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

Answer 1

Answer: (B)

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