1. Problem i: Evaluate $$6^9 \left( \frac{3}{2} - 4^3 \right) 8^3 \times 16 \times 2$$. 2. Calculate powers: $$6^9 = 10077696$$, $$4^3 = 64$$, $$8^3 = 512$$. 3. Compute inside parentheses: $$\frac{3}{2} - 64 = 1.5 - 64 = -62.5$$. 4. Multiply all terms: $$10077696 \times (-62.5) \times 512 \times 16 \times 2$$. 5. Stepwise multiplication: $$10077696 \times (-62.5) = -629855999.9999999 \approx -629856000$$, $$-629856000 \times 512 = -322642176000$$, $$-322642176000 \times 16 = -5162274816000$$, $$-5162274816000 \times 2 = -10324549632000$$. Final answer i: $$-10324549632000$$. 6. Problem ii: Find the sum of algebraic expressions: $$a + 3b - 6c,\ -3a - b + 2c,\ 4a + 7b - 6c,\ -5a + 4b - 3c$$. 7. Sum by adding like terms: $$ (a - 3a + 4a - 5a) + (3b - b + 7b + 4b) + (-6c + 2c - 6c - 3c)$$ $$ = (-3a) + (13b) + (-13c)$$ $$ = -3a + 13b - 13c$$. 8. Problem b i): Multiply $a - 2b - 7c$ by $-2a - 3b - 4c$. 9. Multiply each term: $$ (a)(-2a) + (a)(-3b) + (a)(-4c) + (-2b)(-2a) + (-2b)(-3b) + (-2b)(-4c) + (-7c)(-2a) + (-7c)(-3b) + (-7c)(-4c) $$ $$= -2a^2 - 3ab - 4ac + 4ab + 6b^2 + 8bc + 14ac + 21bc + 28c^2$$ 10. Combine like terms: $$-2a^2 + ( -3ab + 4ab ) + ( -4ac + 14ac ) + 6b^2 + (8bc + 21bc) + 28c^2$$ $$ = -2a^2 + ab + 10ac + 6b^2 + 29bc + 28c^2$$. 11. Problem b ii): Find values of $X^0$ and $Y^0$. 12. Any number to zero power is 1: $$X^0 = 1,$$ $$Y^0 = 1$$ for $X, Y \neq 0$. 13. Problem a (Q4): Given $$a^2 = \frac{b^2 - c^2}{b^2}$$ solve for $b$. 14. Multiply both sides by $b^2$: $$a^2 b^2 = b^2 - c^2$$. 15. Bring $b^2$ terms together: $$a^2 b^2 - b^2 = -c^2$$, $$b^2 (a^2 - 1) = -c^2$$. 16. Divide both sides: $$b^2 = \frac{-c^2}{a^2 - 1} = \frac{c^2}{1 - a^2}$$. 17. Take square root: $$b = \pm \frac{c}{\sqrt{1 - a^2}}$$. 18. Problem b (Q4): Solve equation $$\frac{x}{4} - \frac{x+6}{5} = \frac{x+3}{2}$$. 19. Multiply whole equation by $20$ (LCM of 4, 5, 2): $$5x - 4(x+6) = 10(x+3)$$. 20. Expand: $$5x - 4x - 24 = 10x + 30$$, $$x - 24 = 10x + 30$$. 21. Bring $x$ terms to one side: $$x - 10x = 30 + 24$$, $$-9x = 54$$. 22. Solve for $x$: $$x = \frac{54}{-9} = -6$$. 23. Problem c (Q4): Simplify $$\frac{4a^2 b}{2a}$$. 24. Simplify numerator and denominator: $$\frac{4a^2 b}{2a} = 2ab$$. 25. Problem d i (Q4): Evaluate using laws of indices: $$\frac{a^5 \times a^2}{a^7 \times a^5}$$. 26. Sum powers in numerator and denominator: Numerator: $a^{5+2} = a^7$, Denominator: $a^{7+5} = a^{12}$. 27. Divide powers: $$a^7 \div a^{12} = a^{7-12} = a^{-5} = \frac{1}{a^5}$$. 28. Problem d ii (Q4): Evaluate $$(3ab^2)^2$$. 29. Square each factor: $$3^2 \times a^2 \times (b^2)^2 = 9a^2 b^4$$. 30. Problem e (Q4): In triangle CDE, $CD = 14.83$, $CE = 28.31$, angle $D=90^\circ$. Find $DE$. 31. Right triangle, apply Pythagoras theorem: $$DE = \sqrt{CE^2 - CD^2} = \sqrt{28.31^2 - 14.83^2}$$. 32. Calculate squares: $$28.31^2 = 801.5761,$$ $$14.83^2 = 219.8089$$. 33. Subtract and square root: $$DE = \sqrt{801.5761 - 219.8089} = \sqrt{581.7672} \approx 24.12$$. 34. Problem a (Q5): Evaluate $$\frac{a^3 b^2 c^4}{abc^2}$$ for $a=3$, $b=\frac{1}{8}$, $c=2$. 35. Simplify variables: $$\frac{a^3}{a} = a^2 = 9,$$ $$\frac{b^2}{b} = b = \frac{1}{8},$$ $$\frac{c^4}{c^2} = c^2 = 4$$. 36. Multiply values: $$9 \times \frac{1}{8} \times 4 = \frac{9 \times 4}{8} = \frac{36}{8} = 4.5$$. 37. Problem b i (Q5): Factorize $$2ax - 3ay + 2bx - 3by$$. 38. Group terms: $$(2ax + 2bx) + (-3ay - 3by) = 2x(a + b) - 3y(a + b) = (a + b)(2x - 3y)$$. 39. Problem b ii (Q5): Factorize $$(3p - a)^2 = (3p - a)(3p - a) = 9p^2 - 6ap + a^2$$. 40. Problem b iii (Q5): Evaluate $$(2x + 3y)(x - 4y)$$. 41. Multiply terms: $$2x \times x + 2x \times (-4y) + 3y \times x + 3y \times (-4y) = 2x^2 - 8xy + 3xy - 12y^2$$. 42. Combine like terms: $$2x^2 - 5xy - 12y^2$$. 43. Problem c i (Q5): Convert decimal $53.90625$ to binary. 44. Integer part 53 in binary: $110101$. 45. Fractional part $0.90625$ multiply by 2 repeatedly: $0.90625 \times 2 = 1.8125$ (1), $0.8125 \times 2 = 1.625$ (1), $0.625 \times 2 = 1.25$ (1), $0.25 \times 2 = 0.5$ (0), $0.5 \times 2 = 1.0$ (1). 46. Fraction binary: $0.11101$. 47. Combined binary number: $$53.90625 = 110101.11101_2$$. 48. Problem c ii (Q5): Sum binary numbers $101001_2$ and $10111_2$. 49. Convert to decimal: $101001_2 = 41$, $10111_2 = 23$. 50. Sum: $41 + 23 = 64$. 51. Convert sum to binary: $64 = 1000000_2$. 52. Problem d i (Q5): Convert binary $10010.1110_2$ to decimal. 53. Integer part $10010_2 = 16 + 0 + 0 + 2 + 0 = 18$. 54. Fraction part: $1\times 2^{-1} + 1\times 2^{-2} + 1\times 2^{-3} + 0\times 2^{-4} = 0.5 + 0.25 + 0.125 + 0 = 0.875$. 55. Total decimal: $$18 + 0.875 = 18.875$$. 56. Problem d ii (Q5): Explain three types of angles with diagrams (conceptual explanation only). (a) Acute angle: less than 90°. (b) Right angle: exactly 90°. (c) Obtuse angle: greater than 90° and less than 180°. Final answers complete.