1 霹雳怡保培南独立中学 SEKOLAH MENENGAH POI LAM (SUWA) FINAL EXAMINATION 2023 ADDITIONAL MATHEMATICS PAPER ONE ___________________________________________ DATE : 06.11.2023 (MONDAY) TIME : 0800 – 1000 (2 HOURS) ____________________________________________ NAME : _____________________ REGISTRATION NUMBER: _____________CLASS : S2AI, S2HE, S2QIN & S2PIN INFORMATION FOR CANDIDATES 1. This question paper consists of two sections: Section A and Section B. 2. Answer all questions in Section A and any two questions from Section B. 3. Write your answers in the spaces provided in the question paper. 4. Show your working. It may help you to get marks. 5. If you wish to change your answer, cross out the answer that you have done. Thenwrite down the new answer. 6. The diagrams in the questions provided are not drawn to scale unless stated. 7. The marks allocated for each question are shown in brackets. 8. A list of formulae is provided on pages 2 and 3. 9. The Upper Tail Probability Q(z) For the Normal Distribution N(0, 1) Table is provided on page 4. 10. You may use a scientific calculator. 11. The total mark for this paper is 80. Do Not Turn Over This Page Until You Are Told To Do So _____________________________________________________________________________________________________________ This document consists of 23 printed pages (including this page) Prepared by : Ms Chai Siew Yin Checked by : Mr Ong Eik Hooi Signature : _________________ Signature : __________________

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5 Section A [64 marks] Answer all questions in this section. 1. (a) The diagram shows the function : → 3 − 5. (i) Find the value of m. [1] (ii) Given = − and = 6 − 9, find the value of pand of q. [2]

6 (b) Given −1 +3 = +1 +2 , and = 2, find the value of m. [3]

7 2. The diagram shows a part of a graph for the total population of a type of bacteriakept in a test tube. The variable x represents the number of hours and y represents thetotal population. Variables x and y are related by the equation = , where pandq are constants. y (16, 100) (6, 10) x (a) Sketch the straight line graph of log10 against x. [2] (b) Based on the graph in part (a), find the value of p and of q. [3]

8 3. (a) Given log10 < 0 , state the range of value a. [1] (b) Given that = 7 and = 7 , simplify 49 + + 7 − in terms of aand b . [3]

9 4. The figure shows a straight line PQ. y Q(n, 10) P(-3, 8) O x It is given that the unit vector in the direction of = +2 29 . (i) Find the value of n and of k. [3](ii) Hence, express in the form of a column vector. [1]

10 5. The figure shows a circle with centre O and radius of 5 cm. Given that OQTCis arectangle with an area of 40 cm2 , find [Use = 3.142 ] (a) the value of , in radians, [2] (b) the area of the sector OAB, [2] (c) the perimeter of the shaded region. [2]

11 6. (a) The diagram shows the probability of the binomial distribution ~4, . Find the value of k and of p. [3] (b) Z is a continuous random variable of a normal distribution. Given that < ℎ = 8 ≤ − ℎ . Find (i) < ℎ , (ii) the value of h. [3]

12 7. (a) Given that 4 2 − 3 − 1 = 0, find the range of values of x if <5. [2](b) Find the possible values of k if the equation 2 + − 5 +4 =0hasreal roots. [3]

13 8. The figure shows the shaded region bounded by the curve = and the x-axisfrom = 5 to = 9. Given the area of the shaded region is 10 units 2 . (a) Find (i) 5 9 , [1] (ii) the value of ℎ() 5 9 if 2ℎ() = . [2] (b) Given the graph passes through the point (1, 96) and () = 3 2 − 24 + 17 , find () in terms of . [2]

14 9. (a) (i) The points 1, − 1 , , and (6, 4) lie on a straight linesuch that : = 2 : 3 where p and q are constants. Find the valueofp and of q. [2] (ii) The straight line = 3 + 12 is parallel to the straight line = + 4 + 9, where r is a constant. Determine the value of r.[1] (b) Given that the points , 2 , 3, 4 and (11, 8) are collinear. Findthevalue of x. [2]

15 10. Given the quadratic function = 4 2 + 18 − 5. (a) By using the completing the square method, express in the formof = + 2 + , where p and q are rational numbers. [2] (b) Find the minimum/maximum point of . [2] (c) State the equation of the curve in the completing the square formwhenthegraph is reflected in the x- axis. [2]

16 11. The figure shows the graph of a quadratic function = 2 −8 + whereaand c are constants. y m O n x (a) State the range of values of x if > 0. [2] (b) Express the range of values of a in terms of c. [2] (c) Prove that += 8 . [2]

17 12. (a) Given that = 2 3 − 2 , find the small change in V when y changes from 2 to 1.99. [2]

18 (b) Ali and his friends are visiting a country. On a particular day, Ali and his friends will go to a theme park if the temperature on that day is not more than40C. According to the weather forecast on that day, the rate of change of thetemperature is = 0.4(5 − ) where is the temperature in Candt isthe time in hours. Determine whether Ali and his friends are able to proceedwith their plan if the temperature at 7.00 a.m. at = 0 22°. [4]

19 Section B [16 marks] Answer any two questions from this section. 13. Given that 3, − 2 and 9 , 9 are the solutions of the simultaneous equations − − 1 = 0 and 4 3− 103= 1. Find the values of k, m, nandp. [8]

20 14. (a) Solve the equation 22−2 ! = 10 ! . [2]

21 (b) A shopping mall has organised a contest. They have shortlisted 20 luckycontestants to be selected to win the main prizes which are three cars : brand X, Y and Z and five consolation prizes worth RM5000 each. Five winners for the consolation prizes are drawn before the three winners for the main prizes are drawn. (i) Find the number of ways in which 5 winners for the consolationprizecan be chosen. [1](ii) Find the number of different ways to select the winners for the mainprizes. [1](iii) The winners’ ceremony will take place in one event. Determine thedifferent ways to seat the 8 winners in a row if (a) there is no restriction, [2] (b) all the main prize winners are not allowed to sit next to eachother. [2]

22 15. (a) Given that A and B are two angles which are in the same quadrant andareinbetween 0 and 360 such that cos = 8 17 and tan =−3 4 . Without using a calculator, find (i) tan ( − ) , [2](ii) cosec 1 2 . [2]

23 (b) Solve the equation sin 2 + sin − tan = 0 for 0 ≤ ≤ 2. [4]