CAT HIGH 2025 AMATH PRELIM P1 MS

1 Name: Index Number: Class: CATHOLIC HIGH SCHOOL Preliminary Examination Secondary 4 (O-Level Programme) Additional Mathematics 4049/01 Paper 1 28 August 2025 2 hours 15 minutes Additional Materials: Answer Booklets A, B and C. READ THESE INSTRUCTIONS FIRST Write your name, index number and class on all the work you hand in. Write in dark blue or black pen. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions in the space provided. If working is needed for any question, it must be shown with the answer. Omission of essential working will result in loss of marks. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 90. For examiner’s use: / 90 This document consists of 19 printed pages and one blank page. SOLUTIONS

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the quadratic equation ax 2 + bx + c = 0 , a ac b bx 2 42 − ± −= Binomial Expansion ( ) nr r nnnnn b b ar nb anb ana b a + +     + +     +     + = + −−− 2 21 21 , where n is a positive integer and ( ) ! ) 1 )...( 1 ( ! ! ! r r n n n r r n n r n + − −=−=      2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B  sin A sin B tan ( A ± B ) = tan tan 1 tan tan AB AB ±  sin 2A = 2 sin A cos A cos 2A = cos2 A – sin2 A = 2cos2 A – 1 = 1 – 2 sin2 A tan 2A = 2 2 tan 1 tan A A− Formulae for ∆ ABC sin sin sin abc ABC= = a 2 = b 2 + c 2 − 2bc cos A ∆ = 2 1 ab sin C

3 1 (i) Express 254 2 xx−− in the form ( ) 2c ax b−+ , where a, b and c are constants and a > 0. [2] ( ) 22 2 54 2 2 2 5 2 17 xx x x x −− = − + +  = − ++ (ii) Sketch the graph of 254 2y xx= −− , indicating clearly the turning point and the y - intercept. [2] (iii) Hence find the range of values of k for which the equation 254 2 xxk−− = has at most one root. [1] 7k ≥

4 2 The function f is defined by ( )f 4cosx ax b= + for 03 x π≤≤ , where a and b are constants. The maximum value of f is 6 and the period of f is 4π . (i) State the amplitude of f. [1] Amplitude = 4 (ii) Write down the value of a and of b. [2] ( )f 4cos Period 4 2 4 1 2 46 2 x ax b a a b b π π π = + = = = += = (iii) Sketch the graph of ( )fyx= for 03 x π≤≤ . [3]

5 3 (i) F