Laplace Transforms Online Test

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Laplace Transforms

Description: This quiz is designed to assess your understanding of Laplace transforms, a mathematical tool used to solve differential equations and analyze linear systems.
Number of Questions: 15
Created by: Aliensbrain Bot
Tags: laplace transforms differential equations linear systems

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The Laplace transform of the function $f(t) = e^{at}$ is:

$F(s) = \frac{1}{s - a}$
$F(s) = \frac{1}{s + a}$
$F(s) = \frac{a}{s - a}$
$F(s) = \frac{a}{s + a}$

Which of the following is a property of the Laplace transform?

Linearity
Time shifting
Frequency shifting
All of the above

The inverse Laplace transform of $F(s) = \frac{1}{s^2 + 4}$ is:

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

The Laplace transform of the function $f(t) = t^2$ is:

$F(s) = \frac{2}{s^3}$
$F(s) = \frac{2}{s^4}$
$F(s) = \frac{2!}{s^3}$
$F(s) = \frac{2!}{s^4}$

The Laplace transform of the function $f(t) = \sin(at)$ is:

$F(s) = \frac{a}{s^2 + a^2}$
$F(s) = \frac{s}{s^2 + a^2}$
$F(s) = \frac{1}{s^2 + a^2}$
$F(s) = \frac{s}{s^2 - a^2}$

The Laplace transform of the function $f(t) = \cos(at)$ is:

$F(s) = \frac{s}{s^2 + a^2}$
$F(s) = \frac{a}{s^2 + a^2}$
$F(s) = \frac{1}{s^2 + a^2}$
$F(s) = \frac{s}{s^2 - a^2}$

The Laplace transform of the function $f(t) = e^{at} \sin(bt)$ is:

$F(s) = \frac{b}{(s - a)^2 + b^2}$
$F(s) = \frac{a}{(s - a)^2 + b^2}$
$F(s) = \frac{1}{(s - a)^2 + b^2}$
$F(s) = \frac{s}{(s - a)^2 + b^2}$

The Laplace transform of the function $f(t) = e^{at} \cos(bt)$ is:

$F(s) = \frac{s - a}{(s - a)^2 + b^2}$
$F(s) = \frac{s + a}{(s - a)^2 + b^2}$
$F(s) = \frac{1}{(s - a)^2 + b^2}$
$F(s) = \frac{s}{(s - a)^2 + b^2}$

The Laplace transform of the function $f(t) = \sinh(at)$ is:

$F(s) = \frac{a}{s^2 - a^2}$
$F(s) = \frac{s}{s^2 - a^2}$
$F(s) = \frac{1}{s^2 - a^2}$
$F(s) = \frac{s}{s^2 + a^2}$

The Laplace transform of the function $f(t) = \cosh(at)$ is:

$F(s) = \frac{s}{s^2 - a^2}$
$F(s) = \frac{a}{s^2 - a^2}$
$F(s) = \frac{1}{s^2 - a^2}$
$F(s) = \frac{s}{s^2 + a^2}$

The Laplace transform of the function $f(t) = t^n$ is:

$F(s) = \frac{n!}{s^{n+1}}$
$F(s) = \frac{n!}{s^n}$
$F(s) = \frac{n!}{s^{n-1}}$
$F(s) = \frac{n!}{s^{n+2}}$

The Laplace transform of the function $f(t) = \frac{1}{t}$ is:

$F(s) = \ln(s)$
$F(s) = \frac{1}{s}$
$F(s) = \frac{1}{s^2}$
$F(s) = \frac{1}{s^3}$

The Laplace transform of the function $f(t) = \delta(t)$ is:

$F(s) = 1$
$F(s) = 0$
$F(s) = \infty$
$F(s) = \frac{1}{s}$

The Laplace transform of the function $f(t) = u(t - a)$ is:

$F(s) = \frac{1}{s} e^{-as}$
$F(s) = \frac{1}{s} e^{as}$
$F(s) = e^{-as}$
$F(s) = e^{as}$

The Laplace transform of the function $f(t) = t u(t - a)$ is:

$F(s) = \frac{1}{s^2} e^{-as}$
$F(s) = \frac{1}{s^2} e^{as}$
$F(s) = e^{-as}$
$F(s) = e^{as}$

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