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4-2.Quadratic Equations and Inequations
hard
मान $S=\left\{x: x \in \mathbb{R} \text { एवं }(\sqrt{3}+\sqrt{2})^{x^2-4}+(\sqrt{3}-\sqrt{2})^{x^2-4}=10 \text { हैं }\right\}$ है। तब $\mathrm{n}(\mathrm{S})$ बराबर है-
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
