Fu Hua 2025 AM Prelim P2 Final

Candidate Name: Class: Index No.: FUHUA SECONDARY SCHOOL Secondary 4 Express/ 5 Normal (Academic) PRELIMINARY EXAMINATION 2025 4E5N Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School Fuhua Secondary School ADDITIONAL MATHEMATICS 4049/02 Paper 2 DATE 29 August 2025 TIME 1000 – 1215 DURATION 2 hours 15 minutes READ THESE INSTRUCTIONS FIRST Write your class, index number and name in the spaces at the top of this page. Write in dark blue or black pen. 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. The use of an approved scientific calculator is expected, where appropriate. 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. The total number of marks for this paper is 90. FOR EXAMINER’S USE PARENT’S SIGNATURE Statements/Workings / 90 Accuracy (incl. Units) Presentation Setter: Mr Liu Yaozhong Ms Tan Tuan Ling Vetter: Ms Lim Fen Niu This document consists of 23 printed pages and 1 blank page.

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c+ + = , 2 4 2 b b acx a −  −= Binomial Expansion ( ) 1 2 2 ... ...12 n n n n n r r nn n na b a a b a b a b b r − − −     + = + + + + + +           , where n is a positive integer and ! ( 1)...( 1) !( )! ! n n n n n r r r n r r  − − +== − 2. TRIGONOMETRY Identities 22sin cos 1AA+= 22sec 1 tanAA=+ 22cosec 1 cotAA=+ sin( ) sin cos cos sinA B A B A B =  cos( ) cos cos sin sinA B A B A B= tan tantan( ) 1 tan tan ABAB AB = sin 2 2sin cosA A A= 2 2 2 2cos 2 cos sin 2cos 1 1 2sinA A A A A= − = − = − 2 2 tantan 2 1 tan AA A= − Formulae for ABC sin sin sin a b c A B C== 2 2 2 2 cosa b c bc A= + − 1 sin2 bc A=

3 1 (a) A polynomial ()Px leaves a remainder of 1 when divided by 1x− and a remainder of 1− when divided by 21x− . Find the remainder when ()Px is divided by 22 3 1xx−+ . [3] [Turn over

4 (b) In the expansion of ( ) 8 131 4p x x −−  , the coefficient of x is equal to the coefficient of x3. Find the value of the constant p. [5]

5 2 Do not use a calculator in this question. (a) (i) By using tan 2A , show that cot 67.5 1 2=− + . [6] (ii) Hence, find the exact value of 2cosec 67.5 . [2] [Turn over

6 (b) Given that ( ) ( ) sin 2 sin 3 AB AB + =− , prove that 5tan tan 0BA+= . [4]

7 3 (a) Show that 1x− is a factor of 322 5 2x x x− + + . Hence, factorise 322 5 2x x x− + + completely. [5] [Turn over

8 (b) Solve the equation ( ) ( ) 2 2 1 12 e 5 e 2e 1 0y y y− − −− + + = . Leave your answers in exact form. [5]

9 4 The graph of siny a bx c=+ has a turning point at π ,04   and the next turning point after this has coordinates 3π ,64   . (a) Find the values of the constants a, b and c. [3] (b) Hence, sketch the graph of siny a bx c=+ for 02 πx . [2] (c) By drawing a suitable straight line on the same axes, find the number of solutions of the equation 3sin 2πa bx x =− . [3] [Turn over

10 5 The diagram shows a point P on a circle. PQ is a tangent to the circle at P. Points A, B and C lie on the circle such that PA bisects angle QPB. QAC is a straight line. The lines QC and PB intersect at D. (a) Prove that AB = AP. [2] D P Q A C B