PHSS Final EM DA 2023 QP + Solutions

1 NAME: ( ) CLASS: TEACHING GROUP: MARKS /100 PEI HWA SECONDARY SCHOOL Diagnostic Assessment 2023 Secondary Four Express / Five Normal (Academic) MATHEMATICS 4052 21 June 2023 2 hours 30 minutes Candidates answer on the Question Paper. READ THESE INSTRUCTIONS FIRST Write your class, teaching group, index number and name 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, glue or correction fluid. Answer all questions. The number of marks is given in the 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 number of 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, unless the question requires the answer in terms of . This question paper consists of 20 printed pages. For Examiner’s Use Category Question No. Correction tape Pencil written Arrows Units Others

2 Mathematical Formulae Compound Interest Total amount = 1 100 n rP+ Mensuration Curved surface area of a cone = rl Surface area of a sphere = 24 r Volume of a cone = hr 2 3 1 Volume of a sphere = 3 3 4 r Area of triangle ABC = Cabsin2 1 Arc length = r , where  is in radians Sector area = 2 2 1 r , where  is in radians Trigonometry C c B b A a sinsinsin == Abccba cos2222 −+= Statistics Mean = f fx   Standard Deviation = 22        −  f fx f fx

3 Answer all the questions. 1 (a) Simplify 2 6 3 2 8x y −   , leaving your answer in positive indices. Answer .………..……………… [2] (b) Given that 2x = 5, find the value of 48xx− + . Answer .………..……………… [2] 2 The following chart shows the sales of a company over the years. State one aspect of the graph that may be misleading and explain how this may lead to a misinterpretation of the graph. ……………………………………………………………………………………............... ……………………………………………………………………………………............... ……………………………………………………………………………………............... [2]

4 3 x is decreased by 50% and then increased by 40%. Find the percentage decrease in x from its original value. Answer .………..………………% [2] 4 Write as a single fraction in its simplest form ( ) 2 35 1 1 x x x +− − . Answer .………..……………… [3] 5 There are 18 boys and x girls in the school’s concert band. It is given that the probability of selecting a boy from the concert band is 17 6 . (a) Find the value of x. Answer x = …..…………….…… [1] (b) Two children are selected at random. Find the probability that a boy and a girl are selected. Answer = …..………..………… [2]

5 (c) Some boys joined the concert band so that the probability of selecting a boy at random now becomes 24 13 . Find the number of boys who joined the concert band. Answer .………..……………… [2] 6 (a) Factorise the expression 210 7 12xx+− . Answer .………..……………… [1] (b) Hence solve the equation ( ) 2 10 1 7 19yy− + = . Answer y = ……..... or ……….. [3] 7 The value of a painting increased from $74 000 in 2018 to $P in 2021. The value increased by 8% every year. Find the value of the painting in 2021. Answer $ ……..……………… [2]

6 8 The braking distance, d, of a car is directly proportional to the square of its speed, s. When the speed is p metres per second, the braking distance is 6 metres. Find the percentage change in the breaking distance when the speed is increased to 300%. Answer .………..………… % [3] 9 Factorise completely (a) 1ab a b− − + , Answer .………..……………… [2] (b) 2 2 224x ax a b− + − . Answer .………..……………… [3] 10 Given that 4 8AB =− and 18 mAC = find (i) AB , Answer .………..……………units [1]

7 (ii) the value of m such that AB and AC are parallel, Answer .………..……………… [2] (iii) the coordinates of B if A is the point (–1, –3). Answer B ( ..….…..., ……..….) [1] 11 The ratio of Abby’s money to Betty’s money is 4 : 7. The ratio of Betty’s money to Claire’s money is 4 : 9. The three girls have $321 altogether. Claire gives some money to Abby and Betty so that each of them has the same amount of money. How much money will Abby receive from Claire? Answer $.………..……………… [3]

8 12 In the diagram, AD and BC are tangents to the circle, centre O, at the points A and B respectively. AOEC and BOFD are straight lines. (a) Show that triangle OAD and triangle OBC are congruent. Give a reason for each statement you make. Answer [3] (b) Two similar solids have volumes V1 and V2 such that 128 125VV= . If their surface area, A1 and A2 respectively, are such that 12A kA= , find the value of k. Answer ..………..……………… [2]

9 13 The diagram shows a semicircle SOR and an isosceles trapezium PQRS, where ( )3PS x y=+ cm, ( )4 3 5QR y x= − + cm and ( )7 3 1RS y x= + − cm. Given that the vertical height of the trapezium is ( )23xy+ cm, find the perimeter of the semicircle SOR, leaving your answer in terms of . Answer .………..………… cm [4]

10 14 A line P is perpendicular to another line Q with equation 4 3 16 0yx− − = . Line P also passes through the point (−15, 10). The product (gradient of P) × (gradient of Q) = −1. (a) Use this information to find the equation of line P. Answer .………..……………… [2] (b) The equation of another line R is 4 3 15 0xy+ − = . Without solving for x and y, explain whether line R intersects the line P. Answer [2]