ACJC 2023 JC1 Promos 9649

ANGLO-CHINESE JUNIOR COLLEGE JC1 PROMOTIONAL EXAMINATION Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 4 October 2023 3 hours Additional Materials: Cover Sheet Answer Paper List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your index number, class and name on the work you hand in. Write in dark blue or black pen on both sides of the paper. 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. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. __________________________________________________________________________________ This document consists of 6 printed pages. [Turn Over

2 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 FURTHER MATHEMATICS 9649/01 1 By writing cos 5 in terms of cos, find exactly all solutions of the equation 5316 20 5 1 0xx x   , stating which roots are repeated. [6] 2 Use mathematical induction to show that   d πes i n 3 2es i n 3d3 n xn x n nx xx   for all positive integers n. [6] 3 (i) Find all solutions of the equation 5 10z  . [2] Let  2 2 11p1zz z zz    . (ii) By writing  p z as a geometric series, determine the solutions of the equation p 0z  . [2] Let 1wz z . (iii) Write  p z as a quadratic expression in w, and hence find the values of w for which p 0z  , giving your answers in surd form. [3] (iv) From (ii) and (iii), determine the exact values of πcos 5 and 3πcos 5 in surd form.[3] 4 A curve 1C has the equation 223( 1) 4 12xy  . (i) Find the polar equation of 1C in the form fr   . [4] Another curve 2C with polar equation 22 c o sr  , where 02 π , intersects 1C at points P and Q. (ii) Sketch 1C and 2C on the same diagram, indicating clearly all axial intercepts. [2] (iii) Find the area of triangle OPQ, where O is the origin. [4]

3 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 FURTHER MATHEMATICS 9649/01 [Turn Over 5 A recurrence relation is given by 21 23 4nn nx xx  . (i) Find the general solution of the recurrence relation in the form   cos sinn nx RA n B n   where A and B are arbitrary real constants and R and  are to be determined exactly. [4] (ii) Show that 6nnx kx  for all non-negative integers n, where k is a constant to be determined. [3] (iii) Determine the values of A and B exactly if the initial conditions are 1 0 2 3x  and 1 1x  . [4] 6 (a) Describe completely, in geomet rical terms, the loci given by 43 i 5z  and 15 i 55 izz   and sketch both loci on the same diagram. Find, in the form iab , the complex numbers representing the points of intersection of the loci, giving the exact values of a and b. [8] (b) Another complex number w satisfies the inequalities 43 i 5w  and 15 i 55 iww   (i) Find the greatest possible value of 82 iw . [3] (ii) Find the range of values of  arg 3iw . [4]

4 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 FURTHER MATHEMATICS 9649/01 7 In a science experiment, a beaker containing 100 ml of water is prepared. Every hour, the following steps take place: 1. First, 10 g of salt are added into the beak er. The solution is stirred thoroughly so that the salt is dissolved and the concentration is uniform. 2. After that, k ml of the solution in the beaker is removed, and k ml of pure water is added into the beaker. Let nu be the amount of salt in the solution in the beaker in grams, after the nth time that steps above have been carried out. (i) Explain why  111 0100 nn kuu    and state the initial conditions of the recurrence relation. [2] (ii) Solve the recurrence relation and express nu in terms of n and k. [3] For parts (iii) and (iv), suppose that 10k . (iii) Find the limiting value L of nu as n tends to infinity. [2] (iv) Determine the smallest value of n such that nu is within 10% of L. [3] (v) Find the range of values of k such that the limiting value of nu as n tends to infinity is less than 50. [3]

5 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 FURTHER MATHEMATICS 9649/01 [Turn Over 8 Parabolic mirrors are used becau se they reflect incoming parallel rays to a single point. In reality, however, it is easier and cheaper to manufacture spherical mirrors, which have a cross-section that is circular instead of parabolic. Suppose the cross-section of a circul ar mirror is given by the equation  222x yr r  , where 0r . In particular, the mirror consists only of the portion of the curve which is concave up. The reflective surface is the conc ave side of the curve, and the light rays initially travel in the negative y-direction. (i) Show that a ray of light which travels along the line x k , where kr , is reflected such that it intersects the y-axis at a distance 1 2 2 2 111 2 kr r     from the origin. [5] We approximate the part of the cross-secti on in the neighbourhood of the origin with a parabola. (ii) Rewrite the equation of the circular cr oss-section near the origin by expressing y in terms of ascending powers of x, up to and including the term in 2x . [4] Denote the answer to (ii) by p x . (iii) Find the focus of the parabola  pyx . [2] (iv) For a spherical mirror, the reflected rays are said to converge approximately at a distance half of the radius from the origin. Comment on the suitability of this approximation. [1]

6 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 FURTHER MATHEMATICS 9649/01 9 A parabola is defined by the parametric equations 2 , 2. x at ya t   The points P and Q on the parabola have parameters p and q respectively. (i) The point R lies on the parabola such that the tangent to the parabola at R is parallel to PQ. Find r, the parameter of R, in terms of p and q. [3] (ii) The point S lies on the parabola such that the tangent to the parabola at S is parallel to PR. A line through S parallel to the axis of the parabola intersects PQ at T. Show that the coordinates of T are   223, 342 aa pq p q  . [4] (iii) The lines PR and ST intersect at X. Show that 2XT SX . [5] (iv) Hence, find the ratio of the areas of PQR and PRS . [3] (v) Deduce that the area of the region bounded by the parabola and PQ is 4 3 times the area of PQR . [2]