1. Problem: Find the identity element of a binary operation * on set A = {a, b, c, d} given by a table. Step 1: Recall the identity element e satisfies e * x = x * e = x for all x in A. Step 2: Check each element a, b, c, d in the table to see if it acts as identity. Step 3: The element that leaves all others unchanged under * is the identity. 2. Problem: Calculate 8 + 80 in base two. Step 1: Convert 8 and 80 to base 2. Step 2: 8 in base 2 is $1000$. Step 3: 80 in base 2 is $1010000$. Step 4: Add $1000_2 + 1010000_2$ by converting to decimal: 8 + 80 = 88. Step 5: Convert 88 back to decimal is 88, so answer is 88 (not in options), check options for closest meaning. 3. Problem: Identify property illustrated by $a*(b*c) = (a*b)*c$. Step 1: This is the associative property. 4. Problem: Farmer paddock with fencing 500 m, width 25 m, length y m. (a) Expression for area: $A = 25 \times y$ (b) Find y: Step 1: Perimeter $P = 2(25 + y) = 500$ Step 2: Solve $2(25 + y) = 500 \Rightarrow 25 + y = 250 \Rightarrow y = 225$ 5. Problem: Find volume $V = \frac{1}{3} \pi r^2 h$ with $r=150$, $h=600$. Step 1: Calculate $V = \frac{1}{3} \pi (150)^2 (600)$ Step 2: $150^2 = 22500$ Step 3: $V = \frac{1}{3} \pi \times 22500 \times 600 = \frac{1}{3} \pi \times 13,500,000 = 4,500,000 \pi$ 6. Problem: Temo saves 10000 at 6% annual interest compounded monthly for 2 years. Step 1: Use $A = P(1 + r)^n$ where $P=10000$, $r=0.06/12=0.005$, $n=24$ Step 2: Calculate $A = 10000 (1 + 0.005)^{24}$ Step 3: $A = 10000 (1.005)^{24} \approx 10000 \times 1.12749 = 11274.9$ 7. Problem: Make $v$ the subject in $k = m v^2$. Step 1: Divide both sides by $m$: $\frac{k}{m} = v^2$ Step 2: Take square root: $v = \pm \sqrt{\frac{k}{m}}$ 8. Problem: TV costs 4200, deposit 12%, 24 payments of 180. (a) Deposit: $0.12 \times 4200 = 504$ (b) Total payments: $24 \times 180 = 4320$ Step 1: Total cost = deposit + payments = $504 + 4320 = 4824$ Step 2: Extra cost = $4824 - 4200 = 624$ STRAND 2: ALGEBRA 1. Problem: $f(x) = \frac{x-3}{x+7}$, find $f(2)$. Step 1: Substitute $x=2$: $f(2) = \frac{2-3}{2+7} = \frac{-1}{9} = -\frac{1}{9}$ 2. Problem: Factorize $x^2 - 9$. Step 1: Recognize difference of squares: $x^2 - 3^2$ Step 2: Factor: $(x - 3)(x + 3)$ 3. Problem: Inequality representing $y = -x + 4$. Step 1: Inequality could be $y \leq -x + 4$ or $y \geq -x + 4$ depending on context. 4. Problem: Simplify $(3x^2)^3$. Step 1: Apply power: $3^3 x^{2 \times 3} = 27 x^6$ 5. Problem: Evaluate $\sum_{n=1}^4 (3 + n)$. Step 1: Calculate terms: $3+1=4$, $3+2=5$, $3+3=6$, $3+4=7$ Step 2: Sum: $4 + 5 + 6 + 7 = 22$ 6. Problem: Expand and simplify $4(2 - x) + 3x(x + 3)$. Step 1: $4 \times 2 - 4x + 3x^2 + 9x = 8 - 4x + 3x^2 + 9x$ Step 2: Combine like terms: $3x^2 + 5x + 8$ 7. Problem: Factorize (a) $9x^2 - 16$ is difference of squares: $(3x - 4)(3x + 4)$ (b) $x^2 - 5x + 6$ factors as $(x - 2)(x - 3)$ 8. Problem: Solve $\frac{3x}{4} = \frac{2x - 3}{3}$. Step 1: Cross multiply: $3 \times 3x = 4(2x - 3)$ Step 2: $9x = 8x - 12$ Step 3: $9x - 8x = -12 \Rightarrow x = -12$ 9. Problem: Solve $(x - 1)(x + 2)(x - 4) = 0$. Step 1: 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 $4 - 3x \geq 1$. Step 1: Subtract 4: $-3x \geq -3$ Step 2: Divide by -3 (reverse inequality): $x \leq 1$ 11. Problem: Geometric sequence 2, 6, 18,... Find $r$, $T_n$, $S_n$. Step 1: Common ratio $r = \frac{6}{2} = 3$ Step 2: General term $T_n = 2 \times 3^{n-1}$ Step 3: Sum $S_n = 2 \frac{3^n - 1}{3 - 1} = 2 \frac{3^n - 1}{2} = 3^n - 1$ 12. Problem: Find inverse of $M = \begin{bmatrix}2 & 0 \\ 0 & 2\end{bmatrix}$. Step 1: Inverse of diagonal matrix is diagonal with reciprocals. Step 2: $M^{-1} = \begin{bmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{bmatrix}$