Sec 3 WA3 Revision Time Trial 2023 Bedok South

Secondary 3 Mathematics WA3 Revision Time Trial — Bedok South (2023) Duration: 47 minutes Score: / 31 Topics tested: Graphs of Functions; Graphs in Practical Situations; Further Trigonometry; Arc Length and Sector Area. Section A: Graphs of Functions [11 marks] 1. The variables x and y are connected by the equation y = 1 − x3 + 3x2. The table below shows some corresponding values of x and y, correct to 1 decimal place where appropriate. x −1 −0.5 0 0.5 1 1.5 2 2.5 3 y 5 1.9 1 p 3 4.4 5 4.1 1 (a) Calculate the value of p, giving your answer correct to 1 decimal place. [1] Answer p = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) Using a scale of 2 cm to represent 0.5 units on a horizontal x-axis for −1 ≤ x ≤ 3 and 2 cm to represent 1 unit on a vertical y-axis for −1 ≤ y ≤ 7, on the graph paper provided on the next page, plot the points given in the table and join them with a smooth curve. [3]

(c) Use your graph to find all the solutions of 1 − x3 + 3x2 = 2. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (d) By drawing a suitable tangent, find the gradient of the curve y = 1 − x3 + 3x2 at the point where x = 2.5. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (e) (i) On the same axes, draw the line y = −2x + 5 for −1 ≤ x ≤ 3. [2] (ii) Hence, use your graph to find the range of values of x such that 1 − x3 + 3x2 ≥ −2x + 5 in the range −1 ≤ x ≤ 3. [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Section B: Graphs in Practical Situations [4 marks] 2. The diagram shows the speed-time graph of a car which travelled from P to S. (a) How long was the car travelling at constant speed? [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) What is the distance travelled by the car from P to S? [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (c) What is the deceleration from Q to R in km/h 2? [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Section C: Further Trigonometry [5 marks] 3. ABD is a triangle with AB = 12 cm, AC = 20 cm, CD = 19 cm, AD = 37 cm and BD = 35 cm. (a) Prove that △ABC is a right-angled triangle. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) Express the following as a fraction in its lowest term. (i) sin ∠ACD [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (ii) cos ∠CAD [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Section D: Arc Length and Sector Area [11 marks] 4. The diagram shows the cross-section of 3 circular tins of the same size with centres P , Q and R. The tins touch one another externally at L, M and N . (a) (i) Show that ∠P QR = π 3 radians. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (ii) Given that the area of △P QR = 6 .93 cm2, show that the length QL is approximately 2.00 cm. [2]

(b) Calculate the shaded area. [3] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm2 (c) The circular tins are bounded together by a piece of rope ABCDEF . Calculate the total length of the rope used. [4] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm — END OF PAPER —

Answer Key Sec 3 WA3 Revision Time Trial — Bedok South (2023) Section A: Graphs of Functions [11 marks] 1(a) At x = 0.5: y = 1 − 0.125 + 0.75 = 1 .625. ∴ p = 1.6 (1 d.p.). [1] 1(b) Axes ruled with the given scale ( x: 2 cm per 0.5 unit from −1 to 3; y: 2 cm per unit from −1 to 7), all nine points plotted and joined with a smooth curve: local minimum at (0, 1), rising to a local maximum at (2, 5), then falling to (3, 1). [3] 1(c) 1 − x3 + 3x2 = 2 is where the curve meets the horizontal line y = 2: x ≈ −0.53, 0.65 and 2.88 (accept ±0.05 each). [2] 1(d) T angent drawn at x = 2.5. Exact gradient: dy dx = −3x2 + 6x = −18.75 + 15 = −3.75 (accept ±0.5). [2] 1(e)(i) Line y = −2x + 5 drawn through e.g. (0, 5), (2, 1) and (3, −1). [2] 1(e)(ii) The curve is on or above the line for 1 ≤ x ≤ 3. [1] Section B: Graphs in Practical Situations [4 marks] 2(a) Constant speed from 1000 to 1012, i.e. 12 minutes. [1] 2(b) Distance = area under graph = 1 2 ( 15 60 + 14 60 ) (60) + 1 2 ( 14 60 + 12 60 ) (15) = 14 .5 + 3.25 = 17.75 km. [2] 2(c) From Q to R the speed falls from 75 to 60 km/h in 2 minutes = 1 30 h. Deceleration = 15 1/30 = 450 km/h2. [1] Section C: Further Trigonometry [5 marks] 3(a) BC = BD − CD = 35 − 19 = 16 cm. AB2 + BC 2 = 12 2 + 162 = 400 and AC2 = 20 2 = 400. Since AB2 + BC 2 = AC2, by the Converse of Pythagoras’ Theorem , △ABC is a right-angled triangle. [2] 3(b)(i) sin ∠ACD = sin(180◦ − ∠ACB ) = sin ∠ACB = 12 20 = 3 5 . [1] 3(b)(ii) By the Cosine Rule in triangle ACD : cos ∠CAD = 202 + 372 − 192 2(20)(37) = 1408 1480 = 176 185 . [2]

Section D: Arc Length and Sector Area [11 marks] 4(a)(i) P Q = QR = RP (each is the sum of two equal radii), so △P QR is equilateral and ∠P QR = 60 ◦ = 180◦ 3 = π 3 radians (shown). [2] 4(a)(ii) Let the radius be r, so P Q = QR = 2r. 1 2 (2r)2 sin π 3 = 6.93 ⇒ 4r2 = 2(6.93) sin(π/3) = 16.004 . . . r = 2.0002 . . . ≈ 2.00 cm (3 s.f.), and QL = r (shown). [2] 4(b) Area of one sector (e.g. QLM ) = 1 2 (2)2 ( π 3 ) = 2.0944 cm2. Shaded area = area of △P QR − 3 sectors = 6.93 − 3(2.0944) = 0.647 cm2 (3 s.f.). [3] 4(c) At each tin the rope wraps an angle of 2π − 2 ( π 2 ) − π 3 = 2π 3 . Each arc = 2 ( 2π 3 ) = 4π 3 cm; each straight section = P Q = 4 cm. T otal length= 3 ( 4π 3 ) + 3(4) = 4 π + 12 = 24.6 cm (3 s.f.). [4]