CCHM 2024 P1 Worked Solution

2024 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4052/01 [Turn over Name: Class: Class Register Number: PRELIMINARY EXAMINATION 2024 SECONDARY 4 MATHEMATICS 4052/01 Paper 1 Friday 23 August 2024 Candidates answer on the Question Paper. 2 hours 15 minutes MARKS SCHEME This document consists of 20 printed pages.

2 2024 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4052/01 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 = nrP      + 1001 lr  2 4 r hr 3 1 2 3 3 4 r Cba sin 2 1 r  2 2 1 r  C c B b A a sin sin sin == Acbcba cos 2 222 −+=   f xf 22      −     f xf f xf

3 2024 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4052/01 [Turn over Answer all the questions. 1 The length of a court is 0.028 km and the width of the court is 16 m. Express the ratio of the length to the width of the court in its simplest form. 28 :16 7 : 4 Answer ………… : ………… [1] ________________________________________________________________________________ 2 Given that 0 180x , find the possible values of x for sin 0.985x= . Give your answers correct to 1 decimal place. 11 sin 0.985 sin 0.985 or 180 sin 0.985 80.1 or 99.9 x x −− = =− = Answer x = ………..…or…………… [2] ________________________________________________________________________________ 3 (a) Simplify ( ) 34 37 4 10 ab ab , leaving your answer in positive index form. ( ) 34 12 3 3 7 3 7 9 4 4 64 10 10 32 5 ab ab a b a b a b = = Answer ………………………………... [2] (b) Given that 23x = , 27y = and 92 49 z = , use the laws of indices to find the value of 13 22. x y z−+ ( ) ( ) 1 13 32 2 3 2 2 2 2 937 49 81 49 x y z x y z−+ =   =   = Answer ………………………………... [2] Law of indices: 1: ( ) nm mnaa = 2: m n m na a a += to award M1 if student 3: m n m na a a −= show understanding of 4: 1 nnaa= either law of indices

4 2024 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4052/01 4 A group of students won $ 226 in a competition and shared the amount equally, leaving $ 4 leftover. Competing for a second time , the same group of students won $ 296. They shared the $296 along with the $4 left over from the first competition equally, with no amount remaining. Assuming that each student received a whole number of dollars in both distributions, find the greatest possible number of students in the group. 222 300 2 111 150 3 37 50 HCF is 6. There are 6 students. Answer ………………………students [2] ________________________________________________________________________________ 5 (a) On 6 June 2020, Elijah invested some money in a bank which pays a simple interest at a rate of 3.5% per annum. He received a total interest of $680.40 on 6 June 2023. Find the amount of money Elijah invested in the bank. Amount of interest earn per year $680.40 3 $226.80 100Amount of money invested $226.803.5 $6480 = = = = Answer $……………………………... [2] (b) Freddy invested $8000 in another bank that paid compound interest at a rate of 1.75% per annum, compounded quarterly. Find the total amount Freddy received at the end of 5 years. Give your answer correct to the nearest cent. 54 1.75 4Total amount=8000 1 100 $8729.87 (nearest cent)  + = Answer $……………………………... [2]

5 2024 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4052/01 [Turn over 6 From the diagrams above, select one of them which best illustrates each of the following statements. (a) 34πyx= . (b) The cost of a project $y is a linear function of x, where x is the number of man-hours required to complete the project. The project has a fixed cost of $100. (c) An object is travelling at a constant speed towards a fixed point O. The distance y (in metres) represents how far the object is from point O at time x (in minutes). Answer (a) Diagram ………………………… [1] (b) Diagram ………………………… [1] (c) Diagram ………………………… [1] ________________________________________________________________________________ 7 Write as a single fraction in its simplest form 42 23 x xx − −+− . ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 2 2 4 3 2 242 2 3 2 3 12 7 2 4 23 98 23 18 23 x x xx x x x x x x x xx xx xx xx xx − − − +− −=+ − + − − + − −= +− −+= +− −−= +− Answer ……………………………........... [2] y x y x y x y x y x Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 O O O O O 4 3 2

6 2024 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4052/01 8 The table shows the distribution of the weights of 40 students. Weight (W kg) 40 50W 50 60W 60 70W 70 80W 80 90W Frequency 6 8 a 11 3 (a) Find the value of a. Answer a = ………………………….[1] (b) Calculate an estimate for (i) the mean weight of the students, Answer ……………………………kg [1] (ii) the standard deviation of the weights. Answer ……………………………kg [1] (c) An error in the weighing machine caused the students’ weights to be recorded 2 kg more than their actual values. Explain how the mean and standard deviation will change after the error is rectified. Answer …………………………………………………………………………………………… …………………………………………………………………………………………… ……………………………………………………………………………………………[1] 12 64.25 11.7 The mean will decrease by 2kg to 62.25kg while the standard deviation remains unchanged.

7 2024 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4052/01 [Turn over 9 Andy bought a limited-edition watch from an online website based in Thailand for 10 460 baht. Andy also paid $15 in Singapore dollars for shipping and GST of 9% on the cost of the watch. The exchange rate between Thai baht (฿) and Singapore dollars (S$) was ฿100=S$P. Andy spent a total of $436 in Singapore dollars. Find the value of P, giving your answer correct to 3 significant figures. 10460Cost of watch in Singapore dollars (without GST) 100 104.6 Cost of watch in Singapore dollars (with GST) 1.09 104.6 114.014 114.014 15 436 3.69(3sf) P P P P P P = = = = += = Answer P =………………………………. [3] 10 The gradient of the line joining the points ( )3 2 ,7a−+ and ( )1, 2a+ is 2 3− . Find the value of a. ( ) 27gradient = 1 3 2 5 4 aa a − + − − + −= −+ ( ) 2Given gradient 3 52 43 15 2 4 7 2 a a a =− − =−−+ − =− − + =− Answer a = …………………………........... [2] 11 Given that 2 2 69 0aa bb− + = , find the value of ab . 2 2 2 2 2 69 0 3920 3 0 3 0 3 3 aa bb aa bb a b a b a b ab − + = − + =  −= −= = = Answer ab =………………………........... [2]

8 2024 Preliminary Exam/CCHMS/Secondary 4/Mathematics/4052/01 12 Factorise completely 2 2 34 16 4 64mn n m n n− − + . ( ) ( ) ( ) ( )( ) ( )( ) 2 2 3 2 24 16 4 64 4 4 4 16 4 4 4 4 4 4 4 1 4 mn n m n n n m n n m n n m n n m n m n n m n m n − − + = − − − = − − − + = − − − Answer ……………………………........... [3] 13 A bag contains 4 black coins, 7 red coins and 11 white coins. Two coins are drawn from the bag at random, one after another without replacement. (a) Find the probability that a white coin will be chosen on the second draw. ( ) ( ) Total number of coins 4 7 11 22 P second coin is white P 1st coin of any co lours, 2nd coin is white 111 22 1 2 = + + = = = = Answer …………………………….