JEE Main - Polynomial rational trigonometric logarithmic and exponential functions Online Test (MCQ Quiz)

1. What is the limit of the function f(x) = (x^2 - 4) / (x - 2) as x approaches 2?

A) 0 B) 2 C) 4 D) Undefined

2. For a function f(x) to be continuous at x = c, which of the following conditions must be met?

A) f(c) must be defined. B) The limit of f(x) as x approaches c must exist. C) The limit of f(x) as x approaches c must equal f(c). D) All of the above.

3. If f(x) = |x|, what is the derivative of f(x) at x = 0?

A) 1 B) -1 C) 0 D) Undefined

4. What is the derivative of the function f(x) = x^n?

A) nx^(n-1) B) nx^n C) x^(n-1) D) (n-1)x^n

5. Evaluate the limit of sin(x)/x as x approaches 0.

A) 0 B) 1 C) infinity D) -1

6. What is the derivative of ln(x) with respect to x?

A) x B) 1/x C) ln(x) D) -1/x

7. If f(x) = e^x, what is the derivative of f(x)?

A) e^x B) x*e^(x-1) C) e D) 1

8. Determine if the function f(x) = x^2 is continuous at x = 3.

A) Yes, because f(3) = 9 and lim (x->3) f(x) = 9. B) No, because the function is a polynomial. C) Yes, because the function is differentiable. D) No, because the limit does not exist.

9. What is the limit of (x^3 - 1) / (x - 1) as x approaches 1?

A) 0 B) 1 C) 3 D) Undefined

10. Find the derivative of f(x) = 5x^4 - 3x^2 + 2x - 7.

A) 20x^3 - 6x + 2 B) 5x^3 - 3x + 2 C) 20x^4 - 6x^2 + 2x D) 20x^3 - 6x^2 + 2

11. Which of the following functions is NOT continuous at x = 0?

A) f(x) = x^2 B) f(x) = sin(x) C) f(x) = 1/x D) f(x) = e^x

12. What is the derivative of tan(x)?

A) sec^2(x) B) -csc^2(x) C) sec(x)tan(x) D) -sec(x)tan(x)

13. Evaluate the limit of (e^x - 1) / x as x approaches 0.

A) 0 B) 1 C) e D) Undefined

14. If f(x) = log_b(x), what is its derivative?

A) 1/x B) 1/(x*ln(b)) C) ln(b)/x D) x*ln(b)

15. Consider the function f(x) = x^2 for x < 0 and f(x) = x for x >= 0. Is f(x) continuous at x = 0?

A) Yes, because both parts approach 0. B) No, because the derivative is different from the left and right. C) Yes, because f(0) = 0 and the limits from both sides are 0. D) No, because the function definition changes at x = 0.

16. What is the limit of (x^2 + 5x + 6) / (x + 2) as x approaches -2?

A) -2 B) -1 C) 0 D) Undefined

17. Find the second derivative of f(x) = x^3.

A) 3x^2 B) 6x C) 3 D) 0

18. Which condition is NOT required for a function f(x) to be differentiable at x = c?

A) f(x) must be continuous at x = c. B) The limit of the difference quotient must exist. C) The derivative from the left must equal the derivative from the right. D) f(c) must be a real number.

19. Evaluate the limit of (1 - cos(x)) / x^2 as x approaches 0.

A) 0 B) 1/2 C) 1 D) Undefined

20. What is the derivative of arcsin(x)?

A) 1/sqrt(1 - x^2) B) -1/sqrt(1 - x^2) C) 1/sqrt(x^2 - 1) D) 1/(1 + x^2)

21. If f(x) = a^x, where a > 0 and a != 1, what is its derivative?

A) x*a^(x-1) B) a^x C) a^x * ln(a) D) a^x / ln(a)

22. Consider the function f(x) = x^2 if x is rational and f(x) = 0 if x is irrational. At which point is f(x) continuous?

A) x = 0 B) x = 1 C) All points D) No points

23. What is the limit of (sqrt(x) - 2) / (x - 4) as x approaches 4?

A) 0 B) 1/4 C) 1/2 D) Undefined

24. Find the derivative of f(x) = (x^2 + 1) / (x - 1) using the quotient rule.

A) (x^2 - 2x - 1) / (x - 1)^2 B) (x^2 + 2x + 1) / (x - 1)^2 C) (2x) / 1 D) (x^2 - 2x + 1) / (x - 1)^2

25. A function f(x) is differentiable at x = c if:

A) lim (h->0) [f(c+h) - f(c)] / h exists. B) f(x) is continuous at x = c. C) f'(c) exists. D) All of the above.

26. Evaluate the limit of tan(x) / x as x approaches 0.

A) 0 B) 1 C) infinity D) -1

27. What is the derivative of arccos(x)?

A) -1/sqrt(1 - x^2) B) 1/sqrt(1 - x^2) C) 1/sqrt(x^2 - 1) D) -1/(1 + x^2)

28. The function f(x) = |x - 2| is:

A) Continuous but not differentiable at x = 2. B) Differentiable at x = 2. C) Neither continuous nor differentiable at x = 2. D) Continuous and differentiable at x = 2.

29. What is the limit of (e^(2x) - 1) / x as x approaches 0?

A) 0 B) 1 C) 2 D) Undefined

30. Find the derivative of f(x) = sin(x^2).

A) cos(x^2) B) 2x*cos(x^2) C) cos(2x) D) 2x*sin(x^2)

31. If lim (x->a) f(x) = L and f(a) = L, then f(x) is:

A) Differentiable at x = a. B) Continuous at x = a. C) Undefined at x = a. D) Discontinuous at x = a.

32. What is the limit of (x^n - a^n) / (x - a) as x approaches a?

A) a^n B) na^(n-1) C) n*a^n D) Undefined

33. Find the derivative of f(x) = arctan(x).

A) 1/(1 + x^2) B) -1/(1 + x^2) C) 1/sqrt(1 - x^2) D) 1/x

34. The function f(x) = x^3 is:

A) Continuous everywhere but not differentiable at x = 0. B) Continuous and differentiable everywhere. C) Discontinuous at x = 0. D) Differentiable only for x > 0.

35. Evaluate the limit of (sin(ax)) / (bx) as x approaches 0, where a and b are non-zero constants.

A) 0 B) 1 C) a/b D) b/a

36. What is the derivative of f(x) = log_a(x^2)?

A) 2/(x*ln(a)) B) 1/(x*ln(a)) C) 2x/ln(a) D) 2/(x*log_a(e))

37. If f(x) = x * |x|, what is f'(0)?

A) 1 B) -1 C) 0 D) Undefined

38. Which of the following is NOT a property of limits?

A) lim (x->c) [f(x) + g(x)] = lim (x->c) f(x) + lim (x->c) g(x) B) lim (x->c) [f(x) * g(x)] = lim (x->c) f(x) * lim (x->c) g(x) C) lim (x->c) [f(x) / g(x)] = [lim (x->c) f(x)] / [lim (x->c) g(x)] if lim (x->c) g(x) != 0 D) lim (x->c) f(g(x)) = f(lim (x->c) g(x)) if f is continuous at lim (x->c) g(x)

39. What is the limit of (x^2) as x approaches infinity?

A) 0 B) 1 C) infinity D) -infinity

40. Find the derivative of f(x) = e^(sin(x)).

A) e^(sin(x)) B) cos(x) * e^(sin(x)) C) sin(x) * e^(cos(x)) D) e^(cos(x))

41. The function f(x) = x^3 - x is:

A) Continuous only at x = 0. B) Differentiable only at x = 0. C) Continuous and differentiable for all real numbers. D) Discontinuous at x = 1 and x = -1.

42. Evaluate the limit of (ln(x)) as x approaches 1.

A) 0 B) 1 C) e D) Undefined

43. What is the derivative of f(x) = x^x?

A) x^x * (1 + ln(x)) B) x * x^(x-1) C) x^x D) x^x * ln(x)

44. Consider the function f(x) = x^2 sin(1/x) for x != 0 and f(0) = 0. Is f(x) differentiable at x = 0?

A) Yes, the derivative is 0. B) No, because sin(1/x) is not defined at x=0. C) Yes, because the limit of the difference quotient exists. D) No, because the function is not continuous at x=0.

45. What is the limit of (3x^2 + 2x - 1) / (x^2 - 5) as x approaches infinity?

A) 0 B) 1 C) 3 D) Undefined

46. Find the derivative of f(x) = cos(3x).

A) -sin(3x) B) -3sin(3x) C) 3cos(3x) D) sin(3x)

47. If a function is differentiable at a point, it must be:

A) Constant B) Zero C) Continuous D) Increasing

48. What is the limit of (x - 5) / (x^2 - 25) as x approaches 5?

A) 0 B) 1/10 C) 1/5 D) Undefined

49. What is the derivative of f(x) = sqrt(x^2 + 1)?

A) x / sqrt(x^2 + 1) B) 1 / (2*sqrt(x^2 + 1)) C) sqrt(2x) D) x

50. The function f(x) = 1/x is:

A) Continuous everywhere. B) Differentiable everywhere. C) Continuous for all x != 0, but not differentiable at x = 0. D) Discontinuous at x = 0.

Polynomial rational trigonometric logarithmic and exponential functions - Online Test

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