Question 6a
The system is time variant because of the following example:
Dealing with the following system
$ X_k[n]=Y_k[n] \, $
where
$ X_k[n]=\delta[n-k]\, $
and
$ Y_k[n]=(k+1)^2 \delta[n-(k+1)] \, $
Consider the input and output of the system when k = 0
$ X_0[n]=\delta[n]\, $
and
$ Y_0[n]=\delta[n-1] \, $
If I time shift the input by $ n_0\, $ , then run it through the system I obtain:
$ X_0[n] \longrightarrow X_0[n-n_0] \longrightarrow Y_1[n]=\delta[n-n_0-1], $
but this isn't equal to running the input through the system, then time shifting the output by $ n_0\, $
$ X_0[n] \longrightarrow Y_1[n]=4\delta[n-1] \longrightarrow Y_1[n-n_0]=4\delta[n-n_0-1]\, $
Question 6b
Assuming this system is linear, an input $ X_0[n]=u[n]\, $ would result in an output $ Y_0[n]=u[n-1]\, $.
