(i)

\begin{align*} b^2 - 4ac & = (b)^2 - 4(2)(3b) \\ & = b^2 - 8(3b) \\ & = b^2 - 24b \\ \\ b^2 - 4ac & > 0 \phantom{00000} [\text{Two real and distinct roots}] \\ b^2 - 24b & > 0 \end{align*}

(ii)

\begin{align*} b^2 - 24b & > 0 \\ b(b - 24) & > 0 \\ \\ b & = 25 \end{align*}

(a)

\begin{align*} px^2 - 6x + p & = 0 \\ \\ b^2 - 4ac & = (-6)^2 - 4(p)(p) \\ & = 36 - 4p^2 \\ \\ b^2 - 4ac & = 0 \phantom{00000} [\text{Real and equal roots}] \\ 36 - 4p^2 & = 0 \\ -4p^2 & = -36 \\ p^2 & = {-36 \over -4} \\ p^2 & = 9 \\ p & = \pm \sqrt{9} \\ p & = \pm 3 \end{align*}

(b)

\begin{align*} 3x^2 + 2x - p & = 0 \\ \\ b^2 - 4ac & = (2)^2 - 4(3)(-p) \\ & = 4 - 12(-p) \\ & = 4 + 12p \\ \\ b^2 - 4ac & > 0 \phantom{00000} [\text{Real and distinct roots}] \\ 4 + 12p & > 0 \\ 12p & > - 4 \\ p & > {-4 \over 12} \\ p & > -{1 \over 3} \end{align*}

(c)

\begin{align*} 2x^2 + 3x +2p & = 0 \\ \\ b^2 - 4ac & = (3)^2 - 4(2)(2p) \\ & = 9 - 8(2p) \\ & = 9 - 16p \\ \\ b^2 - 4ac & \ge 0 \phantom{00000} [\text{Real roots - distinct or equal}] \\ 9 - 16p & \ge 0 \\ -16p & \ge -9 \\ 16p & \le 9 \\ p & \le {9 \over 16} \end{align*}

(d)

\begin{align*} px^2 - x - 4 & = 0 \\ \\ b^2 - 4ac & = (-1)^2 - 4(p)(-4) \\ & = 1 - 4p(-4) \\ & = 1 + 16p \\ \\ b^2 - 4ac & < 0 \phantom{00000} [\text{No real roots}] \\ 1 + 16p & < 0 \\ 16p & < -1 \\ p & < -{1 \over 16} \end{align*}

(a)

\begin{align*} y & = 2x^2 \phantom{000} \text{ --- (1)} \\ y & = 5x - 3 \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ 2x^2 & = 5x - 3 \\ 2x^2 - 5x + 3 & = 0 \\ \\ b^2 - 4ac & = (-5)^2 - 4(2)(3) \\ & = 1 \\ \\ \text{Since } b^2 - 4ac > 0, & \text{ the line and curve intersect at 2 points} \end{align*}

(b)

\begin{align*} y & = x^2 - 1 \phantom{000} \text{ --- (1)} \\ y & = 8x - 17 \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ x^2 - 1 & = 8x - 17 \\ x^2 - 8x + 16 & = 0 \\ \\ b^2 - 4ac & = (-8)^2 - 4(1)(16) \\ & = 0 \\ \\ \text{Since } b^2 - 4ac = 0, & \text{ the line and curve intersect at 1 point} \end{align*}

(c)

\begin{align*} y & = 3x^2 \phantom{000} \text{ --- (1)} \\ y & = 2x - 5 \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ 3x^2 & = 2x - 5 \\ 3x^2 - 2x + 5 & = 0 \\ \\ b^2 - 4ac & = (-2)^2 - 4(3)(5) \\ & = -56 \\ \\ \text{Since } b^2 - 4ac < 0, & \text{ the line and curve do not intersect} \end{align*}

\begin{align*} y & = -6x - 5 \phantom{000} \text{ --- (1)} \\ y & = mx^2 + 2 \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ -6x - 5 & = mx^2 + 2 \\ 0 & = mx^2 + 6x + 2 + 5 \\ 0 & = mx^2 + 6x + 7 \\ \\ b^2 - 4ac & = (6)^2 - 4(m)(7) \\ & = 36 - 28m \\ \\ \text{Since } m & < 1, \\ 28m & < 28 \\ -28m & > -28 \\ -28m + 36 & > -28 + 36 \\ 36 - 28m & > 8 \\ \implies b^2 - 4ac & > 8 \\ \\ \text{Since } b^2 - 4ac > 0 \text{ for } m < 1, & \text{ the line and curve intersects at two points} \end{align*}

(a)

\begin{align*} x^2 + (p + 2)x + 4(p - 1) & = 0 \\ x^2 + (p + 2)x + (4p - 4) & = 0 \\ \\ b^2 - 4ac & = (p + 2)^2 - 4(1)(4p - 4) \\ & = (p)^2 + 2(p)(2) + (2)^2 - 4(4p - 4) \\ & = p^2 + 4p + 4 - 16p + 16 \\ & = p^2 - 12p + 20 \\ \\ b^2 - 4ac & = 0 \phantom{00000} [\text{Real and equal roots}] \\ p^2 - 12p + 20 & = 0 \\ (p - 2)(p - 10) & = 0 \\ \\ p - 2 = 0 \phantom{0} & \text{ or } \phantom{0} p - 10 = 0 \\ p = 2 \phantom{0} & \phantom{00000000} p = 10 \end{align*}

(b)

\begin{align*} 3x^2 & = 2x + p - 1 \\ 0 & = -3x^2 + 2x + (p - 1) \\ \\ b^2 - 4ac & = (2)^2 - 4(-3)(p - 1) \\ & = 4 + 12(p - 1) \\ & = 4 + 12p - 12 \\ & = 12p - 8 \\ \\ b^2 - 4ac & > 0 \phantom{00000} [\text{Real and distinct roots}] \\ 12p - 8 & > 0 \\ 12p & > 8 \\ p & > {8 \over 12} \\ p & > {2 \over 3} \end{align*}

(c)

\begin{align*} x^2 + p^2 & = 3px - 5 \\ x^2 - 3px + p^2 + 5 & = 0 \\ \\ b^2 - 4ac & = (-3p)^2 - 4(1)(p^2 + 5) \\ & = 9p^2 - 4(p^2 + 5) \\ & = 9p^2 - 4p^2 - 20 \\ & = 5p^2 - 20 \\ \\ b^2 - 4ac & = 0 \phantom{00000} [\text{Real and equal/repeated roots}] \\ 5p^2 - 20 & = 0 \\ 5p^2 & = 20 \\ p^2 & = {20 \over 5} \\ p^2 & = 4 \\ p & = \pm \sqrt{4} \\ p & = \pm 2 \end{align*}

(d)

\begin{align*} (x + 1)(2x - 1) & = p - 2 \\ 2x^2 - x + 2x - 1 & = p - 2 \\ 2x^2 + x - 1 & = p - 2 \\ 2x^2 + x + 1 - p & = 0 \\ \\ b^2 - 4ac & = (1)^2 - 4(2)(1 - p) \\ & = 1 - 8(1 - p) \\ & = 1 - 8 + 8p \\ & = 8p - 7 \\ \\ b^2 - 4ac & > 0 \phantom{00000} [\text{Real and distinct/unequal roots}] \\ 8p - 7 & > 0 \\ 8p & > 7 \\ p & > {7 \over 8} \end{align*}

(e)

\begin{align*} p(x + 1)(x - 3) & = x - 4p - 2 \\ p(x^2 - 3x + x - 3) & = x - 4p - 2 \\ p(x^2 - 2x - 3) & = x - 4p - 2 \\ px^2 - 2px - 3p & = x - 4p - 2 \\ px^2 - 2px -x + p + 2 & = 0 \\ px^2 + (-2p - 1)x + (p + 2) & = 0 \\ \\ b^2 - 4ac & = (-2p - 1)^2 - 4(p)(p + 2) \\ & = (-2p)^2 - 2(-2p)(1) + (1)^2 - 4p(p + 2) \\ & = 4p^2 + 4p + 1 - 4p^2 - 8p \\ & = -4p + 1 \\ \\ b^2 - 4ac & < 0 \phantom{00000} [\text{No real roots}] \\ -4p + 1 & < 0 \\ -4p & < - 1 \\ 4p & > 1 \\ p & > {1 \over 4} \end{align*}

\begin{align*} x^2 + (p + 1)x & = 5 - 2p \\ x^2 + (p + 1)x + (2p - 5) & = 0 \\ \\ b^2 - 4ac & = (p + 1)^2 - 4(1)(2p - 5) \\ & = (p)^2 + 2(p)(1) + (1)^2 - 4(2p - 5) \\ & = p^2 + 2p + 1 - 8p + 20 \\ & = p^2 - 6p + 21 \\ & = \left(p - {6 \over 2}\right)^2 - \left(6 \over 2\right)^2 + 21 \\ & = (p - 3)^2 - 9 + 21 \\ & = (p - 3)^2 + 12 \\ \\ \text{For all real values} & \text{ of } p, \\ (p - 3)^2 & \ge 0 \\ (p - 3)^2 + 12 & \ge 12 \\ \implies b^2 - 4ac & \ge 12 \\ \\ \text{Since } b^2 - 4ac > 0 \text{ for all real} & \text{ values of } p, \text{ the roots of the equation are real} \end{align*}

\begin{align*} y & = 2x^2 + x + 2p - 1 \phantom{000} \text{ --- (1)} \\ \\ y + 2x & = p \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ 2x^2 + x + 2p - 1 + 2x & = p \\ 2x^2 + 3x + p - 1 & = 0 \\ \\ b^2 - 4ac & = (3)^2 - 4(2)(p - 1) \\ & = 9 - 8(p - 1) \\ & = 9 - 8p + 8 \\ & = 17 - 8p \\ \\ b^2 - 4ac & > 0 \phantom{00000} [\text{Intersect at two distinct points}] \\ 17 - 8p & > 0 \\ -8p & > -17 \\ 8p & < 17 \\ p & < {17 \over 8} \end{align*}

(a)

\begin{align} y & = kx - 5 \phantom{000} \text{ --- (1)} \\ \\ x^2 & = 2y + 1 \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ x^2 & = 2(kx - 5) + 1 \\ x^2 & = 2kx - 10 + 1 \\ x^2 & = 2kx - 9 \\ x^2 - 2kx + 9 & = 0 \\ \\ [a = 1, b & = -2k, c = 9] \\ \\ b^2 - 4ac & = 0 \\ (-2k)^2 - 4(1)(9) & = 0 \\ 4k^2 - 36 & = 0 \\ 4k^2 & = 36 \\ k^2 & = {36 \over 4} \\ & = 9 \\ k & = \pm \sqrt{9} \\ & = \pm 3 \end{align}

(b)

\begin{align} x + 3y & = k - 1 \\ 3y & = k - 1 - x \\ y & = {k - 1 - x \over 3} \phantom{000} \text{ --- (1)} \\ \\ y^2 & = 2x + 5 \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ \left(k - 1 - x \over 3\right)^2 & = 2x + 5 \\ {(k - 1 - x)^2 \over 9} & = 2x + 5 \\ (k - 1 - x)^2 & = 18x + 45 \\ (k - 1 - x)(k - 1 - x) & = 18x + 45 \\ k^2 - k - kx -k + 1 + x - kx + x + x^2 & = 18x + 45 \\ x^2 + 2x - 2kx + k^2 - 2k + 1 & = 18x + 45 \\ x^2 - 16x - 2kx + k^2 - 2k - 44 & = 0 \\ x^2 + (-16 - 2k)x + (k^2 - 2k - 44) & = 0 \\ \\ [a = 1, b & = -16 - 2k, c = k^2 - 2k - 44] \\ \\ \\ b^2 - 4ac & \ge 0 \phantom{00000} [\text{1 or 2 points of intersection}] \\ (-16 - 2k)^2 - 4(1)(k^2 - 2k - 44) & \ge 0 \\ (-16)^2 - 2(-16)(2k) + (2k)^2 - 4(k^2 - 2k - 44) & \ge 0 \\ 256 + 64k + 4k^2 - 4k^2 + 8k + 176 & \ge 0 \\ 72k + 432 & \ge 0 \\ 72k & \ge -432 \\ k & \ge {-432 \over 72} \\ k & \ge -6 \end{align}

(c)

\begin{align} y & = kx + 2 \phantom{000} \text{ --- (1)} \\ \\ y^2 & = 8x - x^2 \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ (kx + 2)^2 & = 8x - x^2 \\ (kx)^2 + 2(kx)(2) + (2)^2 & = 8x - x^2 \\ k^2 x^2 + 4kx + 4 & = 8x - x^2 \\ k^2 x^2 + x^2 + 4kx - 8x + 4 & = 0 \\ (k^2 + 1)x^2 + (4k - 8)x + 4 & = 0 \\ \\ [a = k^2 + 1, b & = 4k - 8, c = 4] \\ \\ b^2 - 4ac & > 0 \\ (4k - 8)^2 - 4(k^2 + 1)(4) & > 0 \\ (4k)^2 - 2(4k)(8) + (8)^2 - 16(k^2 + 1) & > 0 \\ 16k^2 - 64k + 64 - 16k^2 - 16 & > 0 \\ -64k + 48 & > 0 \\ -64k & > -48 \\ 64k & < 48 \\ k & < {48 \over 64} \\ k & < {3 \over 4} \end{align}

(d)

\begin{align} y & = x + k - 1 \phantom{000} \text{ --- (1)} \\ \\ (y - 1)^2 & = 4x \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ (x + k - 1 - 1)^2 & = 4x \\ (x + k - 2)^2 & = 4x \\ (x + k - 2)(x + k - 2) & = 4x \\ x^2 + kx - 2x + kx + k^2 - 2k - 2x -2k + 4 & = 4x \\ x^2 + 2kx - 4x + k^2 - 4k + 4 & = 4x \\ x^2 + 2kx - 8x + k^2 - 4k + 4 & = 0 \\ x^2 + (2k - 8)x + (k^2 - 4k + 4) & = 0 \\ \\ [a = 1, b & = 2k - 8, c = k^2 - 4k + 4] \\ \\ b^2 - 4ac & < 0 \\ (2k - 8)^2 - 4(1)(k^2 - 4k + 4) & < 0 \\ (2k)^2 - 2(2k)(8) + (8)^2 - 4(k^2 - 4k + 4) & < 0 \\ 4k^2 - 32k + 64 - 4k^2 + 16k - 16 & < 0 \\ -16k + 48 & < 0 \\ -16k & < - 48 \\ 16k & > 48 \\ k & > {48 \over 16} \\ k & > 3 \end{align}

(i)

\begin{align*} y & = x^2 \phantom{000} \text{ --- (1)} \\ y & = mx - m + 1 \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ x^2 & = mx - m + 1 \\ 0 & = -x^2 + mx + (1 - m) \\ \\ b^2 - 4ac & = (m)^2 - 4(-1)(1 - m) \\ & = m^2 + 4(1 - m) \\ & = m^2 + 4 - 4m \\ & = m^2 - 4m + 4 \\ & = (m)^2 - 2(m)(2) + (2)^2 \\ & = (m - 2)^2 \\ \\ b^2 - 4ac & = 0 \phantom{00000} [\text{Intersect at one point}] \\ (m - 2)^2 & = 0 \\ m - 2 & = \sqrt{0} \\ m - 2 & = 0 \\ m & = 2 \end{align*}

(ii)

If $m = 2$, the line is tangent to the curve at the point (1, 1).

\begin{align} 2x^2 + p & = 2(x - 1) \\ 2x^2 + p & = 2x - 2 \\ 2x^2 - 2x + p + 2 & = 0 \\ \\ [a = 2, b & = -2, c = p + 2] \\ \\ b^2 - 4ac & = (-2)^2 - 4(2)(p + 2) \\ & = 4 - 8(p + 2) \\ & = 4 - 8p - 16 \\ & = -8p - 12 \\ & = -{24 \over 3}p - 12 \\ & = -12 \left( {2 \over 3}p + 1 \right) \\ \\ \text{If } p & > -{3 \over 2}, \\ {2 \over 3}p & > -{3 \over 2} \times {2 \over 3} \\ {2 \over 3}p & > -1 \\ {2 \over 3}p + 1 & > -1 + 1 \\ {2 \over 3}p + 1 & > 0 \\ -12 \left({2 \over 3}p + 1\right) & < -12(0) \\ -12 \left({2 \over 3}p + 1\right) & < 0 \\ \implies b^2 - 4ac & < 0 \\ \\ \therefore\text{Equation has } & \text{no real roots for } p > -{3 \over 2} \end{align}

(i)

\begin{align*} y & = (k + 3)x^2 - 3x \phantom{000} \text{ --- (1)} \\ y & = x + k \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ (k + 3)x^2 - 3x & = x + k \\ (k + 3)x^2 - 4x - k & = 0 \\ \\ b^2 - 4ac & = (-4)^2 - 4(k + 3)(-k) \\ & = 16 - (4k + 12)(-k) \\ & = 16 - (-4k^2 - 12k) \\ & = 16 + 4k^2 + 12k \\ & = 4k^2 + 12k + 16 \\ & = 4(k^2 + 3k) + 16 \\ & = 4 \left[ \left(k + {3 \over 2}\right)^2 - \left(3 \over 2\right)^2 \right] + 16 \\ & = 4 [ (k + 1.5)^2 - 2.25] + 16 \\ & = 4(k + 1.5)^2 - 9 + 16 \\ & = 4(k + 1.5)^2 + 7 \\ \\ \text{For all real} & \text{ values of } k, \\ (k + 1.5)^2 & \ge 0 \\ 4(k + 1.5)^2 & \ge 4(0) \\ 4(k + 1.5)^2 & \ge 0 \\ 4(k + 1.5)^2 + 7 & \ge 7 \\ \\ \text{Since } b^2 - 4ac > 0 \text { for all } & \text{real values of } k, \text{ the line will meet the curve at two distinct points} \\ \therefore \text{There are no values of } & k \text{ such that the curve lies completely above the line} \end{align*}

(ii)

Since $b^2 - 4ac > 0$ for all real values of $k$, the line will meet the curve at two distinct points.

\begin{align*} f(x) & = ax^2 + c \phantom{00000} [\text{Generic quadratic function}] \\ \\ \text{Since the graph has} & \text{ a maximum point, } a < 0 \\ \\ \text{Let } a = -1, \phantom{0} f(x) & = -x^2 + c \\ \\ \\ f(x) & = 10 \\ -x^2 + c & = 10 \\ 0 & = x^2 + 10 - c \\ 0 & = x^2 + 0x + (10 - c) \\ \\ b^2 - 4ac & = (0)^2 - 4(1)(10 - c) \\ & = 0 - 4(10 - c) \\ & = 0 - 40 + 4c \\ & = 4c - 40 \\ \\ b^2 - 4ac & < 0 \phantom{00000} [\text{No real roots for } f(x) = 10] \\ 4c - 40 & < 0 \\ 4c & < 40 \\ c & < {40 \over 4} \\ c & < 10 \\ \\ \therefore \text{Let } c = 9, \phantom{0} f(x) & = -x^2 + 9 \end{align*}
(Check out the sketch of the graph on Desmos, where you can see that the quadratic function satisfies the properties listed in the question)

\begin{align*} h & = -11t^2 + 20t + 1 \\ \\ \text{When } & h = 6, \\ 6 & = -11t^2 + 20t + 1 \\ 0 & = -11t^2 + 20t - 5 \\ \\ b^2 - 4ac & = (20)^2 - 4(-11)(-5) \\ & = 180 \\ \\ \text{Since } b^2 - 4ac > 0, & \text{ the equation has two real roots} \\ \therefore \text{It is possible} & \text{ for John to catch the key} \end{align*}

(i)

\begin{align} \text{When } & C = 10, \\ 10 & = 1.2n^2 - 14.4n + 53.7 \\ 0 & = 1.2n^2 -14.4n + 43.7 \\ \\ [a = 1.2, b & = -14.4, c = 43.7] \\ \\ \\ b^2 - 4ac & = (-14.4)^2 - 4(1.2)(43.7) \\ & = -2.4 < 0 \\ \\ \text{Since } b^2 - 4ac < 0, & \phantom{0} \text{equation has no real solutions}. \\ \\ \therefore \text{It is not possible to have } & \text{a cost of production of 10 thousand dollars.} \end{align}

(ii)

\begin{align} \text{When } & C = 80, \\ 80 & = 1.2n^2 - 14.4n + 53.7 \\ 0 & = 1.2n^2 -14.4n - 26.3 \\ \\ [a = 1.2, b & = -14.4, c = -26.3] \\ \\ \\ b^2 - 4ac & = (-14.4)^2 - 4(1.2)(-26.3) \\ & = 333.6 > 0 \\ \\ \text{Since } b^2 - 4ac > 0, & \phantom{0} \text{equation has 2 real solutions}. \\ \\ \therefore \text{It is possible to have } & \text{the cost of production to reach 80 thousand dollars.} \end{align}

(iii)

\begin{align} C & = 1.2n^2 - 14.4n + 53.7 \\ C & = 1.2(n^2 - 12n) + 53.7 \\ C & = 1.2 \left[ n^2 - 12n + \left(12 \over 2\right)^2 - \left(12 \over 2\right)^2 \right] + 53.7 \phantom{000000} [\text{Complete the square}] \\ C & = 1.2 ( n^2 - 12n + 6^2 - 36 ) + 53.7 \\ C & = 1.2 [ (n - 6)^2 - 36 ] + 53.7 \\ C & = 1.2 (n - 6)^2 - 43.2 + 53.7 \\ C & = 1.2 (n - 6)^2 + 10.5 \\ \\ \text{Minimum } & \text{value of } C = 10.5 \phantom{000000} [\text{when } x = 6] \\ \\ \implies \text{Mininum } & \text{cost of production} = 10.5 \text{ thousands dollars} \\ \\ \\ \text{For cost of} & \text{ production to be } \textbf{always} \text{ more than } x \text{ thousand dollars,} \\ \\ & \phantom{.} 0 < x < 10.5 \end{align}

(i)

\begin{align} y & = -0.5t^2 + 7t + k \\ \\ \text{When } t = 10 & \text{ and } y = 26, \\ 26 & = -0.5(10)^2 + 7(10) + k \\ 26 & = -50 + 70 + k \\ 26 + 50 - 70 & = k \\ 6 & = k \\ \\ y & = -0.5t^2 + 7t + 6 \\ \\ \text{When } & t = 0, \\ y & = -0.5(0)^2 + 7(0) + 6 \\ y & = 6 \\ \\ \therefore \text{Initial height} & = 6 \text{ m} \end{align}

(ii)

\begin{align} y & = -0.5t^2 + 7t + 6 \\ \\ \text{When } & y = 30, \\ 30 & = -0.5t^2 + 7t + 6 \\ 0 & = -0.5t^2 + 7t - 24 \\ 0 & = -t^2 + 14t - 48 \\ 0 & = t^2 - 14t + 48 \\ 0 & = (t - 6)(t - 8) \\ \\ t - 6 = 0 \phantom{00}&\text{or}\phantom{000} t - 8 = 0 \\ t = 6 \phantom{00}&\phantom{or000-8} t = 8 \\ \\ \text{Since the coefficient of } t^2 \text{ is negative}, & \phantom{0} \text{the path of the ball resembles the shape of a maximum curve} (\cap). \\ \\ \text{Thus the ball is above 30 m} & \text{ between 6 and 8 seconds.} \\ \\ \therefore 6 < \phantom{.}& t < 8 \end{align}

(iii)

\begin{align*} \text{Time when ball is at maximum height} & = {6 + 8 \over 2} \\ & = 7 \\ \\ \text{When } & t = 7, \\ y & = -0.5(7)^2 + 7(7) + 6 \\ y & = 30.5 \\ \\ \text{Maximum height} & = 30.5 \text{ m} \end{align*}

(iv)

\begin{align*} y & = a(t - p)^2 + q \\ \\ \text{The ball is at} & \text{ maximum height of } 30.5 \text{ m when } t = 7 \\ \\ y & = a(t - 7)^2 + 30.5 \\ \\ \text{When } & t = 0 \text{ and } y = 6, \\ 6 & = a(0 - 7)^2 + 30.5 \\ 6 & = a(49) + 30.5 \\ 6 & = 49a + 30.5 \\ -24.5 & = 49a \\ {-24.5 \over 49} & = a \\ -0.5 & = a \\ \\ \therefore y & = -0.5(t - 7)^2 + 30.5 \end{align*}

(v)

No. The curve that represents the motion of the ball has an initial value of $6$ while the curve of $y = -0.5x^2 + 7x + c$ has an initial value of $c$.
In addition, since time ($t$) cannot be negative, the curve that represents the motion of the ball is only valid for $t \ge 0$.