NYGH 2021-S4EOY-Adv Math with ans

Class Register Number Name End-of-Year Examination 2021 Secondary 4 ADVANCED MATHEMATICS 1 hour Monday 4 October 1245 – 1345 READ THESE INSTRUCTIONS FIRST 1. Write your name, register number and class on all the work you hand in. 2. Write in dark blue or black ink. 3. You may use an HB pencil for any diagrams or graphs. 4. Do not use staples, paper clips, glue or correction tape/fluid. 5. Write your answers and working on the separate writing paper provided, unless otherwise stated. 6. Answer all questions. 7. Omission of essential working will result in loss of marks. 8. The use of an approved scientific calculator is expected, where appropriate. 9. 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 degree to one decimal place. For π, use either your calculator value or 3.142, unless the question requires the answer in terms of π. 10. At the end of the examination, fasten all your work securely together. 11. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 3 printed pages and 1 blank page. Setter: A. Low NANYANG GIRLS' HIGH SCHOOL [Turn over

2 1 It is given that 𝑦 = 3𝑥 √4−𝑥2 . (a) Find ∫ 𝑦 d𝑥. [2] (b) Express y in ascending powers of x up to and including the x5 term and state the range of values of x for which the expansion is valid. [4] 2 The equation of a curve is 3𝑦2 = 4𝑥3 + 5. (a) Find the gradient of the normal to the curve at the point where 𝑦 = √3. [3] (b) Find an expression for d2𝑦 𝑑𝑥2 in terms of x and y. [3] 3 Find the volume generated when the area bounded by the curve y = 3ln 2x and the y-axis from y = 4 to y = 5 is rotated through one revolution about the y-axis. Give your answer in exact form. [3] 4 It is given that u = 2i + 4j + 9k and v = aj + 3ak where a is a constant. (a) Find the acute angle between u and v. [2] (b) Given that u × v is a unit vector, find the possible values of a. [3] 5 The first and third terms of an arithmetic progression are equal to the first and third terms of a geometric progression. The common difference of the arithmetic progression is also equal to the common ratio of the geometric progression. It is known that the ninth term of the arithmeti c progression is equal to the fifth term of the geometric progression. Given that the common ratio of the geometric progression is positive, find (i) the first term and the common difference of the arithmetic progression, [5] (ii) the sum of the first nine terms of the geometric progression giving your answer in the form 𝑎 + 𝑏√3. [3]

3 6 (a) A math club is made up of 6 boys and 6 girls. (i) At the beginning of the year, a photograph of the club members is taken. The boys stand in a row at the back while the girls stand in a row at the front. Find the number of ways the photograph can be taken. [2] (ii) In the middle of the year, the club has to form an executive committee which consists of a president, two vice-presidents and a secretary. Find the number of ways this committee can be formed. [2] (iii) At the end of the year, the boys have to pair up with the girls for a dance. Find the number of ways the pairs can be formed. [2] (b) Box A contains 1 red ball and 2 green balls while Box B contains 5 red balls and 4 green balls. A fair die is rolled. If the number obtained is a multiple of 3, a ball is drawn at random from Box A. If the number is not a multiple of 3, a ball is drawn at random from Box B. A girl rolls the die and draws a ball from one of the boxes. Given that the ball is red, find the probability that the ball is from Box B. [2] 7 Find ∫ e2𝑥 cos 3𝑥 d𝑥. [4] END OF PAPER

4 Answer Key 1(a) −3√4 − 𝑥2 + 𝑐 (b) 3 2 𝑥 + 3 16 𝑥3 + 9 256 𝑥5 + ⋯ , |𝑥| < 2 2(a) − √3 2 (b) 4𝑥𝑦2−4𝑥4 𝑦3 3 16.1 cubic units 4(a) 12.7° (b) 𝑎 = ± 1 7 5(i) The first term is √3. The common difference is √3. (ii) 120 + 121√3 6(a)(i) 518400 (ii) 5940 (iii) 720 (b) 10 13 7 3 13 e2𝑥 sin 3𝑥 + 2 13 e2𝑥 cos 3𝑥 + 𝑐