Basic Mathematics & Logarithm
Exponential Equations — Substitution
nta_pyq_2023_jan
Grade None
Question:
Let $S=\left\{x:x\in\mathbb{R}\text{ and }\left(\sqrt{3}+\sqrt{2}\right)^{x^2-4}+\left(\sqrt{3}-\sqrt{2}\right)^{x^2-4}=10\right\}$. Then $n(S)$ is equal to:
Step-by-Step Solution
Key Concept: Let $t=(\sqrt{3}+\sqrt{2})^{x^2-4}$. Since $(\sqrt{3}-\sqrt{2})=\frac{1}{\sqrt{3}+\sqrt{2}}$, we get $t+\frac{1}{t}=10$, so $t^2-10t+1=0\Rightarrow t=5\pm2\sqrt{6}$.
$x=\pm\sqrt{2},\pm\sqrt{6}$. $n(S)=4$.
Correct Answer: 2