ACJC 2025 Prelim P1

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 16 September 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all the questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. 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 2025 H2 FURTHER MATHEMATICS 9649/01 1 The curve 1C and 2C have equations 3 2cosr =− and 2r= respectively for 02 π . (i) Sketch 1C . [1] (ii) Find the exact area of the region that is inside both 1C and 2C . [4] 2 The function f is defined by ( ) 2 1 2fe xx −=− . The equation ( )f0 x = has only one real root  in the interval ( )0.75,1 . (a) Carry out linear interpolation once on the interval ( )0.75,1 to find an approximation x1 for  , giving your answer to 4 decimal places. By considering ( )f x and  f x  , explain with an aid of a diagram whether x1 is an overestimate or underestimate of the root  . [4] (b) Using the value of x1 found in part (i) as the initial value, apply the Newton-Raphson method to find  correct to 6 decimal places. Justify the accuracy of your answer. [3] 3 (i) Use de Moivre’s theorem to prove that ( ) 2 11 0 sin 1 sinsin sin 1 2 cos     +− = −+= − − + nn r n r r x nxx x n xx where x is a real number not equal to 1 , and  is a real number not equal to πk for any integer k. [4] (ii) By substituting 1 3 π = in part (i), find the sum of the first 4m terms of the series 124 815 07 1 ...− − + − − + ++x xxxxxxx , expressing your answer in the form ( ) ( ) p q x x , where ( )p x and ( )q x are polynomials in x with degrees 61 +m and 2 respectively. [3]

3 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 FURTHER MATHEMATICS 9649/01 [Turn Over 4 The variables x and y are related by the differential equation ( ) ( )dsin cos 1d yx x