Let S = { x ? R : ( 3 + 2 ) x + ( 3 - 2 ) x = 10 } . Then the number of elements in S is

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Solution

Q.
Let $S = {x \in R : {(\sqrt{3} + \sqrt{2})}^{x} + {(\sqrt{3} - \sqrt{2})}^{x} = 10}$ . Then the number of elements in S is [2024]

1 4

2 2

3 0

4 1

Ans.
(2)

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

Let $\sqrt{3} + \sqrt{2} = t$

$t^{x} + \frac{1}{t^{x}} = 10$

Let $t^{x} = y, t h e n y + \frac{1}{y} = 10$

$\Rightarrow y^{2} - 10 y + 1 = 0$

$\Rightarrow y = \frac{10 \pm \sqrt{100 - 4}}{2} = \frac{10 \pm \sqrt{96}}{2} = \frac{10 \pm 4 \sqrt{6}}{2} = 5 \pm 2 \sqrt{6}$

${(\sqrt{3} + \sqrt{2})}^{x} = 5 \pm 2 \sqrt{6}$

$x \ln (\sqrt{3} + \sqrt{2}) = \ln (5 + 2 \sqrt{6}) or \ln (5 - 2 \sqrt{6})$

$x = \frac{\ln (5 + 2 \sqrt{6})}{\ln (\sqrt{3} + \sqrt{2})} or x = \frac{\ln (5 - 2 \sqrt{6})}{\ln (\sqrt{3} + \sqrt{2})}$

So, two real values of $x$ .

$∴$ The number of elements in S is 2.

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