Correct Answer - Option 3 :
\(\frac{{{{\left( {t - 1} \right)}^2}}}{2}u\left( {t - 1} \right)\)
\(y\left( t \right) = \;u\left( {t - 1} \right)*tu\left( t \right)\)
Taking Laplace Transform,
\(Y\left( s \right) = \frac{{{e^{ - s}}}}{s} \times \frac{1}{{{s^2}}}\)
\(Y\left( s \right) = \frac{{{e^{ - s}}}}{{{s^3}}}\)
\(Y\left( s \right)\mathop \to \limits^{ILT} \frac{{{{\left( {t - 1} \right)}^2}}}{2}u\left( {t - 1} \right)\)
[As \(\frac{1}{{{s^2}}}\mathop \to \limits^{ILT} t\;u\left( t \right)\)& shifting in time leads to phase in frequency]