[Tampines] [2023] 4E EM 4052 Prelims P1 QP

This document consists of 23 printed pages and 1 blank page. TAMPINES SECONDARY SCHOOL Secondary Four Express/ Five Normal Academic Preliminary Examination 2023 NAME CLASS REGISTER NUMBER MATHEMATICS 4052/01 24 August 2023 2 hours 15 minutes READ THESE INSTRUCTIONS FIRST Write your name, class and register number on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. 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 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 of 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. [Turn over O

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

3 Answer all the questions 1 Tom invests $4500 at a compound interest of 2.8 % per year for 5 years. Calculate the total value of his investment at the end of 5 years. Answer $ ____________________ [2] 2 Given that 𝑦 = −4(3)𝑥, which of the following represents the graph of y against x? Answer Diagram _____________ [1] x x x x O O O O O O O O y y y y Diagram 1 Diagram 2 Diagram 3 Diagram 4

4 3 Given that 3 × 27𝑛 = 1, find the value of n. Answer: n = __________________ [2] 4 The table below shows the distribution of ages of some students participating in a soccer tournament. Age (in years) 13 14 15 16 17 Number of students 8 5 x 6 2 Given that the median age is 14 years, state the range of values of x. Answer: ____________________ [2]

5 5 Box A contains four cards numbered 1, 4, 8 and 10. Box B contains three cards numbered 2, 5 and 7. One card with number x is drawn from Box A and another card with number y is drawn from Box B. The possibility diagram shows the sum of the numbers x and y. (a) Complete the possibility diagram. + 1 4 8 10 2 3 6 10 12 5 6 13 15 7 8 11 15 [2] (b) Find, in simplest form, the probability that (i) x + y is a prime number. Answer: _____________________ [1] (ii) 1 2 𝑥 + 1 2 𝑦 ≥ 8 1 2. Answer: _____________________ [2]

6 6 In the diagram below, ABD is a straight line, ∠𝐷𝐴𝐶 = 90°, AB = 35 cm and BC = 37 cm. (a) Find AC. Answer: _________________ cm [1] (b) Write as a fraction (i) 𝑠𝑖𝑛 ∠𝐴𝐵𝐶, Answer: _____________________ [1] (ii) 𝑐𝑜𝑠 ∠𝐶𝐵𝐷. Answer: _____________________ [1] B C A D 35 cm 37 cm

7 7 A worker works in a fast-food restaurant for 13 hours. The worker is paid $x for working at normal rate and $y for working at overtime rate. If he works for 10 hours at the normal rate and 3 hours at overtime rate, he will be paid $124. On Saturdays, if he works for 8 hours at the normal rate and 5 hours at overtime rate, he will be paid $133. Form and solve two simultaneous equations to find the normal and overtime rate of pay. Answer: Normal rate $ _____________ Answer: Overtime rate $_________ [4]

8 8 The diagram shows an isosceles triangle ABC. Angle 𝑃𝐴𝑄 = angle 𝐵𝐴𝐶 = angle 𝑆𝐴𝑈. AC = 4x. Angle 𝑃𝐴𝑈 = 90°. APRQ and AUTS are two identical sectors. B is the mid-point of AQ and C is the mid-point of AS. Show that the shaded area can be expressed as (p𝜋 +q)𝑥2 where p and q are constants. Answer: [6] 4x A P C U R T B Q S

9 9 n is a positive integer. Show that, for all n, (6n +1)2 − (6n − 1)2 is a multiple of 24. Answer: [2] 10 Given the diagram above, find the value of x. Answer: x = _________________ [2] A B C F D E 135o xo 2xo 164o 115o

10 11 (a) Express 7−8x + x2 in the form of p + (x + q)2. Answer _____________________ [2] (b) Sketch the graph of 𝑦 = 7 − 8x + x2 on the axes below. Indicate clearly the coordinates of the points where the graph crosses the axes and the minimum point on the curve. Answer [3]