MSHS 2025 AMATH PRELIM P2 QP

This document consists of 21 printed pages and 1 blank page [Turn over For Examiners’ Use Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 / 7 / 7 / 9 / 7 / 12 / 9 / 8 / 11 Q9 Q10 SUBTOTAL / 9 / 11 Statement Presentation Units Rounding Off Class/ Index Number Centre Number/ ‘O’ Level Index Number Name / / MARIS STELLA HIGH SCHOOL PRELIMINARY EXAMINATION SECONDARY FOUR ADDITIONAL MATHEMATICS 4049/2 Paper 2 26 August 2025 Candidates answer on the Question Paper. 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 of the marks for this paper is 90. 90

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax 2 + bx + c = 0, x = a ac b b 2 42 − ± − Binomial expansion (a + b) n = an +       1 n an − 1b +       2 n an − 2b2 + ... +       r n an − r br + ... + bn, where n is a positive integer and       r n = ! !( )! n rn r − = ! ) 1 )...( 1 ( r r n n n+ − − 2. TRIGONOMETRY Identities sin 2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 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) = B A B A tan tan1 tan tan  ± sin2 2sin cosA AA= 22 2 2cos 2 cos sin 2cos 1 1 2sinA AA A A= − = −=− 2 2 tantan 2 1 tan AA A= − Formulae for ∆ABC A a sin = B b sin = C c sin 2 22 2 cosa b c bc A=+− 1 sin2 bc A∆=

3 1 (a) A polynomial f(x) has a remainder of −1 when divided by 2x + 3. (i) Find the remainder when f(x) – 1 is divided by 2x + 3. [1] (ii) Find in terms of f(x), a polynomial which is divisible by 2x + 3. [1]

4 (b) The cubic polynomial g(x) is such that the coefficient of x3 is 2 and the roots of g(x) = 0 are −1, m and 2m, where m is an integer. It is given that g(x) has a remainder of 12 when divided by x – 1. Find an expression for g(x) in descending powers of x. [5]

5 2 The equation of a curve is ln xy x= . (a) Find the coordinates of the stationary point of this curve. Leave your answer in terms of e. [4] (b) Determine the nature of the stationary point. [3]

6 3 The decay of a certain radioactive isotope can be modelled by the exponential equation 0 atN Ne −= after t weeks, where N represents the amount of radioactive isotope, N 0 and a are constants. A sample of this radioactive isotope has a mass of 100.9 g initially. After 2 weeks, it is found that the amount of this sample left is 84.6 g. (a) Calculate the value of a. [3] (b) What percentage of this sample has decayed after 5 weeks? [3]

7 (c) Find the number of weeks when the amount of radioactive isotope decayed first exceeds 60 g. Give your answer correct to the nearest week. [3]

8 4 In the diagram, AB is the diameter of the circle with centre O. DE and BF are tangents to the circle at C and B respectively. DCE and BEF are straight lines. (a) Prove that triangle ABC and triangle AFB are similar. [3]

9 (b) Show that EC = EF. [4]

10 5 (a) Find the integer a and b which satisfy the equation ( ) 1 1 32 21 428 abyyy −  ÷= . [3] (b) Solve the equation ( )3 1log 8 2 log 3x x−= − . [5]