Question

Assertion :Let $\theta \epsilon (\frac{\pi }{4},\frac{\pi }{2})$, then statement 1 $(\cos \theta {)}^{\sin \theta }<(\cos \theta {)}^{\cos \theta }<(\sin \theta {)}^{\cos \theta }$ Reason: The equation $e^{\sin \theta }-e^{-\sin \theta }=4$ has a unique solution.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Given, $e^{\sin \theta }-e^{-\sin \theta }=4$
Let $e^{\sin \theta }=t$.
Hence, $t^2-1=4t$
$t^2-4t+-1=0$
$t=\frac{4\pm 2\sqrt{5}}{2}$
$t=2\pm \sqrt{5}$.
$\sin \theta =log(2\pm \sqrt{5})$.
Now
$\sqrt{5}>2$.Hence $2-\sqrt{5}$ is not possible since input of log cannot be negative.
Hence, $\sin \theta =log(2+\sqrt{5})>1$
Now, $\sin \theta >1$ is not possible.
Hence no solution.