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Cambridge AS & A Level Mathematics (9709), Paper 3 Pure Mathematics 3, May/June 2020. Read both free below, or download them. Done the paper? Get it marked by AI against this mark scheme.
9709_s20_qp_319709_s20_ms_319709/31/M/J/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
Find the set of values of x for which 2 31−2x < 5x . Give your answer in a simplified exact form. [4]
Mark scheme
1 Use law of the logarithm of a product or power M1 Obtain a correct linear inequality in any form, e.g. ln 2 + (1 – 2x) ln 3 < x ln 5 A1 Solve for x M1 ln 6 A1 Obtain x > ln 45 4
Question
(a) Expand 2 − 3x−2 in ascending powers of x, up to and including the term in x2 , simplifying the coefficients. [4] (b) State the set of values of x for which the expansion is valid. [1]
Mark scheme
2(a) 3 −2 M1 State a correct unsimplified version of the x or x2 term of the expansion of (2 – 3x)–2 or 1 − x 2 1 B1 State correct first term 4 3 27 2 A1 + A1 Obtain the next two terms x+ x 4 16 4 2(b) 2 B1 State answer x < , or equivalent 3 1
Question
Express the equation tan 1 + 60Å = 2 + tan 60Å − 1 as a quadratic equation in tan 1, and hence solve the equation for 0Å ≤ 1 ≤ 180Å. [6]
Mark scheme
3 Use tan (A ± B) formula and obtain an equation in tan θ M1 Using tan 60° = 3 , obtain a horizontal equation in tan θ in any correct form A1 Reduce the equation to 3 tan 2 θ + 4 tan θ − 1 = 0 , or equivalent A1 Solve a 3-term quadratic for tan θ M1 Obtain a correct answer, e.g. 12.1° A1 Obtain a second correct answer, e.g. 122.9°, and no others in the given interval A1 6
Question
The curve with equation y = e2x sin x + 3 cos x has a stationary point in the interval 0 ≤ x ≤ π. (a) Find the x-coordinate of this point, giving your answer correct to 2 decimal places. [4] (b) Determine whether the stationary point is a maximum or a minimum. [2]
Mark scheme
4(a) Use product rule M1 Obtain derivative in any correct form e.g. 2e 2 x ( sin x + 3 cos x ) + e 2 x ( cos x − 3sin x ) A1 Equate derivative to zero and obtain an equation in one trigonometric ratio M1 Obtain x = 1.43 only A1 4 4(b) Use a correct method to determine the nature of the stationary point M1 x = 1.42, y′ = 0.06e 2.84 >0 e.g. x = 1.44, y′ = −0.07e 2.88 <0 Show that it is a maximum point A1 2
Question
(a) Find the quotient and remainder when 2x3 − x2 + 6x + 3 is divided by x2 + 3. [3] 2x3 − x2 + 6x + 3 3 (b) Using your answer to part (a), find the exact value of Ô x2 + 3 dx. [5] 1
Mark scheme
5(a) Commence division and reach quotient of the form 2x + k M1 Obtain quotient 2x – 1 A1 Obtain remainder 6 A1 3 5(b) Obtain terms x2 – x B1FT (FT on quotient of the form 2x + k) x M1 Obtain term of the form a tan −1 3 6 x A1FT Obtain term tan −1 3 3 (FT on a constant remainder) Use x = 1 and x = 3 as limits in a solution containing a term of the form a tan −1 ( bx ) M1 1 A1 Obtain final answer π + 6 , or exact equivalent 3 5
Question
A r O 2x rad T B The diagram shows a circle with centre O and radius r . The tangents to the circle at the points A and B meet at T , and angle AOB is 2x radians. The shaded region is bounded by the tangents AT and BT , and by the minor arc AB. The area of the shaded region is equal to the area of the circle. (a) Show that x satisfies the equation tan x = π + x. [3] (b) This equation has one root in the interval 0 < x < 12 π. Verify by calculation that this root lies between 1 and 1.4. [2] (c) Use the iterative formula xn+1 = tan−1 π + xn to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3] Let f x = cos x 1 + sin x
Mark scheme
6(a) State or imply AT = r tan x or BT = r tan x B1 Use correct area formula and form an equation in r and x M1 Rearrange in the given form A1 3 6(b) Calculate the values of a relevant expression or pair of expressions at x = 1 and x = 1.4 M1 Complete the argument correctly with correct calculated values A1 2 6(c) Use the iterative formula correctly at least once M1 Obtain final answer 1.35 A1 Show sufficient iterations to 4 d.p. to justify 1.35 to 2 d.p. or show there is a sign change in the interval (1.345, 1.355) A1 3
Question
. (a) Show that f ′ x < 0 for all x in the interval − 12 π < x < 32 π. [4] 1π (b) Find Ó 2 f x dx. Give your answer in a simplified exact form. [4] 1π 6 y
Mark scheme
7(a) Use quotient or product rule M1 − sin x (1 + sin x ) − cos x ( cos x ) A1 Obtain derivative in any correct form e.g. (1 + sin x ) 2 Use Pythagoras to simplify the derivative M1 Justify the given statement A1 4 7(b) State integral of the form a ln (1 + sin x) *M1 State correct integral ln (1 + sin x) A1 Use limits correctly DM1 4 A1 Obtain answer ln 3 4
Question
A certain curve is such that its gradient at a point x, y is proportional to . The curve passes x x through the points with coordinates 1, 1 and 4, e. (a) By setting up and solving a differential equation, find the equation of the curve, expressing y in terms of x. [8] (b) Describe what happens to y as x tends to infinity. [1]
Mark scheme
8(a) dy y B1 State =k , or equivalent dx x x Separate variables correctly and attempt integration of at least one side M1 Obtain term ln y, or equivalent A1 1 A1 Obtain term −2k , or equivalent x b M1 Use given coordinates to find k or a constant of integration c in a solution containing terms of the form a ln y and , where x ab ≠ 0 Obtain k = 1 and c = 2 A1 + A1 2 A1 Obtain final answer y = exp − + 2 , or equivalent x 8 8(b) State that y approaches eଶ B1FT (FT their c in part (a) of the correct form) 1 ρ ρ ρ ρ
Question
With respect to the origin O, the vertices of a triangle ABC have position vectors OA = 2i + 5k, OB = 3i + 2j + 3k and OC = i + j + k. (a) Using a scalar product, show that angle ABC is a right angle. [3] (b) Show that triangle ABC is isosceles. [2] (c) Find the exact length of the perpendicular from O to the line through B and C. [4] (a) The complex number u is defined by u = 3i a + 2i
Mark scheme
9(a) ( ) ( ) State AB or BA and BC or CB in vector form B1 Calculate their scalar product M1 Show product is zero and confirm angle ABC is a right angle A1 3 9(b) Use correct method to calculate the lengths of AB and BC M1 Show that AB = BC and the triangle is isosceles A1 2 9(c) State a correct equation for the line through B and C, B1 e.g. r = i + j + k + λ (2i + j + 2k) or r = 3i + 2 j + 3k + μ ( −2i − j − 2k ) ρ ρ Taking a general point of BC to be P, form an equation in λ by either equating the scalar product of OP and BC to zero, M1 ρ or applying Pythagoras to triangle OBP (or OCP), or setting the derivative of OP to zero 5 A1 Solve and obtain λ = – 9 1 A1 Obtain answer 2 , or equivalent 3 Alternative method for question 9(c) Use a scalar product to find the projection CN (or BN) of OC (or OB) on BC M1 5 14 A1 Obtain answer CN = or BN = 3 3 Use Pythagoras to find ON M1 1 A1 Obtain answer 2 , or equivalent 3 4
Question
, where a is real. (i) Express u in the Cartesian form x + iy, where x and y are in terms of a. [3] (ii) Find the exact value of a for which arg u* = 13 π. [3] (b) (i) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities z − 2i ≤ z − 1 − i and z − 2 − i ≤ 2. [4] (ii) Calculate the least value of arg z for points in this region. [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
10(a)(i) Multiply numerator and denominator by a – 2i, or equivalent M1 Use i2 = –1 at least once A1 6 3ai A1 Obtain answer 2 + 2 a +4 a +4 3 10(a)(ii) 1 B1 Either state that arg u = − π or express u* in terms of a (FT on u) 3 Use correct method to form an equation in a M1 Obtain answer a = −2 3 A1 3 10(b)(i) Show the perpendicular bisector of points representing 2i and 1 + i B1 Show the point representing 2 + i B1 Show a circle with radius 2 and centre 2 + i B1FT (FT on the position of the point for 2 + i) Shade the correct region B1 4 10(b)(ii) State or imply the critical point 2 + 3i B1 Obtain answer 56.3° or 0.983 radians B1 2
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Mathematics 9709/31 for May/June 2020 is out of 75 marks, with 1 hour 50 minutes to answer it.
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