(a)

\begin{align} b^2 - 4ac & = (6)^2 - 4(1)(-3) \\ & = 48 > 0 \\ \\ \implies \text{Equation } & \text{has two real and distinct roots} \end{align}

(b)

\begin{align} b^2 - 4ac & = (-11)^2 - 4(2)(40) \\ & = -199 \\ \\ \implies \text{Equation } & \text{has no real roots} \end{align}

(c)

\begin{align} -72x^2 + 49x - 5 & = 0 \\ \\ b^2 - 4ac & = (49)^2 - 4(-72)(-5) \\ & = 961 \\ \\ \implies \text{Equation } & \text{has two real and distinct roots} \end{align}

(d)

\begin{align} {9 \over 8}x^2 - 6x + 8 & = 0 \\ \\ b^2 - 4ac & = (-6)^2 - 4\left(9 \over 8\right)(8) \\ & = 0 \\ \\ \implies \text{Equation } & \text{has two real and equal roots} \end{align}

\begin{align} 3x^2 + 2px & = -(18 + p) \\ 3x^2 + 2px + (18 + p) & = 0 \\ \\ b^2 - 4ac & = (2p)^2 - 4(3)(18 + p) \\ & = 4p^2 - 12(18 + p) \\ & = 4p^2 - 216 - 12p \\ \\ b^2 - 4ac & = 0 \phantom{000000} [\text{Equal roots}] \\ 4p^2 - 216 - 12p & = 0 \\ p^2 - 54 - 3p & = 0 \\ p^2 - 3p - 54 & = 0 \\ (p - 9)(p + 6) & = 0 \end{align} \begin{align} p - 9 & = 0 && \text{ or } & p + 6 & = 0 \\ p & = 9 &&& p & = -6 \end{align}

\begin{align} hx^2 - 9x + 4 & = 0 \\ \\ b^2 - 4ac & = (-9)^2 - 4(h)(4) \\ & = 81 - 16h \\ \\ b^2 - 4ac & < 0 \phantom{000000} [\text{No real roots}] \\ 81 - 16h & < 0 \\ -16h & < -81 \\ h & > {-81 \over -16} \\ h & > {81 \over 16} \end{align}

\begin{align} {1 \over 7}hx^2 + 2x + 14 & = 0 \\ \\ b^2 - 4ac & = (2)^2 - 4\left({1 \over 7}h\right)(14) \\ & = 4 - {4 \over 7}h (14) \\ & = 4 - 8h \\ \\ b^2 - 4ac & > 0 \phantom{000000} [\text{Two distinct real roots}] \\ 4 - 8h & > 0 \\ -8h & > -4 \\ h & < {-4 \over -8} \\ h & < {1 \over 2} \end{align}

(i)

\begin{align} -4x^2 + 13x - 5h & = 1 \\ -4x^2 + 13x - 5h - 1 & = 0 \\ \\ b^2 - 4ac & = (13)^2 - 4(-4)(-5h-1) \\ & = 169 + 16(-5h - 1) \\ & = 169 - 80h - 16 \\ & = 153 - 80h \\ \\ b^2 - 4ac & \ge 0 \phantom{000000} [\text{Two (distinct/equal) real roots}] \\ 153 - 80h & \ge 0 \\ -80h & \ge - 153 \\ h & \le {-153 \over -80} \\ h & \le {153 \over 80} \end{align}

(ii)

\begin{align} b^2 - 4ac & = 153 - 80h \\ \\ b^2 - 4ac & > 0 \phantom{000000} [\text{Two distinct real roots}] \\ 153 - 80h & > 0 \\ -80h & > - 153 \\ h & < {-153 \over -80} \\ h & < {153 \over 80} \end{align}

(iii)

\begin{align} b^2 - 4ac & = 153 - 80h \\ \\ b^2 - 4ac & < 0 \phantom{000000} [\text{No real roots}] \\ 153 - 80h & < 0 \\ -80h & < - 153 \\ h & > {-153 \over -80} \\ h & > {153 \over 80} \end{align}

(i)

\begin{align} hx^2 + kx + 6 & = 0 \\ \\ b^2 - 4ac & = k^2 - 4(h)(6) \\ & = k^2 - 24h \\ \\ b^2 - 4ac & = 0 \phantom{000000} [\text{Two equal real roots}] \\ k^2 - 24h & = 0 \\ k^2 & = 24h \end{align}

(ii)

\begin{align} \text{Let } & h = 6, \\ k^2 & = 24(6) \\ k^2 & = 144 \\ k & = \sqrt{144} \\ k & = 12 \\ \\ \text{Possible values: } & h = 6, k = 12 \end{align}

\begin{align} y = (4h + 1)x^2 & - (4h - 1)x + h \\ \\ \text{For minimum curve, } 4h + 1 & > 0 \\ 4h & > - 1 \\ h & > - {1 \over 4} \phantom{000000} [\text{Condition 1}] \\ \\ \\ \text{Let } & y = 0, \\ 0 & = (4h + 1)x^2 - (4h - 1)x + h \\ \\ b^2 - 4ac & = [-(4h - 1)]^2 - 4(4h + 1)(h) \\ & = (-4h + 1)^2 -4h(4h + 1) \\ & = (-4h)^2 + 2(-4h)(1) + (1)^2 - 16h^2 - 4h \phantom{0000000} [(a + b)^2 = a^2 + 2ab + b^2] \\ & = 16h^2 - 8h + 1 - 16h^2 - 4h \\ & = 1 - 12h \\ \\ b^2 - 4ac & \ge 0 \phantom{000000} [\text{1 or 2 real roots since curve intersects } x \text{-axis once or twice}] \\ 1 - 12h & \ge 0 \\ -12h & \ge - 1 \\ h & \le {-1 \over -12} \\ h & \le {1 \over 12} \phantom{000000} [\text{Condition 2}] \\ \\ \\ \text{To satisfy both conditions, } & -{1 \over 4} < h \le {1 \over 12} \end{align}

\begin{align} 5x^2 & - kx + 2k^2 + 8 = 0 \\ \\ b^2 - 4ac & = (-k)^2 - 4(5)(2k^2 + 8) \\ & = k^2 - 20(2k^2 + 8) \\ & = k^2 - 40k^2 - 160 \\ & = -39k^2 - 160 \\ \\ \text{For all real} & \text{ values of } k, \\ k^2 & \ge 0 \\ -39k^2 & \le 0 \\ -39k^2 - 160 & \le -160 \\ \\ \text{Since } b^2 - 4ac < 0 & \text{ for all real values of } k, \\ \text{the equation has no } & \text{real roots for all real values of } k \end{align}

(i)

\begin{align} 3x^2 - 4x + 7 & = 3 \left(x^2 - {4 \over 3}x\right) + 7 \\ & = 3 \left[ x^2 - {4 \over 3}x + \left(2 \over 3\right)^2 - \left(2 \over 3\right)^2 \right] + 7 \\ & = 3 \left[ \left(x - {2 \over 3}\right)^2 - {4 \over 9} \right] + 7 \\ & = 3 \left(x - {2 \over 3}\right)^2 - {4 \over 3} + 7 \\ & = 3 \left(x - {2 \over 3}\right)^2 + {17 \over 3} \\ \\ \text{For all real } & \text{values of } x, \\ \left(x - {2 \over 3}\right)^2 & \ge 0 \\ 3\left(x - {2 \over 3}\right)^2 & \ge 0 \\ 3\left(x - {2 \over 3}\right)^2 + {17 \over 3} & \ge {17 \over 3} \\ \\ \\ \therefore 3x^2 - 4x + 7 & \text{ is always positive} \end{align}

(ii)

\begin{align} & [\text{Curve of } y = 3x^2 - 4x + 7 \text{ lies entirely above } x \text{-axis}] \\ \\ & \text{The equation } 3x^2 - 4x + 7 = 0 \text{ has no real roots} \end{align}

(i) If the curve lies above or below the x-axis, it does not meet the x-axis. Thus discriminant < 0

\begin{align} y & = (k - 3)x^2 + 2kx + (k + 1) \\ \\ b^2 - 4ac & = (2k)^2 - 4(k - 3)(k + 1) \\ & = 4k^2 -4 (k^2 + k - 3k - 3) \\ & = 4k^2 - 4 (k^2 - 2k - 3) \\ & = 4k^2 - 4k^2 + 8k + 12 \\ & = 8k + 12 \\ \\ b^2 - 4ac & < 0 \phantom{000000} [\text{No real roots since curve lies above/below } x \text{-axis}] \\ 8k + 12 & < 0 \\ 8k & < -12 \\ k & < {-12 \over 8} \\ k & < -{3 \over 2} \end{align}

(ii)

\begin{align} y & = (k - 3)x^2 + 2kx + (k + 1) \\ \\ \text{If } & k < -{3 \over 2}, \\ k - 3 & < -{3 \over 2} - 3 \\ k - 3 & < -4{1 \over 2} \\ \\ \implies \text{Curve } & y = (k - 3)x^2 + 2kx + (k + 1) \text{ is a maximum curve } (\cap) \\ \\ \therefore \text{Curve } & \text{lies entirely below the } x \text{-axis} \end{align}

(i)

\begin{align} a & > 0 \phantom{000000} [\text{Minimum curve } \cup \text{; Condition 1}] \\ \\ \\ b^2 - 4ac & = (-7)^2 - 4(a)(c) \\ & = 49 - 4ac \\ \\ b^2 - 4ac & < 0 \phantom{000000} [\text{No real roots since curve lies above } x \text{-axis}] \\ 49 - 4ac & < 0 \\ -4ac & < -49 \\ ac & > {-49 \over -4} \\ ac & > {49 \over 4} \phantom{0000000} [\text{Condition 2}] \end{align}

(ii)

\begin{align} \text{Let } & a = 5 \phantom{000000} [\text{to satisfy condition 1}] \\ \\ ac & > {49 \over 4} \phantom{000000} [\text{condition 2}] \\ \\ 5c & > {49 \over 4} \\ c & > { {49 \over 4} \over 5} \\ c & > 2.45 \\ \\ \\ \text{Possible values: } & a = 5, c = 6 \end{align}

(i)

\begin{align} a & < 0 \phantom{000000} [\text{Maximum curve } \cap \text{; Condition 1}] \\ \\ \\ b^2 - 4ac & = (4)^2 - 4(a)(c) \\ & = 16 - 4ac \\ \\ b^2 - 4ac & < 0 \phantom{000000} [\text{No real roots since curve lies below } x \text{-axis}] \\ 16 - 4ac & < 0 \\ -4ac & < -16 \\ ac & > {-16 \over -4} \\ ac & > 4 \phantom{0000000} [\text{Condition 2}] \end{align}

(ii)

\begin{align} \text{Let } & a = -2 \phantom{000000} [\text{to satisfy condition 1}] \\ \\ ac & > 4 \phantom{000000} [\text{condition 2}] \\ \\ -2c & > 4 \\ c & < {4 \over -2} \\ c & < -2 \\ \\ \\ \text{Possible values: } & a = -2, c = -10 \end{align}

\begin{align} 4x^2 - {x \over h} - 2 & = 0 \\ 4x^2 - {1 \over h}x - 2 & = 0 \\ \\ b^2 - 4ac & = \left(-{1 \over h}\right)^2 - 4(4)(-2) \\ & = {1 \over h^2} + 32 \\ \\ b^2 - 4ac & = 0 \phantom{000000} [\text{Two equal real roots}] \\ {1 \over h^2} + 32 & = 0 \\ {1 \over h^2} & = -32 \\ \\ h^2 \ge 0 \text{ for all } & \text{real values of } h \implies {1 \over h^2} \ne -32 \\ \\ \therefore \text{No values of } h & \text{ for which equation has equal roots} \end{align}

No solutions as I think there's a problem with this question.

If statement B is true (ac is negative), then b² - 4ac is positive (which means statement A is false)