Laplace Transform MCQ Quiz in मराठी - Objective Question with Answer for Laplace Transform - मोफत PDF डाउनलोड करा
Last updated on Mar 13, 2025
Latest Laplace Transform MCQ Objective Questions
Top Laplace Transform MCQ Objective Questions
Laplace Transform Question 1:
What is the Laplace transform of v(t) = sin (10t) u(t) ?
Answer (Detailed Solution Below)
Laplace Transform Question 1 Detailed Solution
Concept:
L [f(t)] = F(s)
L [sin(at) u(t)] ↔
Application:
With v(t) = sin (10t) u(t), the Laplace transform will be:
Important Points
Some common Laplace transforms are:
f(t) |
F(s) |
ROC |
δ (t) |
1 |
All s |
u(t) |
|
Re (s) > 0 |
t |
Re (s) > 0 |
|
tn |
Re (s) > 0 |
|
e-at |
Re (s) > -a |
|
t e-at |
Re (s) > -a |
|
tn e-at |
Re (s) > -a |
|
sin at |
Re (s) > 0 |
|
cos at |
Re (s) > 0 |
Laplace Transform Question 2:
Inverse Laplace transform of
Answer (Detailed Solution Below)
Laplace Transform Question 2 Detailed Solution
Concept:
Calculation:
Given:
After Partial fraction we’ll get;
Now, Taking Laplace inverse of F(s);
Laplace Transform Question 3:
What is Laplace transform of function e-5t cos 4t?
Answer (Detailed Solution Below)
Laplace Transform Question 3 Detailed Solution
Concept:
Let the Laplace transform of function of f(t) is L [f(t)] = F (s)
By using first shifting rule
If L [f(t)] = F (s), then
L [eat f(t)] = F (s – a)
Calculation:
Laplace transform of y(t) = e-3t cos 5t
By applying the property of shifting,
Laplace Transform Question 4:
The Laplace transform of the waveform shown in the figure is
Answer (Detailed Solution Below) -1
Laplace Transform Question 4 Detailed Solution
Solution:
The given periodic function can be expressed as the summation of shifted unit step functions as shown:
Similarly x3(t), x4(t), … will be shifted unit step functions.
x1(t) = u(t) - u(t - 1)
x2(t) = u(t - 2) - u(t - 3)
x(t) can be expressed as:
x(t) = x1(t) + x2(t) + x3(t) + ….
x(t) = u(t) - u(t - 1) + u(t - 2) - u(t - 3)+ … ---(1)
Time-shifting affects the frequency as:
The Laplace transform of x(t) from equation (1), can be written as:
Simplifying the geometric series, we can write:
Given
Comparing this with equation (2) we get:
k = -1
Laplace Transform Question 5:
The solution for the differential equation
With initial conditions
Answer (Detailed Solution Below)
Laplace Transform Question 5 Detailed Solution
Given the differential equation,
Taking the Laplace transform,
s2X(s) – sX(0) – X’(0) = -9X(s)
or s2X(s) – s(1) – (1) = - 9X(s)
or X(s) (s2+9) = s + 1
Laplace Transform Question 6:
Match the following and choose the correct alternative from List - I and List - II.
List - I | List - II | ||
(Time function) | (Laplace transform) | ||
A. | 1 | 1. | 1/s |
B. | t | 2. | 1/s2 |
C. | sin ωt | 3. | |
D. | cos ωt | 4. |
Answer (Detailed Solution Below)
Laplace Transform Question 6 Detailed Solution
Laplace Transform:
The Laplace transform of
1.)
2.)
3.)
4.)
Hence, option 2 is correct.
Laplace Transform Question 7:
Fourier transform and Laplace transform are related through
Answer (Detailed Solution Below)
Laplace Transform Question 7 Detailed Solution
The Fourier transform of a function is equal to its two-sided Laplace transform evaluated on the imaginary axis of the s-plane.
Explanation:
The Fourier transform of x(t) is, denoted by X(jω), is defined as:
The Laplace transform of x(t), denoted by X(s), is defined as:
Where s is a continuous complex variable.
We can also express s as: s = σ + jω
Where σ and ω are the real and imaginary parts of s, respectively
The Laplace transform can be written as:
By comparing the above Laplace and Fourier transform equations, it is clear that Laplace transform of x(t) is equal to the Fourier transform of
When σ = 0 or s = jω, both are identical.
That is, Laplace transform generalizes Fourier transform.
Laplace Transform Question 8:
If F(s) is the Laplace transform of f(t), then how will F(s) be written in partial fraction method?
Answer (Detailed Solution Below)
Laplace Transform Question 8 Detailed Solution
- In the partial fraction method of Laplace transforms, the function F(s) is generally represented as a rational function, which is the ratio of two polynomials:
- The goal of partial fraction decomposition is to express F(s) as a sum of simpler fractions to facilitate the inverse Laplace transform.
- These simpler fractions correspond to known Laplace transform pairs, making it easier to find f(t)" id="MathJax-Element-8-Frame" role="presentation" style="position: relative;" tabindex="0">
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F(s)F(s).
Laplace Transform Question 9:
Determine transfer function if the impulse response is e-2t.
Answer (Detailed Solution Below)
Laplace Transform Question 9 Detailed Solution
Concept:
Transfer function:
It is the ratio of the Laplace transform of the output variable to the input variable with initial conditions zero.
TF =
Initial conditions = 0
TF = C(s) / R(s)
If R(s) = 1 i.e impulse input
TF = C(s)
Impulse response (IR) = c(t) = L-1 [C(s)] = L-1[TF]
TF = L-1[IR]
Hence, TF may be also called the impulse response of the system.
Explanation:
From the property of Laplace transform
Put a = 2,
Hence, T.F = 1 / (s+2)
Additional Information
f(t) | L{f(t)} = F(s) |
1 | |
u(t) | |
sin (at) | |
cos (at) | |
X(s - a) |
Laplace Transform Question 10:
Time shifting in Laplace transform if L{x(t)} = X(s), then L[x(t - t0)] is:
Answer (Detailed Solution Below)
Laplace Transform Question 10 Detailed Solution
Concept:
The Laplace transform F(s) of a function f(t) is defined by:
From the time-shifting property of Laplace transform:
Application:
L{x(t)} = X(s)
L(x(t-t0)) =