JEE (Main) Mathematics - Inverse Trigonometric Functions MCQs (16 to 30)

1. 16. The derivative of \( \sin^{-1}(x) \) is:

   A) \( \frac{1}{\sqrt{1-x^2}} \)
   B) \( \frac{-1}{\sqrt{1-x^2}} \)
   C) \( \frac{1}{1-x^2} \)
   D) \( \frac{-1}{1-x^2} \)
   

   Answer: A) \( \frac{1}{\sqrt{1-x^2}} \)

2. 17. The range of \( \sin^{-1}(x) \) is:

   A) \( [0, \pi] \)
   B) \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \)
   C) \( \left[0, \frac{\pi}{2}\right] \)
   D) \( \left[-\frac{\pi}{2}, \pi\right] \)
   

   Answer: B) \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \)

3. 18. The value of \( \sin^{-1}\left(\frac{1}{2}\right) \) is:

   A) \( 30^\circ \)
   B) \( 45^\circ \)
   C) \( 60^\circ \)
   D) \( 90^\circ \)
   

   Answer: A) \( 30^\circ \)

4. 19. The derivative of \( \cos^{-1}(x) \) is:

   A) \( \frac{1}{\sqrt{1-x^2}} \)
   B) \( \frac{-1}{\sqrt{1-x^2}} \)
   C) \( \frac{1}{1-x^2} \)
   D) \( \frac{-1}{1-x^2} \)
   

   Answer: B) \( \frac{-1}{\sqrt{1-x^2}} \)

5. 20. The principal value of \( \cos^{-1}(-1) \) is:

   A) \( 0 \)
   B) \( \frac{\pi}{2} \)
   C) \( \pi \)
   D) \( \frac{3\pi}{2} \)
   

   Answer: C) \( \pi \)

6. 21. Which of the following is the range of \( \tan^{-1}(x) \)?

   A) \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \)
   B) \( \left[0, \pi\right] \)
   C) \( \left[-\pi, \pi\right] \)
   D) \( \left[0, \frac{\pi}{2}\right] \)
   

   Answer: A) \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \)

7. 22. If \( \sin^{-1}(x) = \theta \), then \( \cos(\theta) \) is:

   A) \( \sqrt{1-x^2} \)
   B) \( \sqrt{1+x^2} \)
   C) \( 1-x^2 \)
   D) \( 1+x^2 \)
   

   Answer: A) \( \sqrt{1-x^2} \)

8. 23. The domain of \( \sin^{-1}(x) \) is:

   A) \( [-1, 1] \)
   B) \( (-\infty, \infty) \)
   C) \( [0, 1] \)
   D) \( [-\infty, \infty) \)
   

   Answer: A) \( [-1, 1] \)

9. 24. The range of \( \tan^{-1}(x) \) is:

   A) \( [-\pi, \pi] \)
   B) \( [0, \frac{\pi}{2}] \)
   C) \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \)
   D) \( [0, \pi] \)
   

   Answer: C) \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \)

10. 25. The function \( \cos^{-1}(x) \) is defined for \( x \) in:

    A) \( [-1, 1] \)
    B) \( (-\infty, \infty) \)
    C) \( [0, 1] \)
    D) \( [0, \pi] \)
    

    Answer: A) \( [-1, 1] \)

11. 26. The identity \( \sin(\sin^{-1}x) \) is equal to:

    A) \( x \)
    B) \( \sqrt{1-x^2} \)
    C) \( 1 - x^2 \)
    D) \( \sqrt{x^2} \)
    

    Answer: A) \( x \)

12. 27. The principal value of \( \tan^{-1}(1) \) is:

    A) \( 0 \)
    B) \( \frac{\pi}{4} \)
    C) \( \frac{\pi}{2} \)
    D) \( \frac{3\pi}{4} \)
    

    Answer: B) \( \frac{\pi}{4} \)

13. 28. The inverse function of \( \sin(x) \) is:

    A) \( \cos^{-1}(x) \)
    B) \( \tan^{-1}(x) \)
    C) \( \sin^{-1}(x) \)
    D) \( \sec^{-1}(x) \)
    

    Answer: C) \( \sin^{-1}(x) \)

14. 29. The function \( \tan^{-1}(x) \) is defined for:

    A) \( x \in [-1, 1] \)
    B) \( x \in (-\infty, \infty) \)
    C) \( x \in [0, \infty) \)
    D) \( x \in [0, 1] \)
    

    Answer: B) \( x \in (-\infty, \infty) \)

15. 30. The range of \( \sin^{-1}(x) \) is:

    A) \( [-\frac{\pi}{2}, \frac{\pi}{2}] \)
    B) \( [0, \frac{\pi}{2}] \)
    C) \( [-\pi, \pi] \)
    D) \( [0, \pi] \)
    

    Answer: A) \( [-\frac{\pi}{2}, \frac{\pi}{2}] \)

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