If $\sqrt{3} ({cos}^{2} x) = (\sqrt{3} - 1) cos x + 1,$ the number of solutions of the given equation when $x \in [0, \frac{π}{2}]$ is

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Solution

$\sqrt{3} ({cos}^{2} x) = (\sqrt{3} - 1) cos x + 1$
Put $cos x = t$
$\Rightarrow \sqrt{3} t^{2} - (\sqrt{3} - 1) t - 1 = 0$
$\Rightarrow t = \frac{(\sqrt{3} - 1) \pm \sqrt{4 + 2 \sqrt{3}}}{2 \sqrt{3}}$
$\Rightarrow t = 1$ or $- \frac{1}{\sqrt{3}}$
$\Rightarrow cos x = 1$ or $- \frac{1}{\sqrt{3}} \to$ rejected as $x \in [0, \frac{π}{2}]$
$\Rightarrow cos x = 1$
$\Rightarrow$ Number of solution $= 1$

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If √3(cos2x)=(√3−1)cosx+1, the number of solutions of the given equation when x∈[0,π2] is

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If $\sqrt{3} ({cos}^{2} x) = (\sqrt{3} - 1) cos x + 1,$ the number of solutions of the given equation when $x \in [0, \frac{π}{2}]$ is