CAT HIGH 2025 AMATH PRELIM P2 QP

1 Catholic High School Preliminary Examinations 2025 [TURN OVER Name: Index Number: Class: CATHOLIC HIGH SCHOOL 2025 Preliminary Examination Secondary 4 (O-Level Programme) Additional Mathematics 4049/02 Paper 2 1 September 2025 2 hours 15 minutes Booklet A Additional materials: Question Booklets 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. Calculators should be used where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer answer to three significant figures. Give answers in degrees to one decimal place. For π , use either your calculator value or 3.142, unless the question requires the answer in terms of π . The number of marks is given in brackets [ ] at the end of each question or part question. The total mark for this paper is 90. For examiners’ use __________________________________________________________________________________________ This booklet consists of 8 printed pages. A 30

2 Catholic High School Preliminary Examinations 2025 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 Catholic High School Preliminary Examinations 2025 [TURN OVER 1 Jane bought an electric car in January 2025. After purchase, the value of the car, $V, diminishes over time and can be modelled by 38000 ktVe −= , where k is a positive constant and t is time measured in years. The value of the car is expected to be $29000 after 3 years of driving. Jane intends to sell her car when its value drops to half of its original value. Showing clear mathematical calculations, find the year that Jane is likely to sell her electric car. [5] ____________________________________________________________________________________

4 Catholic High School Preliminary Examinations 2025 2 (i) Given that the curve 2 21y ax bx=+− lies entirely below the x-axis, determine the conditions that must be applied to the constants a and b. [2] (ii) If a and b are both integers, state an example of the values of a and b which satisfy the conditions found in (i). [2] ____________________________________________________________________________________ 3 (a) The variables x and y are defined such that 2 49log 3log log 3xy−= . (i) Give a reason why x and y must be positive numbers. [1]

5 Catholic High School Preliminary Examinations 2025 [TURN OVER (ii) Express x in terms of y. [5]

6 Catholic High School Preliminary Examinations 2025 (b) Solve the equation ( ) 15 2 5 11xx+− += , leaving non-exact value(s) of x in the form loga b . [4] ____________________________________________________________________________________

7 Catholic High School Preliminary Examinations 2025 [TURN OVER 4 (a) (i) The polynomials ( ) 32P 10x ax x bx= +++ and ( ) 32Q5x x ax x b=+ −+ leave the same remainder when divided by 2x+ . Show that 44ab+= . [2] (ii) If ( )P x′ has a factor of 32x− , find the values of a and b. [4]

8 Catholic High School Preliminary Examinations 2025 (b) Solve the equation 322 4 5 70xxx+ − −= , expressing non-exact solutions in the form 1 2ab± where a and b are constants to be determined. [5] ____________________________________________________________________________________ END OF BOOKLET A

9 Catholic High School Preliminary Examinations 2025 [TURN OVER Name: Index Number: Class: CATHOLIC HIGH SCHOOL 2025 Preliminary Examination Secondary 4 (O-Level Programme) Additional Mathematics 4049/02 Paper 2 1 September 2025 Booklet B 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. For examiners’ use ___________________________________________________________________________________________ This booklet consists of 6 printed pages and 2 blank pages. B 27

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