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Question:

How many distinct positive integer-valued solutions exist to the equation ${(x^{2} - 7 x + 11)}^{(x^{2} - 13 x + 42)}$ = 1?

Explanation:

Given, ${(x^{2} - 7 x + 11)}^{(x^{2} - 13 x + 42)}$ = 1?

This is possible when either (x² - 7x + 11) = 1 or

(x² - 13x + 42) = 0 or

(x² - 7x + 11) = -1 and (x² - 13x + 42) = even number.

Case 1 : x² - 7x + 11 = 1
⇒ x² - 7x + 10 = 0
⇒ (x - 5)(x - 2) = 0
⇒ x = 5 or 2.

Case 2 : (x² - 13x + 42) = 0
⇒ (x - 6)(x - 7) = 0
⇒ x = 6 or 7.

Case 3 : x² - 7x + 11 = -1 and (x² - 13x + 42) = even
⇒ x² - 7x + 12 = 0
⇒ (x - 3)(x - 4) = 0
⇒ x = 3 or 4.

Now, (x² - 13x + 42) is even for all values of x.

∴ We have a total of 6 possible values of x i.e. 5 or 2, 6 or 7 and 3 or 4.

Hence, option (d).

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