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Cambridge AS & A Level Mathematics (9709), Paper 3 Pure Mathematics 3, February/March 2021. Read both free below, or download them. Done the paper? Get it marked by AI against this mark scheme.
9709_m21_qp_329709_m21_ms_329709/32/F/M/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
Solve the equation ln x3 − 3 = 3 ln x − ln 3. Give your answer correct to 3 significant figures. [3]
Mark scheme
1 Use law of the logarithm of a product or power M1 ( ) Obtain a correct equation free of logarithms, e.g. 3 x 3 − 3 = x 3 A1 Obtain x = 1.65 A1 3
Question
The polynomial ax3 + 5x2 − 4x + b, where a and b are constants, is denoted by p x. It is given that x + 2 is a factor of p x and that when p x is divided by x + 1 the remainder is 2. Find the values of a and b. [5]
Mark scheme
2 Substitute x = – 2, equate result to zero and obtain a correct equation, B1 e.g. −8a + 20 + 8 + b = 0 Substitute x = – 1 and equate result to 2 M1 Obtain a correct equation, e.g. – a + 5 + 4 + b = 2 A1 Solve for a or for b M1 Obtain a = 3 and b = – 4 A1 5
Question
By first expressing the equation tan x + 45Å = 2 cot x + 1 as a quadratic equation in tan x, solve the equation for 0Å < x < 180Å. [6]
Mark scheme
3 Use correct trig formulae to obtain an equation in tan x *M1 Using tan 45° = 1 , obtain a horizontal equation in tan x in any form DM1 Reduce the equation to tan 2 x + tan x − 1 = 0 , or 3-term equivalent A1 Solve a 3-term quadratic in tan x, for x M1 Obtain answer, e.g. x = 31.7 ° A1 Obtain second answer, e.g. x = 121.7° , and no other in the interval A1 Ignore answers outside the given interval. 6
Question
The variables x and y satisfy the differential equation dy 1 − cos x = y sin x. dx It is given that y = 4 when x = π. (a) Solve the differential equation, obtaining an expression for y in terms of x. [6] (b) Sketch the graph of y against x for 0 < x < 2π. [1]
Mark scheme
4(a) Separate variables correctly and attempt integration of at least one side M1 Obtain term ln y A1 Obtain term of the form ± ln(1 − cos x) M1 Obtain term ln (1 − cos x ) A1 Use x = π , y = 4 to evaluate a constant, or as limits, in a solution containing M1 terms of the form a ln y and b ln(1 − cos x) Obtain final answer y = 2(1 − cos x) A1 OE 6 4(b) Show a correct graph for 0 < x < 2π with the maximum at x = π B1 FT The FT is for graphs of the form y = a(1 − cos x) , where a is positive. 1
Question
(a) Express 7 sin x + 2 cos x in the form R sin x + !, where R > 0 and 0Å < ! < 90Å. State the exact value of R and give ! correct to 2 decimal places. [3] (b) Hence solve the equation 7 sin 21 + 2 cos 21 = 1, for 0Å < 1 < 180Å. [5] 5a
Mark scheme
5(a) State R = 11 B1 Use trig formulae to find α M1 Obtain α = 37.09° A1 3 5(b) 1 B1 FT The FT is on R. Evaluate sin −1 to at least 2 dp (17.5484°) 11 Use correct method to find a value of θ in the interval M1 Obtain answer, e.g. 62.7° A1 Use a correct method to obtain a second answer M1 Obtain second answer, e.g. 170.2° , and no other in the interval A1 Ignore answers outside the given interval. 5
Question
Let f x = , where a is a positive constant. 2x − a 3a − x (a) Express f x in partial fractions. [3] 2a (b) Hence show that Ó f x dx = ln 6. [4] a ` a ` a ` a ` a 1 2 2 1
Mark scheme
6(a) Carry out a relevant method to determine constants A and B such that M1 5a A B = + ( 2 x − a )( 3a − x ) 2 x − a 3a − x Obtain A = 2 A1 Obtain B = 1 A1 3 6(b) Integrate and obtain terms ln(2 x − a) − ln ( 3a − x ) B1 FT The FT is on the values of A and B. B1 FT Substitute limits correctly in a solution containing terms of the form M1 b ln ( 2 x − a ) and c ln ( 3a − x ) , where bc ≠ 0 Obtain the given answer showing full and correct working A1 4
Question
Two lines have equations r = 3 + s −1 and r = 1 + t −1 . 2 3 4 4 (a) Show that the lines are skew. [5] (b) Find the acute angle between the directions of the two lines. [3]
Mark scheme
7(a) Express general point of a line in component form, e.g. B1 (1 + 2s, 3 – s, 2 + 3s) or (2 + t, 1 – t, 4 + 4t) Equate at least two pairs of components and solve for s or for t M1 2 A1 Obtain correct answer for s or for t (possible answers are –1, 6, for s and 5 1 –3, 4, − for t ) 5 Verify that all three component equations are not satisfied A1 Show that the lines are not parallel and are thus skew A1 5 7(b) Carry out correct process for evaluating the scalar product of the direction M1 vectors Using the correct process for the moduli, divide the scalar product by the M1 product of the moduli and evaluate the inverse cosine of the result Obtain answer 19.1° or 0.333 radians A1 3
Question
The complex numbers u and v are defined by u = −4 + 2i and v = 3 + i. u (a) Find in the form x + iy, where x and y are real. [3] v u (b) Hence express in the form r ei1 , where r and 1 are exact. [2] v In an Argand diagram, with origin O, the points A, B and C represent the complex numbers u, v and 2u + v respectively. (c) State fully the geometrical relationship between OA and BC. [2] (d) Prove that angle AOB = 34 π. [2] e2x + 1
Mark scheme
8(a) Multiply numerator and denominator by 3 – i M1 OE Obtain numerator – 10 + 10i or denominator 10 A1 Obtain final answer – 1 + i A1 3 8(b) State or imply r = 2 B1 FT 3 B1 FT State or imply that θ = π 4 2 8(c) State that OA and BC are parallel B1 State that BC = 2OA B1 2 8(d) u M1 Use angle AOB = arg u − arg v = arg v Obtain the given answer A1 Alternative method for question 8(d) Obtain tan AOB from gradients of OA and OB and the tan( A ± B) formula M1 Obtain the given answer A1 Alternative method for question 8(d) Obtain cos AOB by using the cosine rule or a scalar product M1 Obtain the given answer A1 2
Question
Let f x = , for x > 0. e2x − 1 (a) The equation x = f x has one root, denoted by a. Verify by calculation that a lies between 1 and 1.5. [2] (b) Use an iterative formula based on the equation in part (a) to determine a correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3] (c) Find f ′ x. Hence find the exact value of x for which f ′ x = −8. [6]
Mark scheme
9(a) Calculate the values of a relevant expression or pair of expressions at x = 1 M1 and x = 1.5 Complete the argument correctly with correct calculated values A1 2 9(b) e2 xn + 1 M1 Use the iterative formula xn+1 = , or equivalent, correctly at least e2 xn − 1 once Obtain final answer 1.20 A1 Show sufficient iterations to 4 dp to justify 1.20 to 2 dp, or show there is a A1 sign change in the interval (1.195,1.205) 3 9(c) Use quotient rule M1 Obtain correct derivative in any form A1 Equate derivative to – 8 and obtain a quadratic in e2 x M1 ( ) − 5e + 2 = 0 A1 OE 2 Obtain 2 e 2 x 2x Solve a 3-term quadratic in e 2 x for x M1 1 A1 Obtain answer x = ln 2 , or exact equivalent, only 2 Alternative method for question 9(c) Use quotient rule M1 Obtain correct derivative in any form A1 Equate derivative to – 8, take square roots and obtain a quadratic in ex M1 Obtain 2e2 x − e x − 2 = 0 A1 OE Solve a 3-term quadratic in ex for x M1 1 A1 Obtain answer x = ln 2 , or exact equivalent, only 2 6
Question
y M x O 1 2π The diagram shows the curve y = sin 2x cos2 x for 0 ≤ x ≤ 12 π, and its maximum point M . (a) Using the substitution u = sin x, find the exact area of the region bounded by the curve and the x-axis. [5] (b) Find the exact x-coordinate of M . [6] 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
10(a) State or imply du = cos x dx B1 Using double angle formula for sin2x and Pythagoras, express integral in M1 terms of u and du. ( ) Obtain integral 2 u − u 3 du A1 OE Use limits u = 0 and u = 1 in an integral of the form au 2 + bu 4 , where M1 1 a + b or a + b − 0 a = 1 and b = − ab ≠ 0 2 1 A1 Obtain answer 2 5 10(b) Use product rule M1 Obtain correct derivative in any form A1 Equate derivative to zero and use a double angle formula *M1 Obtain an equation in one trig variable DM1 Obtain 4 sin 2 x = 1 , 4 cos2 x = 3 or 3 tan 2 x = 1 A1 1 A1 Obtain answer x = π 6 6
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Mathematics 9709/32 for February/March 2021 is out of 75 marks, with 1 hour 50 minutes to answer it.
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