Cambridge A Level Mathematics 9709 — 2020 May/June Paper 1 · Variant 2

2 (a) Express the equation 3 cos 1 = 8 tan 1 as a quadratic equation in sin 1. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Hence find the acute angle, in degrees, for which 3 cos 1 = 8 tan 1. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 2(a) 3cos 8tan θ θ = → 8sin 3cos cos θ θ θ = M1 3(1 − 2 sin θ ) = 8sinθ M1 3 sin² θ + 8 sin θ – 3 = 0 A1 3 2(b) (3sinθ – 1)(sinθ + 3) = 0 → sin θ = ⅓ M1 θ = 19.5° A1 2 Question Answer Marks

More questions on Trigonometry

3 A weather balloon in the shape of a sphere is being inflated by a pump. The volume of the balloon is increasing at a constant rate of 600 cm3 per second. The balloon was empty at the start of pumping. (a) Find the radius of the balloon after 30 seconds. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the rate of increase of the radius after 30 seconds. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 3(a) Volume after 30 s = 18000 4 π ³ 18000 3 r = M1 r = 16.3 cm A1 2 3(b) d 4π ² d V r r = B1 d d r t = d d r V × d d V t = 600 4π ²r M1 d d r t = 0.181 cm per second A1 3 Question Answer Marks

More questions on Differentiation

7 A r C r 16π rad O B 2r In the diagram, OAB is a sector of a circle with centre O and radius 2r, and angle AOB = 16π radians. The point C is the midpoint of OA. (a) Show that the exact length of BC is r 5 −2 3. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the exact perimeter of the shaded region. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (c) Find the exact area of the shaded region. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 7(a) BC² = r² + 4r² − 2r.2r × cos π 6       = 5r² − 2r²√3 M1 BC = ( ) 5 2 3 r − A1 2 7(b) Perimeter = 2 6 π + + r r ( ) 5 2 3 r − M1 A1 2 7(c) Area = sector – triangle Sector area = 1 π 4 ² 2 6 r M1 Triangle area = ½ r. 2r sin π 6 M1 Shaded area = π 1 ² 3 2 r   −     A1 3 Question Answer Marks

More questions on Trigonometry

10 The equation of a curve is y = 54x − 2x −7 3. dy d2y (a) Find and . [4] dx dx2 ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the coordinates of each of the stationary points on the curve. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (c) Determine the nature of each of the stationary points. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 10(a) d d y x = 54 – 6(2x – 7)² B2,1 d² d ² y x = −24(2x – 7) (FT only for omission of ‘ 2 × ’ from the bracket) B2,1 FT 4 10(b) ( ) 2 d 0 2 7 9 d = → − = y x x M1 x = 5, y = 243 or x = 2, y = 135 A1 A1 3 10(c) x = 5 d² d ² y x = −72 → Maximum (FT only for omission of ‘ 2 × ’ from the bracket) B1FT x = 2 d² d ² y x = 72 → Minimum (FT only for omission of ‘ 2 × ’ from the bracket) B1FT 2 Question Answer Marks

More questions on Differentiation