Question

If $\alpha$ is the only real root of the equation $x ^3+ bx ^2+ cx +1=0$, then the value of $\tan ^{-1} \alpha+\tan ^{-1}\left(\frac{1}{\alpha}\right)$ is equal to

• A. 0
• B. $\frac{\pi}{2}$
• C. $\frac{-\pi}{2}$
• D. $\frac{3 \pi}{2}$

Answer 1

Answer: (C)

$\Theta f(0)=1$
$\therefore$ Only real root $\alpha<0$
$\therefore \tan ^{-1} \alpha+\left(-\pi+\cot ^{-1} \alpha\right)=-\pi+\frac{\pi}{2}=\frac{-\pi}{2}$