1. 2026 NCHS Prelim Math P1 QP with Ans Keys

1 Name: Register Number: Class: PRELIMINARY EXAMINATION 2026 SECONDARY FOUR EXPRESS MATHEMATICS 4052/01 Paper 1 20 August 2026, Thursday 2 hours 15 minutes Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST Write your name, class and register number on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all 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 with the answer. 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 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 paper consists of 21 printed pages including the cover page. For Marker’s Use Lack annotation/ units/ incorrect properties’ name −1 Accuracy −1 Final answer −1 Use of Pencil for essential working −1

2 Mathematical Formulae Compound interest Total amount = Mensuration Curved surface area of a cone = Surface area of a sphere = Volume of a cone = Volume of a sphere = Area of triangle ABC = Arc length = , where is in radians Sector area = , where is in radians Trigonometry Statistics Mean = Standard deviation = n rP      + 1001 rl 24 r hr 2 3 1 3 3 4 r Cabsin2 1 r  2 2 1 r  C c B b A a sinsinsin == Abccba cos222 2 −+= f fx   22        −  f fx f fx

3 Answer all the questions. 1 Express −𝑥2 − 4𝑥 + 2 in the form −(𝑥 + 𝑎)2 + 𝑏. Answer [2] 2 Given that 812 = 9(27−2𝑥), find x. Answer 𝑥 = [3] 3 Simplify 21𝑎4 4𝑏2𝑐3 ÷ 14𝑎3𝑐−1 12𝑏3 . Answer [2]

4 4 Solve 𝑥 5 = 2𝑥−3 4 . Answer x = [2] 5 (a) Factorise 8𝑝3𝑞 − 18𝑝𝑞3 completely. Answer [2] (b) (i) Expand and simplify (2𝑥 + 5𝑦)2. Answer [1] (ii) Hence or otherwise, given that (2𝑥 + 5𝑦)2 = 4(𝑥2 − 10𝑥 + 25), where y is a constant. Find the value of y. Answer 𝑦 = [1]

5 6 (a) It is given that 𝑝 = 𝑞(𝑟−2) 7−𝑟 . Calculate the value of 𝑝 when 𝑞 = 20.26 and 𝑟 = −8.19. Give your answer to 2 decimal places. Answer p = [1] (b) Rearrange the formula to make r the subject. Answer 𝑟 = [3]

6 7 The lowest common multiple of x and 72 is 1800. Given that x is a perfect square, find the largest possible value of x. Answer x = [2] 8 Alice invests $5000 at a rate of r % per annum compounded monthly. At the end of 13 years, the value of her investment is $7381. Calculate the value of r. Answer 𝑟 = [2]

7 9 A shop imports a smart watch and sells it at a certain price. During a sale, the shop offers a 15% storewide discount. With the 15% discount, the shop still makes a profit of 36% on the cost price of the watch. Given that the selling price of the watch during the sale is $204, calculate the original profit made if the shop did not have the storewide discount. Answer $ [3] 10 Sketch the graph of 𝑦 = −(𝑥 − 2)(𝑥 + 4) on the axes below. Indicate clearly the coordinates of the point s where the graph crosses the axes and the turning point on the curve. [3] 𝑦 𝑥 0

8 11 A courier service uses an electric van to transport standardized parcels from a sorting hub to an automated parcel locker station. The van has two strict load limits that cannot be exceeded. Maximum loading weight limit : 850 kg Maximum loading volume limit : 6 𝐦𝟑 Each individual parcel has the following specifications. Weight : 22 kg Volume : 150000 𝐜𝐦𝟑 (a) The courier driver commented that the van will not be able to fit 30 parcels due to the maximum loading volume limit. Do you agree with the worker? Support your answer with workings. Answer I agree / disagree with the worker. [2] (b) Given that 4 trolleys weighing 3.8 kg each must also be in the van during transport, calculate the maximum number of parcels that can be safely loaded in one trip? Answer parcels [2]

9 12 Solve the inequalities 𝑥−1 2 < 5−2𝑥 3 ≤ 2 − 𝑥−3 4 and represent the solutions on the number line below. [4] 13 The points A and B have coordinates (−3, 2) and (9, −6) respectively. (a) Find 𝐴𝐵⃗⃗⃗⃗⃗ as a column vector. Answer 𝐴𝐵⃗⃗⃗⃗⃗ = [1] (b) By vector method, determine whether the point C (3, −2) lies on the straight line segment AB. Answer [2] 0

10 14 The scale drawing shows the position of the visitor centre at A and two straight cycling paths, AP and AQ in a park. Scale: 1cm to 40 metres (a) A scenic lookout tower is located on a bearing of 065° and is 260 m from A. Mark and label the position of the lookout tower with the letter ‘T’. [1] (b) A straight footpath, within the park , starting from A runs exactly halfway between the two cycling paths AP and AQ. Construct the line representing this footpath till the boundary of the park. [1] (c) A vending machine is to be placed within the park, such that it is equidistant from AP and AQ, and equidistant from A and P. Mark and label the position of the vending machine with the letter ‘X’. [1]