1. **Problem 1(a)(i):** Find $s$ when $t=26.5$, $u=104.3$, and $a=-2.2$ using the formula $s = ut + \frac{1}{2}at^2$. 2. Substitute values: $$s = 104.3 \times 26.5 + \frac{1}{2} \times (-2.2) \times (26.5)^2$$ 3. Calculate each term: $$104.3 \times 26.5 = 2764.95$$ $$\frac{1}{2} \times (-2.2) = -1.1$$ $$-1.1 \times (26.5)^2 = -1.1 \times 702.25 = -772.475$$ 4. Sum the terms: $$s = 2764.95 - 772.475 = 1992.475$$ 5. Express in standard form to 4 significant figures: $$s = 1.991 \times 10^3$$ --- 6. **Problem 1(a)(ii):** Rearrange $s = ut + \frac{1}{2}at^2$ to express $a$ in terms of $u$, $t$, and $s$. 7. Subtract $ut$ from both sides: $$s - ut = \frac{1}{2}at^2$$ 8. Multiply both sides by 2: $$2(s - ut) = at^2$$ 9. Divide both sides by $t^2$: $$a = \frac{2(s - ut)}{t^2}$$ --- 10. **Problem 1(b)(i):** Show that the difference between areas of two rectangles is $62$ cm$^2$ leads to $x^2 + 2x - 63 = 0$. 11. Area of first rectangle: $$(2x + 3)(x - 1) = 2x^2 + 3x - 2x - 3 = 2x^2 + x - 3$$ 12. Area of second rectangle: $$(x + 1)(x - 2) = x^2 - 2x + x - 2 = x^2 - x - 2$$ 13. Difference of areas: $$2x^2 + x - 3 - (x^2 - x - 2) = 2x^2 + x - 3 - x^2 + x + 2 = x^2 + 2x - 1$$ 14. Set equal to 62: $$x^2 + 2x - 1 = 62$$ 15. Rearrange: $$x^2 + 2x - 63 = 0$$ --- 16. **Problem 1(b)(ii):** Factorize $x^2 + 2x - 63$. 17. Find factors of $-63$ that sum to $2$: $9$ and $-7$. 18. Factorization: $$(x + 9)(x - 7)$$ --- 19. **Problem 1(b)(iii):** Solve $x^2 + 2x - 63 = 0$. 20. Set each factor to zero: $$x + 9 = 0 \Rightarrow x = -9$$ $$x - 7 = 0 \Rightarrow x = 7$$ 21. Calculate perimeters: First rectangle perimeter: $$2[(2x + 3) + (x - 1)] = 2(3x + 2) = 6x + 4$$ Second rectangle perimeter: $$2[(x + 1) + (x - 2)] = 2(2x - 1) = 4x - 2$$ 22. Difference between perimeters: $$|6x + 4 - (4x - 2)| = |2x + 6|$$ 23. Substitute $x=7$ (positive root): $$2(7) + 6 = 14 + 6 = 20$$ 24. Final answer: $$20 \text{ cm}$$ --- 25. **Problem 3(a)(i):** Make $p$ the subject of $5p + 7 = m$. 26. Subtract 7: $$5p = m - 7$$ 27. Divide by 5: $$p = \frac{m - 7}{5}$$ --- 28. **Problem 3(a)(ii):** Make $p$ the subject of $y^2 - 2p^2 = h$. 29. Rearrange: $$-2p^2 = h - y^2$$ 30. Divide by $-2$: $$p^2 = \frac{y^2 - h}{2}$$ 31. Take square root: $$p = \pm \sqrt{\frac{y^2 - h}{2}}$$ --- 32. **Problem 4(a):** Simplify $$\frac{x + 3}{x - 2} - \frac{x - 2}{x + 2}$$ 33. Find common denominator: $$(x - 2)(x + 2) = x^2 - 4$$ 34. Rewrite: $$\frac{(x + 3)(x + 2)}{x^2 - 4} - \frac{(x - 2)^2}{x^2 - 4}$$ 35. Expand numerators: $$(x + 3)(x + 2) = x^2 + 5x + 6$$ $$(x - 2)^2 = x^2 - 4x + 4$$ 36. Subtract: $$\frac{x^2 + 5x + 6 - (x^2 - 4x + 4)}{x^2 - 4} = \frac{9x + 2}{x^2 - 4}$$ --- 37. **Problem 4(b):** Solve $$2^{12} \div 2^{k/2} = 32$$ 38. Rewrite 32 as power of 2: $$32 = 2^5$$ 39. Use division rule: $$2^{12 - k/2} = 2^5$$ 40. Equate exponents: $$12 - \frac{k}{2} = 5$$ 41. Solve for $k$: $$12 - 5 = \frac{k}{2} \Rightarrow 7 = \frac{k}{2} \Rightarrow k = 14$$ --- 42. **Problem 5(a):** Find $s$ when $u=5.2$, $t=7$, $a=1.6$. 43. Use formula: $$s = ut + \frac{1}{2}at^2 = 5.2 \times 7 + \frac{1}{2} \times 1.6 \times 7^2$$ 44. Calculate: $$5.2 \times 7 = 36.4$$ $$\frac{1}{2} \times 1.6 = 0.8$$ $$0.8 \times 49 = 39.2$$ 45. Sum: $$s = 36.4 + 39.2 = 75.6$$ --- 46. **Problem 5(b)(i):** Simplify $$3a - 5b - a + 2b = 2a - 3b$$ --- 47. **Problem 5(b)(ii):** Simplify $$\frac{5}{3x} \times \frac{9x}{20} = \frac{5 \times 9x}{3x \times 20} = \frac{45x}{60x} = \frac{3}{4}$$ --- 48. **Problem 5(c)(i):** Solve $$\frac{15}{x} = -3$$ 49. Multiply both sides by $x$: $$15 = -3x$$ 50. Divide by $-3$: $$x = -5$$ --- 51. **Problem 5(c)(ii):** Solve $$4(5 - 3x) = 23$$ 52. Expand: $$20 - 12x = 23$$ 53. Subtract 20: $$-12x = 3$$ 54. Divide by $-12$: $$x = -\frac{1}{4} = -0.25$$ --- 55. **Problem 5(d):** Simplify $$(27x^9)^{2/3}$$ 56. Rewrite: $$27^{2/3} \times (x^9)^{2/3}$$ 57. Calculate powers: $$27^{2/3} = (3^3)^{2/3} = 3^{2} = 9$$ $$x^{9 \times \frac{2}{3}} = x^{6}$$ 58. Final answer: $$9x^6$$ --- 59. **Problem 5(e):** Expand and simplify $$(3x - 5y)(2x + y) = 6x^2 + 3xy - 10xy - 5y^2 = 6x^2 - 7xy - 5y^2$$ --- 60. **Problem 6(a)(i):** Factorise $$5am + 10ap - bm - 2bp = (5a - b)(m + 2p)$$ --- 61. **Problem 6(a)(ii):** Factorise $$15(k + g)^2 - 20(k + g) = 5(k + g)(3(k + g) - 4) = 5(k + g)(3k + 3g - 4)$$ --- 62. **Problem 6(a)(iii):** Factorise $$4x^2 y^4 = (2xy^2)^2$$ --- 63. **Problem 6(b):** Expand and simplify $$(x - 3)(x + 1)(3x - 4)$$ 64. First expand first two: $$(x - 3)(x + 1) = x^2 - 2x - 3$$ 65. Multiply by third: $$(x^2 - 2x - 3)(3x - 4) = 3x^3 - 4x^2 - 6x^2 + 8x - 9x + 12 = 3x^3 - 10x^2 - x + 12$$ --- 66. **Problem 6(c):** Given $$(x + a)^2 = x^2 + 22x + b$$ 67. Expand left: $$x^2 + 2ax + a^2 = x^2 + 22x + b$$ 68. Equate coefficients: $$2a = 22 \Rightarrow a = 11$$ $$a^2 = b \Rightarrow b = 11^2 = 121$$ --- 69. **Problem 7(a):** Simplify $$\frac{x^2 - 25}{x^2 - x - 20}$$ 70. Factor numerator: $$(x - 5)(x + 5)$$ 71. Factor denominator: $$(x - 5)(x + 4)$$ 72. Cancel common factor: $$\frac{x + 5}{x + 4}$$ --- 73. **Problem 7(b):** Write as single fraction $$\frac{x + 5}{x} + \frac{x + 8}{x - 1}$$ 74. Common denominator: $$x(x - 1)$$ 75. Rewrite: $$\frac{(x + 5)(x - 1)}{x(x - 1)} + \frac{(x + 8)x}{x(x - 1)} = \frac{(x + 5)(x - 1) + x(x + 8)}{x(x - 1)}$$ 76. Expand numerator: $$(x + 5)(x - 1) = x^2 - x + 5x - 5 = x^2 + 4x - 5$$ $$x(x + 8) = x^2 + 8x$$ 77. Sum numerator: $$x^2 + 4x - 5 + x^2 + 8x = 2x^2 + 12x - 5$$ 78. Final fraction: $$\frac{2x^2 + 12x - 5}{x(x - 1)}$$ --- 79. **Problem 8(a)(i):** Simplify $$7^5 \times 7^6 = 7^{5+6} = 7^{11}$$ --- 80. **Problem 8(a)(ii):** Simplify $$7^{15} \div 7^5 = 7^{15-5} = 7^{10}$$ --- 81. **Problem 8(a)(iii):** Simplify $$42 + 7 = 49 = 7^2$$ --- 82. **Problem 8(b):** Simplify $$(5x^2 \times 2xy^4)^3 = (10x^3 y^4)^3 = 10^3 x^{9} y^{12} = 1000 x^9 y^{12}$$ --- 83. **Problem 8(c)(i):** Find HCF of $$P = 2^5 \times 3^3 \times 7$$ $$Q = 540 = 2^2 \times 3^3 \times 5$$ 84. HCF is product of lowest powers: $$2^2 \times 3^3 = 4 \times 27 = 108$$ --- 85. **Problem 8(c)(ii):** Find LCM: 86. LCM is product of highest powers: $$2^5 \times 3^3 \times 5 \times 7 = 32 \times 27 \times 5 \times 7 = 30240$$ --- 87. **Problem 8(c)(iii):** Find smallest $R$ so that $P \times R$ is a cube. 88. Prime factorization of $P$: $$2^5 \times 3^3 \times 7^1$$ 89. For cube, powers must be multiples of 3. 90. Adjust powers: - $2^5$: needs $2^{1}$ to make $2^{6}$ - $3^3$: already cube - $7^1$: needs $7^{2}$ to make $7^{3}$ 91. So $$R = 2^1 \times 7^2 = 2 \times 49 = 98$$ --- 92. **Problem 8(d)(i):** Factorise $$x^2 - 3x - 28 = (x - 7)(x + 4)$$ --- 93. **Problem 8(d)(ii):** Factorise $$7(a + 2b)^2 + 4a(a + 2b) = (a + 2b)(7(a + 2b) + 4a) = (a + 2b)(11a + 14b)$$ --- 94. **Problem 8(e):** Solve $$3^{2x - 1} = \frac{1}{9^x} \times 3^{2y - x}$$ 95. Rewrite $\frac{1}{9^x} = 9^{-x} = (3^2)^{-x} = 3^{-2x}$ 96. Equation becomes: $$3^{2x - 1} = 3^{-2x} \times 3^{2y - x} = 3^{-2x + 2y - x} = 3^{2y - 3x}$$ 97. Equate exponents: $$2x - 1 = 2y - 3x$$ 98. Rearrange: $$2y = 2x - 1 + 3x = 5x - 1$$ 99. Solve for $y$: $$y = \frac{5x - 1}{2}$$