving terms from side to side. Confirm 3 4cos 3cos θ θ − with sufficient detail A1 AG 3 7(b) Use identity with 5 18 π θ = M1 Obtain 5 1 2 6 cos π and hence 1 4 3 − A1 2 7(c) Express integrand in form 1 2 (cos3 3cos ) (cos9 3cos3 ) + + + k x x k x x M1 Obtain correct integrand 9cos cos9 − x x A1 OE (allow unsimplified). Integrate to obtain form 3 4 sin sin9 + k x k x M1 Obtain correct 1 9 9sin sin9 − x x A1 Now simplified; condone missing ... + c . 4

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