Question

Let $S=\left\{x: x \in R\right.$ and $\left.(\sqrt{3}+\sqrt{2})^{x^2-4}+(\sqrt{3}-\sqrt{2})^{x^2-4}=10\right\}$. Then $n(S)$ is equal to

• A. 2
• B. 4
• C. 0
• D. 6

Answer 1

Answer: (B)

Let $(\sqrt{3}+\sqrt{2})^{x^2-4}= t$
$ t +\frac{1}{ t }=10$
$\Rightarrow t =5+2 \sqrt{6}, 5-2 \sqrt{6} $
$\Rightarrow (\sqrt{3}+\sqrt{2})^{ x ^2-4}=5+2 \sqrt{6}, 5-2 \sqrt{6} $
$\Rightarrow x ^2-4=2,-2 \text { or } x ^2=6,2 $
$\Rightarrow x =\pm \sqrt{2}, \pm \sqrt{6}$