Question

The signal \( x(t) = (t - 1)^2 u(t - 1) \), where \( u(t) \) is the unit-step function, has the Laplace transform \( X(s) \). The value of \( X(1) \) is_________

• A. ( \frac{1}{e} \)
• B. ( \frac{2}{e} \)
• C. ( 2e \)
• D. ( e^2 \)

Hint:

To calculate the Laplace transform of shifted functions, use the shifting property, and for polynomials, apply standard Laplace transform formulas.

Answer 1

The Correct Option is B

Step 1: Understanding the signal.
The given signal is \( x(t) = (t - 1)^2 u(t - 1) \), where \( u(t - 1) \) is the unit step function, which shifts the signal to start from \( t = 1 \). The Laplace transform of \( x(t) \) can be calculated using the formula for the Laplace transform of shifted functions and polynomial terms.
Step 2: Laplace transform of the signal.
The Laplace transform of \( (t - 1)^2 u(t - 1) \) is: \[ X(s) = \mathcal{L}\{ (t - 1)^2 u(t - 1) \} = \frac{2}{s^3}. \] Step 3: Finding \( X(1) \).
To find \( X(1) \), we substitute \( s = 1 \) into the expression for \( X(s) \): \[ X(1) = \frac{2}{1^3} = \frac{2}{e}. \] Step 4: Conclusion.
The correct answer is (B) \( \frac{2}{e} \).