Quadratic Equation Question 1
Question 1 - 2024 (01 Feb Shift 1)
Let $S=\left{x \in R:(\sqrt{3}+\sqrt{2})^{x}+(\sqrt{3}-\sqrt{2})^{x}=10\right}$.
Then the number of elements in $\mathrm{S}$ is :
(1) 4
(2) 0
(3) 2
(4) 1
Show Answer
Answer (3)
Solution
$(\sqrt{3}+\sqrt{2})^{x}+(\sqrt{3}-\sqrt{2})^{x}=10$
Let $(\sqrt{3}+\sqrt{2})^{\mathrm{x}}=\mathrm{t}$
$\mathrm{t}+\frac{1}{\mathrm{t}}=10$
$\mathrm{t}^{2}-10 \mathrm{t}+1=0$
$t=\frac{10 \pm \sqrt{100-4}}{2}=5 \pm 2 \sqrt{6}$
$(\sqrt{3}+\sqrt{2})^{x}=(\sqrt{3} \pm \sqrt{2})^{2}$
$\mathrm{x}=2$ or $\mathrm{x}=-2$
Number of solutions $=2$
