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For Q2, why do you feel case 1 is not correct? Please explain. Also, what is the definition of the convolution integral and what are the limits of integration in the definition.
However, let's make an observation about case 1. In every step of case 1, it's clear that we are taking the inverse transform of G(s) which will yield g(t). This can be seen by inspection in every formulation. Clearly, G(s) (1) is G(s), and clearly g(t) * δ(t)= g(t)
Unfortunately, you didn't try to answer the questions I posed. Hopefully, you at least thought about those questions.