Cambridge A Level Mathematics 9709 — 2020 Feb/March Paper 3 · Variant 2

1 (a) Sketch the graph of y x . [1] = −2 (b) Solve the inequality x 3x [3] −2 < −4. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 1(a) Make a recognisable sketch graph of 2 = − y x B1 1 1(b) Find x-coordinate of intersection with y = 3x – 4 M1 Obtain 3 2 = x A1 State final answer 3 2 > x only A1 Alternative method for question 1(b) Solve the linear inequality 3 4 2 −> − x x , or corresponding equation M1 Obtain critical value 3 2 = x A1 State final answer 3 2 > x only A1 Alternative method for question 1(b) Solve the quadratic inequality ( ) ( ) 2 2 2 3 4 − < − x x , or corresponding equation M1 Obtain critical value 3 2 = x A1 State final answer 3 2 > x only A1 3

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2 has exactly one root3 (a) By sketching a suitable pair of graphs, show that the equation sec x = −12x in the interval 0 1 [2] ≤x < 2π. ........................................................................................................................................................ ........................................................................................................................................................ (b) Verify by calculation that this root lies between 0.8 and 1. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ @ A 2 (c) Use the iterative formula xn+1 = cos−1 4 −xn to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 3(a) Sketch the graph y = sec x M1 Sketch the graph 1 2 2 = − y x , and justify the given statement A1 2 3(b) Calculate the values of a relevant expression or pair of expressions at x = 0.8 and x = 1 M1 Complete the argument correctly with correct calculated values A1 2 3(c) Use the iterative formula correctly at least once M1 Obtain final answer 0.88 A1 Show sufficient iterations to 4 d.p. to justify 0.88 to 2 d.p., or show there is a sign change in the interval (0.875, 0.885) A1 3

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13π 4 Find x sec2x dx. Give your answer in a simplified exact form. [7] Ó 1 6π ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 4 Integrate by parts and reach tan tan d +  ax x b x x Obtain tan tan d − x x x x A1 Complete the integration, obtaining a term lncos ± x , or equivalent M1 Obtain integral tan lncos + x x x, or equivalent A1 Substitute limits correctly, having integrated twice DM1 Use a law of logarithms M1 Obtain answer 5 1 3 ln3 18 2 π − , or exact simplified equivalent A1 7

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cos 3x sin 3x 5 (a) Show that 2 cot 2x. [4] sin x + cos x = ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ 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........................................................................................................................................................ cos 3x sin 3x (b) Hence solve the equation 4, for 0 x [3] sin x + cosx = < < π. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ 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Mark scheme: 5(a) Express LHS correctly as a single fraction B1 Use ( ) cos ± A B formula to simplify the numerator M1 Use sin 2A formula to simplify the denominator M1 Obtain the given result. A1 4 Question Answer Marks Guidance 5(b) Obtain an equation in tan2x and use correct method to solve for x M1 Obtain answer, e.g. 0.232 A1 Obtain second answer, e.g. 1.80 A1 Ignore answers outside the given interval. 3

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6 The variables x and y satisfy the differential equation dy 1 4y2 + . dx = ex It is given that y 0 when x 1. = = (a) Solve the differential equation, obtaining an expression for y in terms of x. 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(b) State what happens to the value of y as x tends to infinity. 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Mark scheme: 6(a) Separate variables correctly and attempt integration of at least one side B1 Obtain term of the form 1 tan (2 ) − a y M1 Obtain term ( ) 1 1 tan 2 2 − y A1 Obtain term e− − x B1 Use x = 1, y = 0 to evaluate a constant or as limits in a solution containing terms of the form ( ) 1 tan− a by and e±x c M1 Obtain correct answer in any form A1 Obtain final answer ( ) 1 1 tan 2e 2e 2 − − = − x y , or equivalent A1 7 Question Answer Marks Guidance 6(b) State that y approaches ( ) 1 1 tan 2e 2 − , or equivalent B1FT The FT is on correct work on a solution containing e−x . 1

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7 The equation of a curve is x3 3xy2 5. + −y3 = dy x2 y2 (a) Show that + [4] dx = y2 −2xy. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the coordinates of the points on the curve where the tangent is parallel to the y-axis. 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Mark scheme: 7(a) State or imply 2 3 6 + y xy d d y x as derivative of 2 3xy B1 State or imply 2 3y d d y x as derivative of 3y B1 Equate attempted derivative of LHS to zero and solve for d d y x M1 Need to see d d y x factorised out prior to AG Obtain the given answer correctly A1 AG 4 7(b) Equate denominator to zero *M1 Obtain y = 2x, or equivalent A1 Obtain an equation in x or y DM1 Obtain the point (1, 2) A1 State the point ( ) 3 5, 0 B1 Alternatively (1.71, 0). 5

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8 G N F E D B C k j M O A i In the diagram, OABCDEFG is a cuboid in which OA 2 units, OC 3 units and OD 2 units. = = = Unit vectors i, j and k are parallel to OA, OC and OD respectively. The point M on AB is such that MB 2AM. The midpoint of FG is N. = (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. 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(c) Find the position vector of P, the foot of the perpendicular from D to the line through M and N. 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Mark scheme: 8(a) Obtain OM = 2i + j B1 Use a correct method to find  MN M1 Obtain  MN = – i + 2j + 2k A1 3 8(b) Use a correct method to form an equation for MN M1 Obtain r = 2i + j + λ(– i + 2j + 2k), or equivalent A1 2 8(c) Find  DP for a point P on MN with parameter λ, e.g. ( ) 2 ,1 2 , 2 2 λ λ λ − + −+ B1 Equate scalar product of  DP and a direction vector for MN to zero and solve for λ M1 Obtain 4 9 λ = A1 State that the position vector of P is 14 17 8 9 9 9 + + i j k A1 4

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