Cambridge A Level Mathematics 9709 — 2015 May/June Paper 3 · Variant 2

2 Using the substitution giving your answer correct to 4x, solve the equation 4x 42 u = + = 4x+2, 3 significant figures. [4]

Mark scheme: 2 Use laws of indices correctly and solve for u M1 16 Obtain u in any correct form, e.g. u = A1 16 − 1 Use correct method for solving an equation of the form 4 x = a , where a > 0 M1 Obtain answer x = 0.0466 A1 [4]

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3 A curve has equation y cos x cos 2x. Find the x-coordinate of the stationary point on the curve in the interval 0 x 1 giving= your answer correct to 3 significant figures. [6] < < 20,

Mark scheme: 3 EITHER: Use correct product rule M1 Obtain correct derivative in any form, e.g. − sin x cos 2 x − 2 cos x sin 2 x A1 Use the correct double angle formulae to express derivative in cos x and sin x, or cos 2x and sin x M1 OR1: Use correct double angle formula to express y in terms of cos x and attempt differentiation M1 Use chain rule correctly M1 Obtain correct derivative in any form, e.g. − 6 cos 2 x sin x + sin x A1 OR2: Use correct factor formula and attempt differentiation M1 3 1 Obtain correct derivative in any form, e.g. − sin3x − sin x A1 2 2 Use correct trig formulae to express derivative in terms of cos x and sin x, or sin x M1 Equate derivative to zero and obtain an equation in one trig function M1 Obtain 6 cos 2 x = 1 , 6 sin 2 x = 5 , tan 2 x = 5 or 3 cos 2 x = − 2 A1 Obtain answer x = 1.15 (or 65.9°) and no other in the given interval A1 [6] [Ignore answers outside the given interval.] [SR: Solution attempts following the EITHER scheme for the first two marks can earn the second and third method marks as follows: Equate derivative to zero and obtain an equation in tan 2x and tan x M1 Use correct double angle formula to obtain an equation in tan x M1]

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1 x 6 Let I dx. 2 x = Ô 0 − 2 2 2 2 (i) Using the substitution u 2 x, show that I du. [4] −u u = − = Ô 1 (ii) Hence show that I 8 ln 2 [4] = −5.

Mark scheme: 1 6 (i) State or imply d u = − d x , or equivalent B1 2 x Substitute for x and dx throughout M1 ± 2 ( 2 − u 2) Obtain integrand , or equivalent A1 u Show correct working to justify the change in limits and obtain the given answer with no errors seen A1 [4] (ii) Integrate and obtain at least two terms of the form a ln u , bu , and cu 2 M1* Obtain indefinite integral 8 ln u − 8u + u 2 , or equivalent A1 Substitute limits correctly M1(dep*) Obtain the given answer correctly having shown sufficient working A1 [4]

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7 The complex number u is given by u i. = −1 + 4ï3 (i) Without using a calculator and showing all your working, find the two square roots of u. Give your answers in the form a ib, where the real numbers a and b are exact. [5] + (ii) On an Argand diagram, sketch the locus of points representing complex numbers satisfying the relation 1. Determine the greatest value of arg for points on this locus.Ï [4] Ï −u = Ï

Mark scheme: 7 (i) Square x + iy and equate real and imaginary parts to −1 and 4 3 M1 Obtain x 2 −y 2 = −1 and 2 xy = 4 3 A1 Eliminate one unknown and find an equation in the other M1 Obtain x 4 + x 2 − 12 = 0 or y 4 −y 2 − 12 = 0 , or three term equivalent A1 Obtain answers ± ( 3 + )i2 A1 [5] [If the equations are solved by inspection, give B2 for the answers and B1 for justifying them] (ii) Show a circle with centre − 1+ 4 3 in a relatively correct position B1 Show a circle with radius 1 and centre not at the origin B1 Carry out a complete method for calculating the greatest value of arg z M1 Obtain answer 1.86 or 106.4° A1 [4] A Bx + C

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9 The number of organisms in a population at time t is denoted by x. Treating x as a continuous variable, the differential equation satisfied by x and t is dx xe−t , dt = k + e−t where k is a positive constant. (i) Given that x 10 when t 0, solve the differential equation, obtaining a relation between x, k and t. = = [6] (ii) Given also that x 20 when t 1, show that k 1 . [2] e = = = −2 (iii) Show that the number of organisms never reaches 48, however large t becomes. [2]

Mark scheme: 9 (i) Separate variables correctly and attempt integration of one side B1 Obtain term ln x B1 Obtain term of the form a ln( k + e − t ) M1 Obtain term − ln( k + e − t ) A1 Evaluate a constant or use limits x = 10, t = 0 in a solution containing terms a ln( k + e − t ) and bln x M1* Obtain correct solution in any form, e.g. ln x − ln 10 = − ln( k + e − t ) + ln( k + )1 A1 [6] (ii) Substitute x = 20, t = 1 and solve for k M1(dep*) Obtain the given answer A1 [2] (iii) Using e −t → 0 and the given value of k, find the limiting value of x M1 Justify the given answer A1 [2]

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10 The points A and B have position vectors given by OA 2i 3k and OB i j 5k. The line l −−→ −−→ has equation r i j 2k 3i j . = −j + = + + = + + + - + −k (i) Show that l does not intersect the line passing through A and B. [5] (ii) Find the equation of the plane containing the line l and the point A. Give your answer in the form ax by d. [6] + + cÏ =

Mark scheme: 10 (i) Carry out a correct method for finding a vector equation for AB M1 Obtain r = 2 i − j + 3k + λ ( − i + 2 j + 2k ) , or equivalent A1 Equate at least two pairs of components of general points on AB and l and solve for λ or for µ M1 4 3 Obtain correct answer for λ or µ, e.g. λ = 1 or µ = 0; λ = − or µ = ; 5 5 1 3 or λ = or µ = − A1 4 2 Verify that not all three pairs of equations are satisfied and that the lines fail to intersect A1 [5] (ii) EITHER: Obtain a vector parallel to the plane and not parallel to l, e.g. i −2 j + k B1 Use scalar product to obtain an equation in a, b and c, e.g. 3a + b − c = 0 B1 Form a second relevant equation, e.g. a – 2b + c = 0 and solve for one ratio, e.g. a : b M1 Obtain final answer a : b : c = 1 : 4 : 7 A1 Use coordinates of a relevant point and values of a, b and c in general equation and find d M1 Obtain answer x + 4y + 7z = 19, or equivalent A1 OR1: Obtain a vector parallel to the plane and not parallel to l, e.g. i −2 j + k B1 Obtain a second relevant vector parallel to the plane and attempt to calculate their vector product, e.g. ( i − 2 j + k ) × (3i + j − k ) M1 Obtain two correct components A1 Obtain correct answer, e.g. i + 4 j + 7 k A1 Substitute coordinates of a relevant point in x + 4y + 7z = d, or equivalent, and find d M1 Obtain answer x + 4y + 7z = 19, or equivalent A1 OR2: Obtain a vector parallel to the plane and not parallel to l, e.g. i −2 j + k B1 Using a relevant point and second relevant vector, form a 2-parameter equation for the plane M1 State a correct equation, e.g. r = 2i − j + 3k + s ( i − 2 j + k ) + t (3i + j − k ) A1 State 3 correct equations in x, y, z, s and t A1 Eliminate s and t M1 Obtain answer x + 4y + 7z = 19, or equivalent A1 OR3: Using the coordinates of A and two points on l, state three simultaneous equations in a, b, c and d, e.g. a + b + 2c = d, 2a − b + 3c = d and 4a + 2b + c = d B1 Solve and find one ratio, e.g. a : b M1 State one correct ratio A1 Obtain a correct ratio of three of the unknowns, e.g. a : b : c = 1 : 4 : 7, or equivalent A1 Either use coordinates of a relevant point and the found ratio to find the fourth unknown, e.g. d, or find the ratio a : b : c : d M1 Obtain answer x + 4y + 7z = 19, or equivalent A1 OR4: Obtain a vector parallel to the plane and not parallel to l, e.g. i −2 j + k B1 Using a relevant point and second relevant vector, form a determinant equation for the plane M1 x − 2 y + 1 z − 3 State a correct equation, e.g. 1 − 2 1 = 0 A1 3 1 − 1 Attempt to expand the determinant M1 Obtain or imply two correct cofactors A1 Obtain answer x + 4y +7z =19, or equivalent A1 [6]

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