If ^k+5P_k+1= 11(k-3) 2 . ^k+3P_k, then the values of k are: - Vidyadip

MCQ

If $^\text{k}+5\text{P}_\text{k+1}=\frac{11(\text{k}-3)}{2}.\ ^\text{k+3}\text{P}_\text{k}$, then the values of $k$ are:

A
$7$ and $11$
✓
$6$ and $7$
C
$2$ and $11$
D
$2$ and $6$

✓

Answer

Correct option: B.

$6$ and $7$

$^\text{k}+5\text{P}_\text{k+1}=\frac{11(\text{k}-3)}{2}.\ ^\text{k+3}\text{P}_\text{k}$
$\Rightarrow \frac{(\text{k}+5)!}{(\text{k}+5-\text{k}-1)!}=\frac{11(\text{k}-1)}{2}\times \frac{(\text{k}+3)!}{(\text{k}+3-\text{k})!}$
$\Rightarrow \frac{(\text{k}+5)!}{(\text{k}+3)!}=\frac{11(\text{k}-1)}{2}\times\frac{4!}{3!}$
$\Rightarrow (\text{k}+5)(\text{k}+4)=22(\text{k}-1)$
$\Rightarrow \text{k}^2+9\text{k}+20=22\text{k}-22$
$\Rightarrow \text{k}^2-13\text{k}+42=0$
$\Rightarrow \text{k}=6,7$

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