Explanation

(d)
Here $\ ^{n - 1}C_{r} = \left( k^{2} - 3 \right)\ ^{n}C_{r + 1}$
$\Longrightarrow \ ^{n - 1}C_{r} = \left( k^{2} - 3 \right)\frac{n}{r + 1}\ ^{n - 1}C_{r}$
$\Longrightarrow k^{2} - 3 = \frac{r + 1}{n}$
$\left\lbrack since,\ n - 1 \geq r \Longrightarrow \frac{r + 1}{n} \leq 1and\ n,r \geq 0 \right\rbrack$
$\Longrightarrow 0 < k^{2} - 3 \leq 1 \Longrightarrow 3 < k^{2} \leq 4$
$\Longrightarrow k \in \left\lbrack - 2,\ - \sqrt{3} \right) \cup \left( \sqrt{3},2 \right\rbrack$