2023 MGS Emath Prelim Exam P2 Solutions

This question paper consists of 21 printed pages and 3 blank pages. Class Index Number Name : Answers METHODIST GIRLS’ SCHOOL Founded in 1887 PRELIMINARY EXAMINATION 2023 Secondary 4 Thursday MATHEMATICS 4052/02 17 August 2023 Paper 2 2 h 15 min Candidates answer on the Question Paper. INSTRUCTIONS TO CANDIDATES Write your name, class and index 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, glue or correction fluid. Answer all the 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 p , use either your calculator value or 3.142, unless the question requires the answer in terms of p The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 90. 90

2 Methodist Girls’ School Mathematics Paper 2 Sec 4 Preliminary Examination 2023 Mathematical Formulae Compound Interest Total amount = ( ) 100 r 1+ n P Mensuration Curved surface area of a cone = rl Surface area of a sphere = 4 2r Volume of a cone = hr 2 3 1 Volume of a sphere = 3 3 4 r Area of a triangle ABC = 2 1 absinC Arc length = r , where  is in radians Sector area = 21 2 r  , where  is in radians Trigonometry C c B b A a sin sin sin == a 2 = b 2 + c 2 – 2bc cos A Statistics Mean =   f fx Standard deviation = 22         −     f fx f fx

3 Methodist Girls’ School Mathematics Paper 2 Sec 4 Preliminary Examination 2023 1 (a) It is given that 𝑦 = 1 4 𝑧(𝑤2 − 𝑥2). (i) Find z such that 𝑦 = 4, 𝑤 = −1 𝑎𝑛𝑑 𝑥 = 3. 4 = 1 4 𝑧[(−1)2 − 32] 16 = z [ -8 ] z = - 2 Answer ……- 2…………….[2] (ii) Make x the subject of the formula 𝑦 = 1 4 𝑧(𝑤2 − 𝑥2). 𝑦 = 1 4 𝑧(𝑤2 − 𝑥2) 4𝑦 = 𝑧(𝑤2 − 𝑥2) 22 4ywx z−= 22 4yxw z=− 2 4yxw z= − Answer 2 4yxw z= − [2]

4 Methodist Girls’ School Mathematics Paper 2 Sec 4 Preliminary Examination 2023 (b) Simplify 2 36 24 x xx ++− . 2 36 24 x xx ++− = 36 2 (2 )(2 ) x x x x++ − + = 3(2 ) 6 (2 )(2 ) xx xx −+ −+ = 6 3 6 (2 )(2 ) xx xx −+ −+ = 3(2 ) (2 )(2 ) x xx + −+ = 3 (2 ) x− Answer 3 (2 ) x− …………………….………….[3] (c) In 2022, Singapore has a population of 5.64 million. The land area of Singapore is 728.6km². Find the population of Singapore per km², leaving your answer as standard form. Population = 65.64 10 728.6  = 7.74 x 10³ (3sf) Answer 7.74 x 10³ [2]

5 Methodist Girls’ School Mathematics Paper 2 Sec 4 Preliminary Examination 2023 2 (a) A closed cone, of height h cm, is filled with water to half its height as shown in Diagram A. It is then inverted as shown in Diagram B. Find the height of the water level, x cm, in terms of h, in Diagram B. 𝑉𝑤𝑎𝑡𝑒𝑟 𝑉𝑐𝑜𝑛𝑒 = (1 2) 3 = 1 8 𝑉𝑎𝑖𝑟 𝑖𝑛 𝑩 𝑉𝑐𝑜𝑛𝑒 = 7 8 3 7 8 hx h − = 3 7 8 hx h − = 3 7 2h x h−= 3 7 2x h h=− Answer ……………….. [3] h cm Diagram A Diagram B x cm cv cv

6 Methodist Girls’ School Mathematics Paper 2 Sec 4 Preliminary Examination 2023 (b) Janet began her round-island 145 km cycling route from Point A at 6 am, cycling at an average speed of x km/h. (i) Write down the expression, in terms of x, for the time taken by Janet to complete the cycling route. 145 x Answer 145 x h [1] Angel started on the same route 10 minutes later. She cycled at a speed of 2 km/h faster than Janet for the first 20 km and cycled x km/h for the remaining route. (ii) Write down the expression, in terms of x, for the time taken by Angel to complete the cycling route. Answer 20 125 2xx ++ h [1] Angel reached the end of the route at the same time as Janet. (iii) Form an equation in x and show that it reduces to 𝑥2 + 2𝑥 − 240 = 0. Answer [3] 20 125 1 145 26x x x+ + =+ 20 1 20 26xx +=+ 120 2 20 6( 2) x xx ++ =+ 2122 120 240x x x+ = + 2 2 240 0xx+ − =

7 Methodist Girls’ School Mathematics Paper 2 Sec 4 Preliminary Examination 2023 (iv) Solve the equation 𝑥2 + 2𝑥 − 240 = 0. 22 (2) 4(1)( 240) 2(1)x −  − −= = 14.5 or -16.5 (3sf) Answer x = ………………..…….. [3] (v) What time did Janet reach the ending point? Time taken = 145 14.524 = 9 h 59 min Time arrived = 3.59 pm Answer ……….………. [2]

8 Methodist Girls’ School Mathematics Paper 2 Sec 4 Preliminary Examination 2023 3 (a) In a logistics company, each delivery man is paid $x for every correct delivery made, $0 for no delivery but has to pay a penalty of $1 for each damaged delivery. Tom and Jerry made the following deliveries on a particular day. correct no delivery damaged A = Tom Jerry (13 4 3 10 8 𝑦 ) B = ( 𝑥 0 −1 ) (i) Evaluate AB in terms of x and y. (13 4 3 10 8 𝑦 ) ( 𝑥 0 −1 ) = ( 13𝑥 − 3 10𝑥 − 𝑦 ) Answer ……………..………… [2] (ii) Explain what your answer in (a)(i) represents. Rep amount of money Tom and Kerry will be paid respectively for their deliveries on a particular day…………………………………………..… [1] It is given that Tom earned $18 more than Jerry despite both making the same number of deliveries. (iii) Find the values of x and y. 13 3 10 6 y y + = + = 13 3 10 6 18 5 xx x − = − + = Answer x = 5………….. y =6…….. [3]

9 Methodist Girls’ School Mathematics Paper 2 Sec 4 Preliminary Examination 2023 (iv) By using matrix multiplication, calculate the total amount of money the company needs to pay both Tom and Jerry on that particular day. ( ) ( ) 13(5) 31 1 10610(5) 6 − = − Answer $106…………[2] (b) Tom is looking for a new job. He targets to earn an annual salary of $150 000 at the end of 5 years. The job that he is interviewing for is offering an annual salary increment of 3% per annum. What is the minimum starting annual salary should Tom request for to meet his target. He needs to set his salary request in multiples of $1000. Let P be the annual starting salary. 5 3150000 1 100P=+  P = 129391.377 Answer $ 130 000… [3]

10 Methodist Girls’ School Mathematics Paper 2 Sec 4 Preliminary Examination 2023 4 (a) In the diagram, A, B, C and D lie on the circle with centre O. EF and GF are tangents to circles at A and B respectively. It is given that o42DAO= , o33BDC= and o112AOB= Find (i) CBD , 180 112 2OAB − = (base angle of isos  ) = 34  180 34 42BCD = − −  (angles in opp segment) = 104  180 104