Cambridge A Level Mathematics 9709 — 2014 May/June Paper 2 · Variant 1
9709/21/M/J/14 · 4 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 · Solve the inequality 3x x 4
1 Solve the inequality 3x x 4 . [4] −2 ≥ +
Mark scheme: 1 Either State or imply non-modular inequality (3 x − 2 )2 >( x + 4 )2 or corresponding equation or pair of linear equations B1 Attempt solution of 3-term quadratic equation or of 2 linear equations M1 Obtain critical values − 12 and 3 A1 State answer x < − 12 , x > 3 A1 [4] Or Obtain critical value x = 3 from graphical method, inspection, equation B1 Obtain critical value x = − 12 similarly B2 State answer x < − 12 , x > 3 B1 [4] 2
Q2 · Find the gradient of each of the following curves at the point for which x 0
2 Find the gradient of each of the following curves at the point for which x 0. = (i) y 3 sin x tan 2x [3] = + 6 (ii) y [3] = 1 e2x +
Mark scheme: 2 (i) Differentiate to obtain form k1 cos x + k 2 sec 2 2 x M1 Obtain correct second term 2 sec 2 2 x A1 Obtain 3 cos x + 2 sec 2 2 x and hence answer 5 A1 [3] 2 x 2 x −2 (ii) Differentiate to obtain form ke ( 1 + e ) M1 2 x 2 x −2 Obtain correct − 12e ( 1 + e ) or equivalent (may be implied) A1 Obtain –3 A1 [3]
Q5 · 5 (i) Prove that tan cot [3] sin 1 + 1 21
2 5 (i) Prove that tan cot [3] sin 1 + 1 21. (ii) Hence (a) find the exact value of tan 1 cot 1 [2] 80 + 80, 1 20 6 (b) evaluate [3] tan cot d1. Ô 0 1 + 1
Mark scheme: 5 (i) Express left-hand side as a single fraction M1 Use sin 2θ = 2 sin θ cosθ at some point B1 Complete proof with no errors seen (AG) A1 [3] GCE AS LEVEL – May/June 2014 9709 21 2 (ii) (a) State or equivalent B1 sin 1 π 4 Obtain 2 2 or exact equivalent (dependent on first B1) B1 [2] (b) State or imply k sin 2θ for any k B1 Integrate to obtain − 32 cos 2θ B1 Substitute both limits correctly to obtain 3 B1 [3] ( )
Q6 · 6 6 (a) Show that dx ln 125
16 6 6 (a) Show that dx ln 125. [5] 2x = Ô6 −7 (b) Use the trapezium rule with four intervals to find an approximation to 17 log10 x dx, Ó 1 giving your answer correct to 3 significant figures. [3]
Mark scheme: 6 (a) Integrate to obtain form k ln( 2 x − 7 ) M1 Obtain correct 3 ln( 2 x − 7 ) A1 Substitute limits correctly (dependent on first M1) DM1 Use law for logarithm of a quotient or power (dependent on first M1) DM1 Confirm ln 125 following correct work and sufficient detail (AG) A1 [5] (b) Evaluate y at (1), 5, 9, 13, 17 M1 Use correct formula, or equivalent, with h = 4 and five y-values M1 Obtain 13.5 A1 [3] d
What was in this paper
The subtopics covered by these 4 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 2014 May/June, Paper 2 · Variant 1. A higher threshold means an easier paper — the bar moves with how the cohort did.