\begin{align*} 2x - y & = 3 \\ -y & = 3 - 2x \\ y & = 2x - 3 \phantom{0} \text{--- (1)} \\ \\ xy & = x + 2y \phantom{0} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ x(2x - 3) & = x + 2(2x - 3) \\ 2x^2 - 3x & = x + 4x - 6 \\ 2x^2 - 3x & = 5x - 6 \\ 2x^2 - 8x + 6 & = 0 \\ x^2 - 4x + 3 & = 0 \\ (x - 1)(x - 3) & = 0 \end{align*} \begin{align*} x - 1 & = 0 && \text{ or } & x - 3 & = 0 \\ x & = 1 &&& x & = 3 \\ \\ \text{Substitute } & \text{into (1),} &&& \text{Substitute } & \text{into (1),} \\ y & = 2(1) - 3 &&& y & = 2(3) - 3 \\ y & = -1 &&& y & = 3 \end{align*} $$ \therefore x = 1, y = -1 \text{ or } x = 3, y = 3 $$

\begin{align*} y - x + 3 & = 0 \\ y & = x - 3 \phantom{0} \text{--- (1)} \\ \\ y & = x^2 - 4x + 1 \phantom{0} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ x - 3 & = x^2 - 4x + 1 \\ 0 & = x^2 - 5x + 4 \\ 0 & = (x - 1)(x - 4) \end{align*} \begin{align*} x - 1 & = 0 && \text{ or } & x - 4 & = 0 \\ x & = 1 &&& x & = 4 \\ \\ \text{Substitute } & \text{into (1),} &&& \text{Substitute } & \text{into (1),} \\ y & = 1 - 3 &&& y & = 4 - 3 \\ y & = -2 &&& y & = 1 \\ \\ \therefore & \phantom{.} (1, -2) &&& \therefore & \phantom{.} (4, 1) \end{align*}

\begin{align*} \text{Let length} & = x \text{ cm}, \text{ breadth} = y \text{ cm} \\ \\ \text{Area} & = (x)(y) \\ 154 & = xy \phantom{0} \text{--- (1)} \\ \\ \text{Perimeter} & = 2x + 2y \\ 50 & = 2x + 2y \\ -2x & = 2y - 50 \\ 2x & = 50 - 2y \\ x & = 25 - y \phantom{0} \text{--- (2)} \\ \\ \text{Substitute } & \text{(2) into (1),} \\ 154 & = (25 - y)y \\ 154 & = 25y - y^2 \\ y^2 - 25y + 154 & = 0 \\ (y - 11)(y - 14) & = 0 \end{align*} \begin{align*} y - 11 & = 0 && \text{ or } & y - 14 & = 0 \\ y & = 11 &&& y & = 14 \\ \\ \text{Substitute } & \text{into (2),} &&& \text{Substitute } & \text{into (2),} \\ x & = 25 - 11 &&& x & = 25 - 14 \\ x & = 14 &&& x & = 11 \end{align*} $$ \therefore \text{Dimensions: 14 cm by 11 cm} $$

(a)

\begin{align*} \text{Substitute } y = x + {1 \over 2} & \text{ into } y = 2x^2 + 5x, \\ x + {1 \over 2} & = 2x^2 + 5x \\ 0 & = 2x^2 + 4x - {1 \over 2} \\ 0 & = 4x^2 + 8x - 1 \\ \\ b^2 - 4ac & = (8)^2 - 4(4)(-1) \\ & = 80 > 0 \\ \\ \implies 0 = 4x^2 + 8x - 1 & \text{ has two real and distinct roots} \\ \\ \therefore \text{Line meets the } & \text{curve at two distinct points} \end{align*}

(b)

\begin{align*} \text{Substitute } y = x + k & \text{ into } y = 2x^2 + 5x, \\ x + k & = 2x^2 + 5x \\ 0 & = 2x^2 + 4x - k \\ \\ b^2 - 4ac & = (4)^2 - 4(2)(-k) \\ & = 16 + 8k \\ \\ b^2 - 4ac & = 0 \phantom{000000} [\text{Line is tangent to curve, i.e. meets only once}] \\ 16 + 8k & = 0 \\ 8k & = -16 \\ k & = -2 \end{align*}

(c)

\begin{align*} \text{Substitute } y = -2x & \text{ into } y = 3x^2 + m, \\ -2x & = 3x^2 + m \\ 0 & = 3x^2 + 2x + m \\ \\ b^2 - 4ac & = (2)^2 - 4(3)(m) \\ & = 4 - 12m \\ \\ b^2 - 4ac & < 0 \phantom{000000} [\text{Line does not meet curve}] \\ 4 - 12m & < 0 \\ -12m & < - 4 \\ m & > {-4 \over -12} \\ m & > {1 \over 3} \end{align*}

(a)

\begin{align*} px^2 - 2x - 2 & =0 \\ \\ b^2 - 4ac & = (-2)^2 - 4(p)(-2) \\ & = 4 + 8p \\ \\ b^2 - 4ac & = 0 \\ 4 + 8p & = 0 \\ 8p & = -4 \\ p & = {-4 \over 8} \\ p & = -{1 \over 2} \end{align*}

(b)

\begin{align*} 4x^2 - 2x + 1 & = 2x - p \\ 4x^2 - 4x + 1 + p & = 0 \\ \\ b^2 - 4ac & = (-4)^2 - 4(4)(1 + p) \\ & = 16 - 16(1 + p) \\ & = 16 - 16 - 16p \\ & = -16p \\ \\ b^2 - 4ac & > 0 \\ -16p & > 0 \\ p & < {0 \over -16} \\ p & < 0 \end{align*}

(c)

\begin{align*} - x^2 + 3x - 2 + p & = 0 \\ \\ b^2 - 4ac & = (3)^2 - 4(-1)(p - 2) \\ & = 9 + 4(p - 2) \\ & = 9 + 4p - 8 \\ & = 4p + 1 \\ \\ b^2 - 4ac & \ge 0 \phantom{000000} [\text{1 or 2 real roots}] \\ 4p + 1 & \ge 0 \\ 4p & \ge -1 \\ p & \ge -{1 \over 4} \end{align*}

(d)

\begin{align*} -x^2 + 2x & < 5 + p \\ 0 & < x^2 - 2x + 5 + p \phantom{000} [ \implies x^2 - 2x + 5 + p \text{ must be positive} ] \\ \\ b^2 - 4ac & = (-2)^2 - 4(1)(5 + p) \\ & = 4 - 4(5 + p) \\ & = 4 - 20 - 4p \\ & = -16 - 4p \\ \\ b^2 - 4ac & < 0 \phantom{000} [\text{Graphically, } y = x^2 - 2x + 5 + p \text{ lies above } x \text{-axis, so it does not cut } x \text{-axis}] \\ -16 - 4p & < 0 \\ -4p & < 16 \\ p & > {16 \over -4} \\ p & > -4 \end{align*}

\begin{align*} \text{Substitute } & y = mx + 2 \text{ into } y = x^2 + x + 3, \\ mx + 2 & = x^2 + x + 3 \\ 0 & = x^2 + x - mx + 1 \\ 0 & = x^2 + (1 - m)x + 1 \\ \\ b^2 - 4ac & = (1 - m)^2 - 4(1)(1) \\ & = \underbrace{ (1)^2 - 2(1)(m) + (m)^2 }_{ (a - b)^2 = a^2 - 2ab + b^2 } - 4 \\ & = 1 - 2m + m^2 - 4 \\ & = m^2 - 2m - 3 \\ \\ b^2 - 4ac & < 0 \phantom{000000} [\text{Line does not intersect curve}] \\ m^2 - 2m - 3 & < 0 \phantom{0} \text{ (Shown)} \end{align*}

\begin{align*} R & = x (200 - 0.4x) \\ \\ \text{Let } & R = 20 \phantom{.} 000, \\ 20 \phantom{.} 000 & = x(200 - 0.4x) \\ 20 \phantom{.} 000 & = 200x - 0.4x^2 \\ 0 & = -0.4x^2 + 200x - 20 \phantom{.} 000 \\ \\ b^2 - 4ac & = (200)^2 - 4(-0.4)(- 20 \phantom{.} 000) \\ & = 8000 > 0 \\ \\ \implies \text{There are } & \text{two distinct roots for } 0 = -0.4x^2 + 200x - 20 \phantom{.} 000 \\ \\ \therefore \text{Jerry's busi} & \text{ness can generate a revenue of \$} 20 \phantom{.} 000 \end{align*}

(i)

\begin{align*} \text{Substitute } & y = 0 \text{ into } x^2 - 2mx + 4, \\ 0 & = x^2 - 2mx + 4 \\ \\ b^2 - 4ac & = (-2m)^2 - 4(1)(4) \\ & = 4m^2 - 16 \\ \\ b^2 - 4ac & \ge 0 \phantom{000000} [\text{Equation has 1 or 2 real roots}] \\ 4m^2 - 16 & \ge 0 \\ m^2 - 4 & \ge 0 \\ (m + 2)(m - 2) & \ge 0 \phantom{000000} [a^2 - b^2 = (a + b)(a - b)] \end{align*}

\begin{align*} m \le -2 \text{ or } m \ge 2 \end{align*}

(ii)

\begin{align*} \text{Substitute } & y = x + 3 \text{ into } x^2 - 2mx + 4, \\ x + 3 & = x^2 - 2mx + 4 \\ 0 & = x^2 - 2m x - x + 1 \\ 0 & = x^2 + (-2m -1)x + 1 \\ \\ b^2 - 4ac & = (-2m - 1)^2 - 4(1)(1) \\ & = \underbrace{ (-2m)^2 - 2(-2m)(1) + (1)^2 }_{ (a - b)^2 = a^2 - 2ab + b^2 } - 4 \\ & = 4m^2 + 4m + 1 - 4 \\ & = 4m^2 + 4m - 3 \\ \\ b^2 - 4ac & > 0 \phantom{000000} [\text{Equation has 2 real roots}] \\ 4m^2 + 4m - 3 & > 0 \\ (2m - 1)(2m + 3) & > 0 \end{align*}

\begin{align*} m & < - 1.5 \text{ or } m > 0.5 \end{align*}

(iii)

\begin{align*} b^2 - 4ac & = 4m^2 + 4m - 3 \phantom{00000000} [\text{From (ii)}] \\ \\ b^2 - 4ac & < 0 \phantom{000000} [\text{Equation has 2 real roots}] \\ 4m^2 + 4m - 3 & < 0 \\ (2m - 1)(2m + 3) & < 0 \end{align*}

\begin{align*} -1.5 & < m < 0.5 \end{align*}