SST 2025 EMATH PRELIM P2 MS

[Turn over SECONDARY 4 2025 PRELIMINARY EXAMINATION MATHEMATICS Paper 2 4052/02 27 August 2025 (Wednesday) 2 hour 15 minutes CANDIDATE NAME Student’s version CLASS 4 - INDEX NUMBER Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST Write your full name, class and index number in the spaces above. Write in dark blue or black pen in the space provided for each question. You may use a HB pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. The number of marks is given in brackets [ ] at the end of each question or part question. If working is needed for any question, it must be shown in the space below the question. Omission of essential working will result in loss of marks. The total of the marks for this paper is 90. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For p, use either your calculator value or 3.142. This document consists of 24 printed pages including the cover page. For Examiner’s Use Q1 10 Q2 12 Q3 10 Q4 11 Q5 10 Q6 9 Q7 10 Q8 8 Q9 10 Total 90

2 Mathematical Formulae Compound Interest Total amount = P Mensuration Curved surface area of a cone = prl Volume of a cone = pr2h Volume of a sphere = pr3 Area of triangle ABC = ab sin C Arc length = rq, where q is in radians Sector area = r2q, where q is in radians Trigonometry a2 = b2 + c2 - 2bc cos A Statistics Mean = Standard deviation = n r ÷ ø ö ç è æ+ 100 1 31342121 C c B b A a sinsinsin == å å f fx 22 ÷÷ ø ö çç è æ- å å å å f fx f fx Surface area of a sphere = 4pr2

[Turn over 3 Answer all the questions. N1 (N1.7 & N1.8) : Numbers and Operations [10] N2 (N2.2 & N2.4) : Ratio and Proportion N3 (N3.4) : Percentage 1. (a) (i) The number of electric cars produced in country Y by company X in 2015 was 3 857 000. Write this number in standard form. Answer 3.857×10! [1] (ii) In 2005, 2.96 million electric cars were produced in country Y. Calculate the percentage increase in the car production from 2005 to 2015. Answer Percentage increase =".$%&×()!*+.,!×()!+.,!×()!×100% =30.3% (3 significant figures) (iii) In 2015, 84.7% of all cars produced in country Y were non-electric cars. Calculate the total number of cars produced in 2015, giving your answer to the nearest million. Answer Total cars in 2015 =".$%&×()!𝟏𝟎𝟎*𝟖𝟒.𝟕×100 =2.5209×10& =25 million cars (nearest million) Or 25 000 000 (iv) In 2015, there were 420 cars per 1000 people in country Y. Estimate the population of the country in 2015. Give your answer in standard form correct to three significant figures. Answer Population =𝟐.𝟓𝟐𝟎𝟗×𝟏𝟎𝟕5+)×1000 =6.00×10& (3 significant figures) Show computation for percentage increase

4 (b) A design team creates a scale model for a museum exhibit. The diameter of an asteroid is 253 km. In a scale model, the diameter of the asteroid is 11 m. (i) Find the scale used for the model. Give your answer in the form 1 : n. Answer 11 𝑚:253 𝑘𝑚 1 𝑚:23 𝑘𝑚 1:23 000 (ii) Drawn to the same scale as the asteroid, the diameter of the asteroid’s moon is 1.2 m. Find the actual surface area of the moon, leaving your answer in terms of 𝝅. Assume that the moon is spherical. Answer 1 𝑚:23 𝑘𝑚 1.2 𝑚∶27.6 𝑘𝑚 Actual surface area =4𝜋5𝟐𝟕.𝟔𝟐6+ =761.76𝜋 km2 Surface area of a sphere = 4pr2 From page 2 Be careful with conversion to km.

[Turn over 5 N7 (7.6, 7.7, 7.8) : Equations and Inequalities [12] 2. (a) Solve 7(𝑥−2)−3(2𝑥−1)=8(3𝑥+5)−6 . Answer 7𝑥−14−6𝑥+3=24𝑥+40−6 𝑥=−5%+"=−1+++" (b) Solve the inequality −3𝑥+4<−2𝑥−6 . Answer −𝑥<−10 𝑥>10 (c) It is given that 𝑝=𝑟+𝑞−2𝑞−𝑟+ . (i) Find 𝑝 when 𝑟=(+ and 𝑞=−1. 𝑝=5126+(−1)−2(−1)−5126+=−9/4−5/4=95=145 Or 1.8 (ii) Express 𝑟 in terms of 𝑝 and 𝑞. 𝑝(𝑞−𝑟+)=𝑟+𝑞−2 𝑝𝑞−𝑝𝑟+=𝑟+𝑞−2 𝑝𝑟++𝑟+𝑞=𝑝𝑞+2 𝑟+(𝑝+𝑞)= 𝑝𝑞+2 𝑟+=𝑝𝑞+2𝑝+𝑞 𝑟=±C𝑝𝑞+2𝑝+𝑞 Based on context, Answer must be exact and shown as mixed number This is a quadratic relationship. Show substitution of values

6 (d) Solve the equation 𝑥+32𝑥2−8𝑥+6+5𝑥3−𝑥=2𝑥−1 . Give your solutions correct to two decimal places. Answer 𝑥+32(𝑥−3)(𝑥−1)+5𝑥3−𝑥=2𝑥−1 𝑥+32(𝑥−3)(𝑥−1)−5𝑥𝑥−3=2𝑥−1 𝑥+32(𝑥−3)(𝑥−1)−5𝑥(2)(𝑥−1)2(𝑥−3)(𝑥−1)=2(2)(𝑥−3)2(𝑥−3)(𝑥−1) 𝑥+3−10𝑥(𝑥−1)=4(𝑥−3) 10𝑥2−7𝑥−15=0 𝑥=−(−7)±√(−7)2−4(10)(−15)2(10) 𝑥=1.62 or −0.92 (2 d.p.). (2 d.p.) Note: 2𝑥2−8𝑥+6 = 𝟐 (𝑥−3)(𝑥−1) Always check back when using the calculator to factorise. 3−𝑥=−(𝑥−3)

[Turn over 7 N5: finding the value of an unknown quantity in a given formula N7: solving fractional equations that can be reduced to quadratic equations, formulating equations to solve problems [10] 3. Two water taps, X and Y, together can fill a tank in 6 hours. Tap X takes x hours to fill the tank alone. Tap Y takes 25 hours less than the Tap X to fill the tank alone. (a) Write down an expression, in terms of x, for the fraction of the tank that Tap X will fill up in one hour. 𝟏𝒙 (b) Write an expression, in terms of x, for the fraction of the tank that Tap Y will fill up in one hour. 𝟏𝒙 $ 𝟐𝟓 (c) Write down an equation to represent the rate of filling the tank by the 2 taps and show that it reduces to 𝑥+−37𝑥+150=0. 1𝑥+1𝑥−25=16 𝑥+𝑥−25𝑥(𝑥−25)=16 6(2𝑥−25)=𝑥(𝑥−25) 12𝑥−150=𝑥+−25𝑥 𝑥+−25𝑥−12𝑥+150=0 𝑥+−37𝑥+150=0 (shown) Equation that represents given context.

8 (d) Solve the equation 𝑥+−37𝑥+150=0. 𝑥+−37𝑥+150=0 𝑥=−(−37)±.(−37)!−4(1)(150)2(1) 𝑥=32.365 or 𝑥=4.634 𝑥=32.4 or 𝑥=4.63 (3 s.f.). (e) Calculate how long it would take to fill the tank from empty using Tap Y. Give your answer in hours and minutes, correct to the nearest minute. Reject 𝑥=4.63 as the time taken, 𝑥−25, cannot be negative. Time Tap Y takes to fill up the tank completely = 32.365−25 = 7.365 = 7 hours 22 minutes (nearest minute) Must show substitution of values. Important to test both values and justify any rejection of solution.

[Turn over 9 N6 (6.10) : Functions and Graphs [11] N7 (N7.2) : Equations and Inequalities 4. (a) Complete the table of values for 𝑦=𝑥27+2𝑥−5 . Values are given to one decimal place where appropriate. 𝑥 0.2 0.5 1 2 3 4 5 6 𝑦 −1.0 −2.9 −3.4 −3.0 −2.2 −1.0 0.5 [1] Answer 5.0 (1 d.p.) (b) On the grid opposite, draw the graph of 𝑦=𝑥27+2𝑥−5 for 0<𝑥≤6. [2] Answer [points plotted correctly] [smoothness of curve and label] (c) Use your graph to write down an inequality in 𝑥 to describe the range of values where 𝑦<−2. Answer 0.7<𝑥<4.2 (±0.1) (d) (i) On the same grid, draw the graph of 2𝑦+4𝑥=9 for 0<𝑥≤6. [2] Answer [at least 3 plotted points] [graph and label] (ii) Write down the x-coordinates of the points where the line intersects the curve. Answer 0.2, 3.55 (±0.1) (iii) These values of x are the solutions of the equation 2𝑥3+𝐴𝑥2−𝐵𝑥+28=0. Find the value of A and the value of B. 𝑥27+2𝑥−5=−2𝑥+92 [equate both functions] 2𝑥3+28−70𝑥=−28𝑥2+63𝑥 2𝑥3+28𝑥2−133𝑥+28=0 𝐴=28, 𝐵=133

10 FYI: Creating of graphs website 𝑦=−2 𝑦=𝑥+7+2𝑥−5 2𝑦+4𝑥=9