Cambridge A Level Mathematics 9709 — 2007 Oct/Nov Paper 2 · Variant 1
9709/21/O/N/07 · 8 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 · Show that 4 1 1 dx = ln 3
1 Show that 4 1 1 dx = ln 3. [4] 2x + 1 2 1
Mark scheme: 1 1 State indefinite integral of the form k In(2x + 1), where k = , 1 or 2 M1 2 1 State correct integral In(2x + 1) A1 2 Use limits correctly, allow use of limits x = 4 and x = 1 in an incorrect form M1 Obtain given answer A1 [4]
Q2 · The sequence of values given by the iterative formula 2xn 4 xn+1 = + , 3 x2 n with…
2 The sequence of values given by the iterative formula 2xn 4 xn+1 = + , 3 x2 n with initial value x1 = 2, converges to α. (i) Use this iterative formula to determine α correct to 2 decimal places, giving the result of each iteration to 4 decimal places. [3] (ii) State an equation that is satisfied by α and hence find the exact value of α. [2]
Mark scheme: 2 (i) Use the iterative formula correctly at least once M1 Obtain final answer 2.29 A1 Show sufficient iterations to justify its accuracy to 2 d.p. (must be working to 4 d.p.) – 3 iterations are sufficient B1 [3] 2 4 (ii) State equation x = x + 2 , or equivalent B1 3 x Derive the exact answer α (or x) = 3 12 , or equivalent B1 [2]
Q3 · Solve the inequality |y −5| < 1
3 (i) Solve the inequality |y −5| < 1. [2] (ii) Hence solve the inequality |3x −5| < 1, giving 3 significant figures in your answer. [3]
Mark scheme: 3 (i) Obtain critical values 4 and 6 B1 State answer 4 < y < 6 B1 [2] (ii) Use correct method for solving an equation of the form 3x = a, where a > 0 M1 Obtain one critical value, i.e. either 1.26 or 1.63 A1 State answer 1.26 < x <1.63 A1 [3] 2
Q4 · The equation of a curve is y = 2x −tan x, where x is in radians
4 The equation of a curve is y = 2x −tan x, where x is in radians. Find the coordinates of the stationary points of the curve for which −12π < x < 12π. [5]
Mark scheme: 4 State derivative 2 – sec2 x, or equivalent B1 Equate derivative to zero and solve for x M1 1 Obtain x = π , or 0.785 (± 45° gains A1) A1 4 1 Obtain x = – π , (allow negative of first solution) A1√ 4 1 1 Obtain corresponding y-values π – 1 and – π + 1, ± 0.571 A1 [5] 2 2
Q5 · The polynomial 3x3 + 8x2 + ax −2, where a is a constant, is denoted by p(x)
5 The polynomial 3x3 + 8x2 + ax −2, where a is a constant, is denoted by p(x). It is given that (x + 2) is a factor of p(x). (i) Find the value of a. [2] (ii) When a has this value, solve the equation p(x) = 0. [4]
Mark scheme: 5 (i) Substitute x = –2 and equate to zero M1 Obtain answer a = 3 A1 [2] (ii) At any stage state that x = –2 is a solution B1 EITHER: Attempt division by x + 2 and reach a partial quotient of 3x2 + kx M1 Obtain quadratic factor 3x2 + 2x – 1 A1 1 Obtain solutions x = – 1 and x = A1 3 OR: Obtain solution x = –1 by trial or inspection B1 1 Obtain solution x = similarly B2 [4] 3 GCE A/AS LEVEL – October/November 2007 9709 02
Q6 · Express 8 sin θ −15 cos θ in the form R sin(θ −α), where R > 0 and 0◦< α < 90◦, giving…
6 (i) Express 8 sin θ −15 cos θ in the form R sin(θ −α), where R > 0 and 0◦< α < 90◦, giving the exact value of R and the value of α correct to 2 decimal places. [3] (ii) Hence solve the equation 8 sin θ −15 cos θ = 14, giving all solutions in the interval 0◦≤θ ≤360◦. [4]
Mark scheme: 6 (i) State answer R = 17, allow 289 B1 Use trig formula to find α M1 Obtain α = 61.93°, (1.08 radians) A1 [3] (ii) Carry out evaluation of sin −(14/17)1 ≈55.44°, or equivalent M1 Obtain answer 117.4°, (2.06 radians) A1 Carry out correct method for second answer M1 Obtain answer 186.5° and no others in the range (3.255 radians) A1√ [4] [Ignore answers outside the given range.]
Q7 · Prove the identity (cos x + 3 sin x)2 ≡5 −4 cos 2x + 3 sin 2x
7 (i) Prove the identity (cos x + 3 sin x)2 ≡5 −4 cos 2x + 3 sin 2x. [4] (ii) Using the identity, or otherwise, find the exact value of 14π (cos x + 3 sin x)2 dx. [4] 0
Mark scheme: 7 (i) Expand and use sin 2A formula M1 Use cos 2A formula at least once M1 Obtain any correct expression in terms of cos 2x and sin 2x only – can be implied A1 Obtain given answer correctly A1 [4] 3 (ii) State indefinite integral 5x – 2sin 2x – cos 2x B2 2 [Award B1 if one error in one term] Substitute limits correctly – must be correct limits M1 1 Obtain answer (5π – 2), or exact simplified equivalent A1 [4] 4
Q8 · The diagram shows the curve y = x2e−x and its maximum point M
8 The diagram shows the curve y = x2e−x and its maximum point M. (i) Find the x-coordinate of M. [4] (ii) Show that the tangent to the curve at the point where x = 1 passes through the origin. [3] (iii) Use the trapezium rule, with two intervals, to estimate the value of 3 x2e−x dx, 1 giving your answer correct to 2 decimal places. [3]
Mark scheme: 8 (i) Differentiate using product or quotient rule M1 Obtain derivative in any correct form A1 Equate derivative to zero and solve for x M1 Obtain answer x = 2 correctly, with no other solution A1 [4] (ii) Find the gradient of the curve when x = 1, must be simplified, allow 0.368 B1 Form the equation of the tangent when x = 1 M1 Show that it passes through the origin A1 [3] (iii) State or imply correct ordinates 0.36787…, 0.54134…, 0.44808… B1 Use correct formula, or equivalent, correctly with h = 1 and three ordinates M1 Obtain answer 0.95 with no errors seen A1 [3]
What was in this paper
The subtopics covered by these 8 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 2007 Oct/Nov, Paper 2 · Variant 1. A higher threshold means an easier paper — the bar moves with how the cohort did.