Find the number of solutions of the equation $x^{x - 1} + x^{1 - x} = \frac{13}{7} .$

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Solution

The correct option is A
0

If we suppose $x^{x - 1} = a .$
Then, $x^{x - 1} = \frac{1}{a}$
And the equation becomes $a + \frac{1}{a} = \frac{13}{7}$
Now we know that for a real number a :
$a + \frac{1}{a}$ is greater than or equal to 2 "OR" less than or equal to -2.
Therefore $a + \frac{1}{a}$ can never be $\frac{13}{7}$ (less than 2 but greater than -2).
Hence the given equation has no solution i.e., the number of solutions is zero.

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