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Let $(\sin a) x^{2} + (\sin a) x + 1 - \cos a = 0$ The set of values of a for which roots of this equation are real and distinct, is

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a

(0, 2 \tan^{- 1} \frac{1}{4})

b

(0, \frac{2 π}{3})

c

(0, π)

d

(0, 2 π)

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detailed solution

Correct option is A

The roots of the given equation will be real and distinct, iff

$\begin{aligned} \sin^{2} a - 4 \sin a (1 - \cos a) > 0 \\ \Rightarrow & (1 - \cos a) {1 + \cos a - 4 \sin a} > 0 \\ \Rightarrow & 2 \cos^{2} \frac{a}{2} - 8 \sin \frac{a}{2} \cos \frac{a}{2} > 0 \\ \Rightarrow & 2 \cos^{2} \frac{a}{2} (1 - 4 \tan \frac{a}{2}) > 0 \\ \Rightarrow & 4 \tan \frac{a}{2} < 1 \Rightarrow - \frac{π}{2} < \frac{a}{2} < \tan^{- 1} \frac{1}{4} \Rightarrow - π < a < 2 \tan^{- 1} \frac{1}{4} \end{aligned}$

Hence, option (a) is correct.

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