TRB BT Assistant - Laplace Transform of Elementary and Special Functions Online Test (MCQ Quiz)

1. What is the Laplace transform of the constant function f(t) = 1, for t >= 0?

A) s B) 1/s C) 1/(s+1) D) s+1

2. Determine the Laplace transform of the exponential function f(t) = e^(at), for t >= 0, where 'a' is a constant.

A) 1/s B) 1/(s-a) C) 1/(s+a) D) a/s

3. Find the Laplace transform of the sine function f(t) = sin(bt), for t >= 0, where 'b' is a constant.

A) b/s B) s/(s^2 + b^2) C) b/(s^2 + b^2) D) 1/(s^2 + b^2)

4. Calculate the Laplace transform of the cosine function f(t) = cos(bt), for t >= 0, where 'b' is a constant.

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

5. What is the Laplace transform of the function f(t) = t^n, for t >= 0, where 'n' is a non-negative integer?

A) n!/s^(n+1) B) (n+1)!/s^n C) n/s^(n+1) D) 1/(s-n)

6. Find the Laplace transform of the hyperbolic sine function f(t) = sinh(bt), for t >= 0, where 'b' is a constant.

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

7. Determine the Laplace transform of the hyperbolic cosine function f(t) = cosh(bt), for t >= 0, where 'b' is a constant.

A) b/(s^2 + b^2) B) s/(s^2 + b^2) C) s/(s^2 - b^2) D) b/(s^2 - b^2)

8. What is the Laplace transform of the Dirac delta function, delta(t), assuming delta(t) = 0 for t < 0?

A) 0 B) 1 C) s D) 1/s

9. Find the Laplace transform of the unit step function u(t), defined as u(t) = 0 for t < 0 and u(t) = 1 for t >= 0.

A) 1 B) s C) 1/s D) 1/(s+1)

10. Calculate the Laplace transform of the function f(t) = t*e^(at), for t >= 0.

A) 1/(s-a)^2 B) a/(s-a)^2 C) 1/(s^2 - a^2) D) a/(s^2 - a^2)

11. Determine the Laplace transform of f(t) = t*sin(bt), for t >= 0.

A) 2bs/(s^2 + b^2)^2 B) b/(s^2 + b^2)^2 C) s/(s^2 + b^2)^2 D) 2b/(s^2 + b^2)^2

12. What is the Laplace transform of f(t) = t*cos(bt), for t >= 0?

A) (s^2 - b^2)/(s^2 + b^2)^2 B) s/(s^2 + b^2)^2 C) 2s/(s^2 + b^2)^2 D) (s^2 + b^2)/(s^2 + b^2)^2

13. Find the Laplace transform of the Gamma function Gamma(z+1) for z > -1.

A) Gamma(z)/s^(z+1) B) Gamma(z+1)/s^(z+1) C) z!/s^(z+1) D) z/s^(z+1)

14. Calculate the Laplace transform of the function f(t) = t^a, where 'a' is a real number greater than -1.

A) Gamma(a+1)/s^(a+1) B) a!/s^(a+1) C) Gamma(a)/s^a D) a/s^(a+1)

15. What is the Laplace transform of the function f(t) = (e^(at) - e^(bt))/ (a-b), for t >= 0?

A) 1/((s-a)(s-b)) B) 1/((s+a)(s+b)) C) (a-b)/((s-a)(s-b)) D) 1/(s-a) - 1/(s-b)

16. Determine the Laplace transform of the function f(t) = (a*e^(at) - b*e^(bt)) / (a-b), for t >= 0.

A) s/((s-a)(s-b)) B) (a-b)/((s-a)(s-b)) C) 1/((s-a)(s-b)) D) a/(s-a) - b/(s-b)

17. Find the Laplace transform of the function f(t) = sin(bt) * e^(at), for t >= 0.

A) b/((s-a)^2 + b^2) B) (s-a)/((s-a)^2 + b^2) C) b/((s+a)^2 + b^2) D) (s+a)/((s+a)^2 + b^2)

18. Calculate the Laplace transform of the function f(t) = cos(bt) * e^(at), for t >= 0.

A) (s-a)/((s-a)^2 + b^2) B) b/((s-a)^2 + b^2) C) (s+a)/((s+a)^2 + b^2) D) b/((s+a)^2 + b^2)

19. What is the Laplace transform of the function f(t) = t^n * e^(at), for t >= 0, where 'n' is a non-negative integer?

A) n!/(s-a)^(n+1) B) (n+1)!/(s-a)^n C) n!/(s+a)^(n+1) D) 1/(s-a)^(n+1)

20. Determine the Laplace transform of the function f(t) = t^a * e^(bt), for t >= 0, where 'a' is a real number greater than -1.

A) Gamma(a+1)/(s-b)^(a+1) B) Gamma(a)/(s-b)^a C) (a+1)!/(s-b)^(a+1) D) Gamma(a+1)/(s+b)^(a+1)

21. Find the Laplace transform of the function f(t) = 1 - cos(bt), for t >= 0.

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

22. Calculate the Laplace transform of the function f(t) = t - sin(bt), for t >= 0.

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

23. What is the Laplace transform of the function f(t) = (e^(at) - 1)/a, for t >= 0, assuming a is not zero?

A) 1/(a*s*(s-a)) B) 1/(s*(s-a)) C) 1/(a*s*(s+a)) D) 1/(a*(s-a))

24. Determine the Laplace transform of the function f(t) = (t*e^(at))/b, for t >= 0.

A) 1/(b*(s-a)^2) B) a/(b*(s-a)^2) C) 1/(b*(s^2 - a^2)) D) 1/(b*(s-a))

25. Find the Laplace transform of the function f(t) = t^2 * sin(bt), for t >= 0.

A) 2b(3s^2 - b^2)/(s^2 + b^2)^3 B) b(3s^2 - b^2)/(s^2 + b^2)^3 C) 2bs/(s^2 + b^2)^3 D) 2b(s^2 - b^2)/(s^2 + b^2)^3

26. Calculate the Laplace transform of the function f(t) = t^2 * cos(bt), for t >= 0.

A) 2s(s^2 - 3b^2)/(s^2 + b^2)^3 B) s(s^2 - 3b^2)/(s^2 + b^2)^3 C) 2s^2/(s^2 + b^2)^3 D) 2s(s^2 + 3b^2)/(s^2 + b^2)^3

27. What is the Laplace transform of the function f(t) = e^(-at) * cosh(bt), for t >= 0?

A) (s+a)/((s+a)^2 - b^2) B) b/((s+a)^2 - b^2) C) (s+a)/((s+a)^2 + b^2) D) b/((s+a)^2 + b^2)

28. Determine the Laplace transform of the function f(t) = e^(-at) * sinh(bt), for t >= 0.

A) b/((s+a)^2 - b^2) B) (s+a)/((s+a)^2 - b^2) C) b/((s+a)^2 + b^2) D) (s+a)/((s+a)^2 + b^2)

29. Find the Laplace transform of the function f(t) = t^n * sin(bt), where n is a positive integer.

A) (-1)^n * d^n/ds^n [b/(s^2 + b^2)] B) (-1)^n * d^n/ds^n [s/(s^2 + b^2)] C) d^n/ds^n [b/(s^2 + b^2)] D) d^n/ds^n [s/(s^2 + b^2)]

30. Calculate the Laplace transform of the function f(t) = t^n * cos(bt), where n is a positive integer.

A) (-1)^n * d^n/ds^n [s/(s^2 + b^2)] B) (-1)^n * d^n/ds^n [b/(s^2 + b^2)] C) d^n/ds^n [s/(s^2 + b^2)] D) d^n/ds^n [b/(s^2 + b^2)]

31. What is the Laplace transform of the function f(t) = sin(at+b)?

A) (a cos b + s sin b)/(s^2 + a^2) B) (a sin b + s cos b)/(s^2 + a^2) C) (a sin b + s sin b)/(s^2 + a^2) D) (a cos b + s cos b)/(s^2 + a^2)

32. Determine the Laplace transform of the function f(t) = cos(at+b)?

A) (s cos b - a sin b)/(s^2 + a^2) B) (s sin b - a cos b)/(s^2 + a^2) C) (s cos b + a sin b)/(s^2 + a^2) D) (s cos b - a cos b)/(s^2 + a^2)

33. Find the Laplace transform of the function f(t) = t * sinh(bt).

A) 2bs/(s^2 - b^2)^2 B) b/(s^2 - b^2)^2 C) s/(s^2 - b^2)^2 D) 2b/(s^2 - b^2)^2

34. Calculate the Laplace transform of the function f(t) = t * cosh(bt).

A) s/(s^2 - b^2)^2 B) b/(s^2 - b^2)^2 C) 2bs/(s^2 - b^2)^2 D) 2b/(s^2 - b^2)^2

35. What is the Laplace transform of the function f(t) = t^a, where a is a positive integer?

A) a!/s^(a+1) B) (a+1)!/s^a C) a/s^(a+1) D) Gamma(a+1)/s^(a+1)

36. Find the Laplace transform of the function f(t) = delta(t-a), for a > 0.

A) e^(-as) B) e^(as) C) 1 D) s

37. Determine the Laplace transform of the function f(t) = a*delta(t).

A) a B) 0 C) 1 D) s

38. Calculate the Laplace transform of the function f(t) = t*u(t).

A) 1/s^2 B) 1/s C) s D) 2/s^3

39. What is the Laplace transform of the function f(t) = t^2 * u(t).

A) 2/s^3 B) 1/s^2 C) s^2 D) 6/s^4

40. Find the Laplace transform of the function f(t) = e^(at) * u(t).

A) 1/(s-a) B) 1/(s+a) C) 1/s D) a/s

41. Determine the Laplace transform of the function f(t) = sin(bt) * u(t).

A) b/(s^2 + b^2) B) s/(s^2 + b^2) C) 1/(s^2 + b^2) D) b/s

42. Calculate the Laplace transform of the function f(t) = cos(bt) * u(t).

A) s/(s^2 + b^2) B) b/(s^2 + b^2) C) 1/(s^2 + b^2) D) s/b

43. What is the Laplace transform of the function f(t) = t^n * u(t), where n is a non-negative integer?

A) n!/s^(n+1) B) (n+1)!/s^n C) n/s^(n+1) D) 1/(s-n)

44. Find the Laplace transform of the function f(t) = (e^(at) - 1)/(a-1), for t >= 0, assuming a != 1.

A) 1/((s-1)(s-a)) B) 1/(s(s-a)) C) 1/(s(s-1)) D) 1/(s-a)

45. Determine the Laplace transform of the function f(t) = (e^(at) - e^(bt))/(a-b), for t >= 0, assuming a != b.

A) 1/((s-a)(s-b)) B) 1/(s-a) - 1/(s-b) C) (a-b)/((s-a)(s-b)) D) 1/(s-a) + 1/(s-b)

46. Calculate the Laplace transform of the function f(t) = t * e^(-at).

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

47. What is the Laplace transform of the function f(t) = t * sin(at)?

A) 2a*s/(s^2 + a^2)^2 B) a/(s^2 + a^2)^2 C) s/(s^2 + a^2)^2 D) 2a/(s^2 + a^2)^2

48. Find the Laplace transform of the function f(t) = t * cos(at).

A) (s^2 - a^2)/(s^2 + a^2)^2 B) s/(s^2 + a^2)^2 C) 2s/(s^2 + a^2)^2 D) (s^2 + a^2)/(s^2 + a^2)^2

49. Determine the Laplace transform of the function f(t) = t^2 * e^(at).

A) 2/(s-a)^3 B) a/(s-a)^3 C) 2a/(s-a)^3 D) 1/(s-a)^3

Laplace Transform of Elementary and Special Functions - Online Test

Test Results