1. Problem: Analyze voting data for homework-free day, calculate circle dimensions from circumference, find HCF using prime factorization, and solve wage and profit questions for a factory. 1.(a)(i) List of votes: Monday, Friday, Thursday, Friday, Wednesday, Friday, Tuesday, Wednesday, Friday, Monday, Tuesday, Monday, Friday, Monday, Friday. Count votes: - Monday: 4 - Tuesday: 2 - Wednesday: 2 - Thursday: 1 - Friday: 6 1.(a)(ii) Day with most votes is Friday with 6 votes. 1.(b)(i) Circumference formula: $$C = 2\pi r$$ Given $C=44$ cm and $\pi = \frac{22}{7}$, solve for $r$: $$44 = 2 \times \frac{22}{7} \times r$$ $$44 = \frac{44}{7} r$$ Multiply both sides by 7: $$44 \times 7 = 44 r$$ $$308 = 44 r$$ Divide both sides by 44: $$r = \frac{308}{44} = 7$$ cm 1.(b)(ii) Area formula: $$A = \pi r^2$$ Using $r=7$ cm: $$A = \frac{22}{7} \times 7^2 = \frac{22}{7} \times 49 = 22 \times 7 = 154$$ cm² 1.(c) Prime factorization: $$108 = 2^2 \times 3^3$$ $$72 = 2^3 \times 3^2$$ HCF is product of lowest powers of common primes: $$HCF = 2^{\min(2,3)} \times 3^{\min(3,2)} = 2^2 \times 3^2 = 4 \times 9 = 36$$ 1.(d)(i) Number of workers = 98 - 10 workers earn 150 daily - 88 workers earn 85 daily Weekly pay: - For 10 workers: $$10 \times 150 \times 6 = 9000$$ - For 88 workers: $$88 \times 85 \times 6 = 44880$$ Total weekly pay: $$9000 + 44880 = 53880$$ 1.(d)(ii) Daily supplies = 1135 Weekly supplies = $$1135 \times 6 = 6810$$ Total weekly cost = wages + supplies = $$53880 + 6810 = 60690$$ Weekly income = 60128 Profit/loss = income - cost = $$60128 - 60690 = -562$$ (loss) 2. Problem: Complete table and graph linear function, calculate sale price, sum numbers, find irregular pentagon perimeter. 2.(a)(i) Given $y = -3x + 7$ and table: For $x = -1$: $y = -3(-1) + 7 = 3 + 7 = 10$ For $x = 0$: $y = -3(0) + 7 = 0 + 7 = 7$ For $x = 1$: $y = -3(1) + 7 = -3 + 7 = 4$ For $x = 2$: $y = -3(2) + 7 = -6 + 7 = 1$ For $x = 3$: $y = -3(3) + 7 = -9 + 7 = -2$ For $x = 4$: $y = -3(4) + 7 = -12 + 7 = -5$ Completed table: | x | -1 | 0 | 1 | 2 | 3 | 4 | |---|----|---|---|---|---|---| | y | 10 | 7 | 4 | 1 | -2 | -5 | 2.(b) Sale price is a tenth: $$\frac{1}{10} \times 9307 = 930.7$$ 2.(c) Sum: $$2.48 + 3.65 + 701.532 + 102.7 = 810.362$$ Rounded to one decimal place: $$810.4$$ 2.(d) Perimeter of pentagon with sides: $$2x + 8, 2x, x + 7, 10, 5x + 4$$ Sum sides to get perimeter $P$: $$P = (2x + 8) + 2x + (x + 7) + 10 + (5x + 4) = 2x + 8 + 2x + x + 7 + 10 + 5x + 4$$ Combine like terms: $$P = (2x + 2x + x + 5x) + (8 + 7 + 10 + 4) = 10x + 29$$ --- Final answers: - Frequency table and Friday most votes - Radius = 7 cm - Area = 154 cm² - HCF = 36 - Weekly wages = 53880 - Weekly loss = 562 - Completed table for $y = -3x + 7$ - Sale price = 930.7 - Sum rounded = 810.4 - Perimeter = $10x + 29$