Number of integral values of $a$ for which the equation ${cos}^{2} x - sin x + a = 0$ has roots when $x ϵ (0, \frac{π}{2})$ is

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Solution

${cos}^{2} x - sin x + a = 0$
$\Rightarrow {sin}^{2} x + sin x - (a + 1) = 0$
$\Rightarrow sin x = \frac{- 1 \pm \sqrt{4 a + 5}}{2}$
Since, for $x \in (0, \frac{π}{2})$
$0 < sin x < 1$
$\Rightarrow 0 < - 1 \pm \sqrt{4 a + 5} < 2$
$\Rightarrow 1 < 4 a + 5 < 9$
$\Rightarrow - 1 < a < 1$
Hence, number of integral values of $a$ are $1 (a = 0)$

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