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The image contains a set of Laplace transform-related multiple-choice problems. Let's go through the questions step by step, starting with the first one:

### 1. \( \mathcal{L}\{2 + e^{-3t} + \sin 4t\} \)
The Laplace transform of:
- \( 2 \) is \( \frac{2}{s} \),
- \( e^{-3t} \) is \( \frac{1}{s + 3} \),
- \( \sin(4t) \) is \( \frac{4}{s^2 + 16} \).

Summing them up, we get:
\[
\frac{2}{s} + \frac{1}{s + 3} + \frac{4}{s^2 + 16}
\]
The correct answer is **(d)**.

### 2. \( \mathcal{L}\{\sin(3t - 2)\} \)
Using the phase shift property of the Laplace transform:
\[
\mathcal{L}\{\sin(3t)\} = \frac{3}{s^2 + 9}
\]
But due to the time-shift, we apply Euler's identity, which leads to additional terms. The correct answer appears to be **(d)**.

### 3. \( \mathcal{L}\{t \sin(2t)\} \)
Using the formula \( \mathcal{L}\{t \sin(at)\} = \frac{2s}{(s^2 + a^2)^2} \) for \( a = 2 \):
The correct answer is **(b)**.

### 4. \( \mathcal{L}\{e^{3t} \cosh(2t)\} \)
We use the Laplace transform of \( \cosh(at) \), adjusted by the exponential term:
The correct answer is **(b)**.

### 5. \( \mathcal{L}\{t \sin(4t)\} \)
From the known formula for \( t \sin(at) \), the Laplace transform is:
\[
\mathcal{L}\{t \sin(at)\} = \frac{2s}{(s^2 + a^2)^2}
\]
For \( a = 4 \), the correct answer is **(a)**.

### Do you want me to continue with more detailed steps, or do you have any questions about the explanation so far?

Here are 5 related questions:
1. What is the Laplace transform of \( \cos(at) \)?
2. How do phase shifts affect the Laplace transform?
3. What is the inverse Laplace transform of \( \frac{1}{s^2 + a^2} \)?
4. How does the time-shift theorem modify the Laplace transform?
5. What is the Laplace transform of \( t^n e^{at} \)?

**Tip**: Use known Laplace transform tables for common functions to speed up solving problems!

Multiple-choice Laplace transform questions including problems like L(2 + e^(-3t) + sin(4t)), L(sin(3t - 2)), and other related transforms.

This page provides multiple-choice questions and solutions on Laplace transforms, including examples like L(2 + e^(-3t) + sin(4t)), inverse transforms, and shifting theorems. Suitable for university students studying differential equations and complex analysis.

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