Sec 3 WA3 Revision Time Trial 2023 Ahmad Ibrahim

Secondary 3 Mathematics WA3 Revision Time Trial — Ahmad Ibrahim (2023) Duration: 48 minutes Score: / 32 Topics tested: Graphs of Functions; Graphs in Practical Situations; Further Trigonometry; Arc Length and Sector Area. Section A: Graphs of Functions [10 marks] 1. The variables x and y are connected by the equation y = 1 2x2 + 1 10 x2 − 3. The table below shows some corresponding values of x and y, correct to 2 decimal places. x 0.5 1 1.5 2 3 4 5 6 y −0.98 −2.40 −2.55 −2.48 −2.04 p −0.48 0.61 (a) Calculate the value of p. [1] Answer p = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) On the grid on the next page, draw the graph of y = 1 2x2 + 1 10 x2 − 3 for 0.5 ≤ x ≤ 6. [3]

x y 0 1 2 3 4 5 6 1 0.5 −0.5 −1 −1.5 −2 −2.5 −3

(c) By drawing a tangent, find the gradient of the curve at x = 2. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (d) By drawing a suitable straight line on the same axes, solve the equation 1 2x2 + 1 10 x2 − x = 0. [4] Answer x = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Section B: Graphs in Practical Situations [4 marks] 2. The diagram shows the speed-time graph for a motorist’s journey. (a) Find the acceleration of the motorist during the first 10 seconds. [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m/s2 (b) Find the speed of the motorist at the 6 th second. [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m/s (c) Sketch the distance-time graph of the motorist for the journey on the axes below. [2]

Section C: Further Trigonometry [7 marks] 3. In the diagram, AB = 8 cm, CD = 6 cm, BC = CD and ACD is a straight line. (a) Show that angle ABC is a right angle. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) Find the value of (i) sin ∠BCD , [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (ii) cos ∠BCD . [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. In the diagram, AB = 127 km, BC = 256 km and CD = 278 km. Angle ACD = 37 ◦ and angle ADC = 82 ◦. Find angle ABC . [3] Answer Angle ABC = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .◦

Section D: Arc Length and Sector Area [11 marks] 5. OBC is a sector of a circle with centre O and radius 5.5 cm. A is a point on radius OB. Angle AOC = 0.872 radians and angle ACO = 0.942 radians. Calculate (a) OA, [2] Answer OA = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm

(b) the area of the shaded region ABC . [3] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm2 6. A sector of a circle radius 7 cm is removed from an isosceles triangle ABC as shown in the diagram. Angle EAD is 74◦ and BC = 22 cm. (a) Calculate the length of the arc ED. [2] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm

(b) (i) Show that the perpendicular height of the triangle ABC is 14.597 cm, correct to 5 significant figures. [1] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (ii) Hence find the area of BCDE . [3] Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm2 — END OF PAPER —

Answer Key Sec 3 WA3 Revision Time Trial — Ahmad Ibrahim (2023) Section A: Graphs of Functions [10 marks] 1(a) At x = 4: y = 1 2(4)2 + 1 10 (4)2 − 3 = 0 .03125 + 1.6 − 3 = −1.36875. ∴ p = −1.37 (2 d.p.). [1] 1(b) All eight tabulated points plotted and joined with a smooth curve: the curve falls from (0.5, −0.98) to a minimum of about −2.55 near x = 1.5, then rises through (6, 0.61). [3] (Marking scheme graph shown, including the tangent for (c) and the line y = x − 3 for (d).) 1(c) T angent drawn at (2, −2.48). Exact gradient: dy dx = − 1 x3 + x 5 = −0.125 + 0.4 = 0.275 (accept 0.175 to 0.375). [2] 1(d) 1 2x2 + 1 10 x2 − x = 0 ⇔ 1 2x2 + 1 10 x2 − 3 = x − 3. So draw the line y = x − 3 (e.g. through (0.5, −2.5) and (4, 1)). The line meets the curve once in range: x ≈ 0.8 (accept 0.7 to 0.9). [4]