2024 NCHS Prelim Math P1 QP

Name: Register Number: Class: PRELIMINARY EXAMINATION 2024 SECONDARY FOUR EXPRESS MATHEMATICS Paper 1 4052/01 15 August 2024, Thursday 2 hours 15 minutes Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST Write your name, class and register number 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, highlighters, glue or correction fluid. Answer all questions. 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 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, unless the question requires the answer in terms of . The number of marks is given in brackets [ ] at the end of each question or part question. The total marks for this paper is 90. This document consists of 23 printed pages including the cover page. For Marker’s Use

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 −+= Abccba cos222 2 −+= Abccba cos222 2 −+= f fx   22        −  f fx f fx

3 Answer all the questions. 1 State whether each of the following statements is True or False. (a) All natural numbers are either composite or prime number. Answer ……………………………. [1] (b) For all real values of 𝑥, 𝑥3 is always greater than 𝑥2. Answer ……………………………. [1] 2 𝑎 = 2 × 32 × 5 𝑏 = 22 × 5 × 7 Using only prime factorisation, explain why 105𝑏 𝑎 is not a whole number. Answer ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… [2]

4 3 A rope is 44 m long. Jay cuts the rope into two pieces so that the ratio of their lengths is 4:7. He subsequently cuts away the same length of rope from each piece. The new ratio of the remaining lengths of the ropes is 2:5. Calculate the length that Jay had cut away from each piece of rope. Answer …………………………m [3] _________________________________________________________________________________________ 4 Solve 79−𝑥2 − 1 = 0. Find the values of 𝑥. Answer ……………………………. [2] _________________________________________________________________________________________

5 5 It is given that 𝑦3 is inversely proportional to the square of 𝑥. The difference between the value of 𝑦3 when 𝑥 = 3 and 𝑥 = 6 is 5. Find the values of 𝑥 when 𝑦 = 2. Answer ……………………………. [3] _________________________________________________________________________________________ 6 There are 12.044 × 1023 atoms in 24 g of carbon. Find the number of atoms in 5 kg of carbon. Leave your answer in standard form. Answer ……………………………. [2] _________________________________________________________________________________________

6 7 The diagrams show the graphs of six different functions. Diagram A Diagram B Diagram C Diagram D Diagram E Diagram F In each case, select one of the graphs that best represent the equation. (a) 𝑦 = (𝑥2 − 4)(𝑥 + 4) Answer Diagram …………… [1] (b) 𝑦 = − 2 𝑥 Answer Diagram …………… [1] (c) 𝑦 = 𝑥2 − 𝑥3 Answer Diagram …………… [1] ________________________________________________________________________________________ x y x y x y x y x y x y

7 8 Explain whether 𝑥 = 22 and 𝑦 = 1 is a possible solution of the equation 9𝑥 + 2.5𝑦 = 3𝑦 − 2𝑥. Answer ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… [2] 9 On the axes provided, indicating the y-intercept clearly, sketch the graph of (a) 𝑦 = −𝑥, [1] (b) 3𝑦 − 𝑘𝑥 − 9 = 0, where 𝑘 < −3. [2] Answer _________________________________________________________________________________________ O y x

8 10 Jake plays a mind-reading game with his friends. Using algebraic expression, explain clearly why Jake is able to predict that the new number is a multiple of 3. Answer …………………………………………………………………………………… …………………………………………………………………………………… …………………………………………………………………………………… …………………………………………………………………………………… …………………………………………………………………………………… …………………………………………………………………………………… …………………………………………………………………………………… …………………………………………………………………………………… [3] Think of a number between 20 to 100. Add the two digits together. Subtract the sum of the two digits from your original number to get a new number. …. And the new number is a multiple of 3!

9 11 It is given that 2(𝑚𝑛)2 − 2𝑚𝑛 = 3𝑚𝑛 + 3. Find 𝑛 in terms of 𝑚. Answer ……………………………. [2] _________________________________________________________________________________________ 12 Simplify 4 3 + 2 9𝑚 ÷ √81𝑚164 , giving your answer as a single fraction. Answer ……………………………. [3] _________________________________________________________________________________________

10 13 Simplify −9𝑥2+12𝑥−4 3 − 1 𝑥 − 𝑥 2 . Answer ……………………………. [3] _________________________________________________________________________________________