First slide

Theory of equations

Question

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

Moderate

A

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

$(0, \frac{2 π}{3})$
C

$(0, π)$
D

$(0, 2 π)$

Solution

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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