2024 NCHS Prelim Math P2 Suggested Solutions

NCHS 2024 EM Paper 2 Solutions 1 (a) 9𝑎2−4𝑏2 3𝑎𝑝+6𝑎𝑞−4𝑏𝑞−2𝑏𝑝 = (3𝑎+2𝑏)(3𝑎−2𝑏) (3𝑎𝑝+6𝑎𝑞)−(2𝑏𝑝+4𝑏𝑞) = (3𝑎+2𝑏)(3𝑎−2𝑏) 3𝑎(𝑝+2𝑞)−2𝑏(𝑝+2𝑞) = (3𝑎+2𝑏)(3𝑎−2𝑏) (𝑝+2𝑞)(3𝑎−2𝑏) = 3𝑎+2𝑏 𝑝+2𝑞 (b) 𝑥 6−7𝑥−5𝑥2 + 2 5𝑥−3 = 𝑥 −(5𝑥−3)(𝑥+2) + 2 5𝑥−3 = −𝑥+2(2+𝑥) (5𝑥−3)(𝑥+2) = −𝑥+4+2𝑥 (5𝑥−3)(𝑥+2) = 𝑥+4 (5𝑥−3)(𝑥+2) 1(c) − 1 2 < 2𝑦 5 − 1+𝑦 3 ≤ 5 6 − 1 2 < 3(2𝑦)−5(1+𝑦) 15 𝑎𝑛𝑑 3(2𝑦)−5(1+𝑦) 15 ≤ 5 6 − 1 2 < 6𝑦−5−5𝑦 15 6𝑦−5−5𝑦 15 ≤ 5 6 − 1 2 < 𝑦−5 15 𝑦−5 15 ≤ 5 6 −15 < 2(𝑦 − 5) 𝑦 − 5 ≤ 25 2 −15 < 2𝑦 − 10 𝑦 ≤ 17 1 2 −5 < 2𝑦 −2 1 2 < 𝑦 ≤ 17 1 2 2(a) 𝑟 19 = 25 40 𝑟 = 25 40 × 19 = 11.875 𝑉 = 1 3 𝜋(11.875)2(25) = 3691.78 𝑐𝑚3 = 3.69 𝑙𝑖𝑡𝑟𝑒𝑠 (b) Method 1 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟 = 1 3 𝜋(19)2(40) ( ℎ1 ℎ2 ) 3 = 𝑉1 𝑉2 ( ℎ1 40) 3 = 1 3𝜋(19)2(40)−1 3𝜋(11.875)2(25) 1 3𝜋(19)2(40) ℎ1 40 = √ (19)2(40)−(11.875)2(25) (19)2(40) 3 ℎ1 = 40 × √ (19)2(40)−(11.875)2(25) (19)2(40) 3 = 36.437 𝐷𝑒𝑝𝑡ℎ 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 = 40 − 36.437 = 3.56 𝑐𝑚 (3𝑠𝑓) Method 2 𝑟1 19 = ℎ1 40 𝑟1 = 19 40 ℎ1 1 3 𝜋(19)2(40) − 1 3 𝜋(𝑟1)2(ℎ1) = 3691.78 1 3 𝜋(𝑟1)2(ℎ1) = 11429.75 r 19 25 40 h1 40 r1 19 h1 h2

( 19 40 ℎ1) 2 (ℎ1) = 10914.61 ℎ1 3 = 48375.00277 ℎ1 = 36.437 𝐷𝑒𝑝𝑡ℎ 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 = 40 − 36.437 = 3.56 𝑐𝑚 (3𝑠𝑓) 3(a) 108888−100176.96 108888 × 100% = 8% (b) Balance owed = 85 100 × 108888 = $92554.80 Interest = 92554.80×2.98×5 100 = $13790.6652 Monthly instalment = 92554.80+13790.6652 5×12 = $1772.42 (2𝑑𝑝) (c) 𝑃 = 108888 − 18888 = $90000 𝐴 = 960 × 10 × 12 = $115200 𝐴 = 𝑃 (1 + 𝑅 100) 𝑛 115200 = 90000 (1 + 𝑅 100) 10 (1 + 𝑅 100) 10 = 115200 90000 1 + 𝑅 100 = √ 115200 90000 10 𝑅 = 100 ( √ 115200 90000 10 − 1) = 2.499 = 2.50 (3sf) 4(a) 𝑄 = ( 1.20 1.50 0.95 ) (b) 𝑅 = ( 60 68 55 49 56 71 53 70 80 ) ( 1.20 1.50 0.95 ) = ( 226.25 210.25 244.60 ) (c) The elements in R represent the total cost of baking caramel, strawberry and mint cupcakes in outlet A, B and C respectively (d) Method 1 Selling Price = ( 3 0 0 0 2.5 0 0 0 4 ) ( 1.20 1.50 0.95 ) = ( 3.60 3.75 3.80 ) Method 2 Profit for each type of cupcake = ( 2 0 0 0 1.5 0 0 0 3 ) ( 1.20 1.50 0.95 ) = ( 2.40 2.25 2.85 )

Profit earned in a week = ( 60 68 55 49 56 71 53 70 80 ) ( 3.60 3.75 3.80 ) − ( 226.25 210.25 244.60 ) = ( 680 656.20 757.30 ) − ( 226.25 210.25 244.60 ) = ( 453.75 445.95 512.70 ) Total Profit earned in a month = 4 [(1 1 1) ( 453.75 445.95 512.70 )] = 4(1412.40) = (5649.60) Total Profit = $5649.60 Profit earned in a week = ( 60 68 55 49 56 71 53 70 80 ) ( 2.40 2.25 2.85 ) = ( 453.75 445.95 512.70 ) Total Profit earned in a month = 4 [(1 1 1) ( 453.75 445.95 512.70 )] = 4(1412.40) = (5649.60) Total Profit = $