$\therefore\left|\begin{array}{ccc}{a^{2}} & {a^{2} r^{6}} & {x} \\ {a^{2} r^{2}} & {a^{2} r^{3}} & {y} \\ {a^{2} r^{6}} & {a^{2} r^{10}} & {z}\end{array}\right|$
$=a^{2} \times a^{2} r^{6}\left|\begin{array}{ccc}{1} & {1} & {x} \\ {r^{2}} & {r^{2}} & {y} \\ {r^{4}} & {r^{4}} & {z}\end{array}\right|=0$
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| $\text{X}:$ | $2$ | $3$ | $4$ | $5$ |
| $\text{P}(\text{X}):$ | $\frac{5}{\text{k}}$ | $\frac{7}{\text{k}}$ | $\frac{9}{\text{k}}$ | $\frac{11}{\text{k}}$ |
The value of k is:
$(i)$ $f (x)$ is bounded on $a \le x \le b.$
$(ii)$ The equation $f (x) = 0$ has at least one solution in $a < x < b.$
$(iii)$ The maximum and minimum values of $f (x)$ on $a \le x \le b$ occur at points where $f ' (c) = 0$.
$(iv)$ There is at least one point $c$ with $a < c < b$ where $f ' (c) > 0$.
$(v)$ There is at least one point $d$ with $a < d < b$ where $f ' (c) < 0.$