BSS 2023 Maths Sec 4E Prelim MS - P2

Broadrick Secondary School Prelims P2 2023 Elementary Mathematics - Secondary Four Express Mark Scheme Qn Mark Scheme 1(a) = 23 (2 8 4)x x x− − + − = 23 2 8 4x x x− + − + = 22 7 7xx− + + 1(b) = 1 2 3 2 (2 3)( 2) x x x x −++ − + = 11 22xx +++ = 2 2x+ 1(ci) = 22( 4) 4 25x− − + = 2( 4) 9x−+ 1(cii) 2( 4)x− is a square number, which makes it always 0 . Hence adding 9 will always make the expression positive for any value of x. 2(a) 90GAO =  (tan ⊥ radius) reflex 126 2AOC =  (  at centre = 2  s at circum.) = 252º 252 180AOE = − = 72º 180 72 2OAE −= (base  s of isos.  ) = 54º 90 54EAG = − = 36º 2(b) EAF = DCF (  s in same seg.) AFE = CFD (vert. opp.  s) FEA = FDC (  s in same seg.)  Triangle EAF and triangle DCF are similar. (AA) 3(ai) DB = 3a – b 3(aii) EC = – 3 4 (3a – b) + 3a = – 9 4 a + 3 4 b + 3a = 3 4 a + 3 4 b

3(b) AD = 4a – (3a – b) = 4a – 3a + b = a + b EC = 3 4 (a + b) = 3 4 AD Since EC = 3 4 AD , they are parallel vectors. Hence EC is parallel to AD. 3(ci) 1 3 3(cii) 3 7 4(a) 100 x 4(b) 100 0.64x+ 4(c) 100 100 4.60.64xx−= + 100( 0.64) 100 4.6( 0.64) xx xx +− =+ 64 4.6 ( 0.64)xx=+ 24.6 2.944 64 0xx+ − = 24600 2944 64000 0xx+−= 28(575 368 8000) 0xx+ − = 2575 368 8000 0xx+ − = (Shown) 4(d) 2575 368 8000 0xx+ − = 2368 368 4(575)( 8000) 2(575)x −  − −= 368 18535424 1150x −= 3.4237x= or 4.0637x=− (rej.) Cost to buy 20 litres of petrol in Jun 2022 = (3.4237 0.64) 20+ = $81.27 (to nearest cents) Jason is incorrect as it will cost $81.27, which is more than $80, to buy 20 litres of petrol in June 2022. 5(a, c, di, dii) Refer to answer behind 5(b) 054º ( )1 5(e) 44 metres ( )1

5(f) 9tan 44 = 1 9tan 44 − =  11.6 = (to 1 d.p.) 6(a) 180 60AED = − (adj.  s on a st. line, equilateral  ) = 120º 180 120 35ADE = − − (  sum of  ) = 25º 9 sin 25 sin 35 AE = 9sin 25 sin 35AE =  = 6.63131 = 6.63 cm (to 3SF) 6(b) AC = 6.63131 + 9 + 5 = 20.631 cm 2 2 217.5 7 20.631cos 2(17.5)(7)ABC +−= 2 2 2 1 17.5 7 20.631cos 2(17.5)(7)ABC −  +−=  = 106.696º = 106.7º (to 1 d.p.) 6(c) Area of triangle ABC = 1 17.5 7 sin106.6962    = 58.7 cm² (to 3SF) 7(a) 231 3 2(5 ) (5 )3 4 3 h = 11 (5)32h= 1 (5)(3)2h= 15 2h= = 7.5 cm (Shown) 7(b) 1 42 5 7.5 EF = 1 20 2 7.5EF = 8 3= cm Volume of empty space = 2 18 (4)33  = 256 27  cm³ Volume of sand = 21 256(5 )(7.5)3 27 − + 32 (5 )3 = 428.36 cm³ = 428 cm³ (to 3SF)

7(c) 2 20 5   = 16 1 The material needed to make the larger ornament is 16 times more than the original amount. Hence James is not correct. 8(a) 1.5 8(b) Refer to answer behind 8(c) (- 0.8, 1.525) and (0.825, 0.475) ( )0.025 8(d) 3 1 2 22 x x− − + = 3 122 x x− + = Draw y = 2. There is only 1 point of intersection between the line y = 2 and the curve 3 12 xyx= − + . Hence there is only 1 solution. 8(ei) 1 12yx=− + x – 1 0 1 y 1.5 1 0.5 8(eii) 1 12yx=− + --- (1) 3 12 xyx= − + --- (2) Sub (1) into (2), 3 11122 x xx− + =− + 3 1 1 1 022 x xx− + + − = 3 1 022 x x−= 3 02 xx− = 3 0xx−= (Shown) 9(ai) $27.50 9(aii) $33 – $22 = $11 9(aiii) The interquartile range will remain the same since all students’ expenses increase by $5.

9(bi) 9(bii) P(late on at least 1 day) = 441 55− = 9 25 9(biii) P(late for sch) = 1 1 1 1 2 5 2 4 +  = 9 40 10(a) For lowest possible cost of rental, 1. rent seventeen 10-pax round table for 170 pax and two 6-pax round table for the remaining 11 pax 2. rent exactly 181 chairs Cost = 17(25) 2(20) 181(2.4)++ = $899.40 10(b) Lowest cost for 9 required items, including beverage = 1+2.5+2.8+4.0+1.8+1.8+1.5+1.2+0.5 = $17.10 Lowest cost for 10 items with 5% discount = (17.10 + 1.5) x 0.95 = $17.67 Difference = 17.67 – 17.10 = $0.57 Hence, the promotion price is higher by $0.57. *Accept answers if students calculate with handling fee/GST/number of students etc. 1 5 4 5 Late Not late Late Not late Not late Late 4 5 1 5 1 5 4 5

10(c) Fixed cost (GST included): 1. Air Cooler Rental + GST = $450 x 1.08 = $486 2. Party Decoration Package = $2500 Sub-total = $2986 No. of pax 181 ~199 (10%) ~208 (15%) Cost per pax $16.497 $15.005 $14.356 Fixed cost (per pax) (GST included): 3. Cost of food per pax, with 2% handling fee & GST = $17.10 x 1.02 x 1.08 = $18.837 Variable cost: 4. Table and Chairs Rental No. of pax 181 ~199 (10%) ~208 (15%) without GST 17(25) 2(20) 181(2.4)++ = $899.40 20(25) 199(2.4)+ = $977.60 20(25) 201(2.2)+ = $942.20 21(25) 208(2.2)+ = $982.60 19(25) 3(20) 208(2.2)++ = $992.60 with 8% GST) $971.352 $1017.576 $1072.008 Cost per pax $5.367 $5.113 $5.154 No. of pax 181 ~199 (10%) ~208 (15%) Total cost per pax $16.497+ $18.837 + $5.367 = $40.70 (to 2 d.p) $15.005 + $18.837 + $5.113 = $38.96 (to 2 d.p) $14.356 + $18.837 + $5.154 = $38.35 (to 2 d.p) If each student pays $39, it will be enough to cover the costs incurred fully if there is an increase in attendance between 10 to 15%. This is provided the school rents 201 chairs such that the unit cost of renting is reduced and the total cost of renting is lower than that of renting just 199 chairs.