Q. Let $\left( \sin a \right)x^{2} + \left( \sin a \right)x + (1 - \cos a = 0)$. The value of $a$ for which roots of this equation are real and distinct, is
A) $(0,2\tan^{- 1}{1/4})$
B) $(0,\ 2\pi/3)$
C) $(0,\ \pi)$
D) $(0,\ 2\pi)$
Explanation
(a)
Since, $D > 0,\sin^{2}{a - 4\sin{a\left( 1 - \cos a \right) > 0}}$
$\Rightarrow \sin{a > 0}$ or $\left( \sin{a - 4 + 4\cos a} \right) > 0$
$\Rightarrow a \in (0,\ \pi)$ or $\frac{1 - \cos a}{\sin a} < \frac{1}{4}$
$\Rightarrow a \in (0,\ \pi)$ or $a \in \left( 0,2\tan^{- 1}\left( \frac{1}{4} \right) \right)$
$\Rightarrow a \in \left( 0,2\tan^{- 1}\left( \frac{1}{4} \right) \right)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.