ASRJC 2023 JC1 Promos 9649

ASRJC 2023 [Turn over FURTHER MATHEMATICS 9649/01 Paper 1 28 September 2023 (Thursday) 8 – 11 a.m. 3 hours Additional Materials: Li st of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet will be provided with this quest ion paper. You should follow the instructions on the front cover of the booklet. If you need additional answer paper ask the invigi lator for a continuation booklet. Write your name, class, syllabus and paper number on the answer booklet you hand in. Write in dark blue or black pen on both sides of the answer booklet. 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. 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. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. 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. This document consists of 5 printed pages and 3 blank pages. Anderson Serangoon Junior College ANDERSON SERANGOON JUNIOR COLLEGE JC1 Promotional Examination 2023 Higher 2

2 ASRJC 2023 1 A conic has cartesian equation 22 22 ( 12) 113 5 xy  . (a) Show that one of its foci is at the origin. [2] (b) Obtain a polar equation of the conic. [3] 2 A dog bowl is produced by rotating the region bounded by the curve 4ln( 1)yx and the lines 0x , 1x and 1 5y fully about the y-axis. By using a substitution e1yu or otherwise, find the exact volume of the material required to produce the dog bowl. [7] 3 A polar curve has equation 1 cos , where 0r     . ๠e curve has two points where the gradient of the curve is 1 . (a) Show that cos cos 2 sin sin 2    at these two points. [3] (b) Without using a calculator, determine the value of  for each of these two points. [3] 4 (a) It is given that f is a function where f( ) 0x  for 0x . If 1 22f n n r rU nn   , where n is a positive integer and 2 0 f( ) dI xx , explain with the aid of a sketch, why  nUI  for all positive integers n,  UI  . [3] (b) If 1 0 22f n n r rV nn     , state an inequality relating nV and I for 1n . [1] (c) By choosing a suitable function f, determine the exact value of 111 1lim 246 3n nnn n      . [3]

3 ASRJC 2023 [Turn over 5 ๠e complex number z satisfies 44 izz and 1 1i 22 z . (a) Sketch the locus of z. [3] (b) Express, in terms of k, the exact minimum value of izk , where 0k  . [2] (c) Find the range of values of arg( 6)z . [3] (d) If w is a complex number such that ie0wz  for all values of  and z, write down the range of values of w , justifying your answer. [2] 6 (a) ๠e diagram shows the seating plan for passengers in a minibus, which has 17 seats in 4 rows. ๠e back row has 5 seats and the other 3 rows have 2 seats on each side. 11 passengers get on the minibus. (i) How many possible seating arrangements are there for 11 passengers? [1] (ii) How many possible seating arrangements are there if 5 particular people sit in the back row? [2] Of the 11 passengers, 5 are unmarried and the other 6 consists of 3 married couples. (iii) In how many ways can 5 of the 11 passengers be chosen if there must be 2 married couples and 1 other person, who may or may not be married? [2] (b) ๠ere are 8 seats at a round table and they are equally spaced as shown in the diagram below. Find the number of ways 2 women, 1 married couple and a family of 4 people can be seated at the table if (i) the married couple and the family of 4 pe ople must each be seated together, [2] (ii) the married couple must be seated together and the 2 women must not be seated directly opposite each other. [3]

4 ASRJC 2023 7 It is given that ln( 1) 1 sinyx  . (a) Apply Maclaurin’s ๠eorem to find the power series for y, up to and including the term in 2x . [4] (b) Given that x is small such that 3x and higher powers of x can be neglected, verify the correctness of your result in part (a) by using standard series from the List of Formulae (MF26). [2] (c) Use your series from part (b) to estimate 1 1s i n 1 ed x x  in terms of e. [2] (d) Explain with the aid of a diagram and withou t any further calculations, if your answer in (c) is an overestimation or underestimation. [2] 8 (a) Give a sketch of a continuous function that has a real root, but it is not possible to use linear interpolation to find approximations of this root. [1] Consider 2f( ) l n 1x xx , where 0x . It has exactly one real root. Li near interpolation is applied to the initial interval (1, 3), and the first three approximations are 1 1.20228x  , 2 1.33937x  and 3 1.42376x  , given correct to 5 decimal places. (b) Find the next two approximations, giving your answers correct to 2 d ecimal places and showing your workings clearly. Verify that the required accuracy of 2 decimal places has yet to be achieved, without comparing with the actual value of the root. [4] For the same function, Newton-Raphson method is now to be applied to find approximations to the root. (c) Find the exact set of values of the initial value 1u such that the method would converge to the root. [2] (d) Obtain a recurrence relation for 1nu  , the th(1 )n approximation of the root, in terms of nu for positive integers n using Newton-Raphson method. [1] (e) Use the relation in (d) and an initial approximation 1 1u  to find the root, terminating when two consecutive approximations are the same when rounded off to 2 decimal places. [2] 9 A hyperbola H has equation 22 22 1xy ab , where a and b are fixed positive constants. One of its foci is (, 0 )Fc , where 0c . (a) Find, in terms of a and b, the x-coordinate of the foot of perpendicular of F to the asymptote of H with positive gradient. [3] (b) Prove that if a line y mx k is tangent to H, then 22 2 2ka m b . [3] ๠e foot of perpendicular from F to a tangent line of H is denoted by N. (c) Prove that the locus of N as the tangent varies is a circle with radius a and centred at the origin, excluding certain point(s) whose x-coordinates should be stated. [5]

5 ASRJC 2023 [Turn over 10 Tim throws two identical fair dice simultaneously. Each die has six faces: two faces numbered 1, two faces numbered 2 and two faces numbered 3. His score is the sum of the two numbers on the dice. (a) Calculate the probability that Tim obtains a score of at least 3. [2] Tim plays a game with his friend Sam, who has two fair dice numbered in the same way. Both of them throw their dice simultaneously in the game. (b) ๠e random variable X represents the largest number shown on the four dice. (i) Find the probability distribution of X. [3] (ii) Show that 226E( ) 81X  and calculate the value of Var( )X , giving your answer correct to 3 significant figures. [3] (c) Given that 3,X  find the probability that the sum of the numbers shown on the four dice is equal to 8. [4] 11 A bank charges %0  annual interest on a loan of $C taken out by a client. At the end of each year, the interest is added and then a fixed amount, $R is paid o ff by the client. Let the amount owed at the start of year n be .nL (a) Show that 1 ,nnL aL b giving and ab in terms