Question

If $\theta$ and $\phi$ are the roots of the equation $8x^2 + 22x + 5 = 0$ , then

• A. both $sin ^{-1} \theta$ and $sin^{-1} \phi$ are real
• B. both $sec ^{-1} \theta$ and $sec^{-1} \phi$ are real
• C. both $tan ^{-1} \theta$ and $tan^{-1} \phi$ are real
• D. None of these

Answer 1

Answer: (C)

$8x^{3}+22x+5=0 \Rightarrow x=-\frac{1}{4}, \frac{5}{2}$
$\because -1 < -\frac{1}{4}<1$ and $-\frac{5}{2}<-1$
$\therefore sin^{-1}\left(-\frac{1}{4}\right)$ exists but $sin^{-1}\left(-\frac{5}{2}\right)$ does not exist
$sec^{-1}\left(-\frac{5}{2}\right)$ exists but $sec^{-1}\left(-\frac{1}{4}\right)$ does not exist,
$tan^{-1}\left(-\frac{1}{4}\right)$ and $tan^{-1}\left(-\frac{5}{2}\right)$ both exist