GATE PH 2007 • Q26
MCQ • 2 Marks
Section 1: Mathematical Physics
Fourier and Laplace transforms
Subtopic: Laplace Transform Properties & Inverses
Question Statement:
The inverse Laplace transform $\mathcal{L}^{-1}\left\{ \frac{s}{s^2 + 4} \right\}$ is
Options:
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(A) $\cos(2t)$
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(B) $\sin(2t)$
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(C) $\cosh(2t)$
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(D) $\sinh(2t)$
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2007 Q.26)**
**1. Standard Laplace Transform Tables & Principles**:
- Standard formula for cosine:
$\mathcal{L}\{\cos(\omega t)\} = \frac{s}{s^2 + \omega^2}$
- Here $\omega^2 = 4 \implies \omega = 2$.
- Therefore:
$\mathcal{L}^{-1}\left\{ \frac{s}{s^2 + 4} \right\} = \cos(2t)$
**2. Official Verification & Conclusion**:
- **Official Master Answer**: **Option (A)**: $\cos(2t)$.
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