Question

\(\text{Let } S = \{x \in \mathbb{R} : (\sqrt{3} + \sqrt{2})^x + (\sqrt{3} - \sqrt{2})^x = 10\}\).Then the number of elements in \( S \) is:

• A. 4
• B. 0
• C. 2
• D. 1

Answer 1

The Correct Option is C

Consider:

\[ \left( \sqrt{3} + \sqrt{2} \right)^x + \left( \sqrt{3} - \sqrt{2} \right)^x = 10 \]

Let:

\[ \left( \sqrt{3} + \sqrt{2} \right)^x = t \]

Thus:

\[ \left( \sqrt{3} - \sqrt{2} \right)^x = \frac{1}{t} \]

Substitute and simplify:

\[ t + \frac{1}{t} = 10 \]

Multiplying through by \( t \) gives:

\[ t^2 - 10t + 1 = 0 \]

Solving this quadratic equation:

\[ t = \frac{10 \pm \sqrt{100 - 4}}{2} = \frac{10 \pm \sqrt{96}}{2} = \frac{10 \pm 4\sqrt{6}}{2} = 5 \pm 2\sqrt{6} \]

Since:

\[ \left( \sqrt{3} + \sqrt{2} \right)^x > 0, \quad t = 5 + 2\sqrt{6} \]

Thus, the corresponding values of \( x \) are:
\[ x = 2 \quad \text{or} \quad x = -2 \]

Number of solutions = 2.