OPSS Prelims 2024 4E Math P2 QP 11 Aug

1 [Turn Over CANDIDATE NAME CLASS INDEX NUMBER MATHEMATICS Paper 2 Secondary 4 Express / 5 Normal (Academic) Setter: Mrs. Jay 4052/02 19 August 2024 2 hours 15 minutes 90 Marks Additional materials: NIL READ THESE INSTRUCTIONS FIRST Write your name, class and index number on all the work you hand in. Write in dark blue or black pen. Use a pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all questions. If working is needed for any question, it must be shown in the space below the question. Omission of essential working will result in loss of marks. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For , use either your calculator value or 3.142, unless the question requires the answer in terms of . The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 90. This document consists of 23 printed pages. ORCHID PARK SECONDARY SCHOOL Preliminary Examination 2024 For Examiner’s Use Total Calculator Model :

2 Compound interest Total amount = Mensuration Curved surface area of a cone = Surface area of a sphere = Volume of a cone = Volume of a sphere = Area of triangle ABC = Arc length = , where is in radians Sector area = , where is in radians Trigonometry Statistics Mean = Standard deviation = n rP      + 1001 rl 24 r hr 2 3 1 3 3 4 r Cabsin2 1 r  2 2 1 r  C c B b A a sinsinsin == Abccba cos2222 −+=   f fx 22       −     f fx f fx Mathematical Formulae

3 [Turn Over Answer all the questions. 1 (a) (i) Express 2𝑥2 − 6𝑥 − 12 in the form 𝑘(𝑥 − 𝑎)2 − 𝑏, where k is an integer. Answer: …….………………………… [1] (ii) Hence, solve 2𝑥2 − 6𝑥 − 12 = 0. Answer: 𝑥 = …….………… or ……………… [2] (b) Simplify 3𝑥+1 2𝑥2+11𝑥+12 − 1 𝑥+4. Answer: …….………………………… [3]

4 [Turn Over 2 In the diagram, ABCD is a rectangle. Given that AB = (2𝑥 + 𝑦) cm, BC = (𝑥 + 𝑦) cm, CD = (3𝑥 + 5𝑦 + 3) cm and AD = (4𝑥 + 5𝑦 – 7) cm. Find the values of x and y. Answer: 𝑥 = …….………… 𝑦 = ……………… [3] (2𝑥 + 𝑦) (4𝑥 + 5𝑦 – 7) (3𝑥 + 5𝑦 + 3) (𝑥 + 𝑦) 𝐴 𝐵 𝐶 𝐷

5 [Turn Over 3 (a) By using factorisation, solve (𝑥 − 3)(2𝑥 + 8) = −12. Answer: 𝑥 = …….………………………… [3] (b) Solve the equation 4 2𝑥−3 − 3 𝑥+2 = 1. Give your solutions correct to two decimal places. Answer: 𝑥 = …….………………………… [3]

6 [Turn Over 4 The diagrams below show a solid hemisphere and a solid cone. The hemisphere has a radius of 3𝑦 cm. The cone has a radius of 2𝑦 cm and slanted height 𝑙 cm. (a) Show that the total surface area of the solid cone is 2𝜋𝑦(𝑙 + 2𝑦) cm2. [2] (b) The total surface area of the solid hemisphere is equal to the total surface area of the solid cone. Find 𝑙 in terms of 𝑦. Answer: …….………………………… [3] 𝑙 cm 2𝑦 cm (3𝑦) cm

7 [Turn Over (c) The volume of the hemisphere is 729 cm3. Calculate the volume of the cone. Answer: …….………………………… cm3 [4] 5 The scale of a map is 2 cm : 1 km. (a) Write this scale in the form 1 : n . Answer: …….………………………… [1] (b) The area of a park is represented by an area of 456 cm2 on the map. Calculate the actual area of the land in square kilometres. Answer: …….………………………… [2]

8 [Turn Over 6 The points A, B, C, D and E are shown in the diagram below such that ABC and AED are straight lines and BE = √8 cm. (a) Without using a calculator, find (i) sin ABE. Answer: …….………………………… [1] (ii) cos ABE Answer: …….………………………… [1] (b) Using your answer in (a)(i), calculate the area of triangle ABE Answer: …….………………………… [2] (c) Given that triangle ABE and triangle ACD are similar, calculate the area of triangle ACD. Answer: …….………………………… [2] E (4, 3) 3) C (4, 1) B (1, 1) A (-3, 1) D

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10 [Turn Over 7 (a) Complete the table of values for 𝑦 = 𝑥 − 4 𝑥2 . Values are given to one decimal place where appropriate. x −15 −10 −5 −1 1 5 10 15 y −15.0 −5.2 −5 −3 4.8 10.0 15.0 [1] (b) On the grid on page 11, draw the graph of 𝑦 = 𝑥 − 4 𝑥2 for −15 ≤ 𝑥 ≤ 15. The graph has an asymptote at x = 0 (the graph goes near but do not cut through the y-axis). [3] (c) (i) On the same grid, draw the graph of 2𝑦 − 2𝑥 = −7 for −15 ≤ 𝑥 ≤ 15. [2] (ii) Write down the x-coordinates of the points where the line intersects the curve. Answer: 𝑥 = …….………… or ……………… [1] (iii) These values of x in part (ii) are the solutions of the equation 𝐴𝑥2 + 𝐵 = 0. Find the value of 𝐴 and the value of 𝐵. Answer: 𝐴 = …….………… 𝐵 = ……………… [1]