Let S = {x ∈ R:(√3 + √2)^x + (√3 − √2)^x = 10}. Then the number of elements in S is:

Question

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

Answer 1

Correct option is (3) 2

\((\sqrt{3}+\sqrt{2})^{\mathrm{x}}+(\sqrt{3}-\sqrt{2})^{\mathrm{x}}=10\)

Let \((\sqrt{3}+\sqrt{2})^{\mathrm{x}}=\mathrm{t}\)

\(\mathrm{t}+\frac{1}{\mathrm{t}}=10\)

\(\mathrm{t}^{2}-10 \mathrm{t}+1=0\)

\(\mathrm{t}=\frac{10 \pm \sqrt{100-4}}{2}=5 \pm 2 \sqrt{6}\)

\((\sqrt{3}+\sqrt{2})^{\mathrm x}=(\sqrt{3} \pm \sqrt{2})^{2}\)

\(\mathrm{x}=2\ or \ \mathrm{x}=-2\)

Number of solutions = 2