Free Printable Laplace and Inverse Laplace Transform

1.

Laplace Transform transforms a function from

a)

$f\left(t\right)\ to\ F\left(s\right)$

b)

$F\left(s\right)\ to\ f\left(t\right)$

c)

$f\left(t\right)\ to\ f'\left(t\right)$

d)

$f"\left(t\right)\ to\ f'\left(t\right)$

2.

Which one is the correct definition of Laplace Transform?

a)

$F\left(s\right)=\int_0^{\infty}e^{-st}f\left(t\right)dt$

b)

$F\left(s\right)=\int_0^tf\left(t\right)g\left(t-u\right)$

3.

What is Laplace Transform for $e^{-2t}$ ?

a)

$\frac{1}{s-2}$

b)

$\frac{1}{s+2}$

c)

$\frac{2}{s}$

d)

$\frac{1}{s^2}$

4.

What is Laplace Transform for $\sin\ 2t$ ?

a)

$\frac{2}{s^2+2}$

b)

$\frac{s}{s^2+2}$

c)

$\frac{2}{s^2+4}$

d)

$\frac{s}{s^2+4}$

5.

What is Laplace Transform for $6\ \sin\ 2t$ ?

a)

$\frac{2}{s^2+4}$

b)

$\frac{s}{s^2+4}$

c)

$\frac{6}{s^2+4}$

d)

$\frac{12}{s^2+4}$

6.

Define this property: $L\left(f\left(t\right)e^{at}\right)=F\left(s-a\right)$

a)

Linearity Property

b)

First Shifting Property

c)

Convolution Theorem

d)

Second Shifting Property

7.

What is the formula for Laplace First Order Derivative?

a)

$ℒ\left(𝑓′(𝑡)\right)=𝑠𝐹(𝑠)−𝑓(0)$

b)

$ℒ\left(𝑓"(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑠𝑓\left(0\right)−𝑓′\left(0\right)$

c)

$ℒ\left(𝑓′(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑓(0)$

8.

$L\left(e^{3t}t\right)$ Based on first shifting property, what is $a$ value?

a)

1

b)

2

c)

3

d)

4

9.

Find $L\left(e^{3t}t\right)$

a)

$\frac{1}{\left(s^2\right)}$

b)

$\frac{1}{\left(s-3\right)^2}$

c)

$\frac{1}{\left(s-2\right)^2}$

d)

$\frac{1}{\left(s\right)}$

10.

$L\left(e^{2t\ }\sin\ t\right)$ What is a and f(t)?

a)

$a=2,\ f\left(t\right)=e^{2t}$

b)

$a=4,\ f\left(t\right)=\cos\ t$

c)

$a=2,\ f\left(t\right)=\sin\ t$

11.

Find $L\left(e^{2t}\sin\ 2t\right)$

a)

$\frac{1}{\left(s-1\right)^2+1^{ }}$

b)

$\frac{s}{\left(s-2\right)^2+4}$

c)

$\frac{s}{\left(s-4\right)^2+1}$

12.

Find $L\left(t^3-3t^2+5t\right)$

a)

$\frac{6}{s^4}-\left(\frac{6}{s^3}\right)+\frac{5}{s^2}$

b)

$\frac{3}{s^3}-\left(\frac{1}{s^{\left(2\right)}}\right)+\frac{1}{s}$

c)

$\frac{2}{s^3}-\left(\frac{3}{s^{\left(2\right)}}\right)+\frac{5}{s}$

13.

$L\left(t\ \cos\ 6t\right)$ Which Laplace property is use to solve this?

a)

Linearity Property

b)

Derivative of Laplace Transform

c)

First Shifting Property

d)

Second Shifting Property

14.

$L\left(t\ \sin\ 4t\right)$ What is n, f(t) and F(s)?

a)

$n=1,\ f\left(t\right)=\sin\ 4t,\ F\left(s\right)=\frac{s}{s^2+16}$

b)

$n=4,\ f\left(t\right)=t,\ F\left(s\right)=\frac{4}{s^2+16}$

c)

$n=1,\ f\left(t\right)=\cos\ 6t,\ F\left(s\right)=\frac{s}{s^2+14}$

15.

$Find\ L\left(t^{ }\sin\ 4t\right)$

a)

$\frac{2}{s^2+4}$

b)

$-\frac{4s}{\left(s^2+4\right)^2}$

c)

$-\frac{8s}{\left(s^2+16\right)^2}$

16.

$L\left(\sin^2t\right)$ What is $f\left(t\right),\ f'\left(t\right)\ and\ f\left(0\right)$ ?

a)

$f\left(t\right)=\sin^2t,\ f'\left(t\right)=2\ \cos t,\ f\left(0\right)=0$

b)

$f\left(t\right)=\sin^2t,\ f'\left(t\right)=\sin\ 2t,\ f\left(0\right)=0$

c)

$f\left(t\right)=\cos^2t,\ f'\left(t\right)=-\ \sin2t,\ f\left(0\right)=1$

17.

What is the formula for Laplace Second Order Derivative?

a)

$ℒ\left(𝑓′(𝑡)\right)=𝑠𝐹(𝑠)−𝑓(0)$

b)

$ℒ\left(𝑓"(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑠𝑓\left(0\right)−𝑓′\left(0\right)$

c)

$ℒ\left(𝑓"(𝑡)\right)=𝑠^2𝐹(𝑠)−𝑓(0)$

18.

Find $L\left(\cos\ t\right)$

a)

$\frac{1}{s^2+1^{ }}$

b)

$\frac{1}{s^2+4}$

c)

$\frac{s}{s^2+1}$

19.

Find $L\left(e^{3t}\left(t^3-3t^2+5t\right)\right)$

a)

$\frac{2}{\left(s-3\right)^4}-\left(\frac{3}{\left(\left(s-3\right)^3\right)}\right)+\frac{5}{\left(s-3\right)^2}$

b)

$\frac{6}{\left(s-3\right)^4}-\left(\frac{6}{\left(s-3\right)^3}\right)+\frac{5}{\left(s-3\right)^2}$

c)

$\frac{2}{\left(s-3\right)^3}-\left(\frac{3}{\left(s-3\right)^{\left(2\right)}}\right)+\frac{5}{s-3}$

20.

What is the Laplace Transform for $f\left(t\right)=t^3$ ?

a)

$\frac{1}{s^3}$

b)

$\frac{2}{s^3}$

c)

$\frac{6}{s^4}$

d)

$\frac{3}{s^3}$

21.

Inverse Laplace transform, transform

a)

$f\left(t\right)\ to\ F\left(s\right)$

b)

$F\left(s\right)\ to\ f\left(t\right)$

c)

$f'\left(t\ \right)\ to\ f\left(t\right)$

d)

$f\left(t\right)\ to\ f'\left(t\right)$

22.

Find $L^{-1}\left(\frac{s}{s^2+25}\right)$

a)

cos 25t

b)

sin 5t

c)

cos 5t

23.

Find $L^{-1}\left(\frac{1}{s^2+1}\right)$

a)

sin t

b)

sin 5t

c)

cos t

24.

$L^{-1}\left(\frac{6}{\left(s\right)}-\frac{4s}{s^2+25}\right)$ What is the property use to find the Inverse Laplace Transform of this equation?

a)

Second Shifting Property

b)

First Shifting Property

c)

Linearity Property

d)

Convolution theorem

25.

$L^{-1\ }\left(\frac{4s}{s^2+25}+\frac{2}{3\left(s^2+1\right)}\right)$

a)

$4\ \cos\ 25t+\frac{2}{3}\sin\ t$

b)

$\cos\ 5t+\ \sin\ t$

c)

$4\ \cos\ 5t\ +\frac{2}{3}\sin\ t$

26.

The $L^{-1}\left(\frac{4}{\left(s^2+9\right)}\right)$ is $a\ \sin\ 3t$ . What is a?

a)

$\frac{2}{3}$

b)

$\frac{4}{3}$

c)

$3$

27.

$L\left(t^{\left(4\right)}\right)$ ?

a)

$\frac{24}{s^5}$

b)

$\frac{4}{s^5}$

c)

$\frac{1}{s^5}$

28.

$L^{-1}\left(\frac{24}{s^5}\right)$ is $t^4$ . What is $L^{-1}\left(\frac{1}{s^{\left(5\right)}}\right)$ ?

a)

$\frac{24}{s^5}$

b)

$\frac{1}{24}t^4$

c)

$\frac{1}{24}t^5$

29.

$L^{-1}\left(\frac{4}{s^{\left(3\right)}}\right)$

a)

$4t^3$

b)

$2t^2$

c)

$2t^3$

30.

$L^{-1}\left(\frac{3}{s^2+36}\right)$

a)

$\frac{1}{2\ }\sin\ 6t$

b)

$\frac{3}{5}\sin\ 5t$

c)

$\cosh2t$

31.

$L^{-1}\left(F\left(s-5\right)\right)=e^{5t}f\left(t\right)$ Based on first shifting property for Inverse Laplace transform, what is $a$ value?

a)

4

b)

5

c)

3

32.

$L^{-1}\left(\frac{1}{\left(s-3\right)^2}\right)$

a)

$e^{2t}t$

b)

$e^tt^2$

c)

$e^{3t}t$

33.

$L^{-1}\left(\frac{3s}{\left(s+2\right)^4}\right)$

a)

$\frac{1}{2}e^{-t}\left(3t^2-t^{\left(3\right)}\right)$

b)

$e^{-2t}\left(\frac{3}{2}t^2-t^{\left(3\right)}\right)$

c)

$e^{-t}\left(\frac{3}{2}t^2-t^{\left(3\right)}\right)$

34.

$L^{-1}\left(\frac{s}{\left(s-4\right)^2+16}\right)$

a)

$e^{4t}\left(\cos\ 4t+\sin\ 4t\right)$

b)

$e^{2t}\left(\cos\ 2t+\sin\ 2t\right)$

c)

$e^{2t}\left(4\cos\ t+4\sin\ t\right)$

35.

$s^2+2s+5$ Complete the square

a)

$\left(s+4\right)^2+2$

b)

$\left(s+2\right)^2+4$

c)

$\left(s+2\right)^2+16$

36.

$L^{-1}\left(\frac{\left(s+2\right)}{s^2+2s+5}\right)$

a)

$f(t)=e^{-t}\sin(2t)+e^{-t}\cos(2t)$

b)

$f(t)=\frac{1}{2}e^{-t}\sin(2t)+e^{-t}\cos(2t)$

c)

$f(t)=e^{-t}\sin(2t)+\frac{1}{2}e^{-t}\cos(2t)$

37.

Find the partial fraction for $\frac{\left(2s-8\right)}{\left(s-2\right)\left(s-3\right)}$

a)

$\frac{2}{s-2}-\frac{4}{s-3}$

b)

$\frac{3}{s-2}-\frac{2}{s-3}$

c)

$\frac{4}{s-2}-\frac{2}{s-3}$

38.

Find the Inverse Laplace for $\frac{\left(2s-8\right)}{\left(s-2\right)\left(s-3\right)}$

a)

$4e^{-2t}-2e^{-3t}$

b)

$2e^{-2t}-4e^{-3t}$

c)

$e^{-2t}-e^{-3t}$

39.

By convolution theorem, find Laplace Transform for $X\left(s\right)=\frac{1}{s\left(s^2+4\right)}$

a)

$\frac{1}{2\ }\sin\ 2t$

b)

$\frac{1}{4}\cos\ 2t-4$

c)

$\frac{1}{4}\left(1-\cos\ 2t\right)$

40.

By convolution theorem, find F(s) and G(s) $X\left(s\right)=\frac{1}{s\left(s^2+4\right)}$

a)

$F\left(s\right)=\frac{1}{s};G\left(s\right)=\frac{1}{s^2+4}$

b)

$F\left(s\right)=\frac{1}{s^{ }};G\left(s\right)=\frac{1}{s^3+4s}$

c)

$F\left(s\right)=\frac{1}{s^2+4};G\left(s\right)=\frac{1}{s^{ }}$

Laplace and Inverse Laplace Transform

Total questions: 40

Worksheet time: 20mins

Similar Resources on Wayground