NASS 2025 AMATH PRELIM P2 QP

Name: .............................................................. Register no: ........... Class: ........... NGEE ANN SECONDARY SCHOOL O PRELIMINARY EXAMINATION ADDITIONAL MATHEMATICS 4049/02 Paper 2 2 September 2025 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 21 printed pages and 1 blank page.

2 NAS/2025/Prelim/AM-NA/P2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 02 = + +c bx ax, a ac b bx 2 42 − ± −= Binomial Expansion ( ) 1 22 ... ...12 n n n n nr r nnn nab a a b a b a b b r −− −  + = + + ++ ++     , where n is a positive integer and ( ) ( ) ( ) 1 ... 1 ! ! ! ! n nn nrn r r nr r − −+ = = − 2. TRIGONOMETRY Identities 22sin cos 1AA+= 22sec 1 tanAA= + 22cosec 1 cotAA= + ( ) B A B A B Asin cos cos sinsin ±= ± ( )cos cos cos sin sinAB A B A B±=  ( ) B A B AB A tan tan1 tan tantan  ±= ± A A Acos sin2 2sin = 22 2 2cos2 cos sin 2cos 1 1 2sinA AA A A= − = −=− 2 2tantan2 1 t a n AA A= − Formulae for ΔABC sin sin sin abc ABC= = 222 2 cosa b c bc A=+− 1 sin2 bc A∆=

3 NAS/2025/Prelim/AM-NA/P2 1 (a) Without using a calculator, express 1925 108 243 2−+ in the form 3a , where a is an integer. [2] (b) Given that 52p= + , express 2 1 p p + − in the form of 5ab + , where a and b are rational numbers. [3]

4 NAS/2025/Prelim/AM-NA/P2 2 (a) Show that d cos 2 2 d 1 sin 2 1 sin 2 x xx x  =− ++ . [4]

5 NAS/2025/Prelim/AM-NA/P2 (b) Hence find ( ) π 4 0 1 d8 1 sin 2 xx+∫ . [4]

6 NAS/2025/Prelim/AM-NA/P2 3 (a) Express ( ) 32 2 2 12 14 5 1 xxx xx + ++ + in the form ( ) 2 2 . 1 qx rx sp xx +++ + [1]

7 NAS/2025/Prelim/AM-NA/P2 (b) Hence express ( ) 32 2 2 12 14 5 1 xxx xx + ++ + in partial fractions. [5]

8 NAS/2025/Prelim/AM-NA/P2 4 (a) Given that the line 6ym x= − is a tangent to the curve 210 2y x nx m= −+ , show that m cannot be negative. [4]

9 NAS/2025/Prelim/AM-NA/P2 (b) Find the range of values of k for which (i) the graph of 22 61yx x k= − ++ lies completely above the x-axis, [2] (ii) ( ) 2 1kx k x k++ + is always negative. [3]

10 NAS/2025/Prelim/AM-NA/P2 5 A scientist is studying how the concentration of a chemical solution (c, in mg/L) decreases over time (t, in hours) during a natural breakdown process. The table below shows some recorded values of t and c. t 0.8 1.0 1.2 1.4 1.8 c 21.0 15.0 10.8 7.5 2.1 It is believed that c and t are related by the equation bc at t= + , where a and b are constants. (a) Express the given equation in a form suitable for drawing a straight line graph, and using suitable scales, draw the graph for the values given on a graph paper. [4]