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Cambridge AS & A Level Mathematics (9709), Paper 2 Pure Mathematics 2, October/November 2020. Read both free below, or download them. Done the paper? Get it marked by AI against this mark scheme.
9709_w20_qp_219709_w20_ms_219709/21/O/N/20
Each question as it's printed, and the mark scheme's points for it, as text. Diagrams, tables and graphs are in the PDFs above.
Question
Given that ln 2x + 1 − ln x − 3 = 2, find x in terms of e. [4]
Mark scheme
1 Use correct logarithm property to simplify left-hand side M1 Or equivalent method Use correct process to obtain equation without logarithms M1 2x + 1 2 A1 OE Obtain =e x−3 2 A1 OE Obtain x = 3e2 + 1 e −2 4
Question
The polynomial p x is defined by p x = x3 + ax2 + bx + 16, where a and b are constants. It is given that x + 2 is a factor of p x and that the remainder is 72 when p x is divided by x − 2. Find the values of a and b. [5]
Mark scheme
2 Substitute x =−2 and equate to zero *M1 Substitute x = 2 and equate to 72 *M1 Obtain 4a − 2b + 8 = 0 and 4a + 2b − 48 = 0 or equivalents A1 Solve a pair of relevant linear simultaneous equations DM1 Dependent at least one M mark Obtain a = 5, b = 14 A1 5
Question
y A B x O 1 The diagram shows the curve y = 2 + e−2x . The curve crosses the y-axis at the point A, and the point B on the curve has x-coordinate 1. The shaded region is bounded by the curve and the line segment AB. Find the exact area of the shaded region. [5]
Mark scheme
3 Integrate to obtain form ax + b e − 2 x M1 1 A1 Obtain correct 2 x − e −2 x 2 5 1 −2 A1 Apply limits to obtain − e 2 2 Attempt to find area of relevant trapezium M1 5 1 −2 A1 Obtain + e and subtract to obtain e−2 or exact equivalent 2 2 5
Question
(a) Solve the equation 2x − 5 = x + 6. [3] (b) Hence find the value of y such that 21−y − 5 = 2−y + 6 . Give your answer correct to 3 significant figures. [2] 6 + 8xn
Mark scheme
4(a) 2 2 B1 State or imply non-modulus equation (2x − 5) = (x + 6) or pair of linear equations Attempt solution of 3-term quadratic equation or of pair of linear M1 equations 1 A1 Obtain − and 11 3 3 4(b) Apply logarithms and use power law for 2 − y = k where k > 0 M1 from (a) Obtain −3.46 A1 AWRT 2
Question
The sequence of values given by the iterative formula xn+1 = with initial value x1 = 2 converges 8 + xn2 to !. (a) Use the iterative formula to find the value of ! correct to 4 significant figures. Give the result of each iteration to 6 significant figures. [3] (b) State an equation satisfied by ! and hence determine the exact value of !. [2]
Mark scheme
5(a) Use iteration correctly at least once M1 Need to see 3 values including the starting values Obtain final answer 1.817 A1 Answer required to exactly 4 significant figures Show sufficient iterations to 6 significant figures to justify answer A1 or show sign change in interval [1.8165, 1.8175] 3 5(b) 6 + 8x B1 State equation x = or equivalent using α 8 + x2 Obtain 3 6 or exact equivalent B1 2
Question
It is given that 3 sin 21 = cos 1 where 1 is an angle such that 0Å < 1 < 90Å. (a) Find the exact value of sin 1. [2] (b) Find the exact value of sec 1. [2] (c) Find the exact value of cos 21. [2]
Mark scheme
6(a) Use sin 2θ = 2sin θ cosθ B1 1 B1 Obtain sin θ = 6 2 6(b) Use correct identity or identities to find value of secθ M1 Obtain 6 or exact equivalent A1 35 2 6(c) Use correct identity or identities to find value of cos2θ M1 17 A1 Obtain or exact equivalent 18 2
Question
A curve is defined by the parametric equations x = 3t − 2 sin t, y = 5t + 4 cos t, where 0 ≤ t ≤ 2π. At each of the points P and Q on the curve, the gradient of the curve is 52 . (a) Show that the values of t at P and Q satisfy the equation 10 cos t − 8 sin t = 5. [3] (b) Express 10 cos t − 8 sin t in the form R cos t + !, where R > 0 and 0 < ! < 12 π. Give the exact value of R and the value of ! correct to 3 significant figures. [3] (c) Hence find the values of t at the points P and Q. [4] 4x3 + 8x − 4
Mark scheme
7(a) dx dy B1 Obtain = 3 − 2cos t and = 5 − 4sin t dt dt dy 5 M1 Equate expression for to dx 2 Obtain 10cos t − 8sin t = 5 A1 AG – sufficient working to be shown. 3 7(b) State R = 164 or exact equivalent B1 Use appropriate trigonometry to find α M1 Obtain 0.675 with no errors seen A1 AWRT 3 7(c) Carry out correct method to find one value of t M1 Must be using the result from (b) Obtain 0.495 A1 AWRT Carry out correct method to find second value of t M1 Obtain 4.44 A1 AWRT 4
Question
A curve has equation y = f x where f x = . 2x − 1 dy (a) Find an expression for and hence find the coordinates of each of the stationary points of the dx curve y = f x. [5] (b) Divide 4x3 + 8x − 4 by 2x − 1, and hence find Ó f x dx. [5] Additional Page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
Mark scheme
8(a) Differentiate using the quotient rule (or product rule) *M1 (2 x − 1)(12 x2 + 8) − 2(4 x3 + 8x − 4) A1 OE Obtain (2 x − 1)2 Equate first derivative to zero and attempt solution DM1 Obtain (0, 4) A1 Allow if given separately Allow A1 if both x-coordinates are given, but y coordinates are omitted. A1 Obtain 3 , 59 4 8 5 8(b) Carry out division to obtain quotient of form 2 x 2 + kx + m M1 For non-zero constants k, m 9 A1 Obtain correct quotient 2 x 2 + x + 2 1 A1 Obtain remainder 2 3 M1 For non-zero constants k1 , k 2 Integrate to obtain at least k1x and k 2 ln(2 x − 1) terms 2 3 1 2 9 1 A1 Condone absence of ... + c and modulus signs Obtain x + x + x + ln(2 x − 1) as final answer 3 2 2 4 5
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Mark my 9709/21Examiners follow the generic marking principles at the front of the scheme, then each question's points: method (M), accuracy (A) and independent (B) marks, error carried forward where the scheme allows it, and positive marking, so you're credited for what you got right.
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On this page, free: the mark scheme for AS & A Level Mathematics 9709/21, October/November 2020 (9709_w20_ms_21), and the question paper (9709_w20_qp_21), to read or download, with each question's marking points written out underneath.
Mathematics 9709/21 for October/November 2020 is out of 50 marks, with 1 hour 15 minutes to answer it.
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