Question

The equation $(\cos p-1)x^2+\cos px+\sin p=0$, in the variable $x$ has real roots. Then $p$ can take any value in the interval

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

Answer 1

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

For given quadratic to posses real roots, its discriminant should be non- negative

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

We can observe that ${\cos }^2\ge 0,\cos p-1\le 0\forall x\in R$

So for discriminant to be always non-negative $\sin p\ge 0$

and we know $\sin x$ positive in 1st and 2nd quadrant

hence interval of $p$ is $[0,\pi ]$