EJC_9649_2025_Prelim_P1

2025 JC2 H2 Further Mathematics Preliminary Examination Paper 1 [Turn over EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2025 General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CIVICS GROUP INDEX NO. FURTHER MATHEMATICS Paper 1 9649/01 02 September 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae (MF27) READ THESE INSTRUCTIONS FIRST Answer all 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 must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages and 2 blank pages.

2 2025 JC2 H2 Further Mathematics Preliminary Examination Paper 1 1 The matrices A and B are given by 1 1, 1 α βγ β γα γ αβ  = A 1 1 33 3 β γα α βγ γ αβ  = B . (a) Show that det ( )( )( )αββγγα= − −−A . [3] (b) Hence, or otherwise, find det B . [2] 2 (a) Let cos 4z xy π= +  where etx= and ln(1 )yt= + . By finding z x ∂ ∂ and z y ∂ ∂ , evaluate the exact value of d d z t when 0t = . [4] (b) Suppose the function f( , )xy satisfies f (, ) f (, ) 0xy yxxy xy= = for all x and y. What does this imply about how fx and fy depend on x and y? Hence deduce a general form of the function f( , )xy . [2] 3 The diagram shows a shaded region bounded by an arc PQ having cartesian equation 2 ln 24 xxy= − , vertical lines 2x= , 4x= and the x-axis. (a) Show that the exact length of the arc PQ is 16 ln 24+ . [4] (b) Both the arc and the shaded region are rotated completely about the y -axis. (i) Find the area of the surface of revolution formed by arc PQ , correct to 2 decimal places. [3] (ii) Find the volume of the solid of revolution formed by the shaded region, correct to 2 decimal places. [2]

3 2025 JC2 H2 Further Mathematics Preliminary Examination Paper 1 [Turn over 4 Do not use a calculator in answering this question. Use de Moivre’s theorem to show that 42cos5 cos (16cos 20cos 5)θθ θ θ= −+ . [5] Hence, or otherwise, evaluate 2cos 10 π and deduce that 15cos54 π += . [5] 5 The complex number z satisfies the three relations below: • 13 5z−≤ • 8 18 10izz−≥−− • 1111tan arg( ) tan22 z−−−≤≤ (a) (i) Illustrate all three relations clearly on a single Argand diagram. [6]