\begin{align*} 3y & = 4x - 15 \\ y & = {4 \over 3}x - 5 \phantom{000} \text{--- (1)} \\ \\ 8x^2 - 27y^2 & = 45 \phantom{000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ 8x^2 - 27 \left({4 \over 3}x - 5\right)^2 & = 45 \\ 8x^2 - 27 \left[ \left({4 \over 3}x\right)^2 - 2 \left({4 \over 3}x\right)(5) + (5)^2 \right] & = 45 \\ 8x^2 - 27 \left( {16 \over 9}x^2 - {40 \over 3} x + 25 \right) & = 45 \\ 8x^2 - 48x^2 + 360x - 675 & = 45 \\ -40x^2 + 360x - 720 & = 0 \\ 40x^2 - 360x + 720 & =0 \\ x^2 - 9x + 18 & = 0 \\ (x - 3)(x - 6) & = 0 \\ \\ x - 3 = 0 \phantom{0} & \text{ or } \phantom{0} x - 6 = 0 \\ x =3 \phantom{0} & \phantom{0000000} x = 6 \\ \\ \text{Substitute } & x = 3 \text{ into (1),} \\ y & = {4 \over 3}(3) - 5 \\ y & = -1 \\ \\ \text{Substitute } & x = 6 \text{ into (2),} \\ y & = {4 \over 3}(6) - 5 \\ y & = 3 \\ \\ \therefore & \phantom{.} A(3, -1), B(6, 3) \end{align*}

(a)

\begin{align*} {2 \over 2x - 7} - {1 \over x} & = 6 \\ {2x \over x(2x - 7)} - {2x - 7 \over x(2x - 7)} & = 6 \\ {2x - (2x - 7) \over x(2x - 7)} & = 6 \\ {2x - 2x + 7 \over x(2x - 7)} & = 6 \\ {7 \over x(2x - 7)} & = 6 \\ 7 & = 6x(2x - 7) \\ 7 & = 12x^2 - 42x \\ 0 & = 12x^2 - 42x - 7 \\ 0 & = x^2 - {7 \over 2}x - {7 \over 12} \\ 0 & = \left(x - {7 \over 4}\right)^2 - \left(7 \over 4\right)^2 - {7 \over 12} \\ 0 & = \left(x - {7 \over 4}\right)^2 - {175 \over 48} \\ {175 \over 48} & = \left(x - {7 \over 4}\right)^2 \\ \pm \sqrt{175 \over 48} & = x - {7 \over 4} \\ \pm 1.9094 & = x - 1.75 \\ \\ x & = \pm 1.9094 + 1.75 \\ x & = 3.6594 \text{ or } -0.1594 \\ x & \approx 3.66 \text{ or } -0.159 \end{align*}

(b)

\begin{align*} {1 \over 4}x (x + 15) & = (3 - x)^2 \\ {1 \over 4}x^2 + {15 \over 4}x & = (3)^2 - 2(3)(x) + (x)^2 \\ {1 \over 4}x^2 + {15 \over 4}x & = 9 - 6x + x^2 \\ x^2 + 15x & = 36 - 24x + 4x^2 \\ 0 & = 3x^2 - 39x + 36 \\ 0 & = x^2 - 13x + 12 \\ 0 & = (x - 1)(x - 12) \\ \\ x - 1 = 0 \phantom{0} & \text{ or } \phantom{0} x - 12 = 0 \\ x = 1 \phantom{0} & \phantom{00000000} x = 12 \end{align*}

\begin{align*} p^2 x^2 - (p + 2)x + 1 & =0 \\ p^2 x^2 + (- p - 2)x + 1 & =0 \\ \\ b^2 - 4ac & = (-p - 2)^2 - 4(p^2)(1) \\ & = (-p)^2 - 2(-p)(2) + (2)^2 - 4p^2 \\ & = p^2 + 4p + 4 - 4p^2 \\ & = -3p^2 + 4p + 4 \\ \\ b^2 - 4ac & = 0 \phantom{00000} [\text{Two real and equal roots}] \\ -3p^2 + 4p + 4 & = 0 \\ 3p^2 - 4p - 4 & = 0 \\ (3p + 2)(p - 2) & = 0 \\ \\ 3p + 2 = 0 \phantom{00} & \text{ or } \phantom{0} p - 2 = 0 \\ 3p = -2 \phantom{(} & \phantom{0000000} p = 2 \\ p = -{2 \over 3} & \end{align*}

\begin{align*} -{1 \over 3}x^2 - & 2kx + 42 - 27k \\ \\ b^2 - 4ac & = (-2k)^2 - 4\left(-{1 \over 3}\right)(42 - 27k) \\ & = 4k^2 + {4 \over 3}(42 - 27k) \\ & = 4k^2 + 56 - 36k \\ & = 4k^2 - 36k + 56 \\ \\ b^2 - 4ac & < 0 \phantom{00000} [\text{No real roots}] \\ 4k^2 - 36k + 56 & < 0 \\ k^2 - 9k + 14 & < 0 \\ (k - 2)(k - 7) & < 0 \end{align*}

$$ 2 < k < 7 $$

(i)

\begin{align*} y & = px \phantom{000} \text{--- (1)} \\ y & = (1 - q)x^2 - 9 \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ px & = (1 - q)x^2 - 9 \\ 0 & = (1 - q)x^2 - px - 9 \\ \\ b^2 - 4ac & = (-p)^2 - 4(1 - q)(-9) \\ & = p^2 + 36(1 - q) \\ & = p^2 + 36 - 36q \\ \\ b^2 - 4ac & \ge 0 \phantom{00000} [\text{Real and equal/distinct roots since line meets curve}] \\ p^2 + 36 - 36q & \ge 0 \\ p^2 + 36 & \ge 36q \end{align*}

(ii)

$$ p = 1, q = 1 $$

(i)

\begin{align*} y & = 4x^2 + mx + m -3 \\ \\ \text{Let } & y = 0, \\ 0 & = 4x^2 + mx + m - 3 \\ \\ b^2 - 4ac & = (m)^2 - 4(4)(m - 3) \\ & = m^2 - 16(m - 3) \\ & = m^2 - 16m + 48 \\ \\ b^2 - 4ac & = 0 \phantom{00000} [\text{Two real and equal roots, since } x \text{-axis is tangent to curve}] \\ m^2 - 16m + 48 & = 0 \\ (m - 4)(m - 12) & = 0 \\ \\ m - 4 = 0 \phantom{0} & \text{ or } \phantom{0} m - 12 = 0 \\ m = 4 \phantom{0} & \phantom{00000000} m = 12 \end{align*}

(ii)

\begin{align*} y & = 4x^2 + mx + m -3 \\ \\ \text{Let } & x = 0, \\ y & = 4(0)^2 + m(0) + m -3 \\ y & = m - 3 \\ \\ m - 3 & > 0 \phantom{00000} [y \text{-intercept is positive}] \\ m & > 3 \end{align*}

(i)

\begin{align*} y & = (3)x^2 + 6x + 4(3) - 4 \\ y & = 3x^2 + 6x + 8 \phantom{000} \text{--- (1)} \\ \\ y & > 17 \\ 3x^2 + 6x + 8 & > 17 \\ 3x^2 + 6x - 9 & > 0 \\ x^2 + 2x - 3 & > 0 \\ (x + 3)(x - 1) & > 0 \end{align*}

$$ x < -3 \text{ or } x > 1 $$

(ii)

\begin{align*} y & = \left(1 \over 2\right) x^2 + 6x + 4 \left(1 \over 2\right) - 4 \\ y & = {1 \over 2}x^2 + 6x - 2 \phantom{000} \text{--- (1)} \\ \\ y & = 8x - 4 \phantom{000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(2) into (1),} \\ 8x - 4 & = {1 \over 2}x^2 + 6x - 2 \\ 0 & = {1 \over 2}x^2 -2x + 2 \\ 0 & = x^2 - 4x + 4 \\ \\ b^2 - 4ac & = (-4)^2 - 4(1)(4) \\ & =0 \\ \\ \text{Since } b^2 - 4ac = 0, & \text{ line is tangent to the curve} \end{align*}

(iii)

\begin{align*} y & = a x^2 + 6x + 4a - 4 \phantom{000} \text{--- (1)} \\ \\ y & = 8x - 4 \phantom{000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(2) into (1),} \\ 8x - 4 & = ax^2 + 6x + 4a - 4 \\ 0 & = ax^2 - 2x + 4a \\ \\ b^2 - 4ac & = (-2)^2 - 4(a)(4a) \\ & = 4 - 16a^2 \\ \\ b^2 - 4ac & = 0 \\ 4- 16a^2 & = 0 \\ -16a^2 & = -4 \\ a^2 & = {-4 \over -16} \\ a^2 & = {1 \over 4} \\ a & = \pm \sqrt{1 \over 4} \\ a & = \pm {1 \over 2} \\ \\ \therefore \text{Other value of } a & = -{1 \over 2} \end{align*}

(i)

\begin{align*} y & = 4x^2 + (2m + 1)x + c + 9 \phantom{000} \text{--- (1)} \\ \\ y & = mx + c \phantom{000} \text{--- (2)} \\ \\ \text{Substitute } & \text{(2) into (1),} \\ mx + c & = 4x^2 + (2m + 1)x + c + 9 \\ mx + c & = 4x^2 + 2mx + x + c + 9 \\ 0 & = 4x^2 + mx + x + 9 \\ 0 & = 4x^2 + (m + 1)x + 9 \\ \\ b^2 - 4ac & = (m + 1)^2 - 4(4)(9) \\ & = (m)^2 + 2(m)(1) + (1)^2 - 144 \\ & = m^2 + 2m + 1 - 144 \\ & = m^2 + 2m - 143 \\ \\ b^2 - 4ac & = 0 \phantom{00000} [\text{Two real and equal roots since line is tangent to curve}] \\ m^2 + 2m - 143 & = 0 \\ (m + 13)(m - 11) & = 0 \\ \\ m + 13 = 0 \phantom{000.} & \text{ or } \phantom{0} m - 11 = 0 \\ m = - 13 \phantom{0} & \phantom{00000000} m = 11 \\ \\ \therefore \text{Positive value of } m & = 11 \end{align*}

(ii)

\begin{align*} \text{Using } 0 & = 4x^2 + (m + 1)x + 9, \phantom{00000} [\text{from (i)}] \\ \\ 0 & = 4x^2 + (11 + 1)x + 9 \\ 0 & = 4x^2 + 12x + 9 \\ 0 & = (2x)^2 + 2(2x)(3) + (3)^2 \\ 0 & = (2x + 3)^2 \\ 0 & = 2x + 3 \\ -3 & = 2x \\ -{3 \over 2} & = x \\ -1.5 & = x \\ \\ \\ y & = 4x^2 + [2(11) + 1]x + c + 9 \\ y & = 4x^2 + 23x + c + 9 \\ \\ \text{Using } & (-1, -8), \\ -8 & = 4(-1)^2 + 23(-1) + c + 9 \\ -8 & = 4 - 23 + c + 9 \\ 2 & = c \\ \\ \text{Equation of curve: } y & = 4x^2 + 23x + 11 \\ \\ \text{When } & x = -1.5, \\ y & = 4(-1.5)^2 + 23(-1.5) + 11 \\ y & = -14.5 \\ \\ \therefore & \phantom{.} P (-1.5, -14.5) \end{align*}

(iii)

$L$ has gradient -13 (i.e. $m = -13$)

\begin{align*} \text{Height} & \le 2.62 \text{ m} \\ \\ y & \le 2.62 \\ -0.04x^2 + 0.44x + 1.5 & \le 2.62 \\ 0 & \le 0.04x^2 - 0.44x + 1.12 \\ 0 & \le x^2 - 11x + 28 \phantom{00000} [\text{Divide each term by 0.04}] \\ 0 & \le (x - 4)(x - 7) \end{align*}

$$ x \le 4 \text{ or } x \ge 7 $$ $$ \text{Since horizontal distance} \ge 0 \text{ m}, 0 \le x \le 4 \text{ or } x \ge 7 $$

\begin{align*} \text{Height} & = 0.875 \text{ m} \\ y & = 0.875 \\ -0.125(x - 4)^2 + 2 & = 0.875 \\ -0.125(x - 4)^2 & = -1.125 \\ 0.125(x - 4)^2 & = 1.125 \\ (x - 4)^2 & = {1.125 \over 0.125} \\ (x - 4)^2 & = 9 \\ x - 4 & = \pm \sqrt{9} \\ x - 4 & = \pm 3 \\ x & = 3 + 4 \text{ or } - 3 + 4 \\ x & = 7 \text{ or } 1 \\ \\ \therefore \text{Horizontal distance} & = 1 \text{ m or 7 m} \end{align*}