Cambridge A Level Mathematics 9709 — 2016 Oct/Nov Paper 2 · Variant 3
9709/23/O/N/16 · 3 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 · The sequence of values given by the iterative formula 4 2xn xn+1 = x2 + 3 , n with…
1 The sequence of values given by the iterative formula 4 2xn xn+1 = x2 + 3 , n with initial value x1 2, converges to = !. (i) Use this iterative formula to find correct to 3 decimal places. Give the result of each iteration ! to 5 decimal places. [3] (ii) State an equation that is satisfied by and hence find the exact value of [2] !, !.
Mark scheme: 1 (i) Use the iterative formula correctly at least once M1 Obtain final answer 2.289 A1 Show sufficient iterations to justify accuracy to 3 dp or show sign change in interval (2.2885, 2.2895) B1 [3] 4 2 (ii) State equation x = 2 + 3 x or equivalent B1 x Obtain exact value 12 1 3 or 3 12 B1 [2]
Q5 · Y R x O 6 1 The diagram shows the curve y 1 e for 0 The region bounded by the curve and…
5 y R x O 6 1 The diagram shows the curve y 1 e for 0 The region bounded by the curve and the = ?0 + 3x1 ≤x ≤6. lines x 0, x 6 and y 0 is denoted by R. = = = (i) Use the trapezium rule with 2 strips to find an estimate of the area of R, giving your answer correct to 2 decimal places. [3] (ii) With reference to the diagram, explain why this estimate is greater than the exact area of R. [1] (iii) The region R is rotated completely about the x-axis. Find the exact volume of the solid produced. [4]
Mark scheme: 5 (i) State or imply correct ordinates 2, 1 + e, 1 + e 2 or decimal equivalents B1 Use correct formula, or equivalent, correctly with h = 3 and three ordinates M1 Obtain answer 12.25 with no errors seen A1 [3] (ii) Refer to top of each trapezium being above curve or equivalent B1 [1] (1 + e 1 (iii) State or imply volume is 3 x ) dx B1 ∫ π 3 x with or without π M1 Integrate to obtain form k1 x + k 2e 1 3 x ) or x + 3e 1 Obtain correct π( x + 3e 1 3 x A1 Obtain π(3 + 3e 2 ) or exact equivalent A1 [4]
Q6 · A curve has parametric equations x ln t 1 , y t2 ln t
6 A curve has parametric equations x ln t 1 , y t2 ln t. = + = dy (i) Find an expression for in terms of t. [5] dx (ii) Find the exact value of t at the stationary point. [2] (iii) Find the gradient of the curve at the point where it crosses the x-axis. [2]
Mark scheme: 6 (i) State ddxt = t+11 B1 Use product rule for derivative of y M1 Obtain 2t ln t + t or equivalent A1 Use ddyx = ddyt ÷ ddxt M1 Obtain (t + 1)(2t ln t + t ) A1 [5] (ii) Solve 2ln t + 1 = 0 M1 − 12 Obtain t = e A1 [2] (iii) Identify t = 1 only B1 Obtain 2 B1 [2] 3 4
What was in this paper
The subtopics covered by these 3 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 Oct/Nov, Paper 2 · Variant 3. A higher threshold means an easier paper — the bar moves with how the cohort did.