Question

The equation $(\cos \beta -1)x^2+\left(\cos \beta \right)x+\sin \beta =0$ where x is a variable, has real roots if $\beta$ lies in the interval

• A. $(0,2\pi )$
• B. $(-\pi ,0)$
• C. $(-\pi /2,\pi /2)$
• D. $(0,\pi )$

Answer 1

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

For real roots, discriminant must be greater than 0.
${\cos }^{2}p+4\sin p(1-\cos p)\ge 0$

${\cos }^{2}p-4\sin p\cos p+4\sin p\ge 0$

$(\cos p-2\sin p{)}^2-4{\sin }^{2}p+4\sin p\ge 0$

$(\cos p-2\sin p{)}^2+4\sin p(1-\sin p)\ge 0$.

Now, $(\cos p-2\sin p{)}^2\ge 0$ for all x.

$-1\le \sin p\le 1$

$1-\sin p\ge 0$.

Hence for the discriminant to be positive, $\sin p>0$
$p\in (0,\pi )$.