[Presbyterian High] [2023] 4E EM 4052 Prelims P2 MS

PRESBYTERIAN HIGH SCHOOL MATHEMATICS 4052/02 PAPER 2 16 August 2023 Wednesday 2 hours 15 minutes PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL PRESBYTERIAN HIGH SCHOOL 2023 SECONDARY FOUR EXPRESS PRELIMINARY EXAMINATION MARKING SCHEME For Examiner’s Use Qn 1 2 3 4 5 6 7 8 9 10 Marks Deducted Marks Category Accuracy Symbols Others Question No. Setter: Mr Tan Lip Sing Vetter: Mdm Cynthia Chua TOTAL MARKS 90

2 Mathematical Formulae Compound Interest Total amount = nrP      + 1001 Mensuration Curved surface area of a cone = πrl Surface area of a sphere = 24πr Volume of a cone = 21 π3 rh Volume of a sphere = 34 π3 r Area of triangle ABC = C absin2 1 Arc length = θr , where θ is in radians Sector area = θ2 2 1 r , where θ is in radians Trigonometry C c B b A a sin sin sin= = A bc c b acos22 2 2− + = Statistics Mean = ∑ ∑ f fx Standard deviation = 22      − ∑ ∑ ∑ ∑ f fx f fx

3 1 (a) Rearrange the formula 2 2 5 3 dc d += − to make d the subject. Answer d =…….…………………. [3] 2 2 22 22 22 2 2 5 3 ( 3) 5 35 53 ( 1) 5 3 53 1 35 1 dc d cd d cd c d cd d c dc c cd c cd c += − −= + −=+ −= + −=+ += − +=± − M1 M1 A1 (b) Write as a single fraction in its simplest form 2 31 2( 2) xx − −− . Answer ……………………………. [2] ( ) ( ) 2 2 2 2 31 2( 2) 31 2( 2) 32 2 1 2 xx xx x x x x − −− = + −− +−= − += − M1 A1

4 (c) Solve these simultaneous equations. 5 3 14 3 5 18 xy xy += += You must show your working. Answer x=……………………. or y=…………………. [3] ( ) 5 3 14 ......(1) 3 5 18 ......(2) 14 5(1): ......( 3)3 Substitute (3) into 2 : 14 53 5 18 3 9 70 25 54 1 3 xy xy xy xx xx x y += += −= −+= +− = = = M1 A1 A1 (d) Solve the equation 21 1 56 23 x xx − =−− . Answer x=……………………. or x=…………………. [3] 2 2 21 1 56 23 (2 1)(2 3) 5 6 4 6 2 35 6 4 13 9 0 (4 9)( 1) 0 1214 x xx xx x x xx x xx xx x or x − =−− − −=− − − += − − += − −= = = M1 M1 (Factorise) A1

5 2 (a) Before departing London for Singapore, Peter bought 3000 Singapore dollars from the bank. The exchange rate between British pounds (£) and Singapore dollars ($) was £1= $1.71. He also had to pay the bank an additional commission fee of 1.5% for the exchange of currency. Calculate the total amount of pounds, inclusive of commission, he paid the bank. Give your answer correct to the nearest pound. Answer £ …….…………………. [2] Total amount before commission 3000 1.71= = £1754.385965 Total amount inclusive of commission 1754.385965 1.015= × ≈ £1781 M1 A1 (b) Peter bought a laptop while he was in Singapore. He paid $664.20 inclusive of the 8% GST (Goods & Services Tax) for the laptop after getting a discount of A% on the original price. The laptop’s original price is $750 before GST. (i) Find the GST amount paid for the laptop. Answer ……………………………. [2] 108% $664.20 88% $664.20 $49.20108 = = ×= GST amount $49.20= M1 A1 (ii) Calculate the value of A. Answer A= ………………………% [2] Discounted price before GST 664.20 49.20 $615.00= −= 750 615 100%750 135 100% 18%750 A A −= × = ×= M1 A1

6 (c) Mary invests $20 000 in an endowment plan that offers 4% per year compound interest. How much interest will she receive after 10 years? Give your answer correct to the nearest cent. Answer $ …………………………. [2 Total amount after 10 years 10 420000 1 100 $29604.89 = +  = Interest received 29604.89 20000 $9604.89 = − = M1 A1 (d) A map of a province has a scale of 1 : 500 000. (i) The length of an expressway on the map is 25 cm. Calculate the actual length, in kilometres, of the expressway. Answer ……………………… km [1] 1 cm : 5 km 25 cm : 125 km Actual length 125 km= B1 (ii) The area of a reservoir is 180 km2. Calculate the area, in square centimetres, of the reservoir on the map. Answer ……………………… cm2 [2] (1 cm)2 : (5 km)2 1 cm2 : 25 km2 Area on map 2180 7.2 cm25= = M1 A1

7 3 The variables x and y are connected by the equation 2 4 5 xy x= + . The table below shows some values of x and the corresponding values of y correct to 2 decimal places. x 0.5 1 1.5 2 2.5 3 4 5 y 8.05 4.20 3.12 2.80 2.85 3.13 4.20 5.80 (a) On the grid provided, draw the graph of 2 4 5 xy x= + for 0.5 5 x≤≤ . Plot the points given in the table and join them with a smooth curve. [3] Plot all 8 points correctly. Join all points with a smooth curve. B2 (6 or 7 points correct – B1) B1 (b) By drawing a tangent, find the gradient of the curve at 3x= . Answer ……………………………. [2] Draw the correct tangent line at 3x= . 6 0.95 0.72170Gradient −= ≈ − (Accept 0.7 to 0.8) M1 A1 (c) (i) On the same grid, draw the line 17 2yx= − for 05 x≤≤ . [1] Draw correct line 17 2yx= − . B1 (ii) Write down the x-coordinates of the points where this line intersects the curve. Answer x=……………………. or ……………………. [2] 0.6 4.4 (0.55 0.65) (4.35 4.45) x or x= = −− B1, B1

8 (iii) Find the equation, in the form 3220x ax bx c+ + += , which is satisfied by the values of x found in (c)(ii). Answer …………………………………. [2] 2 32 32 4 752 2 40 70 5 2 5 70 40 0 xx x x xx xx x +=− += − + − += M1 A1 (d) Use your graph to find the values of x in the range 05 x≤≤ for which 2 40.2 2 3x x+−= . Answer x=……………………. or ……………………. [2] 2 2 40.2 2 3 4 55 x x x x +−= += Draw the line 5y= , from the graph, 0.8 4.55 (0.75 0.85) (4.5 4.6) x or x= = −− B1, B1

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10 4 (a) The daily average temperature at Town A was recorded for 60 days. The cumulative frequency curve below shows the distribution of the temperatures. (i) Use the curve to estimate (a) the median temperature, Answer ……………………………. C° [1] 28.5° B1 (b) the interquartile range of the temperatures, Answer ……………………………. C° [2] 28.9 28.2 0.7°− °= ° M1, A1 (c) the number of days that Town A had temperatures above 29°C. Answer ……………………………. days [1] 60 48 12 days−= B1 Daily Average Temperature (oC)