Question

The quadratic equation $(\cos p-1)x^2+\left(\cos p\right)x+\sin p=0$
(where $x\in R$) has real roots if $p$ lies in

• A. $(0,2\pi )$
• B. $(-\pi ,0)$
• C. $(-\frac{\pi }{2},\frac{\pi }{2})$
• D. $(0,\pi ]$

Answer 1

The correct option is D $(0,\pi ]$

$(\cos p-1)x^2+\left(\cos p\right)x+\sin p=0$
Given equation has real roots if $D\ge 0$
$\Rightarrow {\cos }^{2}p-4(\cos p-1)\sin p\ge 0$
$\Rightarrow {\cos }^{2}p-4\cos p\sin p+4\sin p\ge 0$
$\Rightarrow (\cos p-2\sin p{)}^2-4{\sin }^{2}p+4\sin p\ge 0$
$\Rightarrow (\cos p-2\sin p{)}^2+4\sin p(1-\sin p)\ge 0$
For the above inequality to be true $\sin p\ge 0$
$\because (\cos p-2\sin p{)}^2\ge 0,1-\sin p\ge 0$ for all values of $p$
$\therefore p\in (0,\pi ](\because \cos p\ne 1)$