SST 2025 EMATH PRELIM P1 MS

[Turn over 1 SECONDARY 4 2025 PRELIMINARY EXAMINATION MATHEMATICS Paper 1 4052/01 26 August 2025 (Tuesday) 2 hour 15 minutes CANDIDATE NAME CLASS S 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 , use either your calculator value or 3.142. This document consists of 28 printed pages including the cover page. For Examiner’s Use Q1 2 Q2 4 Q3 4 Q4 6 Q5 2 Q6 3 Q7 4 Q8 3 Q9 3 Q10 3 Q11 2 Q12 4 Q13 4 Q14 2 Q15 4 Q16 6 Q17 5 Q18 5 Q19 4 Q20 6 Q21 2 Q22 8 Q23 4 Total 90 Solution

2 Mathematical Formulae Compound Interest Total amount = P Mensuration Curved surface area of a cone = rl Surface area of a sphere = 4r2 Volume of a cone = r2h Volume of a sphere = r3 Area of triangle ABC = ab sin C Arc length = r, where  is in radians Sector area = r2, where  is in radians Trigonometry a2 = b2 + c2 − 2bc cos A Statistics Mean = Standard deviation = n r      + 1001 3 1 3 4 2 1 2 1 C c B b A a sinsinsin ==   f fx 22      −     f fx f fx

[Turn over 3 Answer all the questions. 1.  = { integers 𝑥 ∶ 3 < 𝑥 < 18 } 𝑃 = {multiples of 5} 𝑄 = {perfect squares} (a) List the elements in (𝑃 ∪ 𝑄)′ . 𝑃 ∪ 𝑄 = {4, 5, 9, 10, 15, 16} (𝑃 ∪ 𝑄)′ = {6, 7, 8, 11, 12, 13, 14, 17} Answer …………………….……. [1] (b) On the Venn diagram, shade the region which represents (𝑃 ∩ 𝑄) ∪ (𝑃 ∩ 𝑄′). Answer [1]

4 2. (a) Use prime factors to explain why 18 × 63 is not a perfect square. Answer 18 = 2 × 32 63 = 7 × 32 18 × 63 = 2 × 32 × 7 × 32 18 × 63 = 2 × 34 × 7 Since the indices of the prime factors of 18 × 63 are not all even numbers, 18 × 63 is not a perfect square. ……………...…………………………………………………………….….… …………...…………………………………………………………….….…… …………...…………………………………………………………….….…… [2] (b) The number 18𝑝 is a perfect cube. Find the smallest positive integer value of 𝑝. 18 = 2 × 32 18𝑝 = 2 × 32 × 𝑝 𝑝 = 22 × 3 𝑝 = 12 Answer 𝑝 =……………….……. [2]

[Turn over 5 3. (a) Given that 2𝑚−1 × 3𝑚 = 1, find the value of 36𝑚 . 2𝑚−1 × 3𝑚 = 1 2𝑚 × 3𝑚 = 2 6𝑚 = 2 36𝑚 = 22 36𝑚 = 4 Answer 36𝑚 = …………….….……. [2] (b) Simplify . (2𝑥𝑦)−2 12 ÷ 𝑥3 4𝑦2 = 1 48𝑥2𝑦2 × 4𝑦2 𝑥3 = 1 12𝑥5 Answer …………………….……. [2] (2𝑥𝑦)−2 12 ÷ 𝑥3 4𝑦2

6 4. (a) Solve the equation 𝑥2 − 4𝑥 − 11 = 0 by completing the square. Give your answer correct to 2 decimal places. 𝑥2 − 4𝑥 − 11 = 0 (𝑥2 − 4𝑥 + 22 − 22) − 11 = 0 (𝑥 − 2)2 − 4 − 11 = 0 (𝑥 − 2)2 = 15 𝑥 − 2 = ±√15 𝑥 = 2 ± √15 𝑥 = 5.87 (2 𝑑. 𝑝. ) or 𝑥 = −1.87 (2 𝑑. 𝑝. ) [ Answer x = ………… or x = ….….……. [3] (b) (i) Factorise completely 4𝑥2 + 5𝑥 − 6. 4𝑥2 + 5𝑥 − 6 = (4𝑥 − 3)(𝑥 + 2) Answer ………………..…………….……. [1] (ii) Hence, factorise completely 4(2𝑡 + 2)2 + 5(2𝑡 + 2) − 6. Write your answer as simply as possible. 4(2𝑡 + 2)2 + 5(2𝑡 + 2) − 6 = [4(2𝑡 + 2) − 3][(2𝑡 + 2) + 2] = (8𝑡 + 5)(2𝑡 + 4) = 2(8𝑡 + 5)(𝑡 + 2) Answer ………………..…………….……. [2]

[Turn over 7 5. It is given that y is directly proportional to x2. Find the percentage increase in y if x is increased by 25% of its original value. 𝑦 = 𝑘𝑥2, where k is a constant 𝐿𝑒𝑡 𝑛𝑒𝑤 𝑥 = 𝑥1 𝐿𝑒𝑡 𝑛𝑒𝑤 𝑦 = 𝑦1 𝑥1 = 1.25𝑥 𝑦1 = 𝑘(1.25𝑥)2 𝑦1 = 1.5625𝑘𝑥2 Percentage increase in y = 𝑦1−𝑦 𝑦 × 100% = 1.5625𝑘𝑥2 − 𝑘𝑥2 𝑘𝑥2 × 100% = 𝑘𝑥2(1.5625 − 1) 𝑘𝑥2 × 100% = 56.25 % Answer ………………..….… % [2] 6. Express as a single fraction in its simplest form 4(𝑥 − 2) 𝑥2 − 4 − 2(3𝑥 − 1) 3𝑥2 + 5𝑥 − 2 . 4(𝑥 − 2) 𝑥2 − 4 − 2(3𝑥 − 1) 3𝑥2 + 5𝑥 − 2 = 4(𝑥 − 2) (𝑥 − 2)(𝑥 + 2) − 2(3𝑥 − 1) (3𝑥 − 1)(𝑥 + 2) = 4 𝑥 + 2 − 2 𝑥 + 2 = 2 𝑥 + 2 Answer ………………..…………….……. [3]

8 7. In the sequence, the difference between any two consecutive terms is the same number. 17 x y z 9 … (a) Find the values of 𝑥, 𝑦 and 𝑧. 𝑇1 = 17 𝑇2 = 𝑥 = 17 + 1(−2) = 15 𝑇3 = 𝑦 = 17 + 2(−2) = 13 𝑇4 = 𝑧 = 17 + 3(−2) = 11 𝑇5 = 23 = 17 + (5 − 1)(−2) = 9 Alternative method Common difference = 2 x = 17 – 2 = 15 y = 17 – 4 = 13 z = 17 – 6 = 11 Answer 𝑥 = …………….….…… 𝑦 = …………….………. 𝑧 = ………………..….… [2] (b) Write down an expression for the nth term of the sequence. 𝑇𝑛 = 17 + (−2)(𝑛 − 1) 𝑇𝑛 = 19 − 2𝑛 Answer ………………...………….……. [1] (c) Explain why −317 is a term of this sequence. Answer 19 − 2𝑛 = −317 𝑛 = 168 Since n is a positive integer, −317 is the 168th term of the sequence. OR Since n is a positive integer, −317 is a term of the sequence. …………...…………………………………………………………...…….….…… [1]

[Turn over 9 8. Sketch the graph of 𝑦 = −(2𝑥 − 1)(𝑥 + 3) on the axes below. Indicate clearly the coordinates of the turning point and where the graph crosses the axes. [3]

10 9. Diana has two outlets that sell donuts of three flavours. The number of donuts sold in the outlets over a two-hour period is given by the matrix D. Blueberry Chocolate Apple D = ( 75 86 48 108 56 36 ) 𝑂𝑢𝑡𝑙𝑒𝑡 1 𝑂𝑢𝑡𝑙𝑒𝑡 2 (a) Each blueberry donut is sold at $3.50. Each chocolate donut is sold at $4.50. Each apple donut is sold at $2.50. Represent the selling price in a 3 × 1 column matrix P. P = ( 3.5 4.5 2.5 ) Answer P = …………………….……. [1] (b) Evaluate the matrix M = DP. M = DP = ( 75 86 48 108 56 36 ) ( 3.5 4.5 2.5 ) = ( (75 × 3.5) + (86 × 4.5) + (48 × 2.5) (108 × 3.5) + (56 × 4.5) + (36 × 2.5)) = (769.5 720 ) Answer M = …………………….……. [1] (c) Explain what each element in matrix M represents. 769.5 represents the total amount in dollars collected by Diana from selling donuts of the 3 flavours in outlet 1 over a two-hour period. 720 represents the total amount collected in dollars by Diana from selling donuts