Hua Yi Sec Prelim 2024 A Math Paper 1 QP

[Turn Over HUA YI SECONDARY SCHOOL PRELIMINARY EXAM 2024 4-G3 / 5-G2 NAME CLASS INDEX NUMBER ADDITIONAL MATHEMATICS 4049/01 PAPER 1 (XX) 22 September 2024 2 hour 15 minutes Candidates answer on the Question Paper No Additional Materials is required. READ THESE INSTRUCTIONS FIRST Write your Name, Class, and Index Number in the spaces at the top of this page. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue, or correction fluid. Answer all the questions. The number of marks is given in the brackets [ ] at the end of each question or part question. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. The total number of marks for this paper is 90. 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. Gives answers in degrees to one decimal place. For π, use either your calculator value or 3.142. This document consists of 21 printed pages and 1 blank page. © HYSS 2024 No part of this document may be reproduced in any form or transmitted in any form or by any means without the prior permission of Hua Yi Secondary School. Setter: Ms Lee Hui Ling

2 2024_4-G3 / 5-G2_PRELIMS_ ADDITIONAL MATHEMATICS _PAPER 1 [Turn Over

3 2024_4-G3 / 5-G2_PRELIMS_ ADDITIONAL MATHEMATICS _PAPER 1 [Turn Over 1 A cuboid has a base area of ( 7 + 4√5)cm2 and a volume of ( 16 + 18√5)cm3. Find, without using a calculator, the height of the cuboid, in cm, in the form (𝑎 + 𝑏√5), where 𝑎 and 𝑏 are integers. [3]

4 2024_4-G3 / 5-G2_PRELIMS_ ADDITIONAL MATHEMATICS _PAPER 1 [Turn Over 2 The curve 5𝑥 − 𝑥𝑦 = 20 and the line 𝑥 − 2𝑦 − 3 = 0 intersects at the points 𝐴 and 𝐵. Find the 𝑦-coordinate of 𝐴 and of 𝐵. [3]

5 2024_4-G3 / 5-G2_PRELIMS_ ADDITIONAL MATHEMATICS _PAPER 1 [Turn Over 3 (a) Express 12𝑥 − 13 − 3𝑥2 in the form 𝑎(𝑥 + 𝑏)2 + 𝑐 and hence state the coordinates of the turning point of the curve 𝑦 = 12𝑥 − 13 − 3𝑥2. [4] (b) State the range of 𝑘 such that 𝑦 = 𝑘 will intersect the curve 𝑦 = 12𝑥 − 13 − 3𝑥2. [1]

6 2024_4-G3 / 5-G2_PRELIMS_ ADDITIONAL MATHEMATICS _PAPER 1 [Turn Over 4 Integrate 4 5𝑥+1 − 6 𝑥3 with respect to 𝑥. [2] 5 Express 7𝑥2−17𝑥+1 (𝑥2+1)(2−3𝑥) in partial fractions. [5]

7 2024_4-G3 / 5-G2_PRELIMS_ ADDITIONAL MATHEMATICS _PAPER 1 [Turn Over 6 Given that 𝑥5 + 𝑎𝑥3 + 𝑏𝑥2 − 3 ≡ (𝑥2 − 1)Q(𝑥) − 𝑥 − 2, where Q(𝑥) is a polynomial, (a) state the degree of Q(𝑥), [1] (b) show that 𝑎 = −2 and 𝑏 = 1, [3] (c) find the polynomial Q(𝑥). [3]

8 2024_4-G3 / 5-G2_PRELIMS_ ADDITIONAL MATHEMATICS _PAPER 1 [Turn Over 7 The sketch above shows part of the graph of 𝑦 = 𝑎 sin ( 𝑥 𝑏) + 𝑐, where 𝑥 is in degrees. (a) Explain why 𝑐 = 1. [2] (b) State the value of 𝑏. [1] (c) Find the value of 𝑚 and explain how you get it. [2] ( 𝑚, 4)

9 2024_4-G3 / 5-G2_PRELIMS_ ADDITIONAL MATHEMATICS _PAPER 1 [Turn Over 8 The figure shows the curve 𝑦 = 𝑥2 and the point 𝑅(1,0). The variable point 𝑃(𝑝, 0) moves along the 𝑥 axis and 𝑃𝑄 is vertical. It is given that p decreases at the rate of 1.2 units per second. (a) Show that the area of the triangle 𝑃𝑄𝑅, 𝐴 units2, is 𝐴 = 1 2 𝑝2 − 1 2 𝑝3. [2] (b) Find the rate at which 𝐴 is increasing at the instant when 𝑝 = −7. [4] y x 𝑦 = 𝑥2 P (p, 0 ) Q R ( 1, 0 )

10 2024_4-G3 / 5-G2_PRELIMS_ ADDITIONAL MATHEMATICS _PAPER 1 [Turn Over 9 From a rectangular piece of metal of width 2m and length 6m, two squares of side x m and two rectangles of sides x m and ( x + y) m are removed as shown. The metal is then folded about the dotted lines to give a closed box with height x m. (a) Show that the volume of the box, V m3, is given by 𝑉 = 2𝑥3 − 8𝑥2 + 6𝑥. [3] 6 m x + y m 2 m x x x x