2023 MGS Emath Prelim Exam P1 Solutions

This question paper consists of 22 printed pages and 2 blank pages. Class Index Number Name : __________________________________________ METHODIST GIRLS’ SCHOOL Founded in 1887 PRELIMINARY EXAMINATION 2023 Secondary 4 TUESDAY MATHEMATICS 4052/01 15 August 2023 Paper 1 2 hours 15 minutes Candidates answer on the Question Paper. INSTRUCTIONS TO CANDIDATES Write your class, name 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 SOLUTIONS

Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2023 Page 2 of 24 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 = 2 1 ab sin C Arc length = r , where  is in radians Sector area = 2 2 1 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

Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2023 Page 3 of 24 Answer all the questions. 1 It is given that M is a prime number and is a digit of the number 508 M45. If the number, correct to 3 significant figures, is 509 000, find the maximum value of M. Answer M = ……….7………. [1] 2 Simplify ( ) 2 3 2 3 2x+− . ( ) ( ) 2 2 2 2 3 2 3 2 3 2 9 12 4 3 18 24 8 18 24 11 x xx xx xx +− = + − + = + − + = − + Answer …….…… 218 24 11xx−+ ……..... [2] 3 Solve the equation 2 31 497 x  = . 2 3 322 2 2 3 1 497 77 77 22 3 1 3 x x x x x − −  = = = −= =− Answer x = …… 1 3− ………. [2]

Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2023 Page 4 of 24 4 (a) Factorise 236 aayy + completely. ( ) 236 32 aa aa yy yy + =+ Answer …… ( )32aayy + ……..… [1] (b) Factorise 20 15 3 4pq sq sr pr− + − completely. ( ) ( ) ( ) ( ) ( )( ) 20 15 3 4 5 4 3 3 4 5 4 3 4 3 4 3 5 pq sq sr pr q p s r s p q p s r p s p s q r − + − = − + − = − − − = − − Answer ………… ( )( )4 3 5p s q r−− ……..… [2] 5 Express 2 2 3 8 3 5 3 93 n n n nn − − − +−+ as a single fraction in its simplest form. ( )( ) ( )( ) ( )( ) ( )( ) 2 2 3 8 3 5 3 93 3 1 3 53 3 3 3 3 1 3 53 3 3 3 3 1 5 3 3 24 3 n n n nn nn n n n n nn n n n n nn n n n − − − +−+ +− −=+ − + + − + − −=+ − + + − − + −= + −= + Answer ……… 24 3 n n − + …..… [3]

Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2023 Page 5 of 24 6 Given that x satisfies the inequality ( ) ( )13 2 2 3 524x x x− −  − − , find the smallest value of x if x is an integer. ( ) ( ) ( ) ( ) 13 2 2 3 524 2 4 2 3 2 3 20 2 4 8 6 9 20 2 6 29 8 8 37 4.625 x x x x x x x x x xx x x − −  − − − −  − − − +  − − − − − − − −  Answer x =………5……..… [3] 7 It is given that 90 180A   and 7sin 25A= . (a) Find the value of A . 1 7180 sin 25 180 16.260... 163.739... 163.7 (1dp) A − = −  = −  = = Answer ………… 163.7 …..… [1] (b) Without the use of the calculator, find the numerical value of cos A . 2225 7 24 24cos 25 x A =− = =− Answer ……………… 24 25− ……..… [2]

Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2023 Page 6 of 24 8 The time, t seconds, taken to download a file from a computer is inversely proportional to the internet connect speed, v Mbps. When the speed is 250 Mbps, the time taken is 4 seconds. (a) Sketch the graph of against tv in the axes below. [1] . (b) Find the time taken to download a file when the speed is 160 Mbps. 4 250 1000 1000 1000 160 6.25 kt v k t v = = = = = = Answer …….…6.25……. sec [2] (c) If the speed is increased by 25%, find the percentage change in the time taken. 1 1 4 5 4 5 1Percentage change = 100%5 20% kt v kt v tt = = = − =− Answer ……… 20− ……% [2] t 0

Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2023 Page 7 of 24 9 (a) On the Venn Diagram, shade the the region which represents ' Y'X  . [1] (b) From the Venn diagram above, (i) write, in set notation that illustrates the relationship between A and B. Answer BA [1] (ii) write down the set notation that represents the shaded region. Answer …… ' AB ..……… [1] 10 It is given that { : is an integer and 0 15}x x x=   , {P= x : x is a prime number} and {Q= x : x is divisible by 3 or 5}. List the elements in ' PQ . ' { 1, 4,6,8,9,10,12,14,15} {3,5,6,9,10,12,15} ' {6,9,10,12,15} P Q PQ = = = Answer {………6, 9, 10, 12, 15…………} [2] X Y A B

Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2023 Page 8 of 24 11 The graph shows the price of 1 kg of apples in Country X from 2010 – 2020. (a) State the misleading feature of the graph. Answer The vertical axis does not start at zero [1] (b) Explain how this feature affects the reader’s interpretation of the graph. Answer The graph misleads the reader thinking that the price of apples in 2020 is thrice that of 2010’s when it is only about twice. [1] 1.5 2 2.5 3 3.5 4 4.5 5 2010 2015 2020 Price/kg ($) Year

Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2023 Page 9 of 24 12 Consider the pattern, 𝐿1: 32 − 22 = 3 + 2 = 5 𝐿2: 42 − 32 = 4 + 3 = 7 𝐿3: 52 − 42 = 5 + 4 = 9 𝐿4: 62 − 52 = 6 + 5 = 11 : : : : : : 𝐿𝑛: … … … … … … … … … .. (i) Write down Ln of the pattern. [1] ( ) ( ) 22 : 2 1 2 1 2 3 nL n n n n n+ − + = + + + = + (ii) Explain, with calculations, why 1137 is a term of the sequence 5, 7, 9, 11, …. [2] 2 3 1137 567 n n += = n is an integer. Hence 1137 is the 567th term of the sequence. (iii) Find the exact value of L1 + L2 + L3 + … + L99. 22101 2 10197−= Answer ……10197…………[1]

Methodist Girls’ School Mathematics Paper 1 Sec 4 Preliminary Examination 2023 Page 10 of 24 13 The numbers 720 and 1575, written as the products of their prime factors are 4 2 2 2720 2 3 5 and 1575 3 5 7.=   =  