- Home
- Standard 11
- Mathematics
4-2.Quadratic Equations and Inequations
hard
Let $S=\left\{ x : x \in R \text { and }(\sqrt{3}+\sqrt{2})^{ x ^2-4}+(\sqrt{3}-\sqrt{2})^{ x ^2-4}=10\right\} \text {. }$ Then $n ( S )$ is equal to
A
$2$
B
$4$
C
$6$
D
$0$
(JEE MAIN-2023)
Solution
Let $(\sqrt{3}+\sqrt{2})^{ x ^2-4}= t$
$t +\frac{1}{ t }=10$
$\Rightarrow \quad t =5+2 \sqrt{6}, 5-2 \sqrt{6}$
$\Rightarrow \quad (\sqrt{3}+\sqrt{2})^{ x ^2-4}=5+2 \sqrt{6}, 5-2 \sqrt{6}$
$\Rightarrow \quad x ^2-4=2,-2 \quad \text { or } x ^2=6,2$
$\Rightarrow \quad x =\pm \sqrt{2}, \pm \sqrt{6}$
Standard 11
Mathematics
