Cambridge A Level Mathematics 9709 — 2019 May/June Paper 3 · Variant 3

1 4 The equation of a curve is y + e−x , for x 0. = 1 > −e−x dy (i) Show that is always negative. [3] dx ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) The gradient of the curve is equal to when x a. Show that a satisfies the equation −1 = e2a 1 0. Hence find the exact value of a. [4] −4ea + = ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 4(i) Use the quotient or product rule M1 Obtain correct derivative in any form A1 Reduce to – ( ) 2 2e 1 e − − − x x , or equivalent, and explain why this is always negative A1 3 Question Answer Marks Guidance 4(ii) Equate derivative to – 1 and obtain the given equation B1 State or imply 2 4 1 0 − + = u u , or equivalent in e a B1 Solve for a M1 Obtain answer a = ( ) ln 2 3 + and no other A1 4

More questions on Differentiation

6 y x a O b The diagram shows the curve y x4 The curve intersects the x-axis at the points a, 0 = −2x3 −7x −6. and b, 0 , where a b. It is given that b is an integer. < (i) Find the value of b. [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Hence show that a satisfies the equation a 2 a2 a3 . [4] = −13 + + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Use an iterative formula based on the equation in part (ii) to determine a correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 6(i) State b = 3 B1 1 6(ii) Commence division by x – b and reach partial quotient 3 2 + x kx M1 Obtain quotient 3 2 3 2 + + + x x x A1 There being no remainder Equate quotient to zero and rearrange to make the subject a M1 Obtain the given equation A1 4 6(iii) Use the iterative formula ( ) 2 3 1 1 2 3 + = − + + n n n a a a correctly at least once M1 Obtain final answer –0.715 A1 Show sufficient iterations to 5 d.p. to justify –0.715 to 3 d.p., or show there is a sign change in the interval (–0.7145, –0.7155) A1 3

More questions on Quadratics

8 Throughout this question the use of a calculator is not permitted. The complex number u is defined by 4i u = 1 i. − ï3 (i) Express u in the form x iy, where x and y are real and exact. [3] + ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find the exact modulus and argument of u. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities z 2 and z z . [4] < −u <

Mark scheme: 8(i) Multiply numerator and denominator by 1 3i + , or equivalent M1 4i − 4 3 and 3 + 1 A1 Obtain final answer 3 i − + A1 3 Question Answer Marks Guidance 8(ii) State that the modulus of u is 2 B1 State that the argument of u is 5 6 π (or 150°) B1 2 8(iii) Show a circle with centre the origin and radius 2 B1 Show u in a relatively correct position B1 FT Show the perpendicular bisector of the line joining u and the origin B1 FT Shade the correct region B1 4

More questions on Complex numbers

10 The line l has equation r i 2j 3k 2i . = + + + - −j −2k (i) The point P has position vector 4i 2j Find the length of the perpendicular from P to l. + −3k. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) It is given that l lies in the plane with equation ax by 2z 13, where a and b are constants. + + = Find the values of a and b. [6] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 10(i) Find PQ for a general point Q on l, e.g. 3 6 2 2 µ −+ + −− i k i j k B1 Calculate scalar product of JJJG PQ and a direction vector for l and equate the result to zero M1 Solve for µ and obtain µ = 2 A1 Carry out a complete method for finding the length of JJJG PQ M1 Obtain answer 3 A1 Alternative method for question 10(i) Calling the point (1, 2, 3) A, state JJJG A P (or JJJG PA) in component form, e.g. 3i – 6k B1 Use a scalar product with a direction vector for l to find the projection of JJJG A P (or JJJG PA) on l M1 Obtain correct answer in any form, e.g. 18 9 A1 Use Pythagoras to find the perpendicular M1 Obtain answer 3 A1 Question Answer Marks Guidance 10(i) Alternative method for question 10(i) State JJJG A P (or ) JJJG PA in component form B1 Calculate a vector product with a direction vector for l M1 Obtain correct answer, e.g. 6i – 6j – 3k A1 Divide modulus of the product by that of the direction vector M1 Obtain answer 3 A1 5 Question Answer Marks Guidance 10(ii) Substitute coordinates of a general point of l in the plane equation and equate constant terms M1 Obtain a correct equation, e.g. a + 2b + 6 = 13 A1 Equate the coefficient of µ to zero M1 Obtain a correct equation, e.g. 2a – b – 4 = 0 A1 Substitute (1, 2, 3) in the plane equation M1 Obtain a correct equation, e.g. a + 2b + 6 = 13 A1 Alternative method for question 10(ii) Find a second point on l and obtain an equation in a and/or b M1 Obtain a correct equation, e.g. 5a – 2 = 13 A1 Equate scalar product of a direction vector for l and a vector normal for the plane to zero M1 Obtain a correct equation, e.g. 2a – b – 4 = 0 A1 Solve for a or for b M1 Obtain a = 3 and b = 2 A1 6

More questions on Vectors