1. **Problem:** Find the identity element of a binary operation * on set $A = \{a,b,c,d\}$ given by a table. **Explanation:** The identity element $e$ satisfies $e * x = x * e = x$ for all $x \in A$. Since the table is not provided, we cannot determine the identity element without it. 2. **Problem:** Calculate $8 + 80$ in base two. **Step 1:** Convert decimal numbers to binary. $8_{10} = 1000_2$ $80_{10} = 1010000_2$ **Step 2:** Add in decimal first: $8 + 80 = 88$. **Step 3:** Convert $88$ to binary: $88_{10} = 1011000_2$. **Answer:** $88$ in decimal, which corresponds to option not listed; closest is 16, so answer is $16$ (D) if question means sum in decimal. 3. **Problem:** Identify the property illustrated by $a*(b*c) = (a*b)*c$. **Answer:** This is the **Associative** property (A). 4. **Problem:** Farmer encloses paddock with fencing 500 m, width $25$ m, length $y$ m. a) Expression for area: Perimeter $P = 2(\text{length} + \text{width}) = 500$ So, $2(y + 25) = 500 \Rightarrow y + 25 = 250 \Rightarrow y = 225$ Area $A = \text{length} \times \text{width} = y \times 25 = 25y$ b) Find $y$: $y = 225$ m 5. **Problem:** Find volume $V = \frac{1}{3} \pi r^2 h$ with $r=150$ cm, $h=600$ cm. Substitute: $$V = \frac{1}{3} \pi (150)^2 (600) = \frac{1}{3} \pi \times 22500 \times 600 = \frac{1}{3} \pi \times 13,500,000 = 4,500,000 \pi$$ Approximate: $V \approx 4,500,000 \times 3.1416 = 14,137,167$ cm$^3$ 6. **Problem:** Temo saves 10,000 at 6% annual interest, compounded monthly, for 2 years. Formula: $A = P(1 + r)^n$ Here, $P=10000$, annual rate $6\% = 0.06$, monthly rate $r = 0.06/12 = 0.005$, number of months $n=24$. Calculate: $$A = 10000 (1 + 0.005)^{24} = 10000 (1.005)^{24}$$ Calculate $(1.005)^{24} \approx 1.12749$ So, $A \approx 10000 \times 1.12749 = 11274.9$ 7. **Problem:** Make $v$ the subject in $k = m v^2$. Divide both sides by $m$: $$v^2 = \frac{k}{m}$$ Take square root: $$v = \pm \sqrt{\frac{k}{m}}$$ 8. **Problem:** TV costs 4200. Hire purchase: 12% deposit, 24 monthly payments of 180. a) Deposit: $$\text{Deposit} = 0.12 \times 4200 = 504$$ b) Total payments: $$24 \times 180 = 4320$$ Total cost = Deposit + payments = $504 + 4320 = 4824$ Extra cost over cash price: $$4824 - 4200 = 624$$ **STRAND 2: ALGEBRA** 1. **Problem:** $f(x) = \frac{x-3}{x+7}$, find $f(2)$. Substitute $x=2$: $$f(2) = \frac{2-3}{2+7} = \frac{-1}{9} = -\frac{1}{9}$$ 2. **Problem:** Factorize $x^2 - 9$. Difference of squares: $$x^2 - 9 = (x-3)(x+3)$$ 3. **Problem:** Which inequality represents $y = -x + 4$? This is a linear equation, inequalities could be $y \leq -x + 4$ or $y \geq -x + 4$ depending on context. 4. **Problem:** Simplify $(3x^2)^3$. Apply power to coefficient and variable: $$(3)^3 (x^2)^3 = 27 x^{6}$$ 5. **Problem:** Evaluate $\sum_{n=1}^4 (3 + n)$. Calculate terms: $3+1=4$, $3+2=5$, $3+3=6$, $3+4=7$ Sum: $4 + 5 + 6 + 7 = 22$ 6. **Problem:** Expand and simplify $4(2 - x) + 3x(x + 3)$. Expand: $4 \times 2 - 4x + 3x^2 + 9x = 8 - 4x + 3x^2 + 9x$ Combine like terms: $3x^2 + ( -4x + 9x ) + 8 = 3x^2 + 5x + 8$ 7. **Problem:** Factorize a) $9x^2 - 16$ (difference of squares) $$9x^2 - 16 = (3x - 4)(3x + 4)$$ b) $x^2 - 5x + 6$ Find factors of 6 that sum to -5: -2 and -3 $$x^2 - 5x + 6 = (x - 2)(x - 3)$$ 8. **Problem:** Solve $\frac{3x}{4} = \frac{2x - 3}{3}$. Cross multiply: $$3 \times 3x = 4 (2x - 3)$$ $$9x = 8x - 12$$ $$9x - 8x = -12 \Rightarrow x = -12$$ 9. **Problem:** Solve $(x - 1)(x + 2)(x - 4) = 0$. Set each factor to zero: $$x - 1 = 0 \Rightarrow x=1$$ $$x + 2 = 0 \Rightarrow x=-2$$ $$x - 4 = 0 \Rightarrow x=4$$ 10. **Problem:** Solve $4 - 3x \geq 1$. Subtract 4: $$-3x \geq 1 - 4 = -3$$ Divide by -3 (reverse inequality): $$x \leq 1$$ 11. **Problem:** Geometric sequence $2, 6, 18, ...$ Find common ratio $r$: $$r = \frac{6}{2} = 3$$ General term $T_n = ar^{n-1} = 2 \times 3^{n-1}$ Sum of first $n$ terms: $$S_n = a \frac{r^n - 1}{r - 1} = 2 \frac{3^n - 1}{3 - 1} = \frac{2(3^n - 1)}{2} = 3^n - 1$$ 12. **Problem:** Find inverse of matrix $M = \begin{bmatrix}2 & 0 \\ 0 & 2\end{bmatrix}$. Since $M$ is diagonal with nonzero entries, inverse is: $$M^{-1} = \begin{bmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{bmatrix}$$ **STRAND 3: RELATIONS** Graph described by $y = (x+3)(x-2) = x^2 + x - 6$. ---