Laplace transforms and Fourier Transforms 1

Fourier and Laplace transforms - 1
Welcome to the Fourier and Laplace transforms revision assessment 1.
There are 22 questions in total in different topics. You may need your calculator at hand.
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1 of 22

For a suitably defined s find Laplace transform of

2 {cos}^{2} (11 t)

\frac{2 (s^{2} + 242)}{s^{3} + 484 s}

\frac{484}{s^{3} + 484 s}

\frac{2 (s^{2} + s + 484)}{s^{2} + 484}

\frac{2 (s^{2} + s + 484)}{s^{3} + 484}

None of these!

I don't know!

2 of 22

For suitably defined s, find the correct expression for the Laplace transform of

e^{- 4 x} sin (7 x)

\int_{0}^{\infty} e^{- (4 - s) x} sin (7 x) d x

\int_{0}^{\infty} e^{- s x} sin (7 x) d x

\int_{0}^{\infty} e^{- (-4 - s) x} sin (7 x) d x

\int_{0}^{\infty} e^{- (4 + s) x} sin (7 x) d x

None of these!

I don't know!

3 of 22

The Laplace transform of a function f(x) is defined as:

ℒ [f (x)] = \int_{0}^{\infty} e^{- s x} [f (x)] d x = F (s)

,

where s is the Laplace transform parameter.

For suitably defined s, find the Laplace transform of

f = 15 cos (5 x) + 6 sin (8 x)

F (s) = \frac{75}{s^{2} + 25} + \frac{6}{s^{2} + 8^{2}}

F (s) = \frac{15 s}{s^{2} + 5^{2}} + \frac{6}{s^{2} + 8^{2}}

F (s) = \frac{15 s}{s^{2} - 5^{2}} + \frac{48}{s^{2} - 8^{2}}

F (s) = \frac{15 s}{s^{2} + 5^{2}} + \frac{48}{s^{2} + 8^{2}}

None of these!

I don't know!

4 of 22

The Laplace transform of a function f(x) is defined as:

ℒ [f (x)] = \int_{0}^{\infty} e^{- s x} f (x) d x

,

where s is the Laplace transform parameter defined in a suitable domain.

Find Laplace transform of f(t) from the following options where,

f (t) = e^{- 20 t} sin [4 (2 - t)]

\frac{4 s e^{- 40}}{s^{2} + 40 s + 416}

\frac{4 e^{- 40}}{s^{2} + 40 s + 416}

\frac{4 e^{40}}{s^{2} + 40 s + 416}

\frac{4 e^{- 40}}{s^{2} - 40 s + 416}

None of these!

I don't know!

5 of 22

If the Laplace transform of f(t) is F(s) then

ℒ [t^{n} f (t)] = {(- 1)}^{n} \frac{d^{n} F (s)}{d s}

.

For suitably defined s, find the Laplace transform of f(t) from the following options, where

f (t) = t cos (5 t)

\frac{25 - s^{2}}{{(s^{2} + 25)}^{2}}

\frac{10 s}{{(s^{2} + 25)}^{2}}

\frac{s^{2} + 25}{{(s^{2} - 25)}^{2}}

\frac{s^{2} - 25}{{(s^{2} + 25)}^{2}}

None of these!

I don't know!

6 of 22

For a suitably defined domain of s, find the inverse Laplace transform of

\frac{33}{s - 3} + \frac{3}{s^{9}}

33 e^{3 t} + \frac{3 t^{9}}{8!}

33 e^{3 t} + \frac{3 t^{8}}{8!}

33 e^{- 3 t} + \frac{3 t^{8}}{8!}

33 e^{- 3 t} + \frac{3 t^{9}}{9!}

None of these!

I don't know!

7 of 22

The Laplace transform of a function f(t) has been given as

\frac{s + 16}{s^{2} - 27 s + 162}

.

For a suitably defined domain of s, find the correct form of f(t) from the following options.

\frac{1}{27} (-7 e^{18 t} - 34 e^{9 t})

\frac{1}{9} (34 e^{18 t} - 25 e^{9 t})

\frac{1}{9} (27 e^{18 t} - 25 e^{9 t})

\frac{1}{9} (34 e^{- 18 t} - 25 e^{- 9 t})

None of these!

I don't know!

8 of 22

The convolution theorem for inverse Laplace transforms states that

$\begin{matrix} ℒ^{- 1} [F (s) G (s)] = \int_{0}^{t} f (x) g (t - x) d x \end{matrix}$ ,

where

$ℒ^{- 1} [F (s)] = f (t) and ℒ^{- 1} [G (s)] = g (t)$ .

For a suitably defined domain of s, find the inverse Laplace transform of

$\frac{s}{{(s^{2} + 9)}^{2}}$ ,

from the following options:

$\frac{t}{2} sin (3 t)$

$\frac{t}{6} sin (3 t) - \frac{1}{12} cos (3 t)$

$\frac{1}{6} (1 - cos (6 t))$

$\frac{t}{9} sin (3 t)$

None of these!

I don't know!

9 of 22

The Laplace inverse of a function can be obtained by using the properties/theorems of Laplace inverse.

Choose the appropriate value of the following expression, where t > 3π.

ℒ^{- 1} [\frac{e^{- 3 π s}}{s^{2} + 49}]

\frac{1}{7} sin (7 t - 21 π)

sin (7 t - 3 π)

\frac{1}{7} sin (7 t + 21 π)

\frac{1}{7} sin (7 t - 3 π)

None of these!

I don't know!

10 of 22

Ingrid is trying to find out the Laplace inverse of :

\frac{9}{s^{3} (s^{2} + 36)}

She may have made a mistake.

Please input the line number where a mistake first occurs, or input 0 if there is no mistake.

Ingrid's solution

We know that

line 1 ...

ℒ^{- 1} [\frac{1}{(s^{2} + 36)}] = \frac{1}{6} sin (6 t)

and ℒ^{- 1} [\frac{F (s)}{s}] = \int_{0}^{t} F (u) d u

line 2 ...

∴ ℒ^{- 1} [\frac{1}{s (s^{2} + 36)}] = \frac{1}{6} [1 - cos (6 t)]

line 3 ...

ℒ^{- 1} [\frac{1}{s^{2} (s^{2} + 36)}] = \frac{1}{6} [t - \frac{sin (6 t)}{6}]

line 4 ...

ℒ^{- 1} [\frac{9}{s^{3} (s^{2} + 36)}] = \frac{9}{6} [\frac{t^{2}}{2} + \frac{cos (6 t)}{36} - \frac{1}{36}]

The error is in line ...

11 of 22

The function f(t) has Laplace transform defined as

ℒ [f (t)] = \int_{0}^{\infty} e^{- s t} f (t) d t

,

where s is the Laplace transform parameter.

Use this to integrate the following expression

\int_{0}^{\infty} \frac{e^{- 5 t} sin (4 t)}{5 t} d t

\frac{π}{20}

\frac{π}{4}

\frac{π}{16}

\frac{5 π}{4}

None of these!

I don't know!

12 of 22

Which of the following expressions can be evaluated by using the first shift property?

(Maximum score is 3, each wrong answer gets -1 mark)

ℒ [e^{2 t} sin (2 t)]

ℒ [sin (2 x) sinh (2 x)]

ℒ [sinh (2 x) + cos (2 x)]

ℒ [t^{5} e^{- 2 t}]

ℒ [2 e^{- 2 t} + t^{5}]

ℒ [sinh (2 x) + cosh (2 x)]

13 of 22

Find the solution of the IVP

(D^{2} + 49) y = cos (5 t)

,

where

\begin{matrix} y (0) = 1 \\ y (\frac{π}{2}) = - 1 \end{matrix}

y = \frac{23}{24} cos (7 t) - \frac{1}{24} sin (7 t) + \frac{1}{24} cos (5 t)

y = - \frac{1}{24} cos (7 t) - sin (7 t) + \frac{1}{24} cos (5 t)

y = \frac{7}{24} cos (7 t) - 7 sin (7 t) + \frac{1}{24} cos (5 t)

y = \frac{5}{24} cos (7 t) + sin (7 t) + \frac{1}{24} 5 cos (5 t)

None of these!

I don't know!

14 of 22

Find the solution for the following IVP

[t D^{2} + (1 - 5 t) D - 5] y = 0

,

where

\begin{matrix} y (0) = 5 \\ y^{'} (0) = 25 \end{matrix}

y = - 5 e^{5 t}

y = e^{5 t}

y = 5 e^{5 t}

y = - 5 e^{- 5 t}

None of these!

I don't know!

15 of 22

The solution of the differential equation

19 \frac{d y}{d t} + 133 y = 76 e^{t}

is given as:

P e^{t} + Q e^{R t}

,

where

y (0) = 2

.

Find the appropriate values of P, Q and R.

P = \frac{11}{8}, Q = \frac{3}{2}, R = 6

P = \frac{1}{2}, Q = \frac{5}{4}, R = 6

P = \frac{11}{8}, Q = \frac{5}{4}, R = 6

P = \frac{1}{2}, Q = \frac{3}{2}, R = -7

None of these!

I don't know!

16 of 22

Christine is trying to find the solution of the following problem :

A mass of 8 grams moves on the x-axis and is attracted towards the origin O with a force numerically equal to 800 x. If it rests initially at the position x=11, find its position at time t. Consider the damping force numerically equal to 160 times the instantaneous velocity.

Christine may have made a mistake.

Please input the line number where the mistake is, or input 0 if there is no mistake.

Please input the your confidence about the answer according to the following scale:
1=I am very unsure
2=I am not quite sure
3=I am absolutely sure

SOLUTION

$The equation of motion is obtained as$

line 1 ... $x^{''} + 20 x^{'} + 100 x = 0$

Taking the Laplace transform:

$ℒ [x^{''}] + 20 ℒ [x^{'}] + 100 ℒ [x] = 0$

$Writing ℒ [x] = \bar{x}$

line 2 ... $s^{2} \bar{x} - s x (0) - x^{'} (0) + 20 (s \bar{x} - x (0)) + 100 \bar{x} = 0$

line 3 ... $\bar{x} = \frac{11}{s + 10} + \frac{110}{{(s + 10)}^{2}}$

Taking the Laplace inverse:

line 4 ... $x = 11 e^{10 t} + 110 t e^{10 t}$

The error is in line .............

The level of my confidence is.............

17 of 22

An inductor of 5 henrys is in series with a resistor of 10 ohms and an e.m.f. of 8 volts.

Assuming that at time t=0, the current is zero, find the current at time t=2.

Input your answer correct to 3 decimal places.

amps

18 of 22

Find Fourier cosine transform of

f (x) = e^{- 8 x}

- \frac{α}{64 + α^{2}}

\frac{8}{64 + α^{2}}

- \frac{8 α}{64 + α^{2}}

\frac{1}{64 - α^{2}}

None of these!

I don't know!

19 of 22

For suitably defined α find Fourier transform of

f (x) : = {\begin{matrix} 261 - x^{2}, | x | < 16 \\ 0, | x | > 16 \end{matrix}

(\frac{10}{α} + \frac{4}{α^{3}}) sin (16 α) - \frac{64}{α^{2}} cos (16 α)

(\frac{5}{α} - \frac{2}{α^{3}}) sin (16 α) - \frac{32}{α^{2}} cos (16 α)

(\frac{10}{α} - \frac{4}{α^{3}}) sin (16 α) + \frac{64}{α^{2}} cos (16 α)

(\frac{5}{α} + \frac{2}{α^{3}}) cos (16 α) - \frac{16}{α^{2}} sin (16 α)

None of these!

I don't know!

20 of 22

Linda obtained Fourier sine transform of

f (x) = {\begin{matrix} x, & 0 < x < 21 \\ 33 - x, & 21 < x < 27 \\ 0, & x > 27 \end{matrix}

\frac{a_{1} cos (21 α)}{α} + \frac{a_{2} cos (27 α)}{α} + \frac{a_{3} sin (21 α)}{α^{2}} + \frac{a_{4} sin (27 α)}{α^{2}}

.

Find the numerical value of

a_{1}

that she obtained. Here

a_{1}, a_{2}, a_{3}, a_{4}

are positive or negative integers.

Don't forget to put negative sign if your answer is negative integer.

21 of 22

For well defined α find

ℱ_{s}^{- 1} [6 e^{- 8 π α}]

\frac{12 x}{π (8 π^{2} + x^{2})}

\frac{6 x}{π (16 π^{2} + x^{2})}

\frac{6 x}{π (64 π^{2} + x^{2})}

\frac{12 x}{π (64 π^{2} + x^{2})}

None of these!

I don't know!

22 of 22

Asfia obtained the solution of

\int_{0}^{\infty} f (x) sin (α x) d x = {\begin{matrix} 20 - 4 α, 0 \leq α \leq 20 \\ 0, α > 20 \end{matrix}

\frac{P_{1} cos (20 x)}{π x} + \frac{P_{2}}{π x} + \frac{P_{3} sin (20 x)}{π x^{2}}

,

where

P_{1}, P_{2}

are integers and α is well defined.

Find the numerical value of

P_{1}

that she obtained.

Don't forget to put negative sign if your answer is negative integer.

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