The number of real solutions of $(x^{2} - 7 x + 11)^{x^{2} - 11 x + 30} = 1$ is

3

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4

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6

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No solution

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Solution

The correct option is A $3$
$(x^{2} - 7 x + 11)^{x^{2} - 11 x + 30} = 1$
For real solutions,
$(x^{2} - 7 x + 11) = 1$ or $x^{2} - 11 x + 30 = 0$
$=> x^{2} - 7 x + 10 = 0$ or $=> x^{2} - 11 x + 30 = 0$
$=> x^{2} - 5 x - 2 x + 10 = 0$ or $=> x^{2} - 6 x - 5 x + 30 = 0$
$=> x (x - 5) - 2 (x - 5) = 0$ or $=> x (x - 6) - 5 (x - 6) = 0$
$=> (x - 5) (x - 2) = 0$ or $=> (x - 6) (x - 5) = 0$
$=> x = 5, x = 2$ or $=> x = 6, x = 5$
Thus, number of real solutions are $3$
Hence, option 'A' is correct.

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