2024 NCHS Prelim Math P1 Suggested Solutions

Nan Chiau High School 2024 Secondary 4 Mathematics Mathematics Prelim Examination Paper 1 Solutions Qn Solution 1a False 1b False 2 105𝑏 𝑎 = (3×5×7)(22×5×7) 2×32×5 = 22×3×52×72 2×32×5 = 2×5×72 3 Numerator 2 × 5 × 72 does not include the factor 3 to reduce 105𝑏 3 to a whole number. 3 Length of shorter portion of rope = 4 11 ×44 = 16 cm Length of longer portion of rope = 28 cm Let 𝑥 be the length rope cut. 16−𝑥 28−𝑥 = 2 5 5(16 − 𝑥) = 2(28 − 𝑥) 80 − 5𝑥 = 56 − 2𝑥 24 = 3𝑥 𝑥 = 8 4 79−𝑥2 − 1 = 0 79−𝑥2 = 1 7(3−𝑥)(3+𝑥) = 70 𝑥 = 3 𝑜𝑟 𝑥 = −3 5 𝑦3 = 𝑘 𝑥2 𝑘 32 − 𝑘 62 = 5 𝑘 9 − 𝑘 36 = 5 𝑘 12 = 5 𝑘 = 60 𝑦3 = 60 𝑥2 23 = 60 𝑥2 𝑥2 = 7.5 𝑥 = ±2.74

6 24 g → 12.044 × 1023 atoms 1 g → 0.5.01833 × 1023 atoms 5000g → 0.5.01833 × 1023 × 5000 = 2509.167 × 1023 = 2.51 × 1026 7a Diagram A 7b Diagram B 7c Diagram D 8 9𝑥 + 2.5𝑦 = 3𝑦 − 2𝑥 11𝑥 = 0.5𝑦 22𝑥 = 𝑦 or 𝑥 𝑦 = 1 22 Hence, 𝑥 = 22 and 𝑦 = 1 is not a solution, as 22(22) ≠ (1) Or LHS = 9(22) + 2.5(1) = 200.5 RHS = 3(1) − 2(22) = −41 Since LHS ≠RHS, 𝑥 = 22 and 𝑦 = 1 is not a solution. 9 3𝑦 − 𝑘𝑥 − 9 = 0 3𝑦 = 𝑘𝑥 + 9 𝑦 = 𝑘 3 𝑥 + 3, where 𝑘 < −3 3𝑦 − 𝑘𝑥 − 9 = 0 𝑦 = −𝑥 3 y x

10 Let the first digit be 𝑥 and second digit be 𝑦 A number between 20 to 100 Let the number be 10𝑥 + 𝑦 Add the two digits together 𝑥 + 𝑦 Subtract the sum of the two digits from your original number to get a new number (10𝑥 + 𝑦) − (𝑥 + 𝑦) = 9𝑥 The new number is a multiple of 3 Since the new number is a multiple of 9 and 3 is a factor of 9, hence the new number is a multiple of 3. 11 2(𝑚𝑛)2 − 2𝑚𝑛 = 3𝑚𝑛 + 3 2𝑚2𝑛2 − 5𝑚𝑛 − 3 = 0 (𝑚𝑛 − 3)(2𝑚𝑛 + 1) = 0 𝑚𝑛 = 3 or 𝑚𝑛 = − 1 2 𝑛 = 3 𝑚 or 𝑛 = − 1 2𝑚 12 4 3 + 2 9𝑚 ÷ √81𝑚164 = 4 3 + 2 9𝑚 ÷ 3𝑚4 = 4 3 + 2 9𝑚 × 1 3𝑚4 = 4 3 + 2 27𝑚5 = 36𝑚5+2 27𝑚5 13 −9𝑥2+12𝑥−4 3 − 1 𝑥 − 𝑥 2 = −(3𝑥−2)2 3 − 1 2𝑥−𝑥 2 = −(3𝑥−2)2 3 − 2 𝑥 = −(3𝑥−2)2 3𝑥−2 𝑥 = −𝑥(3𝑥 − 2) 14 𝑦 = 1 5 𝑥2 − 2𝑥 + 7 ------------ [1] 𝑦 = 𝑥 ----------- [2] Sub eqn[1] into eqn[2] 𝑥 = 1 5 𝑥 2 − 2𝑥 + 7 1 5 𝑥2 − 3𝑥 + 7 = 0 𝑥 = 3±√(−3)2−4(1 5)(7) 2(1 5) 𝑥 = 2.89 or