Cambridge A Level Mathematics 9709 — 2018 May/June Paper 2 · Variant 1

1 Solve the equation 3e2x 27 0, giving your answers in the form k ln 3. [5] −82ex + = ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 1 M1 Either directly or using substitution e = x u Obtain 1 3 e , e 27 = = x x A1 1 3 e , e 27 = = x x may be implied if e = x u is stated Use correct process at least once for solving e = x c where 0 > c M1 Obtain ln3 − from a correct solution A1 Condone use of e = x x Obtain 3ln3 from a correct solution A1 5 Question Answer Marks Guidance 2 Either State or imply equation ln ln ln ln = + y A B x B1 Equate gradient of line to ln B M1 Obtain ln 1.6486... = B and hence 5.2 = B A1 Substitute appropriate values to find ln A M1 Obtain ln 1.2809... = A and hence 3.6 = A A1 Or State or imply equation ln ln ln ln = + y A B x B1 Use given coordinates to obtain a correct equation B1 Equations are 4.908 ln 2.2ln = + A B and 11.008 ln 5.9ln A B = + Use given coordinates to obtain a second correct equation and attempt to solve both equations simultaneously to obtain at least one of the unknowns ln A or ln B M1 Obtain ln 1.6486... = B and hence 5.2 = B A1 Obtain ln 1.2809... = A and hence 3.6 = A A1

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4 y M x O P 5 ln x The diagram shows the curve with equation y 2x 1. The curve crosses the x-axis at the point P = and has a maximum point M. + (i) Find the gradient of the curve at the point P. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ x 0.5 (ii) Show that the x-coordinate of the point M satisfies the equation x . [2] + ln x = ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Use an iterative formula based on the equation in part (ii) to find the x-coordinate of M correct to 4 significant figures. Show the result of each iteration to 6 significant figures. 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Mark scheme: 4(i) Use quotient rule or equivalent M1 Obtaining two terms in numerator and 2 (2 1) + x in denominator for a quotient Obtain correct 5 2 (2 1) 10ln (2 1) + − + x x x x or equivalent, or ( ) ( ) 1 2 5 2 1 10ln 2 1 − − + − + x x x x or equivalent A1 Obtaining one term with ( ) 1 2 1 − + x oe and a second term with ( ) 2 2 1 − + x oe for a product Condone poor use of brackets if recovered later Substitute 1 = x to obtain 15 9 or 5 3 or equivalent, www A1 3 4(ii) Equate numerator to zero and attempt relevant arrangement M1 For M1, need to see at least one line of working after either 5 10 10ln 0 + − = x x or their numerator (which must have at least 2 terms, one involving ln x ) = 0 Confirm 0.5 ln + = x x x A1 AG; necessary detail needed 2 4(iii) Use iteration process correctly at least once M1 Obtain final answer 3.181 A1 Show sufficient iterations to 6 sf to justify answer or show sign change in interval (3.1805, 3.1815) A1 3

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5 The parametric equations of a curve are x 2 cos 3 sin y 3 cos = 21 + 1, = 1 for 0 1 < 1 < 20. (i) Find the gradient of the curve at the point for which 1 radian. [4] 1 = ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find the value of sin at the point on the curve where the tangent is parallel to the y-axis. [3] 1 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 5(i) Obtain d d 4sin2 3cos θ θ θ = − + x B1 Use d d d d d d / θ θ = y y x x in terms of θ or with 1 already substituted M1 Obtain or imply d 3sin d 4sin 2 3cos θ θ θ − = − + y x A1 Substitute 1 to obtain 1.25 A1 Or greater accuracy 1.252013… 4 5(ii) Equate denominator of first derivative to zero M1 Use sin2 2sin cos θ θ θ = A1 Obtain 3 8 sinθ = A1 3

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