CAT HIGH 2025 AMATH PRELIM P1 QP

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 Booklet A consists of 21 printed pages.

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] (ii) Sketch the graph of 254 2y xx= −− , indicating clearly he 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] y x

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] (ii) Write down the value of a and of b. [2] (iii) Sketch the graph of ( )fyx= for 03 x π≤≤ . [3] y x

5 3 (i) Find the range of values of k for which the line 2y kx= + intersects the curve 2xy y x−= at two distinct points. [4] (ii) Hence state the possible value(s) of k for which the line 2y kx= + is a tangent to the curve 2xy y x−= . [1]

6 4 Solve the equation 8 11 2 xx− −= − , giving your answer in the form a bc+ , where a, b and c are integers. [4]

7 5 Water is being poured, at a constant rate of 42 m 3/s, into an empty container. After t seconds, the depth of water is h m and the volume, V m3, of the water in the container is given by ( ) 352 3 hV π += . (i) Find the rate of change of the depth of water in terms of h and π. [3] (ii) As the depth of water in the container rises, determine with a reason, whether the rate of change of h will increase or decrease. [1]

8 6 The diagram shows a solid prism with a triangular base, ABC , with angle 60BAC = ° and ( )23AB = + cm. The height of the prism is 20 3   cm and its volume is ( ) 325 10 3 cm+ . Without using a calculator , express the length of AC in the form ( ) 3ab+ where a and b are constants. [5] A B C cm 60 °

9 7 (i) Express ( ) ( ) 2 83 23 4 x xx − ++ in partial fractions. [4] (ii) Differentiate ( ) 2ln 4x + with respect to x. [1]

10 (iii) Use your answers to parts (i) and (ii) to find ( ) ( ) 1 20 83 d 23 4 x x xx − ++∫ . [5]