Fu Hua 2025 4E5N AM Prelims P1 - QP

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 4049/01 Paper 1 DATE 26 August 2025 TIME 1050 – 1305 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: Ms Gao Conger Vetter: Ms Lim Fen Niu This document consists of 24 printed pages.

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 [Turn over 1 Express 32 2 4 24 41 4 21 x x x xx − − + −− in partial fractions. [5]

4 2 (a) In a quiz question, the quadratic equation ( ) 22 1 8 6 0k x x x+ − − = is given. Find the range of values of the constant k, for which the equation has no real roots. [4]

5 [Turn over (b) Another question in the quiz is shown below. Find the range of values of the constant k, for which the curve ( ) 262y k x x= − − lies completely above the line 6y x k=− . Jeremy stated that the range of values of k found in part (b) is the same as that in part (a). Is Jeremy correct? Explain your answer clearly. [2]

6 3 Solve the simultaneous equations. [4] 2 10xy+= 32 2yx+=

7 [Turn over 4 In triangle ABC, ( )72BC =− cm and angle ABC 30= . The area of triangle ABC is ( )4 7 2− cm2. Without using a calculator, express the length of AB in the form ( )14pq+ cm, where p and q are constants, and state the value of p and of q. [4]

8 5 A circle C1, with centre C, has equation 22 8 2 17 0x x y y− + + − = . (a) Find the coordinates of C and the radius of the circle. [3] (b) Determine whether the point ( )1, 5S −− lies inside, outside or on the circle. Show your working clearly. [2]

9 [Turn over (c) Given that ( )7, 6P − , ( )9, 2Q and R are three points on the circle, and PQ is perpendicular to QR, find the coordinates of R. [3] (d) Another circle C2, is a reflection of C1 in the x-axis. Write down the coordinates of the centre of C2. [1]

10 6 (a) Prove that sin sin 2cotsec 1 sec 1 xx xxx +=+− . [4]