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4-2.Quadratic Equations and Inequations
hard
Let $\mathrm{S}=\left\{x \in R:(\sqrt{3}+\sqrt{2})^x+(\sqrt{3}-\sqrt{2})^x=10\right\}$. Then the number of elements in $\mathrm{S}$ is :
A
$4$
B
$0$
C
$2$
D
$1$
(JEE MAIN-2024)
Solution
$(\sqrt{3}+\sqrt{2})^{\mathrm{x}}+(\sqrt{3}-\sqrt{2})^{\mathrm{x}}=10$
$\text { Let }(\sqrt{3}+\sqrt{2})^{\mathrm{x}}=\mathrm{t}$
$\mathrm{t}+\frac{1}{\mathrm{t}}=10$
$\mathrm{t}^2-10 \mathrm{t}+1=0$
$\mathrm{t}=\frac{10 \pm \sqrt{100-4}}{2}=5 \pm 2 \sqrt{6}$
$(\sqrt{3}+\sqrt{2})^{\mathrm{x}}=(\sqrt{3} \pm \sqrt{2})^2$
$\mathrm{x}=2 \text { or } \mathrm{x}=-2$
$\text { Number of solutions }=2$
Standard 11
Mathematics
