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Q.
The number of solutions of the equation $2sin^{-1} \left(\sqrt{x^{2} - x + 1}\right) + cos^{-1} \left(\sqrt{x^{2} - x}\right) = \frac{3\pi}{2}$ is
Inverse Trigonometric Functions
Solution:
$sin^{-1}\sqrt{x}, cos^{-1}\sqrt{x}$ are defined for $x \le 1$ and $x \ge 0$
$\therefore \sqrt{x^2 - x + 1} \le 1$ and $\sqrt{x^2 - x} \ge 0$
$\Rightarrow x^2 - x \le 0$ and $x^2 - x \ge 0$
$\Rightarrow x^2 - x = 0$
$\Rightarrow x = 1, 0$
$\therefore $ There are two solutions, both satisfies the equation.