EJC_9649_2025_Prelim_P1_Solutions

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 Solution (a) Cofactor expansion along the first column: 2 2 d e t ( ) () ( ) () () () () () () ( ) ( )( ) ( )( ( ) ( )) ( )( )( ) β αβ γ γα α αβ γ βγ α γα β βγ α β αβ γαβ αβ α γβ γ α β αβ γ αγ βγ αβγ βγ α βγ αββγγα = −− ++− = − −− −+− + =−−−++ =− −− − = − −− A (b) det 3( )( )( )αββγγα= −− − −B 2 Solution (a) sin 4 z y xyx π∂ = −+ ∂  , sin 4 z x xyy π∂ = −+ ∂  ddd ddd 1sin e sin 4 41 t z zx zy t xt yt y xy x xy t ππ ∂∂= +∂∂  = − +⋅ − +⋅  +  When 0t = , 1, 0xy= = Therefore d2 sind 42 z t π= −= − (b) f (, ) 0xy xy = implies fx is independent of y f (, ) 0yx xy = implies fy is independent of x So f must take the form of f ( , ) g( ) h( )xy x y= + for some functions g and h

3 2025 JC2 H2 Further Mathematics Preliminary Examination Paper 1 [Turn over 3 Solution (a) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 21 11 4 44 2 4 42 2d 1 d4 22 44 2222 11 11 24 2422 4 22 11 442 4 21 42 4 ,1 when 2 4 ,42 4 1 24 2 Length of arc 1 d 1 d 1 d d 2d d d , ln x xx y xx xx xx x x xx xx x PQ x xx x xx x xx x xx xx x + = += + ≤≤ = + = +− = + −+ = ++ = ++ = + = +  = + ⌠ ⌠ ⌡⌡ ⌠⌠⌡⌡ ⌠⌡ ⌠⌡ ∫  ( ) ( ) 4 2 221 1 14 1 2 4 42 4 4 2 ln4 ln2 6 (ln ) 6 ln2. (shown)  = −+ − = + = + (bi) ( ) 4 2d d 2 4 2 2 Area of surface of revolution generated by arc about the -axis 2π1 d 121 d 4 120.4277... 120.43 (2 d.p.) y x PQ y xx xx x xπ = + = +−  = = ⌠⌡ ⌠ ⌡ (bii) ( ) 2 4 2 4 ln 242 Volume of solid of revolution generated by shaded region abt. -axis 2 πd 2 π d, 177.9648... 177.96 (2 d.p.) xx y xy x xx = = − = = ∫