Algebra
Inverse Trig & Equations
MMTS_Full_Test_02
Grade 12
Question:
If the equation $x^3+bx^2+cx+1=0$ ($b<c$) has only one real root $\alpha$. Then the value of $2\tan^{-1}(\text{cosec}\,\alpha)+\tan^{-1}(2\sin\alpha\sec^2\alpha)$ is
$-\dfrac{\pi}{2}$
$-\pi$
$\dfrac{\pi}{2}$
$\pi$
Step-by-Step Solution
Key Concept: $b<c$ means $\alpha<-1$; use addition formula for arctan
Since $b<c$ and Vieta's: $\alpha<-1$. Expression $=\pi$.
Correct Answer: 4