1. **Factorize completely:** (a)(i) Factorize $2r^2 - 8r$. - Step 1: Identify the greatest common factor (GCF) of the terms, which is 2r. - Step 2: Factor out 2r: $$2r^2 - 8r = 2r(r - 4)$$ (a)(ii) Factorize $3x^2 - 5x - 2$. - Step 1: Multiply the coefficient of $x^2$ (3) by the constant term (-2): $3 \times (-2) = -6$. - Step 2: Find two numbers that multiply to -6 and add to -5: these are -6 and 1. - Step 3: Rewrite the middle term: $$3x^2 - 6x + x - 2$$ - Step 4: Factor by grouping: $$3x(x - 2) + 1(x - 2)$$ - Step 5: Factor out the common binomial: $$(3x + 1)(x - 2)$$ 2. **Make C the subject and calculate C:** (b)(i) Given $$F = \frac{9}{3}C + 32$$, simplify and solve for C. - Step 1: Simplify $$\frac{9}{3} = 3$$, so $$F = 3C + 32$$. - Step 2: Subtract 32 from both sides: $$F - 32 = 3C$$. - Step 3: Divide both sides by 3: $$C = \frac{F - 32}{3}$$. (b)(ii) Given $$F = 113$$, calculate $$C$$. - Step 1: Substitute $$F = 113$$ into the formula: $$C = \frac{113 - 32}{3} = \frac{81}{3} = 27$$. 3. **Ticket sales problem:** (c)(i)(a) Number of tickets sold at $10 each: - Total tickets = 500 - Tickets sold at $6 each = $$x$$ - Tickets sold at $10 each = $$500 - x$$ (c)(i)(b) Total amount collected: - Amount from $6 tickets: $$6x$$ - Amount from $10 tickets: $$10(500 - x)$$ - Total amount: $$6x + 10(500 - x)$$ (c)(ii) Given total amount collected is 4108, find $$x$$. - Step 1: Write the equation: $$6x + 10(500 - x) = 4108$$ - Step 2: Expand: $$6x + 5000 - 10x = 4108$$ - Step 3: Combine like terms: $$-4x + 5000 = 4108$$ - Step 4: Subtract 5000 from both sides: $$-4x = 4108 - 5000 = -892$$ - Step 5: Divide both sides by -4: $$x = \frac{-892}{-4} = 223$$ **Final answers:** - (a)(i) $$2r(r - 4)$$ - (a)(ii) $$(3x + 1)(x - 2)$$ - (b)(i) $$C = \frac{F - 32}{3}$$ - (b)(ii) $$C = 27$$ - (c)(i)(a) $$500 - x$$ - (c)(i)(b) $$6x + 10(500 - x)$$ - (c)(ii) $$x = 223$$