NASS 2026 AM-O S4E5N Prelim P2 (QP)

Name: .............................................................. Register no: ........... Class: ........... NGEE ANN SECONDARY SCHOOL O PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS 4049/02 Paper 2 31 August 2026 2 hours 15 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your name, register number and class 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 Total /90 Checked by student: ………………………………. Date: ………………… This document consists of 25 printed pages and 1 blank page.

2 NAS/2026/Prelim/AM-O/P2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02  cbxax , a acbbx 2 42 Binomial Expansion   1 2 2 ... ...1 2 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 rn r r n r r          2. TRIGONOMETRY Identities 2 2sin cos 1A A  2 2sec 1 tanA A  2 2cosec 1 cotA A    BABABA sincoscossinsin   cos cos cos sin sinA B A B A B     BA BABA tantan1 tantantan   AAA cossin22sin  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 ΔABC sin sin sin a b c A B C  2 2 2 2 cosa b c bc A   1 sin2 bc A

3 NAS/2026/Prelim/AM-O/P2 1 The simultaneous equations 2 6y x px   and 32 2y x  have a common solution for which 3x . Find the value of p, and hence solve the simultaneous equations. [5]

4 NAS/2026/Prelim/AM-O/P2 2 (a) (i) In an architectural design, a support beam is modelled by the line y = 2x + k and a parabolic arch is modelled by y = x2 − 4x + 7. The beam is to touch, but not cut, the arch. Find the value of k. [3] (ii) The beam is then raised vertically by 3 units from this tangent position. Without drawing or referring to a graph, determine the number of points at which the beam intersects the arch. [2]

5 NAS/2026/Prelim/AM-O/P2 (b) Represent the solution set of   2 2 1 4 8x x   on a number line. [3]

6 NAS/2026/Prelim/AM-O/P2 3 A cubic polynomial is given as   3 2P x x ax bx c    . The graph of  Py x cuts the y-axis at  0, 2 . The remainder when  P x is divided by 1x is 3 , and 2x is a factor of  P x . (a) Find the values of a, b and c. [4]

7 NAS/2026/Prelim/AM-O/P2 (b) Hence solve the equation  P 0x  , leaving your answers in exact form. [3]

8 NAS/2026/Prelim/AM-O/P2 4 The amount of medicine remaining in a patient’s bloodstream t hours after it is taken is M mg. It is modelled by the equation ln ln16 ln 25 3 M t      . (a) Show that the model can be written in the form ln 2380 t M e   . [2]

9 NAS/2026/Prelim/AM-O/P2 (b) Find the time taken for the medicine to decrease to 20 mg. [3] (c) A second dose should be taken only when the amount of medicine remaining in the bloodstream has fallen to less than 15 mg. A patient takes the medicine at 07 00. The patient asks whether the second dose can be taken at 14 20. Use the model to determine whether the second dose can be taken at 14 20. Give a reason for your answer. [3]

10 NAS/2026/Prelim/AM-O/P2 5 The graph of  cosy a bx c  has adjacent minimum points at  0, 1 and  4π, 1 . It also has a maximum point at  2π,5 . (a) Find the values of a, b and c. [3]