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

2 Solve the inequality x 2x [4] −2 > −3.

Mark scheme: 2 EITHER: State or imply non-modular inequality ( x − 2 ) 2 > ( 2 x − 3) 2 , or corresponding equation B1 Solve a 3-term quadratic, as in Q1. M1 5 Obtain critical value x = A1 3 5 State final answer x < only A1 3 OR1: State the relevant critical linear inequality ( 2 − x ) > ( 2 x − 3) , or corresponding equation B1 Solve inequality or equation for x M1 5 Obtain critical value x = A1 3 5 State final answer x < only A1 3 OR2: Make recognisable sketches of y = 2x – 3 and y = x − 2 on a single diagram B1 Find x-coordinate of the intersection M1 5 Obtain x = A1 3 5 State final answer x < only A1 4 3

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3 Solve the equation cot 2x cotx 3 for x [6] + = 0Å < < 180Å.

Mark scheme: 3 Use correct tan 2A and cot A formulae to form an equation in tan x M1 Obtain a correct equation in any form A1 Reduce equation to the form tan2 x + 6 tan x − 3 = 0 , or equivalent A1 Solve a three term quadratic in tan x for x, as in Q1. M1 Obtain answer, e.g. 24.9° (24.896) A1 Obtain second answer, e.g. 98.8 (98.794) and no others in the given interval A1 6 [Ignore outside the given interval. Treat answers in radians as a misread.] Radian answers 0.43452, 1.7243

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e2x 4 The curve with equation y has one stationary point. Find the exact values of the coordinates = 4 e3x of this point. + [6]

Mark scheme: 4 Use correct quotient or product rule M1 Obtain correct derivative in any form A1 Equate derivative to zero and obtain a horizontal equation M1 Carry out complete method for solving an equation of the form a3e x = b , or ae 5 x = be 2 x M1 Obtain x = ln 2 , or exact equivalent A1 1 Obtain y = , or exact equivalent A1 6 3 d x 3 d y 3

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5 The parametric equations of a curve are x a cos4t, y a sin4t, = = where a is a positive constant. dy (i) Express in terms of t. [3] dx (ii) Show that the equation of the tangent to the curve at the point with parameter t is x sin2t y cos2t a sin2t cos2t. + = (iii) Hence show that if the tangent meets the x-axis at P and the y-axis at Q, then OP OQ a, + = where O is the origin. [2]

Mark scheme: d x 3 d y 35 (i) State = − 4 a cos t sin t , or = 4 a sin t cos t B1 d t d t d y d y d x Use = ÷ M1 d x d t d t d y Obtain correct expression for in a simplified form A1 3 d x (ii) Form the equation of the tangent M1 Obtain a correct equation in any form A1 Obtain the given answer A1 3 (iii) State the x-coordinate of P or the y-coordinate of Q in any form B1 Obtain the given result correctly B1 2 ∫

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a 6 It is given that x cosx dx 0.5, where 0 a 1 Ó 0 = < < 20. 1.5 (i) Show that a satisfies the equation sin a −cosa . [4] = a (ii) Verify by calculation that a is greater than 1. [2] (iii) Use the iterative formula P1.5 −cosan Q sin−1 an+1 = an to determine the value of a correct to 4 decimal places, giving the result of each iteration to 6 decimal places. [3]

Mark scheme: 6 (i) Integrate and reach ± x sin x m ∫ sin x dx M1* Obtain integral x sin x + cos x A1 Substitute limits correctly, must be seen since AG, and equate result to 0.5 M1(dep*) Obtain the given form of the equation A1 4 (ii) EITHER: Consider the sign of a relevant expression at a = 1 and at another relevant value, π e.g. a = 1.5 Y M1 2 OR: Using limits correctly, consider the sign of [x sin x + cos x ]0a − 5.0 , or compare the value of [x sin x + cos x ]a0 with 0.5, for a =1 AND for another relevant value, π e.g a = 1.5 Y . M1 2 Complete the argument, so change of sign, or above and below stated, both with correct calculated values A1 2 (iii) Use the iterative formula correctly at least once M1 Obtain final answer 1.2461 A1 Show sufficient iterations to 6 d.p. to justify 1.2461 to 4 d.p., or show there is a sign change in the interval (1.24605, 1.24615) A1 3

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7 The number of micro-organisms in a population at time t is denoted by M. At any time the variation in M is assumed to satisfy the differential equation dM k cos 0.02t , dt = ïM where k is a constant and M is taken to be a continuous variable. It is given that when t 0, M 100. = = (i) Solve the differential equation, obtaining a relation between M, k and t. [5] (ii) Given also that M 196 when t 50, find the value of k. [2] = = (iii) Obtain an expression for M in terms of t and find the least possible number of micro-organisms. [2]

Mark scheme: 7 (i) Separate variables correctly and integrate one side B1 Obtain term 2 M , or equivalent B1 Obtain term 50 k sin( .0 02t ) , or equivalent B1 Evaluate a constant of integration, or use limits M = 100, t =0 in a solution with terms of the form a M and b sin( .0 02t ) M1* Obtain correct solution in any form, e.g. 2 M = 50 k sin( .002t ) + 20 A1 5 (ii) Use values M =196, t =50 and calculate k M1(dep*) Obtain answer k = 0.190 A1 2 (iii) State an expression for M in terms of t, e.g. M = ( .4 75 sin( .0 02t ) + 10 2) M1(dep*) State that the least possible number of micro-organisms is 28 or 27.5 or 27.6 (27.5625) A1 2 i

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8 The complex number 1 is denoted by u. −i (i) Showing your working and without using a calculator, express i u in the form x iy, where x and y are real. [2] + (ii) On an Argand diagram, sketch the loci representing complex numbers satisfying the equations Ï and 2. [4] Ï −u = Ï Ï −i = (iii) Find the argument of each of the complex numbers represented by the points of intersection of the two loci in part (ii). [3]

Mark scheme: i 8 (i) EITHER: Substitute for u in and multiply numerator and denominator by 1 + i M1 u 1 1 Obtain final answer − + i , or equivalent A1 2 2 OR: Substitute for u, obtain two equations in x and y and solve for x or for y M1 1 1 Obtain final answer − + i , or equivalent A1 2 2 2 (ii) Show a point representing u in a relatively correct position B1 Show the bisector of the line segment joining u to the origin B1 Show a circle with centre at the point representing i B1 Show a circle with radius 2 B1 4 1 (iii) State argument − π , or equivalent, e.g. 270° B1 2 State or imply the intersection in the first quadrant represents 2 + i B1 State argument 0.464, (0.4636)or equivalent, e.g. 26.6° (26.5625) B1 3

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9 Two planes have equations x 3y 4 and 2x y 5. The planes intersect in the straight + −2Ï = + + 3Ï = line l. (i) Calculate the acute angle between the two planes. [4] (ii) Find a vector equation for the line l. [6]

Mark scheme: 9 (i) State or imply a correct normal vector to either plane, e.g. i + 3 j − 2k , or 2i + j + 3k B1 Carry out correct process for evaluating the scalar product of two normal vectors M1 Using the correct process for the moduli, divide the scalar product of the two normals by the product of their moduli and evaluate the inverse cosine of the result M1 Obtain answer 85.9° or 1.50 radians A1 4 (ii) EITHER: Carry out a complete strategy for finding a point on l M1 Obtain such a point, e.g. (0, 2, 1) A1 EITHER: State two equations for a direction vector ai + bj + ck for l, e.g. a + 3b – 2c = 0 and 2a + b + 3c = 0 B1 Solve for one ratio, e.g. a : b M1 Obtain a : b : c = 11 : −7 : −5 A1 State a correct answer, e.g. r = 2 j + k + λ (11i − 7 j − 5k ) A1  22 3  OR1: Obtain a second point on l, e.g.  , ,0  B1  7 −7  Subtract position vectors and obtain a direction vector for l M1 Obtain 22i − 14 j − 10k , or equivalent A1 State a correct answer, e.g. r = 2 j + k + λ ( 22i − 14 j − 10k ) A1 OR2: Attempt to find the vector product of the two normal vectors M1 Obtain two correct components A1 Obtain 11 i − 7 j − 5k , or equivalent A1 State a correct answer, e.g. r = 2 j + k + λ (11 i − 7 j − 5k ) A1 OR3: Express one variable in terms of a second M1 Obtain a correct simplified expression, e.g. x = ( 22 − 11 y /) 7 A1 Express the same variable in terms of the third M1 Obtain a correct simplified expression, e.g. x = (11 − 11z /) 5 A1 Form a vector equation for the line M1  7 5  State a correct answer, e.g. r = 2j + k + λ  i − j − k  A1  11 11  OR4: Express one variable in terms of a second M1 Obtain a correct simplified expression, e.g. y = ( 22 − 7 x /) 11 A1 Express the third variable in terms of the second M1 Obtain a correct simplified expression, e.g. z = (11 − 5 x /) 11 A1 Form a vector equation for the line M1  7 5  State a correct answer, e.g. r = 2j + k + λ  i − j − k  A1 6  11 11  [The marks are dependent on all M marks being earned.] A B C

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