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The image contains questions on **Laplace Transforms** and **Inverse Laplace Transforms**. I will help you with the solutions and explanations for each.

### Laplace Transforms:
1. The value of \( L[1] \) is:
   \[
   \text{Answer: } \frac{1}{s}
   \]

2. The value of \( L[a^t] \) is:
   \[
   \text{Answer: } \frac{1}{s - a}
   \]

3. The value of \( L[\cos(at)] \) is:
   \[
   \text{Answer: } \frac{s}{s^2 + a^2}
   \]

4. The value of \( L[e^{at}] \) is:
   \[
   \text{Answer: } \frac{1}{s - a}
   \]

5. The value of \( L[t^6] \) is:
   \[
   \text{Answer: } \frac{6!}{s^7} \quad \left( \text{since } L[t^n] = \frac{n!}{s^{n+1}} \right)
   \]

6. The value of \( L[e^{at}t^n] \) is:
   \[
   \text{Answer: } \frac{n!}{(s - a)^{n+1}}
   \]

7. If \( L[f(t)] = F(s) \), then \( L[f(at)] \) is:
   \[
   \text{Answer: } \frac{1}{a}F\left( \frac{s}{a} \right)
   \]

8. If \( L[f(t)] = F(s) \) and \( f(t) \) is continuous, then \( L[f'(t)] \) is:
   \[
   \text{Answer: } sF(s) - f(0)
   \]

9. If \( L[f(t)] = F(s) \), then \( L[t f(t)] \) is:
   \[
   \text{Answer: } -F'(s)
   \]

10. If \( L[f(t)] = F(s) \), then \( L\left[ \int_0^t f(\tau) d\tau \right] \) is:
    \[
    \text{Answer: } \frac{F(s)}{s}
    \]

11. If \( L[f(t)] = F(s) \), then \( L\left[ \int_0^\infty f(t) dt \right] \) is:
    \[
    \text{Answer: } \frac{F(s)}{s^2}
    \]

### Inverse Laplace Transforms:
1. The value of \( L^{-1} \left[ \frac{1}{s^2 + a^2} \right] \) is:
   \[
   \text{Answer: } \frac{\sin(at)}{a}
   \]

2. The value of \( L^{-1} \left[ \frac{s+a}{(s+a)^2 + 1} \right] \) is:
   \[
   \text{Answer: } e^{-at} \cos(t)
   \]

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Here are 5 additional related questions to practice:
1. What is the Laplace Transform of \( \sin(at) \)?
2. Compute the Laplace Transform of \( t^n e^{bt} \).
3. Find the inverse Laplace Transform of \( \frac{s}{s^2 + 4} \).
4. What is the Laplace Transform of \( e^{-2t} \sin(3t) \)?
5. Derive the inverse Laplace of \( \frac{1}{(s-2)(s+3)} \).

### Tip:
When dealing with inverse Laplace transforms involving partial fractions, always break down complex expressions into simpler fractions for easier computation.

The image contains one-mark questions related to Laplace Transforms and Inverse Laplace Transforms.

This page contains detailed solutions to one-mark questions on Laplace and Inverse Laplace Transforms, covering core formulas like L[1] = 1/s, L[e^{at}], L[cos(at)], and integration properties.

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