Cambridge A Level Mathematics 9709 — 2016 May/June Paper 2 · Variant 2
9709/22/M/J/16 · 5 questions · 50 marks · ≈56 min
The question paper and its mark scheme, free to read here and free to download. This is Cambridge’s own paper, exactly as it was sat.
Question paper4 pages
Mark scheme5 pages
Answers below. Sit the paper first if you are practising.
Questions as text
Q1 · X 1 Given that 53x 74y, use logarithms to find the value of correct to 4 significant figures
x 1 Given that 53x 74y, use logarithms to find the value of correct to 4 significant figures. [3] y =
Mark scheme: 1 Use power law for logarithms correctly at least once M1 Obtain 3 x log5 = 4 y log7 or 3 x ln5 = 4 y ln7 or equivalent A1 Obtain 1.612 A1 [3]
Q2 · Find the quotient and remainder when 2x3 3 is divided by x2 5
2 (i) Find the quotient and remainder when 2x3 3 is divided by x2 5. [3] −7x2 −9x + −2x + (ii) Hence find the values of the constants p and q such that x2 5 is a factor of 2x3 px q. −2x + −7x2 + +[2]
Mark scheme: 2 (i) Carry out division, or equivalent, at least as far as quotient 2x + k M1 Obtain quotient 2 x − 3 A1 Obtain remainder −25 x + 18 A1 [3] (ii) Subtract remainder of form ax + b ( ab ≠ 0 ) from 2 x 3 − 7 x 2 − 9 x + 3 or multiply their quotient by x 2 − 2 x + 5 M1 Obtain p = 16 and q = −15 A1 [2] 2 2
Q3 · Solve the equation 3u 1 2u
3 (i) Solve the equation 3u 1 2u . [3] + = −5 (ii) Hence solve the equation 3 cotx 1 2 cotx for 0 x 1 giving your answer correct to 3 significant figures. + = −5 < < 20, [2]
Mark scheme: 3 (i) State or imply non-modular equation (3u + 1) 2 = (2u − 5) 2 or corresponding pair of linear equations B1 Attempt solution of 3-term quadratic equation or of 2 linear equations M1 Obtain −6 and 54 A1 [3] (ii) Evaluate tan −1 1k for at least one of their solutions k from part (i) M1 Obtain 0.896 A1 [2]
Q5 · 5 The equation of a curve is y 6xe 3x
1 5 The equation of a curve is y 6xe 3x. At the point on the curve with x-coordinate p, the gradient of the curve is 40. = @ A 20 (i) Show that p 3 ln . [4] p 3 = + (ii) Show by calculation that 3.3 p 3.5. [2] < < (iii) Use an iterative formula based on the equation in part (i) to find the value of p correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
Mark scheme: 1 3 x 13 x5 (i) Use product rule to obtain form k1e + k 2 xe *M1 1 3 x 13 x Obtain correct 6e + 2 xe A1 Equate first derivative to 40 and obtain equation without e present, dep *M DM1 Confirm p = 3ln p20+ 3 or x = 3ln x20+ 3 A1 [4] (ii) Consider sign of p − 3ln p20+ 3 at 3.3 and 3.5 or equivalent M1 Complete argument correctly with appropriate calculations A1 [2] (iii) Carry out iterative process correctly at least once M1 Obtain final answer 3.412 A1 Show sufficient iterations to justify accuracy to 3 dp or show sign change in interval (3.4115, 3.4125) B1 [3] 2
Q7 · Y P Q x O M The diagram shows the curve with parametric equations x 2 y 1 3 cos 2t, =…
7 y P Q x O M The diagram shows the curve with parametric equations x 2 y 1 3 cos 2t, = −cost, = + for 0 t The minimum point is M and the curve crosses the x-axis at points P and Q. < < 0. dy (i) Show that cos t. [4] dx = −12 (ii) Find the coordinates of M. [2] (iii) Find the gradient of the curve at P and at Q. [4]
Mark scheme: dx dy 7 (i) State dt = sin t and dt = −6sin2t B1 Use sin2t = 2sin t cos t B1 d y Form expression for d x in terms of t M1 Confirm −12cost A1 [4] (ii) Identify 12π as value of t B1 Obtain (2, − 2) B1 [2] (iii) Identify cos2t = − 13 B1 Attempt to find value of t (or of cost ) for at least one of the two points M1 Obtain 0.955 (or 1 ) or 2.186 (or − 1 ) A1 3 3 Obtain − 12 or − 4 3 or −6.93 and 12 or 4 3 or 6.93 A1 [4] 3 3
What was in this paper
The subtopics covered by these 5 questions, and how many questions each got. Open one in a new tab to see every Cambridge question on it.
What you needed in this session
Cambridge’s own grade thresholds for 2016 May/June, Paper 2 · Variant 2. A higher threshold means an easier paper — the bar moves with how the cohort did.