Quadratic Equations
Exponential Equation — Substitution
nta_pyq_2024_jan
Grade 11
Question:
Let $S=\left\{x\in\mathbb{R}:(\sqrt{3}+\sqrt{2})^x+(\sqrt{3}-\sqrt{2})^x=10\right\}$. Then the number of elements in $S$ is:
Step-by-Step Solution
Key Concept: Let $t=(\sqrt{3}+\sqrt{2})^x$. Then $(\sqrt{3}-\sqrt{2})^x=1/t$ (since $(\sqrt3+\sqrt2)(\sqrt3-\sqrt2)=1$). Equation becomes $t+1/t=10\Rightarrow t^2-10t+1=0\Rightarrow t=5\pm2\sqrt6=(\sqrt3\pm\sqrt2)^2$. So $x=2$ or $x=-2$.
$x=2$ or $x=-2$. Number of elements $=2$.
Correct Answer: 3