1 霹雳怡保培南独立中学 SM POI LAM (SUWA) IPOH FINAL EXAMINATION 2021 ADDITIONAL MATHEMATICS PAPER ONE DATE: 19 NOV 2021 (FRIDAY) TIME: 1035 - 1235 (2 hours) NAME：______________________ REG NO: _________________ CLASS： S1ZI, S1AI, S1QIN, S1HE INSTRUCTIONS • Answer all questions. • Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs. • Write your name and registration number in the space provided. • Write your answer to each question in the space provided. • Do not use an erasable pen or correction fluid. • You should use a calculator where appropriate. • You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator. • Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question. INFORMATION • The total mark for this paper is 80. • The number of marks for each question or part question is shown in brackets [ ]. This paper consists of 15 pages, including this cover page. Prepared by ： Leow Sook Wan ………..…………….. (Ms Leow Sook Wan) Checked by： Ong Eik Hooi ……………………….. (Mr Ong Eik Hooi) Do Not Turn This Page Until You Are Told To Do So
2 1. The line = − 5 meets the curve 2 + 2 + 2 − 35 = 0 at the points A and B. Find the exact length of AB. [5]
3 2. (i) Write down the amplitude of 4 sin 3 − 1. [1] (ii) Write down the period of 4 sin 3 − 1. [1] (iii) On the axes below, sketch the graph of = 4 sin 3 − 1 for −90° ≤ ° ≤ 90° [3]
4 3. (a) Given that = and = , find, in terms of p and q, (i) 2 , [2] (ii) ( 3 ), [2] (iii) + . [1] (b) Using the substitution = 3 , or otherwise, solve 3 − 3 1+2 + 4 = 0. [3]
5 4. (a) A 5-digit code is to be chosen from the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9. Each digit may be used once only in any 5-digit code. Find the number of different 5-digit codes that may be chosen if (i) there are no restrictions, [1] (ii) the code is divisible by 5, [1] (iii) the code is even and greater than 70 000. [3]
6 4. (b) A team of 6 people is to be chosen from 8 men and 6 women. Find the number of different teams that may be chosen if (i) there are no restrictions, [1] (ii) there are no women in the team, [1] (iii) there are a husband and wife who must not be separated. [3]
7 5. Differentiate with respect to x (i) (1 + 4) 10 [4] (ii) 4−5 [4]
8 6. NOT TO SCALE The diagram shows a shape consisting of two circles of radius 3 cm and 4 cm with centres A and B which are 5 cm apart. The circles intersect at C and D as shown. The lines AC and BC are tangents to the circles, centres B and A respectively. Find (a) the angle CAB in radians, [2]
9 6. (b) the perimeter of the whole shape, [4] (c) the area of the whole shape. [4]
10 7. The first three terms of the binomial expansion of (2 − ) are 64 − 16 + 100 2 . Find the value of each of the integers , and . [7]
11 8. The functions and are defined, for > 1 , by () = 9√ − 1 , () = 2 + 2. (i) Find an expression for −1 () , stating its domain. [3] (ii) Find the exact value of (7). [2] (iii) Solve () = 5 2 + 83 − 95. [3]
12 9. Solve the equation (a) 2| | = 1 for − ≤ ≤ radians, [3] (b) 3 (2 + 15°) = 1 for 0° ≤ ≤ 180°, [3]
13 9. (c) 32 = 2 − 7 + 1 for 0° ≤ ≤ 360°. [4]
14 10. In this question all lengths are in metres. NOT TO SCALE A water container is in the shape of a triangular prism. The diagrams show the container and its cross-section. The cross-section of the water in the container is an isosceles triangle ABC, with angle ABC = angle BAC = 30. The length of AB is x and the depth of water is h. The length of the container is 5. (i) Show that = 2√3 ℎ and hence find the volume of water in the container in terms of h. [3]
15 10. (ii) The container is filled at a rate of 0.5 3 per minute. At the instant when h is 0.25 m, find (a) the rate at which h is increasing, [4] (b) the rate at which x is increasing. [2] -End of paper-