JEE Main - Inverse trigonometric functions and their properties Online Test (MCQ Quiz)

1. What is the principal value range of the inverse trigonometric function arcsin(x)?

A) [0, π] B) [-π/2, π/2] C) (0, π) D) (-π/2, π/2)

2. The principal value range of arccos(x) is:

A) [0, π] B) [-π/2, π/2] C) (0, π) D) (-π/2, π/2)

3. What is the domain of the inverse trigonometric function arctan(x)?

A) [-1, 1] B) (-∞, ∞) C) [0, π] D) (-∞, 0) U (0, ∞)

4. The principal value range of arctan(x) is:

A) [-π/2, π/2] B) (0, π) C) (-π/2, π/2) D) [0, π]

5. What is the value of arcsin(1/2)?

A) π/6 B) π/4 C) π/3 D) π/2

6. Find the value of arccos(-1/2).

A) π/3 B) 2π/3 C) π/2 D) π

7. What is the value of arctan(√3)?

A) π/6 B) π/4 C) π/3 D) π/2

8. Evaluate arcsin(-1).

A) π/2 B) -π/2 C) 0 D) π

9. The value of arccos(0) is:

A) 0 B) π/2 C) π D) 3π/2

10. What is the value of arctan(0)?

A) π/2 B) π C) 0 D) -π/2

11. Which property relates arcsin(x) and arccos(x)?

A) arcsin(x) + arccos(x) = π/2 B) arcsin(x) - arccos(x) = π/2 C) arcsin(x) + arccos(x) = π D) arcsin(x) * arccos(x) = 1

12. What is the value of tan(arctan(x))?

A) x B) 1/x C) 0 D) undefined

13. What is the value of sin(arcsin(x))?

A) x B) 1/x C) 0 D) undefined

14. What is the value of cos(arccos(x))?

A) x B) 1/x C) 0 D) undefined

15. The property 2arctan(x) = arctan(2x / (1 - x^2)) is valid when:

A) x < 1 B) x > 1 C) x = 1 D) x = 0

16. The property 2arctan(x) = π + arctan(2x / (1 - x^2)) is valid when:

A) x < 1 B) x > 1 C) x = 1 D) x = 0

17. The property 2arctan(x) = -π + arctan(2x / (1 - x^2)) is valid when:

A) x < -1 B) x > 1 C) x = -1 D) x = 0

18. What is the value of arctan(1) + arctan(2) + arctan(3)?

A) π B) 3π/2 C) π/2 D) 2π

19. If x > 0, then sin⁻¹(x) + cos⁻¹(x) = ?

A) π/2 B) π C) 0 D) 2π

20. If x < 0, then sin⁻¹(x) + cos⁻¹(x) = ?

A) π/2 B) π C) 0 D) 2π

21. What is the value of tan(arcsin(x))?

A) x / √(1 - x²) B) √(1 - x²) / x C) 1 / √(1 - x²) D) √(1 - x²)

22. What is the value of cos(arctan(x))?

A) 1 / √(1 + x²) B) x / √(1 + x²) C) √(1 + x²) D) 1 / (1 + x²)

23. The expression arctan(x) + arctan(y) is equal to:

A) arctan((x+y)/(1-xy)) B) arctan((x+y)/(1+xy)) C) arctan((x-y)/(1+xy)) D) arctan((x-y)/(1-xy))

24. The expression arctan(x) - arctan(y) is equal to:

A) arctan((x+y)/(1-xy)) B) arctan((x-y)/(1+xy)) C) arctan((x+y)/(1+xy)) D) arctan((x-y)/(1-xy))

25. The principal value of arccot(x) lies in the interval:

A) (0, π) B) [0, π] C) (0, π] D) [0, π)

26. What is the value of arccot(1)?

A) π/4 B) π/2 C) π D) 0

27. What is the relationship between arccot(x) and arctan(x)?

A) arccot(x) + arctan(x) = π/2 B) arccot(x) - arctan(x) = π/2 C) arccot(x) + arctan(x) = π D) arccot(x) = 1/arctan(x)

28. The value of 2arcsin(x) is equal to:

A) arccos(1 - 2x²) B) arcsin(2x√(1-x²)) C) arccos(2x² - 1) D) arcsin(x√2)

29. The value of 2arccos(x) is equal to:

A) arccos(2x² - 1) B) arccos(1 - 2x²) C) 2arccos(√x) D) arccos(x²)

30. What is the value of arcsin(sin(2π/3))?

A) 2π/3 B) π/3 C) π/6 D) π

31. What is the value of arccos(cos(4π/3))?

A) 4π/3 B) 2π/3 C) π/3 D) π

32. What is the value of arctan(tan(5π/4))?

A) 5π/4 B) π/4 C) 3π/4 D) π

33. The domain of arcsin(x) is:

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

34. The domain of arccos(x) is:

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

35. The domain of arctan(x) is:

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

36. The range of arccsc(x) is:

A) [-π/2, π/2] B) [-π/2, 0) U (0, π/2] C) [0, π] D) (0, π)

37. The range of arcsec(x) is:

A) [0, π] B) [0, π/2) U (π/2, π] C) (-π/2, π/2) D) [0, π/2]

38. What is the value of arcsin(x) + arcsin(y)?

A) arcsin(x√(1-y²) + y√(1-x²)) B) arcsin(x√(1+y²) + y√(1+x²)) C) arcsin(x√(1-y²) - y√(1-x²)) D) arcsin(x+y)

39. What is the value of arccos(x) + arccos(y)?

A) arccos(xy - √(1-x²)√(1-y²)) B) arccos(xy + √(1-x²)√(1-y²)) C) arccos(x+y) D) arccos(xy)

40. What is the value of arcsin(x) - arcsin(y)?

A) arcsin(x√(1-y²) - y√(1-x²)) B) arcsin(x√(1+y²) - y√(1+x²)) C) arcsin(x√(1-y²) + y√(1-x²)) D) arcsin(x-y)

41. What is the value of arccos(x) - arccos(y)?

A) arccos(xy + √(1-x²)√(1-y²)) B) arccos(xy - √(1-x²)√(1-y²)) C) arccos(x-y) D) arccos(x+y)

42. The value of arcsin(1/3) + arcsin(4/5) is:

A) arcsin(63/65) B) arcsin(5/8) C) arcsin(17/15) D) arcsin(1/2)

43. The value of arccos(12/13) + arccos(4/5) is:

A) arccos(33/65) B) arccos(48/65) C) arccos(16/65) D) arccos(1/1)

44. If tan⁻¹(x) + tan⁻¹(y) = π/2, then:

A) xy = 1 B) x + y = 1 C) xy = -1 D) x = y

45. What is the value of sin(2 arctan(3/4))?

A) 24/25 B) 7/25 C) 12/25 D) 4/5

46. What is the value of cos(2 arctan(1/2))?

A) 3/5 B) 4/5 C) 1/5 D) 2/5

47. What is the value of tan(2 arctan(1/3))?

A) 3/4 B) 2/3 C) 1/3 D) 1/2

48. The expression cot⁻¹(x) is equal to:

A) tan⁻¹(1/x) for x > 0, π + tan⁻¹(1/x) for x < 0 B) tan⁻¹(1/x) C) π - tan⁻¹(1/x) D) π/2 - tan⁻¹(x)

49. What is the value of tan(arcsin(3/5))?

A) 3/4 B) 4/3 C) 5/3 D) 3/5

50. What is the value of sin(arccos(5/13))?

A) 12/13 B) 5/13 C) 13/12 D) 13/5

Inverse trigonometric functions and their properties - Online Test

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