Cambridge A Level Mathematics 9709 — 2022 May/June Paper 3 · Variant 1

2 (a) Expand 2 in ascending powers of x, up to and including the term in x4, simplifying the −x2 −2 coefficients. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) State the set of values of x for which the expansion is valid. [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 2(a) State a correct unsimplified version of the x or the 4 x term of the expansion of   2 2 2 x   or 2 2 1 1 2 x         M1 2 2 2 1 2. 3 1 2 ... 4 2 2 2 x x                   Symbolic binomial coefficients are not sufficient for the M1. State correct first term 1 4 B1 Accept 2-2 . Obtain the next two terms 2 4 1 3 4 16 x x  A1 A1 A1 for each one correct ISW. Full marks for   2 4 3 1 4 4 1 x x   ISW. SC allow M1 A1 A1 for 1 4 and 2 4 3 4 1 x x   SOI. SC allow M1 A1 for 2 4 3 4 1 x x   4 2(b) State answer 2 x  B1 Or 2 2 x    . 1

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3 Solve the equation 2 cot 2x 3 cot x 5, for x [6] + = 0Å < < 180Å. ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 3 2 1 tan 3 5 tan tan x x x    Obtain a correct linear equation in any form A1 2 1 tan 3 5 tan x x    Reduce equation to a 3-term quadratic A1 2 tan 5tan 4 0 x x    , or 3-term equivalent Solve a 3-term quadratic in tan x and obtain a value of x DM1 Obtain answer, e.g. x = 35.1 A1 Obtain second answer, e.g. x = 99.9, and no other in   0 , 180   A1 Ignore answers outside   0 , 180  . Treat answers in radians   0.612,1.74 as a misread. Alternative method for question 3 Use correct formulae for sin2x and cos2x to form an equation in sinx and cosx *M1 Obtain cos sin 4 5 sin cos x x x x   A1 Reduce equation to a 3-term quadratic A1 2 tan 5tan 4 0 x x    , or 3-term equivalent Solve a 3-term quadratic in tan x and obtain a value of x DM1 Obtain answer, e.g. x = 35.1 A1 Obtain second answer, e.g. x = 99.9, and no other in   0 , 180   A1 Ignore answers outside   0 , 180  . Treat answers in radians   0.612,1.74 as a misread. 6

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3 27 6 Let I dx. 2 = Ô0 9 x2 + 1 4π (a) Using the substitution x 3 tan show that I [4] = 1, = Ó0 cos21 d1. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Hence find the exact value of I. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 6(a) State or imply 2 d 3sec d x    Substitute throughout for x and dx M1 Obtain any correct form in terms of  A1 e.g.   2 2 2 81sec d 9 9tan      Justify change of limits and obtain π 4 2 0 cos d   correctly A1 AG 4 6(b) Obtain indefinite integral of the form cos 2 d a b     , where 0 ab *M1 Obtain 1 1 sin 2 2 4    A1 Use correct limits correctly in an expression containing and sin 2 p q   where 0 pq  DM1   π 1 8 4 0   Obtain answer   1 π 2 8  A1 Or exact equivalent e.g. 1 1 π 8 4  . 4

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2 2i 7 The complex number u is defined by u , where a is a positive integer. −a 1 2i = + (a) Express u in terms of a, in the form x iy, where x and y are real and exact. [3] + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ It is now given that a 3. = (b) Express u in the form where r 0 and giving the exact values of r and [2] rei1, > −π < 1 ≤π, 1. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (c) Using your answer to part (b), find the two square roots of u. Give your answers in the form rei1, where r 0 and giving the exact values of r and [3] > −π < 1 ≤π, 1. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 7(a) Multiply numerator and denominator by 1 – 2i, or equivalent M1 At least one multiplication completed. Obtain correct numerator     1 2 2 2 2i a a    A1 OE Obtain final answer 1 2 2 2 2i 5 5 a a    A1 OE Alternative method for question 7(a) Multiply i x y  by 1 2i  and compare real and imaginary parts M1 Obtain 2 2 x y   and 2 2 x y a  A1 Obtain final answer 1 2 2 2 2i 5 5 a a    A1 OE 3 7(b) Obtain r = 2 B1 FT Obtain 3 π 4  B1 2 7(c) Use correct method to find r or  M1 State answer 3 πi 8 2e  A1 FT State answer 5πi 8 2e A1 FT 3

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8 The equation of a curve is x3 y3 2xy 8 0. + + + = dy (a) Express in terms of x and y. [4] dx ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ The tangent to the curve at the point where x 0 and the tangent at the point where y 0 intersect atthe acute angle = = !. (b) Find the exact value of tan [5] !. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 8(a) State or imply 2 3y d d y x as derivative of 3y State or imply 2 2 y x  d d y x as derivative of 2xy B1 Complete differentiation and equate attempted derivative to zero and solve for d d y x M1 Obtain answer  2 2 3 2 3 2 x y y x   A1 4 8(b) Find gradient at either (0, – 2) or (– 2, 0) M1 Obtain answers 1 3 and 3 A1 A1 Use   tan A B  formula to find tan M1 Obtain answer 4 tan 3  A1 5

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9 G F M D E k O C j i A N B In the diagram, OABCDEFG is a cuboid in which OA 2 units, OC 4 units and OG 2 units. Unit vectors i, j and k are parallel to OA, OC and OG respectively.= The =point M is the midpoint= of DF. The point N on AB is such that AN 3NB. = (a) Express the vectors OM and MN in terms of i, j and k. [3] −−−¿ −−−¿ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find a vector equation for the line through M and N. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ 53 . [4](c) Show that the length of the perpendicular from O to the line through M and N is ? 6 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 9(a) Obtain 2 2 OM   i j k Use a correct method to find MN  M1 e.g. MO OA AN      or MO ON    Obtain 2 MN  i j k  A1 Accept any notation. 3 9(b) Use a correct method to form an equation for MN M1 Allow without ...  r Obtain r = 2i + 3j + 𝜆 (i + j – 2k) A1 FT OE e.g.   2 2 2        r i j k i j k Must have ...  r Follow their answers to part 9(a). 2 9(c) State O P  for a general point P on MN in component form, e.g. (2 + λ, 3 + , 2 )    B1 Equate scalar product of O P  and a direction vector for MN to zero and solve for 𝜆 M1 Obtain λ = 5 6  A1 OE e.g. 1 6  Obtain 53 6 correctly A1 AG e.g. from    2 2 2 7 13 5 6 6 3   4

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10 y x O a π The curve y x sin x has one stationary point in the interval 0 x where x a (see diagram). = < < π, = (a) Show that tan a 2a. [4] = −1 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Verify by calculation that a lies between 2 and 2.5. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (c) Show that if a sequence of values in the interval 0 x given by the iterative formula 1 converges, then it converges to a,<the root< π of the equation in part (a). [2] xn+1 = π −tan−1 2xn ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (d) Use the iterative formula given in part (c) to determine a correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 10(a) Use correct product rule M1 Condone incorrect / missing chain rule Obtain correct derivative in any form A1 e.g. d cos sin d 2 sin y x x x x x   or 2 d 2 2 sin cos d y y x x x x x   Equate derivative to zero and obtain an equation in tan x or tan a M1 Obtain 1 2 tana a  correctly A1 AG 4 10(b) Calculate the value of a relevant expression or pair of expressions at a = 2 and a = 2.5 M1 Must be working in radians At least one correct Complete the argument correctly with correct calculated values A1 e.g. 1 2.18 and 1.25 0.747    2 10(c) State a suitable equation, e.g. 1 1 π tan 2 x x         B1 A correct equation without subscripts or quote   tan tan      Using   tan A B  formula, or otherwise, rearrange this as 1 tan 2 x x  B1 Complete argument correctly 2 Question Answer Marks Guidance 10(d) Use the iterative process correctly at least once M1 Must be working in radians Obtain answer a = 2.29 A1 Show sufficient iterations to 4 dp to justify 2.29 to 2 dp or show there is a sign change in the interval (2.285, 2.295) A1 e.g. 2.25, 2.2974, 2.2871, 2.2893, 2.2888, … 3

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