Cambridge A Level Mathematics 9709 — 2015 May/June Paper 3 · Variant 1

1 Use logarithms to solve the equation 25x giving the answer correct to 3 significant figures. = 32x+1, [4]

Mark scheme: 1 Use law for the logarithm of a power at least once *M1 Obtain correct linear equation, e.g. 5 xIn 2 = ( 2 x + )1 In3 A1 Solve a linear equation for x M1 dep *M Obtain x = 0.866 A1 [4]

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3 Show that, for small values of x2, 2 1 1 6x2 3 −2x2 −2 − + ≈kx4, where the value of the constant k is to be determined. [6]

Mark scheme: 3 Either Obtain correct (unsimplified) version of x 2 or x 4 term in 1( −x2 2) −2 M1 Obtain 1 + 4 x 2 A1 Obtain … + 12x 4 A1 2 Obtain correct (unsimplified) version of x 2 or x 4 term in 1( + 6 x 2) 3 M1 Obtain 1 + 4 x 2 − 4 x 4 A1 Combine expansions to obtain k = 16 with no error seen A1 2 Or Obtain correct (unsimplified) version of x 2 or x 4 term in 1( + 6 x 2) 3 M1 Obtain 1 + 4 x 2 A1 Obtain … − 4x 4 A1 Obtain correct (unsimplified) version of x 2 or x 4 term in 1( −x2 2) −2 M1 Obtain 1 + 4 x 2 + 12 x 4 A1 Combine expansions to obtain k = 16 with no error seen A1 [6]

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4 The equation of a curve is y 3 cos 2x 7 sin x 2. = + + Find the x-coordinates of the stationary points in the interval 0 Give each answer correct to 3 significant figures. ≤x ≤0. [7]

Mark scheme: 4 Differentiate to obtain form a sin 2 x + b cos x M1 Obtain correct − 6 sin 2 x + 7 cos x A1 Use identity sin 2 x = 2 sin x cos x B1 Solve equation of form c sin x cos x + d cos x = 0 to find at least one value of x M1 Obtain 0.623 A1 Obtain 2.52 A1 1 Obtain 1.57 or π from equation of form c sin x cos x + d cos x = 0 A1 2 Treat answers in degrees as MR – 1 situation [7] 2 2

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5 (a) Find 4 tan22x dx. [3] Ó + 1 1 20 sin x (b) Find the exact value of + 60 dx. [5] Ô 1 sin x 40

Mark scheme: 5 (a) Use identity tan 2 2 x = sec 2 2 x − 1 B1 Obtain integral of form ax + b tan 2 x M1 1 Obtain correct 3x + tan 2x , condoning absence of + c A1 [3] 2 1 1 (b) State sin x cos π + cos x sin π B1 2 6 1 cos x sin 16 π Simplify integrand to cos π + or equivalent B1 6 sin x Integrate to obtain at least term of form a In(sin x ) *M1 Apply limits and simplify to obtain two terms M1 dep *M 1 1 Obtain π 3 −1 ln( ) or equivalent A1 [5] 8 2 2 2 

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6 The straight line l1 passes through the points 0, 1, 5 and 2, 1 . The straight line l2 has equation −2, r 7i j k i 2j 5k . = + + + - + + (i) Show that the lines l1 and l2 are skew. [6] (ii) Find the acute angle between the direction of the line l2 and the direction of the x-axis. [3]

Mark scheme:   6 (i) Obtain ±  − 3  as direction vector of 1l B1   − 4  State that two direction vectors are not parallel B1 Express general point of 1l or 2l in component form, e.g. ( 2 λ , 1 − 3λ , 5 − 4 λ ) or ( 7 + µ , l + 2 µ , 1 + 5 µ ) B1 Equate at least two pairs of components and solve for λ or for µ M1 Obtain correct answers for λ and µ A1 Verify that all three component equations are not satisfied (with no errors seen) A1 [6]  1   1      (ii) Carry out correct process for evaluating scalar product of  2  and  0  M1     5 0     Use correct process for finding modulus and evaluating inverse cosine M1 Obtain 79 5. ° or 1.39 radians A1 [3] dy

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9 y M x O The diagram shows the curve y and its maximum point M. = x2e2−x (i) Show that the x-coordinate of M is 2. [3] 2 (ii) Find the exact value of Ó 0 x2e2−x dx. [6]

Mark scheme: 9 (i) Use product rule to find first derivative M1 Obtain 2 xe 2 −−x x 2 e 2 − x A1 Confirm x = 2 at M A1 [3] (ii) Attempt integration by parts and reach ± x 2 e 2 −±x ∫ 2 xe 2 − x dx *M1 Obtain − x 2 e 2 −+x ∫ 2 xe 2 − x dx A1 Attempt integration by parts and reach ± x 2 e 2 − x ± 2 xe 2 − x ± 2e 2 − x *M1 Obtain − x 2 e 2 − x − 2 xe 2 − x − 2e 2 − x A1 Use limits 0 and 2 having integrated twice M1 dep *M Obtain 2e 2 − 10 A1 [6] dx 2 dy 2

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10 y O x P The diagram shows part of the curve with parametric equations x 2 ln t 2 , y t3 2t 3. = + = + + (i) Find the gradient of the curve at the origin. [5] (ii) At the point P on the curve, the value of the parameter is p. It is given that the gradient of the curve at P is 12. 1 (a) Show that p [1] = 3p2 2 −2. + (b) By first using an iterative formula based on the equation in part (a), determine the coordinates of the point P. Give the result of each iteration to 5 decimal places and each coordinate of P correct to 2 decimal places. [4]

Mark scheme: dx 2 dy 210 (i) Obtain = and = 3t + 2 B1 dt t + 2 dt dy dy dx Use = ÷ M1 dx dt dt dy 1 2 Obtain = (3t + 2)(t + 2) A1 dx 2 Identify value of t at the origin as –1 B1 5 Substitute to obtain as gradient at the origin A1 [5] 2 1 1 (ii) (a) Equate derivative to and confirm p = − 2 B1 [1] 2 2 3 p + 2 (b) Use the iterative formula correctly at least once M1 Obtain value p = − .1924 or better (–1.92367…) A1 Show sufficient iterations to justify accuracy or show a sign change in appropriate interval A1 Obtain coordinates (–5.15, –7.97) A1 [4]

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