Cambridge A Level Mathematics 9709 — 2019 May/June Paper 1 · Variant 1

k 6 1 The term independent of x in the expansion of 2x + , where k is a constant, is 540. x (i) Find the value of k. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) For this value of k, find the coefficient of x2 in the expansion. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 1(i) Ind term = ( ) 3 3 2 k x x   ×    × 6C3 B2,1,0 = 540 → k = 1½ B1 3 1(ii) Term, in x² is ( ) 2 4 2 k x x   ×    × 6C2 B1 All correct – even if k incorrect. 15 × 16 × k² = 540 (or 540 2 x ) B1 FT For 240k² or 240 2 2 k x 2

More questions on Series

2 The line 4y = x + c, where c is a constant, is a tangent to the curve y2 = x + 3 at the point P on the curve. (i) Find the value of c. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find the coordinates of P. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 2(i) Eliminates x or y → ² 4 3 0 y y c or ( ) ² 2 16 ² 48 0 x c x c + − + − = M1 Uses ² 4 b ac = → 4c – 28 = 0 M1 Uses discriminant = 0. (c the only variable) Any valid method (may be seen in part (i)) c = 7 A1 Alternative method for question 2(i) 1 1 4 2 ( 3) dy dx x = = + M1 Solving M1 c = 7 A1 3 2(ii) Uses c = 7, y² − 4y + 4 = 0 M1 Ignore (1,–2), c=-9 (1, 2) A1 2

More questions on Differentiation

3 A sector of a circle of radius r cm has an area of A cm2. Express the perimeter of the sector in terms of r and A. [4] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 3 Uses A = ½r²θ M1 Uses area formula. θ = 2 ² A r A1 P r r rθ = + + B1 2 2 A P r r = + A1 Correct simplified expression for P. 4

More questions on Circular measure

4 y C h, 3h D B 0, 2 x O A 4, 0 The diagram shows a trapezium ABCD in which the coordinates of A, B and C are 4, 0 , 0, 2 and h, 3h respectively. The lines BC and AD are parallel, angle ABC = 90Å and CD is parallel to the x-axis. (i) Find, by calculation, the value of h. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Hence find the coordinates of D. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 4(i) Gradient of AB = −½ → Gradient of BC = 2 M1 Use of m1.m2 = −1 for correct lines Forms equation in h 3 2 2 h h − = M1 Uses normal line equation or gradients for h. h = 2 A1 Alternative method for question 4(i) Vectors AB.BC=0 M1 Use of vectors AB and BC Solving M1 h = 2 A1 Alternative method for question 4(i) Use of Pythagoras to find 3 lengths M1 Solving M1 h = 2 A1 3 4(ii) y coordinate of D is 6, (3 × ‘their’ h) 6 0 2 4 x − = − → x = 7 → D (7, 6) B1 FT Vectors: AD.AB=0 M1 A1 Must use y = 6 Realises the y values of C and D are equal. Uses gradient or line equation to find x. 3

More questions on Coordinate geometry

1 2 1 −sin x 6 (i) Prove the identity −tan x  [4] cosx 1 + sin x. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ @ A2 1 1(ii) Hence solve the equation −tan 2x = for 0 ≤x ≤0. [3] cos 2x 3 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 6(i) LHS = 2 1 s c c   −     1 1 ² s s c − − 2 1 1 1 s s s − − = − B1 B1 correctly 1– s2 as the denominator = ( )( ) ( )( ) 1 1 1 1 s s s s − − − + M1 Factors and correct cancelling www 1 sin 1 sin x x − + AG A1 4 Question Answer Marks Guidance 6(ii) Uses part (i) to obtain 1 sin2 1 sin2 x x − + = 1 3 → sin 2x = ½ M1 Realises use of 2x and makes sin2x the subject x = 12 π A1 Allow decimal (0.262) (or) x = 5 12 π A1 FT for ½π – 1st answer. Allow decimal (1.31) 12 π and 5 12 π only, and no others in range. SC sinx=½ → 6 π 5 6 π B1 3

More questions on Trigonometry

7 G F D E M C B k j O i A The diagram shows a three-dimensional shape in which the base OABC and the upper surface DEFG are identical horizontal squares. The parallelograms OAED and CBFG both lie in vertical planes. The point M is the mid-point of AF. Unit vectors i and j are parallel to OA and OC respectively and the unit vector k is vertically upwards. −−→ −−→ The position vectors of A and D are given by OA = 8i and OD = 3i + 10k. −−→ −−−→ (i) Express each of the vectors AM and GM in terms of i, j and k. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Use a scalar product to find angle GMA correct to the nearest degree. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 7(i) AM = 1.5i + 4j + 5k GM JJJJG = 6.5i – 4j − 5k 3 7(ii) AM JJJJG . GM JJJJG = 9.75 −16 – 25 = −31.25 M1 Use of 1 2 1 2 1 2 x x y y z z + + on AM and GM AM JJJJG . GM JJJJG = √(1.5²+4²+5²) × √( 6.5²+4²+5²) cos GMA M1 M1 M1 for product of 2 modulii M1 all correctly connected Equating → Angle GMA = 121° A1 4

More questions on Vectors

8 (a) The third and fourth terms of a geometric progression are 48 and 32 respectively. Find the sum to infinity of the progression. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Two schemes are proposed for increasing the amount of household waste that is recycled each week. Scheme A is to increase the amount of waste recycled each month by 0.16 tonnes. Scheme B is to increase the amount of waste recycled each month by 6% of the amount recycled in the previous month. The proposal is to operate the scheme for a period of 24 months. The amount recycled in the first month is 2.5 tonnes. For each scheme, find the total amount of waste that would be recycled over the 24-month period. [5] Scheme A ....................................................................................................................................... ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ Scheme B ....................................................................................................................................... ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 8(a) ar² = 48, ar³ = 32, r = ⅔ or a = 108 M1 Solution of the 2 eqns to give r (or a). A1 (both) r = ⅔ and a = 108 A1 S∞ = 108 1 3 = 324 A1 FT Needs correct formula and r between −1 and 1. 3 8(b) Scheme A a = 2.50, d = 0.16 Sn = 12(5 + 23×0.16) M1 Correct use of either AP Sn formula. Sn = 104 tonnes. A1 Scheme B a = 2.50, r = 1.06 B1 Correct value of r used in GP. = ( ) 24 2.5 1.06 1 1.06 1 − − M1 Correct use of either Sn formula. Sn = 127 tonnes. A1 5

More questions on Series

d2y 10 A curve for which = 2x −5 has a stationary point at 3, 6 . dx2 (i) Find the equation of the curve. [6] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find the x-coordinate of the other stationary point on the curve. [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (iii) Determine the nature of each of the stationary points. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 10(i) integrating → d d y x = x² − 5x (+c) B1 = 0 when x = 3 M1 Uses the point to find c after ∫ = 0. c = 6 A1 integrating again → ( ) ³ 5 ² 6 3 2 x x y x d = − + + B1 FT Integration again FT if a numerical constant term is present. use of (3, 6) M1 Uses the point to find d after ∫ = 0. d = 1½ A1 6 Question Answer Marks Guidance 10(ii) d d y x = x² − 5x + 6 = 0 → x = 2 B1 1 10(iii) x = 3, d²y 1 d ²x = and/or +ve Minimum. x = 2, d²y 1 d ²x =− and/or −ve Maximum B1 www May use shape of ‘ 3 x + ’ curve or change in sign of dy dx B1 www SC: 3 x = , minimum, 2 x = , maximum, B1 2

More questions on Differentiation

11 y 3 y = 1 + 4x P 2, 1 x O Q 3 The diagram shows part of the curve y = and a point P 2, 1 lying on the curve. The 1 + 4x normal to the curve at P intersects the x-axis at Q. (i) Show that the x-coordinate of Q is 16 . [5] 9 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find, showing all necessary working, the area of the shaded region. [6] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 11(i) 3 × −½ × ( ) 3 2 1 4x − + d d y x = 3 × −½ × ( ) 3 2 1 4x − + × 4 B1 Must have ‘× 4’ If x = 2, m = 2 9 − , Perpendicular gradient = 9 2 M1 Use of m1.m₂ = − 1 Equation of normal is ( ) 9 1 2 2 y x −= − M1 Correct use of line eqn (could use y=0 here) Put y = 0 or on the line before → 16 9 A1 AG 5 Question Answer Marks Guidance 11(ii) Area under the curve = 2 0 3 1 4x + ∫ dx = 3 1 4 1 2 x + ÷ 4 B1 B1 Correct without ‘÷4’. For 2nd B1, ÷4’. Use of limits 0 to 2 → 4½ − 1½ M1 Use of correct limits in an integral. 3 A1 Area of the triangle = ½ × 1 × 2 9 = 1 9 or attempt to find 2 16/9 9 8 2 x dx   −     ∫ M1 Any correct method. Shaded area = 3 − 1 9 = 2 8 9 A1 6

More questions on Integration