Question

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

• A. 2
• B. 4
• C. 0
• D. 6

Answer 1

Answer: (B)

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