Cambridge A Level Mathematics 9709 — 2021 Oct/Nov Paper 2 · Variant 1

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

Mark scheme: 1 Integrate to obtain 2 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 . [2] = = −3 (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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4 The curve with equation y xe2x has a minimum point M. = + 5e−x (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 Obtain 2 2 e 2 e 5e− + − x x x x A1 Equate first derivative to zero and multiply by ex 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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