Question

Assertion(A): The range of f(x)=3sin^(-1)⁡x+5π/2, where xϵ[-1,1],is [π,4π]. Reason(R): The range of the principal value branch of sin^(-1)⁡x is [-π,0].

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is not the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

Answer: (C)

A is true but R is false

Answer 2

Assertion (A):

Given:

$$f(x)=3\sin^{-1}x+\frac{5\pi}{2}, \quad x\in[-1,1]$$

The principal value range of $\sin^{-1}x$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.

• Minimum value of $f(x)$: $$3\left(-\frac{\pi}{2}\right)+\frac{5\pi}{2}=-\frac{3\pi}{2}+\frac{5\pi}{2}=\pi$$
• Maximum value of $f(x)$: $$3\left(\frac{\pi}{2}\right)+\frac{5\pi}{2}=\frac{3\pi}{2}+\frac{5\pi}{2}=\frac{8\pi}{2}=4\pi$$

Thus, the range of $f(x)$ is $[\pi,4\pi]$ and Assertion (A) is true.

Reason (R):

The statement claims that the range of the principal branch of $\sin^{-1}x$ is $[-π, 0]$, which is incorrect; the correct range is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.

Conclusion:

Assertion (A) is true, and Reason (R) is false.