Sec3 WA3 Revision Time Trial 2023 Bedok View

Secondary 3 Mathematics WA3 Revision Time Trial — Bedok View (2023) Duration: 66 minutes Score: / 44 Topics tested: Graphs of Functions; Further Trigonometry; Applications of 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 = x3 5 − x + 2. Some corresponding values of x and y are given in the table below. x −4 −3 −2 −1 0 1 2 3 y −6.8 p 2.4 2.8 2 1.2 1.6 4.4 (a) Find the value of p. [1] Answer p = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) On the grid on the next page, draw the graph of y = x3 5 − x + 2 for −4 ≤ x ≤ 3. [3]

x y 0−4 −3 −2 −1 1 2 3 4 2 4 6 8 −2 −4 −6

(c) By drawing a suitable tangent on the same grid, find the gradient of the curve when x = 2. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (d) (i) On the same grid, draw the graph of 2y = −x + 2. [1] (ii) Using your graph, state the x-coordinate of the intersection point between the curve and the line. [1] Answer x = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (iii) The roots of the equation 2x3 + ax2 + bx + c = 0 is given by the x-value of the intersection point in part (d)(ii). Find the values of a, of b and of c. [3] Answer a = . . . . . . . . . . . . . . . . . . . . . . . . . ,b = . . . . . . . . . . . . . . . . . . . . . . . . . ,c = . . . . . . . . . . . . . . . . . . . . . . . . .

Section B: Further Trigonometry [11 marks] 2. Given that 7 sin x = 3, find two possible values for angle x, where 0◦ ≤ x ≤ 180◦. [2] Answer x = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .or . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. In the diagram below, DE = 15 cm, DF = 17 cm, DG = 25 cm, EF = 8 cm and EF GH is a straight line. (a) Show that triangle DEF is a right-angled triangle. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) Find angle DGF . [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .◦

(c) Find the length of F G. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm (d) If angle DF G = θ, find, without the use of a calculator, (i) cos θ, [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (ii) sin(90◦ − ∠DF E). [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Section C: Applications of Trigonometry [11 marks] 4. The following figure shows a park ABCD . AD = 570 m, AC = 610 m and BC = 420 m. B is east of A and BCD is a straight line. Angle ACB = 120 ◦. (a) Calculate the length of AB. [3] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m (b) Calculate angle ADC . [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .◦

(c) Calculate the bearing of D from B. [3] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .◦ (d) A drone is spotted 70 m directly above C. Find the greatest angle of elevation of the drone from a point along AB. [3] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .◦

Section D: Arc Length and Sector Area [11 marks] 5. The diagram shows two circles that touch at C. A, B and C are points on the bigger circle with centre X. C, D and E are points on the smaller circle with centre Y . BCE and AXCY D are straight lines. (a) Given that angle BCX = 30 ◦, find (i) angle CXB , [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .◦ (ii) angle XAB . [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .◦ (b) What type of triangle is triangle XAB ? [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) The radius of the larger circle is 9 cm and the radius of the smaller circle is 6 cm. (i) Find the area of the minor sector XAB , leaving your answer in terms of π. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm2 (ii) Calculate the percentage of the shaded areas to the total area of the two circles. [6] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . % — END OF PAPER —

Answer Key Sec 3 WA3 Revision Time Trial — Bedok View (2023) Section A: Graphs of Functions [11 marks] 1(a) At x = −3: y = −27 5 + 3 + 2 = −0.4. [1] 1(b) All eight points plotted and joined with a smooth curve: rising from (−4, −6.8) to a local maxi- mum near (−1.3, 2.9), dipping to a local minimum near (1.3, 1.1), then rising to (3, 4.4). [3] (Marking scheme graph shown, including the tangent for (c) and the line for (d)(i).) 1(c) T angent drawn at x = 2. Exact gradient: dy dx = 3x2 5 − 1 = 12 5 − 1 = 1.4 (accept 1.2 to 1.8). [2] 1(d)(i) Line 2y = −x + 2, i.e. y = − 1 2 x + 1, drawn through (0, 1) and (2, 0). [1] 1(d)(ii) From the graph: x ≈ −2.15 (accept ±0.1). [1] 1(d)(iii) x3 5 − x + 2 = − 1 2 x + 1 ×10: 2x3 − 10x + 20 = −5x + 10 ⇒ 2x3 − 5x + 10 = 0 ∴ a = 0, b = −5, c = 10. [3] Section B: Further Trigonometry [11 marks]