Discussion
Match List I with List II.
| List I | List II |
|---|---|
| (a) Laplace transform of sinωt | (i) \(\frac{s}{s^{2}+\omega^{2}}\) |
| (b) Laplace transform of cosωt | (ii) \(\frac{\omega}{s^{2}+\omega^{2}}\) |
| (c) Laplace transform of sinhωt | (iii) \(\frac{s}{s^{2}-\omega^{2}}\) |
| (d) Laplace transform of coshωt | (iv) \(\frac{\omega}{s^{2}-\omega^{2}}\) |
Choose the most appropriate answer from the options given below :
Answer: D
The correct matching is (a)-(ii), (b)-(i), (c)-(iv), (d)-(iii).
The standard Laplace transforms are:
- \(\sin\omega t \rightarrow \frac{\omega}{s^{2}+\omega^{2}}\), item (ii)
- \(\cos\omega t \rightarrow \frac{s}{s^{2}+\omega^{2}}\), item (i)
- \(\sinh\omega t \rightarrow \frac{\omega}{s^{2}-\omega^{2}}\), item (iv)
- \(\cosh\omega t \rightarrow \frac{s}{s^{2}-\omega^{2}}\), item (iii)
A quick check: the sine-type functions start at zero, so their transforms carry \(\omega\) in the numerator, while cosine-type functions start at one and carry \(s\). The hyperbolic pair has a minus sign in the denominator. Option (a)-(i), (b)-(ii) is the common trap of pairing sine with \(s\).
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