Question

The range of values of 9 in the interval $[0, \pi / 2]$ such that the equation $(\sin \theta)x^2 + 2(\cos \theta )x + \sin \theta = 0$ has two distinct real roots is given by

• A. $ 0 \leq \theta \leq \pi/ 4$
• B. $ 0 < \theta < \pi/ 4$
• C. $ 0 \leq \theta < \pi/ 4$
• D. $ 0 < \theta \leq \pi/ 4$

Answer 1

Answer: (B)

$(\sin \theta ) x^2 + 2 (\cos \theta)x + \sin \theta = 0$
For two distinct real roots; $d > 0$
$i.e., b^2 - 4ac > 0$
$\Rightarrow \, 4 \cos^2 \theta - 4 \sin \theta ·\sin \theta > 0$
$\Rightarrow \, \cos^2 \theta - \sin^2 \theta > 0 \Rightarrow \, \cos 2 \theta > 0$
$\Rightarrow \, 0 < 2 \theta < \frac{\pi}{2} \, \Rightarrow 0 < \theta < \frac{\pi}{4} $