2024 SJI Prelim P2

ST JOSEPH’S INSTITUTION PRELIMINARY EXAMINATION 2024 (YEAR 4) CANDIDATE NAME CLASS INDEX NUMBER MATHEMATICS Paper 2 Candidates answer on the Question Paper. 4052/02 15 August 2024 2 hours 15 minutes (08:30 – 10:45) READ THESE INSTRUCTIONS FIRST Write your class, index number and name on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use paper clips, 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 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 π, use either your calculator value or 3.142, unless the question requires the answer in terms of π. The number of marks is given in brackets [ ] at the end of each question or part question. The total mark for this paper is 90. This document consists of 20 printed pages.

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 = Arc length = , where is in radians Sector area = , where is in radians Trigonometry Statistics Mean = Standard deviation = 1 100 n rP    rl 24 r 21 3 r h 34 3 r ABC 1 sin2 ab C r  21 2 r   sin sin sin a b c A B C  2 2 2 2 cosa b c bc A   fx f   22fx fx f f          

3 [TURN OVER 1 (a) In 2024, apples cost $x per kilogram and oranges cost $y per kilogram. The total cost of 5kg of apples and 10kg of oranges is $40. (i) Show that 8 2 y x  . Answer [2] (ii) In 2023, the cost, per kilogram, of apples were 9% lower than in 2024 while the cost of oranges, per kilogram were 9% higher than in 2024. Find an expression, in terms of x and y, for the total cost of 3kg of apples and 2kg of oranges in 2023. Give your answer in its simplest form. Answer ………………………...... [3]

4 (b) Write as a single fraction in its simplest form 2 3 12 4 x x x x x    . Answer …………………….......... [4]

5 [TURN OVER 2 The diagram shows a sequence made by using grey and white squares. (a) Draw Pattern 2 in the space above. [1] (b) Complete the table for the next three pattern in the sequence. Pattern Number 4 5 6 Number of grey squares 18 Number of white squares 32 [2] (c) Find an expression, in terms of k, for the number of grey squares in Pattern k. Answer ………………….............. [2] (d) Determine if any pattern can be formed with 100 grey squares. Show your working. Answer [2]

6 3 In triangle ACD, B is the midpoint of AC and E is the midpoint of AD. AB 6a + 3b and DC  5a + 2b. (a) Express AD , as simply as possible, in terms of a and b. Answer AD ……………… [3] (b) Show that triangles AEB and ADC are similar. Answer [2]

7 [TURN OVER (c) S is a point on AE such : 2 : 3AS SE  . It is given that  1 53 26SC k a b . Find the value of k. Answer k  …………………. [2] (d) Find the numerical value of area of triangle area of AEB EBCD . Answer ………………………… [2] (e) Use vectors to show that EB is parallel to DC . Answer [2]

8 4 A tank is a closed cylinder of radius 5 m and height 8 m. It is made of negligible thickness and stands with its base horizontal. (a) The tank contains fuel to a depth of 3 m. Find correct to the nearest cubic metre, the volume of the fuel in the tank. Answer …………………… m3 [1] (b) 20 000 litres of fuel are added. Find the increase in depth of the fuel in the tank. Give your answer in cm. (1 litre = 1000 cm3) Answer …………………… cm [3] (c) The outer curved surface and the top of the tank are to be painted. The paint is sold in tins, each of which contains 5 litres. One litre of paint can cover 7 m2. Calculate the number of tins required. Answer …………………… tins [3]

9 [TURN OVER (d) The tank is completely emptied of all fuel and then partially filled with water. It is now placed with its curved surface on a horizontal floor. Diagram 1 shows the dimensions of the tank. Diagram 2 shows a circular end of the fuel tank. O is the centre of the circle and D is vertically below O. AB represents the level of the water surface and 2angle 3AOB  . Calculate the area of the inside surface of the tank which is not in contact with the water. Answer …………………………. m2 [5] 8 m 10 m B water Diagram 1 Diagram 2 2 3 B A

10 5 (a) (i) Convert 14 km/h to m/s. Answer …………………………. m/s [1] (ii) Haley rode a bike at a speed of 14 km/h for 1 hour 20 minutes from Kensington Town to Brood City. After a 30 minutes rest at Brood City, she continued her 25 km journey to Bristol Plains at a speed of 10 km/h. Calculate Haley’s average speed, in km/h, for the entire journey. Answer ………………………... km/h [4]