Question

Assertion :The minimum value of $f\left(\theta \right)=\left|\frac{2i}{3-i{e}^{i\theta }}\right|$ is $\frac{1}{\sqrt{2}}$ Reason: The maximum value of $f\left(\theta \right)=\left|\frac{2i}{3-i{e}^{i\theta }}\right|$ is $1$

• 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. Both Assertion and Reason are incorrect

Answer 1

The correct option is D Both Assertion and Reason are incorrect

$f\left(\theta \right)=\left|\frac{2i}{3-i{e}^{i\theta }}\right|$
$\Rightarrow f\left(\theta \right)=\frac{2}{|3+\sin \theta +i\cos \theta |}$
$\Rightarrow f\left(\theta \right)=\frac{2}{\sqrt{{(3+\sin \theta )}^2+{\cos }^2\theta }}=\frac{2}{\sqrt{10+6\sin \theta }}$
Since, $-1\le \sin \theta \le 1$
Therefore, $\frac{1}{2}\le \frac{2}{\sqrt{10+6\sin \theta }}\le \frac{1}{8}$
Ans: D