(a)

\begin{align*} 3(2x - 1) - 4(x - 7) & = 6x - 3 - 4x + 28 \\ & = 2x + 25 \end{align*}

(b)

\begin{align*} 14 - 3(5 - 4x) + 6x & = 14 - 15 + 12x + 6x \\ & = -1 + 18x \end{align*}

(c)

\begin{align*} 7(2y + 3) - 4(3 - y) & = 14y + 21 - 12 + 4y \\ & = 18y + 9 \end{align*}

(d)

\begin{align*} 9(5p - 6) + 4(7 - 13p) & = 45p - 54 + 28 - 52p \\ & = - 7p - 26 \end{align*}

(e)

\begin{align*} 5 - 3(q + r) - 6(3r - 2q) & = 5 - 3q - 3r - 18r + 12q \\ & = 5 + 9q - 21r \end{align*}

(f)

\begin{align*} (a + 2b)^2 - (a - 2b)^2 & = \underbrace{ (a)^2 + 2(a)(2b) + (2b)^2 }_{ (a + b)^2 = a^2 + 2ab + b^2 } - [ \underbrace{ (a)^2 - 2(a)(2b) + (2b)^2 }_{ (a - b)^2 = a^2 - 2ab + b^2 } \\ & = a^2 + 4ab + 4b^2 - (a^2 - 4ab + 4b^2) \\ & = a^2 + 4ab + 4b^2 - a^2 + 4ab - 4b^2 \\ & = 8ab \end{align*}

(a)

\begin{align*} 5x^2 - 20x^2y & = 5x^2 (1 - 4y) \end{align*}

(b)

\begin{align*} x^2 - 4xy + 4y^2 & = (x - 2y)(x - 2y) \\ & = (x - 2y)^2 \end{align*}

(c)

\begin{align*} (3x + 4y)^2 - 9z^2 & = (3x + 4y)^2 - (3z)^2 \\ & = (3x + 4y + 3z)(3x + 4y - 3z) \phantom{000000} [a^2 - b^2 = (a + b)(a - b)] \end{align*}

(d)

\begin{align*} 5p^2 + 11p + 2 & = (5p + 1)(p + 2) \end{align*}

(e)

\begin{align*} 6q^2 - 31q + 35 & = (3q - 5)(2q - 7) \end{align*}

(a)

\begin{align*} 2[3a - 2(3a - 1) + 4(a + 1)] & = 2(3a - 6a + 2 + 4a + 4) \\ & = 2(a + 6) \\ & = 2a + 12 \end{align*}

(b)

\begin{align*} 8(x - y) - [x - y - 3(y - z - x)] & = 8x - 8y - (x - y - 3y + 3z + 3x) \\ & = 8x - 8y - (4x - 4y + 3z) \\ & = 8x - 8y - 4x + 4y - 3z \\ & = 4x - 4y - 3z \end{align*}

(c)

\begin{align*} 2b(c - a) - [3c(a - b) - 3a(b + c)] & = 2bc - 2ab - (3ac - 3bc - 3ab - 3ac) \\ & = 2bc - 2ab - (-3bc - 3ab) \\ & = 2bc - 2ab + 3bc + 3ab \\ & = 5bc + ab \end{align*}

(d)

\begin{align*} 3(a - c) - \{ 5(2a - 3b) - [5a - 7(a - b)] \} & = 3a - 3c - [ 10a - 15b - (5a - 7a + 7b) ] \\ & = 3a - 3c - [ 10a - 15b - (-2a + 7b) ] \\ & = 3a - 3c - ( 10a - 15b + 2a - 7b ) \\ & = 3a - 3c - ( 12a - 22b ) \\ & = 3a - 3c - 12a + 22b \\ & = -9a + 22b - 3c \end{align*}

(a)

\begin{align*} {25 a^2 \over b^2 c} \times {bc^3 \over 100a^3} \div {15 \over c^2} & = {25 a^2 b c^3 \over 100 a^3 b^2 c} \times {c^2 \over 15} \\ & = {25 a^2 b c^5 \over 1500 a^3 b^2 c} \phantom{000000} [a^m \times a^n = a^{m + n} ] \\ & = { c^4 \over 60 a b} \end{align*}

(b)

\begin{align*} {8 a^5 b^2 c \over (-2ab)^2 } & = { 8a^5 b^2 c \over 4a^2 b^2 } \phantom{000000} [ (-2ab)^2 = (-2ab) \times (-2ab)] \\ & = { 2 a^3 c \over 1} \\ & = 2a^3 c \end{align*}

(c)

\begin{align*} {2a^2 b^3 \over 3b} \div { (2a)^2 \over 15ab^2 } & = { 2 a^2 b^2 \over 3} \times { 15ab^2 \over (2a)^2 } \\ & = { 2 a^2 b^2 \over 3} \times { 15 ab^2 \over 4a^2 } \phantom{000000} [ (2a)^2 = 2a \times 2a ] \\ & = { 30 a^3 b^4 \over 12 a^2 } \\ & = { 5 a b^4 \over 2} \end{align*}

(d)

\begin{align*} {a - 1 \over a - b} \div {1 - a \over a^2 - b^2 } & = { a - 1 \over a - b} \times {a^2 - b^2 \over 1 - a} \\ & = { (a - 1)(a^2 - b^2) \over (a - b)(1 - a)} \\ & = { - (1 - a)(a + b)(a - b) \over (a - b)(1 - a)} \\ & = -(a + b) \end{align*}

(e)

\begin{align*} {2x^2 + 11x + 15 \over x^2 - 9} & = { (2x + 5)(x + 3) \over x^2 - 3^2 } \\ & = { (2x + 5)(x + 3) \over (x + 3)(x - 3) } \phantom{000000} [ a^2 - b^2 = (a + b)(a - b)] \\ & = { 2x + 5 \over x - 3} \end{align*}

(a)

\begin{align*} x^2 + 3y + xy + 3x & = x^2 + xy + 3x + 3y \\ & = x(x + y) + 3(x + y) \\ & = (x + y)(x + 3) \end{align*}

(b)

\begin{align*} ab - bc - ac + c^2 & = b(a - c) - c(a - c) \\ & = (a - c)(b - c) \end{align*}

(c)

\begin{align*} ax - kx - ah + kh & = ax - ah - kx + kh \\ & = a(x - h) - k (x - h) \\ & = (x - h)(a - k) \end{align*}

(d)

\begin{align*} 20 ac - 4ad - 15kc + 3kd & = 4a (5c - d) - 3k (5c - d) \\ & = (5c - d)(4a - 3k) \end{align*}

(e)

\begin{align*} 6a^2 + 3ab - 8ka - 4kb & = 3a(2a + b) - 4k(2a + b) \\ & = (2a + b)(3a - 4k) \end{align*}

(a)

\begin{align*} ax^2 + bx + c & = 0 \\ bx & = - ax^2 - c \\ b & = {-ax^2 - c \over x} \phantom{00000} [\text{Can stop here}] \\ b & = - {ax^2 + c \over x} \end{align*}

(b)

\begin{align*} {1 \over a} + {b \over 2} + {3 \over c} & = k \\ {2c \over 2ac} + { abc \over 2ac} + {6a \over 2ac} & = k \\ {2c + abc + 6a \over 2ac} & = {k \over 1} \\ 2c + abc + 6a & = 2ac k \phantom{000000} [\text{Cross-multiply}] \\ 2c + abc - 2ack & = - 6a \\ c(2 + ab - 2ak) & = - 6a \\ c & = {-6a \over 2 + ab - 2ak} \phantom{000000} [\text{Can stop here}] \\ c & = {6a \over -(2 + ab - 2ak)} \\ c & = {6a \over -2 -ab + 2ak} \end{align*}

(c)

\begin{align*} \sqrt{ 4x^2 - 5k } & = 2x + 3 \\ 4x^2 - 5k & = (2x + 3)^2 \\ 4x^2 - 5k & = \underbrace{ (2x)^2 + 2(2x)(3) + (3)^2 }_{ (a + b)^2 = a^2 + 2ab + b^2 } \\ 4x^2 - 5k & = 4x^2 + 12x + 9 \\ 4x^2 - 4x^2 - 12x & = 9 + 5k \\ -12x & = 9 + 5k \\ x & = {9 + 5k \over -12} \\ x & = -{9 + 5k \over 12} \end{align*}

(d)

\begin{align*} v^2 & = u^2 + 2as \\ - u^2 & = 2as - v^2 \\ u^2 & = v^2 - 2as \\ u & = \pm \sqrt{ v^2 - 2as} \end{align*}

(e)

\begin{align*} x & = \sqrt[3]{a \over b - a} \\ x^3 & = {a \over b - a} \\ {x^3 \over 1} & = {a \over b - a} \\ x^3 (b - a) & = a \phantom{000000} [\text{Cross-multiply}] \\ bx^3 - ax^3 & = a \\ -ax^3 - a & = - bx^3 \\ ax^3 + a & = bx^3 \\ a(x^3 + 1) & = bx^3 \\ a & = {bx^3 \over x^3 + 1} \end{align*}

(f)

\begin{align*} V & = {2\pi \over 3} (h^3 - d^3) \\ 3V & = 2\pi (h^3 - d^3) \\ {3V \over 2\pi} & = h^3 - d^3 \\ {3V \over 2\pi} + d^3 & = h^3 \\ \\ h & = \sqrt[3]{ {3V \over 2\pi} + d^3 } \phantom{000000} [\text{Can stop here}] \\ & = \sqrt[3]{ {3V \over 2\pi} + {2\pi d^3 \over 2\pi} } \\ & = \sqrt[3] { {3V + 2\pi d^3 \over 2\pi } } \end{align*}

(a)

\begin{align*} {3 \over 4} + {x - 3 \over 2x} & = {3x \over 4x} + {2x - 6 \over 4x} \\ & = {3x + 2x - 6 \over 4x} \\ & = {5x - 6 \over 4x} \end{align*}

(b)

\begin{align*} {2y + 3 \over 9y^2 - 1} - {5 \over 3y - 1} & = {2y + 3 \over (3y)^2 - (1)^2} - {5 \over 3y - 1} \\ & = {2y + 3 \over (3y + 1)(3y - 1)} - {5 \over 3y - 1} \phantom{000000} [ a^2 - b^2 = (a + b)(a - b)] \\ & = {2y + 3 \over (3y + 1)(3y - 1)} - {5(3y + 1) \over (3y + 1)(3y - 1)} \\ & = {2y + 3 - 5(3y + 1) \over (3y + 1)(3y - 1)} \\ & = {2y + 3 - 15y - 5 \over (3y + 1)(3y - 1)} \\ & = {-13y - 2 \over (3y + 1)(3y - 1)} \phantom{000000} [\text{Can stop here}] \\ & = - {13y + 2 \over (3y + 1)(3y - 1)} \end{align*}

(c)

\begin{align*} {3a \over a - 3} + {2 \over a + 4} & = {3a(a + 4) \over (a - 3)(a + 4} + {2(a - 3) \over (a - 3)(a + 4)} \\ & = {3a(a + 4) + 2(a - 3) \over (a - 3)(a + 4)} \\ & = {3a^2 + 12a + 2a - 6 \over (a - 3)(a + 4)} \\ & = {3a^2 + 14a - 6 \over (a - 3)(a + 4)} \end{align*}

(d)

\begin{align*} {2 \over p^2 + 4p - 5} - {1 \over p - 1} & = {2 \over (p - 1)(p + 5)} - {p + 5 \over (p - 1)(p + 5)} \\ & = {2 - p - 5 \over (p - 1)(p + 5)} \\ & = {- p - 3 \over (p - 1)(p + 5)} \\ & = - {p + 3 \over (p - 1)(p + 5)} \end{align*}

(e)

\begin{align*} {5x \over 2x - y} + {y \over 3x - y} & = {5x(3x - y) \over (2x - y)(3x - y)} + {y(2x - y) \over (2x - y)(3x - y)} \\ & = {5x(3x - y) + y(2x - y) \over (2x - y)(3x - y)} \\ & = {15x^2 - 5xy + 2xy - y^2 \over (2x -y)(3x - y)} \\ & = {15x^2 - 3xy - y^2 \over (2x - y)(3x - y)} \end{align*}

(a)

\begin{align*} x^2 - 4 & = x^2 - 2^2 \\ & = (x + 2)(x - 2) \phantom{000000} [a^2 - b^2 = (a + b)(a - b)] \\ \\ {3 \over x + 2} - {x - 5 \over (x + 2)(x - 2)} + {1 \over x - 2} & = {3(x - 2) \over (x + 2)(x - 2} - {x - 5 \over (x + 2)(x - 2)} + {x + 2 \over (x + 2)(x - 2)} \\ & = {3(x - 2) - (x - 5) + x + 2 \over (x + 2)(x - 2)} \\ & = {3x - 6 - x + 5 + x + 2 \over (x + 2)(x - 2)} \\ & = {3x + 1 \over (x + 2)(x - 2)} \end{align*}

(b)

\begin{align*} x^2 - 1 & = -(1 - x^2) \\ & = -(1^2 - x^2) \\ & = -(1 + x)(1 - x) \phantom{000000} [a^2 - b^2 = (a + b)(a - b)] \\ \\ {1 \over 1 - x} + {2 \over 1 + x} + {2 \over -(1 + x)(1 - x)} & = {1 + x \over (1 + x)(1 - x)} + {2(1 - x) \over (1 + x)(1 - x)} - {2x \over (1 + x)(1 - x)} \\ & = {1 + x + 2(1 - x) - 2x \over (1 + x)(1 - x)} \\ & = {1 + x + 2 - 2x - 2x \over (1 + x)(1 - x)} \\ & = {3 - 3x \over (1 + x)(1 - x)} \\ & = {3(1 - x) \over (1 + x)(1 - x)} \\ & = {3 \over 1 + x} \end{align*}

(c)

\begin{align*} 2x^2 + x - 6 & = (2x - 3)(x + 2) \\ \\ {1 \over 2x - 3} - {2 \over x + 2} - {2x - x^2 \over (2x - 3)(x + 2)} & = {x + 2 \over (2x - 3)(x + 2)} - {2(2x - 3) \over (2x - 3)(x + 2)} - {2x - x^2 \over (2x - 3)(x + 2)} \\ & = {x + 2 - 2(2x - 3) - (2x - x^2) \over (2x - 3)(x + 2)} \\ & = {x + 2 - 4x + 6 - 2x + x^2 \over (2x - 3)(x + 2)} \\ & = {x^2 - 5x + 8 \over (2x - 3)(x + 2)} \end{align*}

(d)

\begin{align*} x^2 - x - 2 & = (x - 2)(x + 1) \\ \\ {5 \over x - 2} - {3x + x^2 \over (x - 2)(x + 1)} + {x \over x + 1} & = {5(x + 1) \over (x - 2)(x + 1)} - {3x + x^2 \over (x - 2)(x + 1)} + {x(x - 2) \over (x - 2)(x + 1)} \\ & = {5(x + 1) - (3x + x^2) + x(x - 2) \over (x - 2)(x + 1)} \\ & = {5x + 5 - 3x - x^2 + x^2 - 2x \over (x - 2)(x + 1)} \\ & = {5 \over (x - 2)(x + 1)} \end{align*}

(i)

\begin{align*} {3 \over a} & = {2 \over b} + {1 \over c} \\ {3 \over a} & = {2c \over bc} + {b \over bc} \\ {3 \over a} & = {2c + b \over bc} \\ 3bc & = a(2c + b) \\ 3bc & = 2ac + ab \\ 3bc - ab & = 2ac \\ b(3c - a) & = 2ac \\ b & = {2ac \over 3c - a} \end{align*}

(ii)

\begin{align*} b & = {2ac \over 3c - a} \\ & = {2 \left(2{1 \over 5}\right)(-3) \over 3(-3) - \left(2{1 \over 5}\right)} \\ & = 1 {5 \over 28} \end{align*}

(a)

\begin{align*} V & = {b^2 \over 2} (a + 3h) \\ 2V & = b^2 (a + 3h) \\ {2V \over b^2} & = a + 3h \\ -3h & = a - {2V \over b^2} \\ 3h & = {2V \over b^2} - a \\ h & = {1 \over 3} \left( {2V \over b^2} - a \right) \\ h & = {2V \over 3b^2} - {1 \over 3}a \end{align*}

(b)(i)

\begin{align*} V & = {b^2 \over 2} (a + 3h) \\ & = {(1.2)^2 \over 2} [ 0.2 + 3(1.5) ] \\ & = 3.384 \text{ m}^3 \end{align*}

(b)(ii)

\begin{align*} h & = {2V \over 3b^2} - {1 \over 3}a \\ & = {2(12) \over 3(0.4)^2} - {1 \over 3}(2.5) \\ & = 49 {1 \over 6} \text{ m} \end{align*}

(b)(iii)

\begin{align*} V & = {b^2 \over 2} (a + 3h) \\ 15 & = {b^2 \over 2} [0.25 + 3(2.4)] \\ 15 & = {b^2 \over 2} (7.45) \\ {15 \over 7.45} & = {b^2 \over 2} \\ 2(15) & = 7.45 b^2 \\ 30 & = 7.45b^2 \\ \\ b^2 & = {30 \over 7.45} \\ b & = \pm \sqrt{30 \over 7.45} \\ b & \approx \pm 2.01 \end{align*}

(i)

\begin{align*} {x^2 \over a^2} - {y^2 \over b^2} & = 1 \\ {b^2 x^2 \over a^2 b^2} - {a^2 y^2 \over a^2 b^2} & = 1 \\ {b^2 x^2 - a^2 y^2 \over a^2 b^2 } & = {1 \over 1} \\ b^2 x^2 - a^2 y^2 & = a^2 b^2 \phantom{000000} [\text{Cross-multiply}] \\ -a^2 y^2 & = a^2 b^2 - b^2 x^2 \\ a^2 y^2 & = b^2 x^2 - a^2 b^2 \\ y^2 & = {b^2 x^2 - a^2 b^2 \over a^2} \\ y & = \pm \sqrt{ b^2 x^2 - a^2 b^2 \over a^2 } \end{align*}

(ii)

\begin{align*} y & = \pm \sqrt{ b^2 x^2 - a^2 b^2 \over a^2 } \\ & = \pm \sqrt{ (2)^2 (-5)^2 - (3)^2 (2)^2 \over (3)^2 } \\ & = \pm \sqrt{ 64 \over 9 } \\ & = \pm 2 {2 \over 3} \end{align*}

(i)

\begin{align*} \text{Fraction of pool (A)} & = {1 \over x} \end{align*}

(ii)

\begin{align*} \text{Fraction of pool in two hours (B)} & = {1 \over x + 1.5} \times {2 \over 1} \\ & = {2 \over x + 1.5} \end{align*}

(iii)

\begin{align*} \text{Fraction of pool in one hour (A & B)} & = {1 \over x} + {1 \over x + 1.5} \end{align*}

(a)

\begin{align*} & {2 \over 12}, {5 \over 12}, {8 \over 12}, {11 \over 12}, \underline{ {14 \over 12} = {7 \over 6}, {17 \over 12} } \end{align*}

(b)

\begin{align*} 0, -1, -8, -27, -64, ... & = (0)^3, (-1)^3, (-2)^3, (-3)^3, (-4)^3, ... \\ & = (1 - 1)^3, (1 - 2)^3, (1 - 3)^3, (1 - 4)^3, (1 - 5)^3, ... \\ \\ T_n & = (1 - n)^3 \end{align*}

(a)

\begin{align*} \text{Common difference, } d & = {42 - 78 \over 4 } \\ & = -9 \\ \\ a & = 78 - 9 = 69 \\ b & = 69 - 9 = 60 \\ c & = 60 - 9 = 51 \end{align*}

(b)

\begin{align*} T_n & = a + (n - 1)(d) \phantom{000000} [a: \text{ first term}] \\ & = 78 + (n - 1)(-9) \\ & = 78 - 9(n - 1) \\ & = 78 - 9n + 9 \\ & = 87 - 9n \end{align*}

(c)

\begin{align*} T_n & < 0 \\ 87 - 9n & < 0 \\ -9n & < -87 \\ n & > {-87 \over -9} \\ n & > 9 {2 \over 3} \\ \\ \text{First } n & = 10 \\ \\ T_{10} & = 87 - 9(10) \\ & = -3 \end{align*}

(i)

\begin{align*} u_5 & = 7^2 - 5 + 3 = 47 \\ \\ u_6 & = 8^2 - 6 + 3 = 61 \end{align*}

(ii)

\begin{align*} u_n & = (n + 2)^2 - n + 3 \end{align*}

(iii)

\begin{align*} u_n & = (n + 2)^2 - n + 3 \\ & = \underbrace{ (n)^2 + 2(n)(2) + (2)^2 }_{ (a + b)^2 = a^2 + 2ab + b^2 } - n + 3 \\ & = n^2 + 4n + 4 - n + 3 \\ & = n^2 + 3n + 7 \\ \\ \text{Let } u_k & = 161 \\ k^2 + 3k + 7 & = 161 \\ k^2 + 3k + 7 - 161 & = 0 \\ k^2 + 3k - 154 & = 0 \\ (k - 11)(k + 14) & = 0 \end{align*} \begin{align*} k - 11 & = 0 && \text{ or } & k + 14 & = 0 \\ k & = 11 &&& k & = -14 \text{ (Reject, since } k > 0) \end{align*}

(iv)

\begin{align*} u_p & = p^2 + 3p + 7 \\ \\ u_{p + 1} & = (p + 1)^2 + 3(p + 1) + 7 \\ & = \underbrace{ (p)^2 + 2(p)(1) + (1)^2 }_{ (a + b)^2 = a^2 + 2ab + b^2 } + 3p + 3 + 7 \\ & = p^2 + 2p + 1 + 3p + 3 + 7 \\ & = p^2 + 5p + 11 \\ \\ u_{p + 1} - u_p & = p^2 + 5p + 11 - (p^2 + 3p + 7) \\ & = p^2 + 5p + 11 - p^2 - 3p - 7 \\ & = 2p + 4 \end{align*}

(v)

\begin{align*} \text{From (v), } u_{p + 1} - u_p & = 2p + 4 \\ & = 2(p + 2) \\ \\ \text{Since } u_{p + 1} - u_p \text{ is not } \text{divisible by }& 3, \text{ two consecutive terms cannot have difference of } 3 \end{align*}

(a)

\begin{align*} \text{Total no. of children} & = x + 1 + y + 2 \\ & = x + y + 3 \\ \\ \text{Total age of children} & = q(x + y + 3) \\ & = qx + qy + 3q \\ \\ \text{Total age of boys} & = p(x + 1) \\ & = px + p \\ \\ \text{Total age of girls} & = qx + qy + 3q - (px + p) \\ & = qx + qy + 3q - px - p \\ \\ \text{Average age of girls} & = { qx + qy + 3q - px - p \over y + 2} \end{align*}

(b)

\begin{align*} \text{Sum of age in 2 years' time} & = x - 3 - 3 \phantom{000000} [\text{Minus 3 for each person}] \\ & = x - 6 \\ \\ \text{Vani's age in 2 years' time} & = {x - 6 \over 3} \phantom{000000000} [\text{Vani: 1 unit, Siti: 2 units}] \\ \\ \text{Siti's age in 2 years' time} & = {x - 6 \over 3} \times 2 \\ & = {2(x - 6) \over 3} \\ \\ \text{Siti's present age} & = {2(x - 6) \over 3} - 2 \\ & = {2x - 12 \over 3} - {2 \over 1} \\ & = {2x - 12 \over 3} - {6 \over 3} \\ & = {2x - 18 \over 3} \end{align*}

(i)

\begin{align*} S_3 & = (1 + 2 + 3)^2 = 36 \\ \\ S_4 & = (1 + 2 + 3 + 4)^2 = 100 \\ \\ S_5 & = (1 + 2 + 3 + 4 + 5)^2 = 225 \end{align*}

(ii)

\begin{align*} S_1 & = \left[ {1 \over 2}(1)(2) \right]^2 = 1^2 \\ S_2 & = \left[ {1 \over 2}(2)(3) \right]^2 = 3^2 \\ S_3 & = \left[ {1 \over 2}(3)(4) \right]^2 = 3^2 \\ \\ S_n & = \left[ {1 \over 2}n(n + 1) \right]^2 \end{align*}

(iii)

\begin{align*} S_k & = \left[ {1 \over 2}k(k + 1) \right]^2 \\ 44 \phantom{.} 100 & = \left[ {1 \over 2}k(k + 1) \right]^2 \\ \sqrt{ 44 \phantom{.} 100 } & = {1 \over 2}k (k + 1) \\ 210 & = {1 \over 2}k(k + 1) \\ 210 & = {1 \over 2}k^2 + {1 \over 2}k \\ 420 & = k^2 + k \\ 0 & = k^2 + k - 420 \\ 0 & = (k - 20)(k + 21) \end{align*} \begin{align*} k - 20 & = 0 && \text{ or } & k + 21 & = 0 \\ k & = 20 &&& k & = -21 \text{ (Reject, since } k > 0) \end{align*}

(a)

\begin{align*} a & = 8 \phantom{0000000000} [\text{Cubes at the corners}] \\ \\ b & = 8 + 6 + 6 + 4 = 24 \\ \\ c & = 4 \times 6 = 24 \\ \\ d & = 2^3 = 8 \phantom{000000} [\text{'Inner' cube after removing outside layer}] \end{align*}

(b)

$$ \text{No. of cubes} = 8 \phantom{0000000000} [\text{Cubes at the corners}] $$

(c)

\begin{align*} \text{No. of cubes not painted} & = 3^3 = 27 \end{align*}

(d)(i)

\begin{align*} \text{Pattern for 2 faces painted} & = \underbrace{0}_{n = 2}, 12, 24, ... \\ \\ T_n & = a + (n - 1)d \phantom{000000} [a \text{: first term, } d \text{: difference}] \\ & = -12 + (n - 1)(12) \\ & = -12 + 12(n - 1) \\ & = -12 + 12n - 12 \\ & = 12n - 24 \\ & = 12(n - 2) \end{align*}

(d)(ii)

\begin{align*} \text{Pattern for 1 face painted} & = \underbrace{0}_{n = 2}, 6, 24, ... \\ & = 6(0)^2, 6(1)^2, 6(2)^2, ... \\ & = 6(2 - 2)^2, 6(3 - 2)^2, 6(4 - 2)^2, ... \\ \\ T_n & = 6(n - 2)^2 \end{align*}

(d)(iii)

\begin{align*} \text{Pattern for 0 face painted} & = \underbrace{0}_{n = 2}, 1, 8, ... \\ & = 0^3, 1^3, 2^3, ... \\ & = (2 - 2)^3, (3 - 2)^3, (4 - 2)^3, ... \\ \\ T_n & = (n - 2)^3 \end{align*}

(i)

\begin{align*} l & = 40 + 20 = 60 \\ \\ m & = 5^2 = 25 \\ \\ n & = (5 + 1)^2 = 36 \end{align*}

(ii)

\begin{align*} T & = S + P - 1 \end{align*}

(iii)

\begin{align*} T & = S + P - 1 \\ 364 & = 169 + P - 1 \\ 364 - 169 + 1 & = P \\ 196 & = P \end{align*}

(iv)

\begin{align*} \sqrt{112} & = 10.583 \\ \\ 112 & \text{ is not a perfect square} \end{align*}

(v)

\begin{align*} T & = S + P - 1 \\ & = n^2 + (n + 1)^2 - 1 \\ & = n^2 + (n)^2 + 2(n)(1) + (1)^2 - 1 \\ & = n^2 + n^2 + 2n + 1 - 1 \\ & = 2n^2 + 2n \\ \\ \text{Let } 4442 & = 2n^2 + 2n \\ 0 & = 2n^2 + 2n - 4442 \\ 0 & = n^2 + n - 2221 \\ \\ n & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-1 \pm \sqrt{(1)^2 - 4(1)(-2221)} \over 2(1)} \\ & = {-1 \pm \sqrt{8885} \over 2} \\ & = 46.63 \text{ or } -47.63 \\ \\ \text{Since } n \text{ is not an} & \text{ integer, }4442 \text{ cannot appear in the } T \text{ column} \end{align*}

(a)

\begin{align*} p & = 4, q = 5 \\ \\ \text{As no. of sides increase by } & 1, \text{ no. of diagonals increase by 1} \end{align*}

(b)

\begin{align*} r & = 9 + 5 = 14 \\ \\ s & = 14 + 6 = 20 \end{align*}

(c)

\begin{align*} v & = n - 3 \end{align*}

(d)(i)

\begin{align*} 0 & = {0 \times 3 \over 2} \\ 2 & = {4 \times 1 \over 2} \\ 5 & = {2 \times 5 \over 2} \\ 9 & = {3 \times 6 \over 2} \\ \\ d & = {vn \over 2} \end{align*}

(d)(ii)

\begin{align*} d & = {vn \over 2} \\ \\ \text{Since } & v = n - 3, \\ d & = { (n - 3)n \over 2} \\ d & = { n(n - 3) \over 2 } \end{align*}

(d)(iii)

\begin{align*} d & = { n(n - 3) \over 2 } \\ & = {30(30 - 3) \over 2} \\ & = 405 \end{align*}

(i)

\begin{align*} p & = {256 \over 2} = 128 \end{align*}

(ii)

\begin{align*} 1024 & = 2^{10} = 2^{11 - 1} \\ 512 & = 2^{9} = 2^{11 - 2} \\ 256 & = 2^8 = 2^{11 - 3} \\ \\ T_n & = 2^{11 - n} \end{align*}

(iii)

\begin{align*} T_k & = 2^{11 - k} \\ {1 \over 4} & = 2^{11 - k} \\ {1 \over 2^2} & = 2^{11 - k} \\ 2^{-2} & = 2^{11 - k} \phantom{000000} \left[ a^{-n} = {1 \over a^n} \right] \\ \\ \therefore -2 & = 11 - k \\ k & = 11 + 2 \\ k & = 13 \end{align*}