Question

Match List-I with List-II. List-I: A. \(e^{at}\), B. \(\cos at\), C. \(\sin at\), D. \(\cosh at\).

• A. A-I, B-II, C-III, D-IV
• B. A-II, B-I, C-III, D-IV
• C. A-II, B-I, C-IV, D-III
• D. A-I, B-II, C-IV, D-III

Hint:

Remember the standard Laplace transforms: \(L\{e^{at}\}=\frac{1}{s-a}\), \(L\{\cos at\}=\frac{s}{s^2+a^2}\), \(L\{\sin at\}=\frac{a}{s^2+a^2}\), and \(L\{\cosh at\}=\frac{s}{s^2-a^2}\).

Answer 1

The Correct Option is C

Concept:
Laplace transform is used to convert a function of time \(f(t)\) into a function of complex variable \(s\), written as: \[ F(s)=L\{f(t)\} \]

Step 1: Laplace transform of \(e^{at}\).
The standard formula is: \[ L\{e^{at}\}=\frac{1}{s-a} \] So, \[ A \rightarrow II \]

Step 2: Laplace transform of \(\cos at\).
The standard formula is: \[ L\{\cos at\}=\frac{s}{s^2+a^2} \] So, \[ B \rightarrow I \]

Step 3: Laplace transform of \(\sin at\).
The standard formula is: \[ L\{\sin at\}=\frac{a}{s^2+a^2} \] So, \[ C \rightarrow IV \]

Step 4: Laplace transform of \(\cosh at\).
The standard formula is: \[ L\{\cosh at\}=\frac{s}{s^2-a^2} \] So, \[ D \rightarrow III \] Therefore: \[ A-II,\ B-I,\ C-IV,\ D-III \] \[ \therefore \text{Correct Answer is (C)} \]