Peicai Prelims 4049 4E AMath P2 2021 solutions

[Turn over PEICAI SECONDARY SCHOOL SECONDARY 4 EXPRESS PRELIMINARY EXAMINATION 2021 CANDIDATE NAME CLASS REGISTER NUMBER ADDITIONAL MATHEMATICS 4049/02 Paper 2 27 August 2021 2 hours 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your register number, class and name 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, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an approved scientific calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. 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 20 printed pages. Setter: Mrs Ho Thuk Lan SOLUTIONS

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the quadratic equation ax 2 + bx + c = 0, Binomial expansion , where n is a positive integer and 2. TRIGONOMETRY Identities Formulae for ∆ABC

3 [Turn over 1 Solve the equation for . [5] x = 0.253, 1.57, 2.89 2 The diagram shows two circles and with centres P and Q respectively. Both circles and is tangent to the x axis at R (1.5, 0). The radius of is 6 units. (i) Find the coordinates of the centre, P of circle . [1] P = (ii) Find the equation of the circle . [1] (iii) Given that the area of is 18 times the area of , find the coordinates of the centre, Q of . [3] Q = (1.5, ) P(1.5,6) x Q R(1.5, 0) R a d i u s

4 3 (i) Using , show that . [3] (ii) State the amplitude and period of . [2] Amplitude = 2 Period = 120 (iii) Sketch the graph of for . [3]

5 [Turn over 4 A glass of hot water was left to cool in the fridge. The temperature, T C, of the water decreases with time, t minutes. The table shows the measured values of T and t. t (min) 10 20 30 40 50 T (C) 60 37 23 14 9 It is known that t and T are related by the equation where a and k are constants. (i) Explain clearly how a and k can be calculated when a graph of ln T against t is drawn. [2] k = gradient ln a is the ln T – intercept, c ln a = c a = t (min) 10 20 30 40 50 ln T (C) 4.09 3.61 3.14 2.64 2.20 (ii) On the grid below, plot ln T against t for the given data and draw a straight line graph. [2] t ln T 5 4 3 2 4.55

6 Use your graph to estimate (iii) the value of a and of k. [3] k = 0.04725 ln a is the ln T – intercept, c a = (iv) the temperature of the water at the beginning of the experiment. [1] Temperature = 94.6C 5 A bus travelling on a straight road passes a traffic light at junction P with speed 11 m/s and, 2 minutes later, passes another traffic light at junction Q with speed 21 m/s. During the journey from P to Q, the acceleration, , where k is a constant and t seconds is the time after passing P. (i) Show that . [5] Speed = 11 m/s 2 min later Speed = 21 m/s P Q At P, t = 0, v = 11 At Q, t = 2 min, v = 21m/s (ii) Find the distance between P and Q. [4] At P, t = 0, s = 0 c = 0 10 20 30 40 50

7 [Turn over Distance = 3080 m 6 The diagram shows a circle passing through the points A, B and C. The point B lies on the line SC. ST is a tangent to the circle at A. The points E and F lie on AB and AC respectively. Given that EF is parallel to ST, show that BCFE is a cyclic quadrilateral. [5] BAS = ACB (Alternate segment Theorem) M1 [Alt segment] BAS = AEF (Alternate angles, EF//ST) M1 [Alternate s] BEF = 180 AEF (Adjacent angles on a straight line) BEF = 180 BAS = 180 ACB ACB + BEF = ACB + 180 ACB ACB + BEF =180 M1 CAT = ABC (Alternate segment Theorem) CAT = EFA (Alternate angles, EF//ST) EFC = 180 EFA (Adjacent angles on a straight line) EFC = 180 CAT = 180 ABC ABC + EFC = ABC + 180 ABC ABC + EFC =180 M1 Since ACB + BEF =180 (angles in opposite segments) M1 and ABC + EFC =180 (angles in opposite segments) Therfore BCFE is a cyclic quadrilateral. A B C S T E F

8 7 The equation of a polynomial given is by . (i) Find the remainder when f(x) is divided by x + 1. [1] Remainder = 8 (ii) Show that 3x 1 is a factor of f(x). [1] Since , therefore 3x 1 is a factor of f(x). (iii) Show that the equation f(x) = 0 has only one real root. [4]

9 [Turn over or Since . Hence f(x) = 0 has only one real root. A1 (iv) Use your answers to parts (ii) and (iii) to solve the equation . [4] y = 1.58 (3SF)

10 8 A contractor was given 240m of fencing for a playground, He designed the playground ABCDEF consisting of a rectangle ABCF and an isosceles trapezium CDEF. Given that AB = 12 m, DE = 6 m, CD = EF = 5x m and BC = y m. (i) The contractor used 240 m of fencing to enclose the playground. Express y in terms of x. [1] (ii) Show that the enclosed area, A , of the playground is given by . [2] Height of trapezium = Area = Area = 6 E D 5x 5x C F y A B 12