Binomial Theorem 2 Question 5
6.
If ${ }^{n-1} C_{r}=\left(k^{2}-3\right){ }^{n} C_{r+1}$, then $k$ belongs to
(a) $(-\infty, -2)$
(b) $(2, \infty)$
(c) $(-\sqrt{3}, \sqrt{3})$
(d) $(-\sqrt{3}, 2)$
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Answer:
Correct Answer: 6. (d)
Solution:
- Given, ${ }^{n-1} C_{r}=\left(k^{2}-3\right){ }^{n} C_{r+1}$
$\Rightarrow \quad{ }^{n-1} C_{r}=\left(k^{2}-3\right) \frac{n}{r+1}{ }^{n-1} C_{r}$
$\Rightarrow \quad k^{2}-3=\frac{r+1}{n}$
[since, $n \geq r \Rightarrow \frac{r+1}{n} \leq 1$ and $n, r>0$ ]
$\Rightarrow \quad 0<k^{2}-3 \leq 1 \Rightarrow 3<k^{2} \leq 4$
$\Rightarrow \quad k \in[-2,-\sqrt{3}) \cup(\sqrt{3}, 2]$
