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 function $\sec^{-1} x$ is defined for $|x| \geq 1$, so $\sec^{-1} \left( \frac{\sqrt{3}}{2} \right)$ is not defined, as $\frac{\sqrt{3}}{2}$ lies within the interval $(-1, 1)$, where the secant function is not defined. Hence, the set of values of $\sec^{-1} \left( \frac{\sqrt{3}}{2} \right)$ is indeed a null set. Thus, both the assertion and the reason are correct.