The number of integral solution(s) of ${cos}^{- 1} (4 x^{3} - 12 x^{2} + 11 x - \frac{5}{2}) = \frac{π}{3}$ is

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Solution

The correct option is C $1$
Given : ${cos}^{- 1} (4 x^{3} - 12 x^{2} + 11 x - \frac{5}{2}) = \frac{π}{3}$
$\Rightarrow 4 x^{3} - 12 x^{2} + 11 x - \frac{5}{2} = cos \frac{π}{3}$
$\Rightarrow 4 x^{3} - 12 x^{2} + 11 x - \frac{5}{2} = \frac{1}{2}$
$\Rightarrow 4 x^{3} - 12 x^{2} + 11 x - 3 = 0$
By observation, $x = 1$ satisfied above equation. Hence, $(x - 1)$ must be a factor.
$\Rightarrow (x - 1) (4 x^{2} - 8 x + 3) = 0$
$\Rightarrow (x - 1) (2 x - 3) (2 x - 1) = 0$
$\Rightarrow x = 1, \frac{3}{2}, \frac{1}{2}$
$∴$ Only one integral solution.

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