Question

$^{n-1}C_r = (k^2 - 8) ^{n}C_{r+1}$ if and only if:

• A. $2\sqrt{2} < k ≤ 3$
• B. $2\sqrt{3} < k ≤ 3\sqrt{2}$
• C. $2\sqrt{3} < k <3 \sqrt{3}$
• D. $2\sqrt{2} < k < 2\sqrt{3}$

Answer 1

Answer: (A)

$2\sqrt{2} < k ≤ 3$

Answer 2

Given: $n−1C_r = (k^2 − 8) ^nC_{r+1}$
We know: $n−1C_r = (k^2 − 8) ^nC_{r+1}$
For this expression to hold, $k^2 − 8$ must be positive:
$k^2 − 8 > 0 \Rightarrow k > 2\sqrt{2} \text{ or } k < -2\sqrt{2}$
Thus, $k \in (-\infty, -2\sqrt{2}) \cup (2\sqrt{2}, \infty)$
Next, we check the range $-3 \le k \le 3$ to satisfy the constraint. Combining both conditions: $k \in [2\sqrt{2}, 3]$