Question

Assertion (A): The set of values of $\sec^{-1} \left( \frac{\sqrt{3}}{2} \right)$ is a null set.
Reason (R): $\sec^{-1}x$ is defined for $x \in \mathbb{R} - (-1, 1)$.

• A. Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

The inverse secant function is only defined for $|x| \geq 1$. If the argument is within the interval $(-1, 1)$, the inverse secant is not defined.

Answer 1

The Correct Option is A

The domain of $\sec^{-1} x$ is $|x| \geq 1$. Since $\frac{\sqrt{3}}{2}$ is in the interval $(-1, 1)$, $\sec^{-1} \left( \frac{\sqrt{3}}{2} \right)$ is undefined. Consequently, the set of values for $\sec^{-1} \left( \frac{\sqrt{3}}{2} \right)$ is empty. Therefore, both the assertion and the reason are correct.