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Cambridge AS & A Level Mathematics (9709), Paper 3 Pure Mathematics 3, October/November 2021. Read both free below, or download them. Done the paper? Get it marked by AI against this mark scheme.
9709_w21_qp_339709_w21_ms_339709/33/O/N/21
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
Find the quotient and remainder when 2x4 + 1 is divided by x2 − x + 2. [3]
Mark scheme
1 Commence division and reach partial quotient of the form 2x 2 + kx M1 Obtain quotient 2 x 2 + 2 x − 2 A1 Obtain remainder −6 x + 5 A1 3
Question
(a) Sketch the graph of y = 2x − 3. [1] (b) Solve the inequality 2x − 3 < 3x + 2. [3]
Mark scheme
2(a) Show a recognizable sketch graph of y = 2 x − 3 B1 1 2(b) Find x-coordinate of intersection with y = 3x + 2 M1 1 A1 Obtain x = 5 1 A1 State final answer x > only 5 Alternative method for Question 2(b) Solve the linear inequality 3 − 2 x < 3 x + 2 , or corresponding equation M1 1 A1 Obtain critical value x = 5 1 A1 State final answer x > only 5 Alternative method for Question 2(b) Solve the quadratic inequality ( 2 x − 3) < ( 3 x + 2 ) , or corresponding 2 2 M1 equation 1 A1 Obtain critical value x = 5 1 A1 State final answer x > only 5 3
Question
Solve the equation 4x−2 = 4x − 42 , giving your answer correct to 3 decimal places. [4] π
Mark scheme
3 Use laws of indices correctly and solve for 4 x M1 256 A1 Obtain correct solution in any form, e.g. 4 x = 15 Use a correct method for solving an equation of the form 4 x = a , where M1 a>0 Obtain answer 2.047 A1 4
Question
Find the exact value of Ó x sin 12 x dx. [5] 1π 3
Mark scheme
4 1 1 *M1 Commence integration and reach ax cos x + b cos x dx 2 2 1 1 A1 OE Obtain −2 x cos x + 2 cos x dx 2 2 1 1 A1 OE Complete integration obtaining −2 x cos x + 4sin x 2 2 Use limits correctly, having integrated twice DM1 3 A1 Obtain answer 2 + π, or exact equivalent 3 5
Question
Solve the equation sin 1 = 3 cos 21 + 2, for 0Å ≤ 1 ≤ 360Å. [5]
Mark scheme
5 Use double angle formula and obtain an equation in sin θ M1 Reduce to 6sin 2θ + sin θ − 5 = 0 , or 3-term equivalent A1 Solve a 3-term quadratic in sin θ and calculate θ M1 Obtain answer, e.g. 56.4° A1 Obtain second and third answers, e.g. 123.6° and 270° and no others in the A1 Ignore answers outside the interval. Treat answers in given interval radians as a misread. 5
Question
(a) By first expanding cos x − 60Å, show that the expression 2 cos x − 60Å + cos x can be written in the form R cos x − !, where R > 0 and 0Å < ! < 90Å. Give the exact value of R and the value of ! correct to 2 decimal places. [5] (b) Hence find the value of x in the interval 0Å < x < 360Å for which 2 cos x − 60Å + cos x takes its least possible value. [2]
Mark scheme
6(a) Use cos ( A − B ) formula and obtain an expression in terms of sin x and cos x M1 Collect terms and reach 2 cos x + 3 sin x A1 State R = 7 A1 Use trig formula to find α M1 Obtain α = 40.89° A1 5 6(b) Use correct method to find x M1 Obtain answer x = 220.9 ° A1 2
Question
The equation of a curve is ln x + y = x − 2y. dy x+y−1 (a) Show that = . [4] dx 2 x + y + 1 (b) Find the coordinates of the point on the curve where the tangent is parallel to the x-axis. [3]
Mark scheme
7(a) Use chain rule to differentiate LHS *M1 1 dy A1 Obtain 1 + x + y dx dy dy DM1 Equate derivative of LHS to 1 – 2 and solve for dx dx Obtain the given answer correctly A1 4 7(b) State x + y = 1 B1 Obtain or imply x – 2y = 0 B1 2 1 B1 Obtain coordinates x = and y = 3 3 3 ρ
Question
D N k C j B O M i A In the diagram, OABCD is a pyramid with vertex D. The horizontal base OABC is a square of side 4 units. The edge OD is vertical and OD = 4 units. The unit vectors i, j and k are parallel to OA, OC and OD respectively. The midpoint of AB is M and the point N on CD is such that DN = 3NC. (a) Find a vector equation for the line through M and N . [5] (b) Show that the length of the perpendicular from O to MN is 31 82. [4] 1
Mark scheme
8(a) State OM = 4i + 2 j B1 ρ Use a correct method to find ON M1 Obtain answer 3j + k A1 Use a correct method to find a line equation for MN M1 Obtain answer r = 3j + k + λ(4i – j – k), or equivalent A1 5 8(b) Taking a general point P on MN, form an equation in λ by either equating a M1 ρ relevant scalar product to zero or equating the derivative of OP to zero or using Pythagoras in triangle OPM or OPN 2 A1 OE Obtain λ = 9 Use correct method to find OP M1 Obtain the given answer correctly A1 Alternative method to Question 8(b) Use a scalar product to find the projection of OM (or ON) on MN M1 14 4 A1 Obtain answer or 18 18 Use Pythagoras to obtain the perpendicular M1 Obtain the given answer correctly A1 4
Question
Let f x = . 9 − x x (a) Find the x-coordinate of the stationary point of the curve with equation y = f x. [4] 4 (b) Using the substitution u = x, show that Ó f x dx = 13 ln 5. [6] 0
Mark scheme
9(a) Use quotient or product rule M1 Obtain correct derivative in any form A1 Equate derivative to zero and solve for x M1 Obtain answer x = 3 A1 4 9(b) du 1 B1 State = , or dx = 2 xdu , or 2u du = dx dx 2 x 2 B1 Substitute and obtain integrand 9 − u2 Use given formula for the integral or integrate relevant partial fractions M1 1 3+u A1 OE Obtain integral ln 3 3−u Use limits u = 0 and u = 2 correctly M1 Obtain the given answer correctly A1 6
Question
A large plantation of area 20 km2 is becoming infected with a plant disease. At time t years the area infected is x km2 and the rate of increase of x is proportional to the ratio of the area infected to the area not yet infected. dx When t = 0, x = 1 and = 1. dt (a) Show that x and t satisfy the differential equation dx 19x = . [2] dt 20 − x (b) Solve the differential equation and show that when t = 1 the value of x satisfies the equation x = e0.9+0.05x . [5] (c) Use an iterative formula based on the equation in part (b), with an initial value of 2, to determine x correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3] (d) Calculate the value of t at which the entire plantation becomes infected. [1]
Mark scheme
10(a) dx x M1 State or imply equation of the form =k dt 20 − x Obtain k = 19 A1 AG 2 10(b) Separate variables and integrate at least one side M1 Obtain terms 20 ln x – x and 19t, or equivalent A1 A1 Evaluate a constant or use t = 0 and x = 1 as limits in a solution containing M1 terms a ln x and bt Substitute t = 1 and rearrange the equation in the given form A1 AG 5 10(c) Use xn +1 = e0.9 + 0.05 xn correctly at least once M1 Obtain final answer x = 2.83 A1 Show sufficient iterations to 4 decimal places to justify 2.83 to 2 d.p. or show A1 there is a sign change in the interval (2.825, 2.835) 3 10(d) Set x = 20 and obtain answer t = 2.15 B1 1
Question
The complex number − 3 + i is denoted by u. (a) Express u in the form r ei1 , where r > 0 and −π < 1 ≤ π, giving the exact values of r and 1. [2] (b) Hence show that u6 is real and state its value. [2] (c) (i) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities 0 ≤ arg z − u ≤ 14 π and Re z ≤ 2. [4] (ii) Find the greatest value of z for points in the shaded region. Give your answer correct to 3 significant figures. [2] 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
11(a) State or imply r = 2 B1 5 B1 State or imply θ = π 6 2 11(b) Use a correct method for finding the modulus or argument of u 6 M1 Show correctly that u 6 is real and has value -64 A1 2 11(c)(i) Show half lines from the point representing − 3 + i B1 Show correct half lines B1 Show the line x = 2 in the first quadrant B1 Shade the correct region B1 4 11(c)(ii) Carry out a correct method to find the greatest value of z M1 Obtain answer 5.14 A1 2
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Mathematics 9709/33 for October/November 2021 is out of 75 marks, with 1 hour 50 minutes to answer it.
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