Cambridge A Level Mathematics 9709 — 2021 Feb/March Paper 3 · Variant 2

4 The variables x and y satisfy the differential equation dy 1 y sin x. −cosx dx = It is given that y 4 when x = = π. (a) Solve the differential equation, obtaining an expression for y in terms of x. 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(b) Sketch the graph of y against x for 0 x [1] < < 2π.

Mark scheme: 4(a) Separate variables correctly and attempt integration of at least one side M1 Obtain term ln y A1 Obtain term of the form ln(1 cos ) ± − x M1 Obtain term ( ) ln 1 cos − x A1 Use π = x , y = 4 to evaluate a constant, or as limits, in a solution containing terms of the form ln a y and ln(1 cos ) − b x M1 Obtain final answer 2(1 cos ) = − y x A1 OE 6 Question Answer Marks Guidance 4(b) Show a correct graph for 0 2π x < < with the maximum at x = π B1 FT The FT is for graphs of the form (1 cos ) = − y a x , where a is positive. 1

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5 (a) Express 7 sin x 2 cos x in the form R sin x , where R 0 and State the exact + + ! > 0Å < ! < 90Å. value of R and give correct to 2 decimal places. 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(b) Hence solve the equation 7 sin 2 cos 1, for [5] 21 + 21 = 0Å < 1 < 180Å. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 5(a) State 11 = R B1 Use trig formulae to find α M1 Obtain 37.09 α = ° A1 3 5(b) Evaluate 1 1 sin 11 −      to at least 2 dp (17.5484°) B1 FT The FT is on R. Use correct method to find a value of θ in the interval M1 Obtain answer, e.g. 62.7° A1 Use a correct method to obtain a second answer M1 Obtain second answer, e.g. 170.2° , and no other in the interval A1 Ignore answers outside the given interval. 5

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7 Two lines have equations r 3 s and r 1 t . = + −1 = + −1 2 3 4 4 (a) Show that the lines are skew. 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(b) Find the acute angle between the directions of the two lines. 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Mark scheme: 7(a) Express general point of a line in component form, e.g. (1 + 2s, 3 – s, 2 + 3s) or (2 + t, 1 – t, 4 + 4t) B1 Equate at least two pairs of components and solve for s or for t M1 Obtain correct answer for s or for t (possible answers are –1, 6, 2 5 for s and –3, 4, 1 5 − for t ) A1 Verify that all three component equations are not satisfied A1 Show that the lines are not parallel and are thus skew A1 5 7(b) Carry out correct process for evaluating the scalar product of the direction vectors M1 Using the correct process for the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result M1 Obtain answer 19.1° or 0.333 radians A1 3

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8 The complex numbers u and v are defined by u 2i and v 3 i. = −4 + = + u (a) Find in the form x iy, where x and y are real. [3] v + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ u (b) Hence express are exact. [2] in the form rei1, where r and 1 v ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ In an Argand diagram, with origin O, the points A, B and C represent the complex numbers u, v and 2u v respectively. + (c) State fully the geometrical relationship between OA and BC. 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(d) Prove that angle AOB 3 [2] = 4π. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 8(a) Multiply numerator and denominator by 3 – i M1 OE Obtain numerator – 10 + 10i or denominator 10 A1 Obtain final answer – 1 + i A1 3 8(b) State or imply r = 2 B1 FT State or imply that 3 π 4 θ = B1 FT 2 8(c) State that OA and BC are parallel B1 State that BC = 2OA B1 2 Question Answer Marks Guidance 8(d) Use angle AOB = arg arg arg u u v v − = M1 Obtain the given answer A1 Alternative method for question 8(d) Obtain tan AOB from gradients of OA and OB and the tan( ) ± A B formula M1 Obtain the given answer A1 Alternative method for question 8(d) Obtain cos AOB by using the cosine rule or a scalar product M1 Obtain the given answer A1 2

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10 y M x O 1 2π The diagram shows the curve y sin 2x cos2x for 0 and its maximum point M. = ≤x ≤12π, (a) Using the substitution u sin x, find the exact area of the region bounded by the curve and the = x-axis. 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(b) Find the exact x-coordinate of M. 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Mark scheme: 10(a) State or imply du = cos x dx B1 Using double angle formula for sin2x and Pythagoras, express integral in terms of u and du. M1 Obtain integral ( ) 3 2 d  − u u u A1 OE Use limits u = 0 and u = 1 in an integral of the form 2 4 + au bu , where 0 ≠ ab M1 a + b or a + b − 0 1 1 and 2 a b   = = −     Obtain answer 1 2 A1 5 10(b) Use product rule M1 Obtain correct derivative in any form A1 Equate derivative to zero and use a double angle formula *M1 Obtain an equation in one trig variable DM1 Obtain 2 4 sin 1 = x , 2 4 cos 3 = x or 2 3 tan 1 = x A1 Obtain answer 1 π 6 x = A1 6

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