The sum of all distinct roots of the equation $(x^{2} - 11 x + 29)^{(x^{2} - 8 x + 7)} = 1$ is

12

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16

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17

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19

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Solution

The correct option is C $17$
There are three possible cases:
Case $1$ :
$x^{2} - 11 x + 29 = 1$
$\Rightarrow x^{2} - 11 x + 28 = 0 \Rightarrow x = 4, 7$

Case $2$ :
$x^{2} - 8 x + 7 = 0 \Rightarrow x = 1, 7$

Case $3$ :
$x^{2} - 11 x + 29 = - 1$
$\Rightarrow x^{2} - 11 x + 30 = 0 \Rightarrow x = 5, 6$
For $x = 5, LHS = (- 1)^{- 8} = 1 = RHS$
For $x = 6, LHS = (- 1)^{- 5} = - 1 \neq RHS$

Therefore, distinct roots are $1, 4, 5, 7$
Sum of distinct roots $= 4 + 7 + 1 + 5 = 17$

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