2025 Prelim 4E5N Presbyterian High School P2 Solutions

Name: Index No.: Class: PRESBYTERIAN HIGH SCHOOL ADDITIONAL MATHEMATICS 4049/02 Paper 2 26 August 2025 Tuesday 2 hours 15 min PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL 2025 SECONDARY FOUR EXPRESS / FIVE NORMAL (ACADEMIC) PRELIMINARY EXAMINATIONS DO NOT OPEN THIS QUESTION PAPER UNTIL YOU ARE TOLD TO DO SO. INSTRUCTIONS TO CANDIDATES Write your name, index number and class in the spaces provided above. 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. Write your answers in the spaces provided below 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 Qn 1 2 3 4 5 6 7 8 9 10 11 Marks Deducted Marks Setter: Ms Sabrina Tan Vetter: Mr Tan Lip Sing This question paper consists of 22 printed pages and 0 blank page. Category Accuracy Units Notations Others Question No. TOTAL MARKS 90 WORKED SOLUTIONS

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0,ax bx c+ + = 2 4 2 b b acx a −  −= Binomial expansion 1 2 2( ) ... ... , 12 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 2. TRIGONOMETRY Identities 22sin cos 1AA+= 22sec 1 tanAA=+ 22cosec 1 cotAA=+ sin( ) sin cos cos sinA B A B A B =  cos( ) cos cos sin sinA B A B A B= tan tantan( ) 1 tan tan ABAB AB = sin 2 2sin cosA A A= 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 sin sin sin a b c A B C== 2 2 2 2 cosa b c bc A= + − 1 sin2 bc A= ! ( 1)...( 1) ( )! ! ! n n n n n r r n r r r  − − +== − ABC

3 1 The equation of a curve is . Find the x-coordinates of the points at which the gradient of the curve is 7.5. [4]

4 2 The temperature of water in a tea cup, T °C, t minutes after boiling water is poured into the empty cup can be modelled by the formula . 70 – 80 °C 80 °C 90 – 95 °C 100 °C The diagram above shows the ideal brewing temperature for various teas at a particular teahouse. At this teahouse, green tea leaves are placed into the tea cup for brewing at the ideal temperature when . Determine whether the teahouse’s standards for brewing tea would be adhered to if black tea leaves are placed into the teacup when . [4] When t = 6, T = 80, When t = 2, T = 92.6 °C Since the temperature of the water would be within the ideal range of 90 – 95 °C, the standards for brewing black tea would be adhered to. White Green Black Herbal Alternative:

5 3 It is given that and . (a) Find f(x). [4] (b) Find the exact value of . [2]

6 4 (a) The time taken, t in minutes, to download a particular gaming software is related to internet speed, x Mbps. The variables x and t are related by the formula , where a and b are positive constants. The data below shows some measured values of x and t. x (Mbps) 50 100 200 500 1000 t (minutes) 23 13 7.3 3.2 1.6 (i) Plot against x and draw a straight line graph. [2] 0.313 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 200 400 600 800 1000 x

7 (ii) Use the graph to estimate the value of a and of b, correct to 2 significant figures. [4] Gradient = (iii) Use the graph to estimate the time taken for John to download the gaming software if the speed of his internet is 240 Mbps. [2] From the graph, (iv) Suggest an alternative set of variables to be plotted on each axis to draw a straight line to represent the formula. [1] An alternative would be to plot xt against t OR t against xt

8 (b) The diagram shows part of a straight line graph drawn to represent the equation . Given that the line passes through the point (2.5, 8.1) and has a gradient of 3, find the value of p correct to the nearest integer. [3] When lg y = 8.1, lg x = 2.5, lg x lg y (2.5, 8.1) 0

9 5 (a) Find the range of values of the constant k for which has no real roots. [4] Since there are no x-intercepts, (b) Hence, explain why is always positive. [2] Since the discriminant < 0 when k = –1 and the coefficient of x2 , i.e. 1, is positive, is always positive

10 6 (a) By using a suitable substitution, or otherwise, solve the equation . [4] (b) Sketch the graph of on the given axes. Label any axial intercepts. [1] 1 x y O