Sec 3 WA3 Revision Time Trial 2023 Beatty

Secondary 3 Mathematics WA3 Revision Time Trial — Beatty (2023) Duration: 48 minutes Score: / 32 Topics tested: Graphs of Functions; Graphs in Practical Situations; Further Trigonometry; Applications of Trigonometry. Section A: Graphs of Functions [8 marks] 1. The variables x and y are connected by the equation y = 3 2 x2 + 1 2 x3. Some corresponding values of x and y are given in the following table. x −4 −3.5 −3 −2.5 −1 0 1 1.5 2 y −8 a 0 1.55 1 0 2 5.06 10 (a) Find the value of a. [1] Answer a = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) On the graph paper provided on the next page, draw the graph of y = 3 2 x2 + 1 2 x3 using a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis. [3]

(c) Use your graph to write down the range of values of x for which y > 1. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (d) (i) On the same axes, draw the line with gradient −2 that passes through the point (2, 1). [1] (ii) Write down the x-coordinate of the point of intersection of the line from d(i) and the graph of y = 3 2 x2 + 1 2 x3. [1] Answer x = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Section B: Graphs in Practical Situations [5 marks] 2. The following diagram shows the speed-time graph of a car during a journey. The car accelerated from rest at 1.25 m/s 2 to a speed of v m/s in 20 seconds. It travelled at this speed until t seconds before it came to a stop at t = 80 seconds. The total distance travelled for the whole journey was 1.45 km. The distance travelled by the car during a certain time interval is equal to the area bounded by the graph and the time-axis in that time interval. (a) Find the maximum speed of the journey. [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m/s (b) Calculate the average speed, in km/h, of the car for the whole journey. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . km/h (c) Find the value of t. [2] Answer t = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Section C: Further Trigonometry [6 marks] 3. (a) The sine of an angle is 0.8711. Give two possible values for the angle. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .or . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) In triangle P QR, P Q = 9 cm, QR = 40 cm and P R = 41 cm. (i) Show that angle P QR is right-angled. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (ii) Write down the values of (a) sin ∠P RQ, [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) cos ∠P RS. [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Section D: Applications of Trigonometry [13 marks] 4. ABC is a triangular plot of land. It is given that AB = 93 m, AC = 56 m and ∠BAC = 105 ◦. A is due south of C. (a) Find BC . [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m (b) Find the bearing of C from B. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .◦

(c) A tree of height 25 m is planted at A. A man walks from B to C. Calculate the largest angle of elevation of the top of the tree from the man. [3] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .◦ 5. ABCDEF GH is a cube of side 6 cm. The point X lies on the side GH such that GX : XH = 1 : 3 . (a) Find the length of EX . [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm

(b) Find ∠AXE . [4] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .◦ — END OF PAPER —

Answer Key Sec 3 WA3 Revision Time Trial — Beatty (2023) Section A: Graphs of Functions [8 marks] 1(a) At x = −3.5: y = 3 2 (−3.5)2 + 1 2 (−3.5)3 = 18.375 − 21.4375 = −3.0625. ∴ a = −3.06 (3 s.f.). [1] 1(b) Axes ruled with the given scale ( x: 2 cm per unit from −4 to 2; y: 1 cm per unit from −8 to 10), all nine points plotted and joined with a smooth curve: local maximum ≈ (−2, 2), local minimum at (0, 0). [3] (Marking scheme graph shown, including the line for (d)(i).) 1(c) y = 1 line drawn; curve above it for: −2.75 < x < −1 and x > 0.8 (accept −2.8 to −2.7 for the left end, 0.75 to 0.85 for the right). [2] 1(d)(i) Line through (2, 1) with gradient −2, i.e. y = −2x + 5 (e.g. through (0, 5) and (2, 1)), drawn on the same axes. [1] 1(d)(ii) From the graph: x ≈ 1.1 (accept 1.05 to 1.15). [1]

Section B: Graphs in Practical Situations [5 marks] 2(a) v 20 = 1.25 ⇒ v = 25 m/s. [1] 2(b) Average speed = 1450 80 = 18.125 m/s = 18.125 × 3.6 = 65.25 km/h. [2] 2(c) Area under graph (trapezium) = 1 2 [ 80 + (t − 20) ] (25) = 1450 80 + t − 20 = 116 ⇒ t = 56. [2] Section C: Further Trigonometry [6 marks] 3(a) sin θ = 0.8711 ⇒ θ = 60.6◦ or 180◦ − 60.6◦ = 119.4◦. ∴ θ = 60.6◦ or 119.4◦ (1 d.p.). [2] 3(b)(i) P R2 = 41 2 = 1681 and P Q2 + QR2 = 9 2 + 402 = 1681. Since P R2 = P Q2 + QR2, by the Converse of Pythagoras’ Theorem , triangle P QR is right-angled and ∠P QR = 90 ◦. [2] 3(b)(ii)(a) sin ∠P RQ = P Q P R = 9 41 . [1] 3(b)(ii)(b) ∠P RS = 180 ◦ − ∠P RQ (adjacent ∠s on straight line QRS) cos ∠P RS = − cos ∠P RQ = − QR P R = − 40 41 . [1]