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Cambridge AS & A Level Mathematics (9709), Paper 2 Pure Mathematics 2, February/March 2024. Read both free below, or download them. Done the paper? Get it marked by AI against this mark scheme.
9709_m24_qp_229709_m24_ms_229709/22/F/M/24
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
Use logarithms to solve the equation 3 4 x + 3 = 5 2 x + 7 . Give your answer correct to 3 significant figures. [4]
Mark scheme
1 Apply logarithms to both sides and apply power law at least once *M1 Obtain (4 x + 3)ln3 = (2 x + 7)ln5 A1 Or equivalent with x not in a power. Attempt solution of linear equation DM1 Obtain 6.78 A1 Or greater accuracy. 4
Question
(a) Sketch the graph of y = 3x - 7 , stating the coordinates of the points where the graph meets the axes. [2] (b) Hence find the set of values of the constant k for which the equation 3x - 7 = k (x - 4) has exactly two real roots. [2]
Mark scheme
2(a) Draw V-shaped graph with vertex on positive x-axis B1 State ( 73 , 0) and (0, 7) B1 Allow if only 7 and 7 shown on relevant axes. 3 2 2(b) State or imply that gradient of left-hand part of graph is –3 B1 State −3 < k < 0 B1 Using < and not ⩽. 2
Question
The polynomial p (x) is defined by p (x) = 6 x 3 + ax 2 + 3x - 10, where a is a constant. It is given that (2 x - 1) is a factor of p (x) . (a) Find the value of a and hence factorise p (x) completely. [5] (b) Solve the equation p (cosec i) = 0 for - 90° 1 i 1 90° . [2]
Mark scheme
3(a) Substitute x = 12 , equate to zero and attempt solution M1 Obtain a = 31 A1 Divide by 2 x − 1 at least as far as 3x 2 + mx M1 Or equivalent (for example by inspection, …). Obtain 3x 2 + 17 x + 10 A1 Obtain (2 x − 1)(3x + 2)( x + 5) A1 5 3(b) Attempt solution of sinθ = k where k is valid constant from answer to part (a) M1 Obtain −11.5 A1 Or greater accuracy. 2
Question
y O 6 x The diagram shows the curve with equation y = 1 + e 0.5x . The shaded region is bounded by the curve and the straight lines x = 0, x = 6 and y = 0 . (a) Use the trapezium rule with three intervals to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures. [3] (b) The shaded region is rotated completely about the x-axis. Find the exact volume of the solid produced. [4]
Mark scheme
4(a) B1 Or decimal equivalents Use y-values 2, 1 + e, 1 + e2 , 1 + e3 Use correct formula, or equivalent, with h = 2 M1 Obtain 15.7 A1 Or greater accuracy. 3 4(b) State or imply that volume is π (1 + e0.5 x ) dx B1 Implied if π appears only at the end. Integrate to obtain form k1 x + k2 e0.5 x M1 Where k1k2 ≠ 0 with or without π . Obtain π( x + 2e0.5 x ) or x + 2e0.5 x A1 Obtain 4π + 2π e3 A1 Or exact equivalent. 4
Question
y P O x x3 The diagram shows part of the curve with equation y = . At the point P, the gradient of the curve x+2 is 6. 3 (a) Show that the x-coordinate of P satisfies the equation x = 12 x + 12 . [4] (b) Show by calculation that the x-coordinate of P lies between 3.8 and 4.0 . [2] (c) Use an iterative formula, based on the equation in part (a), to find the x-coordinate of P correct to 3 significant figures. Show the result of each iteration to 5 significant figures. [3]
Mark scheme
5(a) Differentiate using quotient rule *M1 OE ( x + 2)3x2 − x3 A1 OE Obtain ( x + 2)2 Equate first derivative to 6 and simplify at least as far as x3 = ... DM1 Confirm x = 3 12 x + 12 A1 Answer given – necessary detail needed. 4 5(b) Consider sign of x − 3 12 x + 12 , or equivalent, for 3.8 and 4.0 M1 Obtain −0.06... and 0.08... , or equivalents, and justify conclusion A1 Answer given – necessary detail needed. 2 5(c) Use iterative process correctly at least once M1 Obtain final answer 3.88 A1 Answer required to exactly 3 sf. Show sufficient iterations to 5 sf to justify answer or show sign change in the A1 interval [3.875, 3.885] 3
Question
y A B O x M The diagram shows the curve with parametric equations x = 1+ t, y = (ln t + 2) (ln t - 3), for 0 1 t 1 25. The curve crosses the x-axis at the points A and B and has a minimum point M. dy 4 ln t - 2 (a) Show that = . [4] dx t (b) Find the exact gradient of the curve at B. [2] (c) Find the exact coordinates of M. [3]
Mark scheme
6(a) dy M1 Or chain rule if expression expanded before Use product rule to find dt differentiation. dy 2ln t 1 A1 Obtain = − dt t t dy dx M1 Divide attempt at by attempt at dt dt dy 4ln t − 2 A1 Answer given – necessary detail needed. Confirm = dx t 4 6(b) State or imply t = e3 at point B B1 −3 B1 Obtain gradient 10e 2 or exact equivalent 2 6(c) Equate first derivative to zero and obtain value for lnt or t M1 1 A1 Or exact equivalent. Obtain x-coordinate 1 + e 4 Obtain y-coordinate − 25 4 A1 Or equivalent. 3
Question
(a) Prove that sin 2i (a cot i + b tan i) / a + b + (a - b) cos 2i, where a and b are constants. [4] 1 y r 6 (b) Find the exact value of sin 2i (5 cot i + 3 tan i) di . [3] 1 r 12 (c) Solve the equation sin 23 a `2 cot 13 a + 7 tan 13 aj = 11 for - r 1 a 1 r . [3] Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown.
Mark scheme
7(a) Use correct identity for sin 2θ or for cot θ B1 Express left-hand side in terms of sinθ and cosθ M1 Attempt to express k1 cos 2 θ + k2 sin 2 θ in terms of cos2θ only M1 Confirm a + b + (a − b)cos2θ A1 Answer given – necessary detail needed. 4 7(b) Obtain integrand 8 + 2cos2θ B1 Integrate to obtain 8θ + sin 2θ B1 Apply limits correctly to obtain 23 π + 12 3 − 12 B1 Or exact equivalent. 3 7(c) Use identity to obtain 5cos 23 α = −2 B1 Obtain α = 2.97 B1 Or greater accuracy. Obtain α = −2.97 B1 Or greater accuracy; and no others between − π and π. 3
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Mathematics 9709/22 for February/March 2024 is out of 50 marks, with 1 hour 15 minutes to answer it.
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