Part a
System: $ X_{k}[n]=\delta[n-k] \to Y_{k}[n] = (k+1)^2 \delta [n-(k+1)] $
Time-delay: $ X_{k}[n]=\delta[n-k] \to X_{k}[n-N]=\delta[n-N-k] $
$ X_{k}[n] \to timedelay \to sys \to Z_{k}[n]=(k+1)^2 \delta [n-N-(k+1)] $
$ X_{k}[n] \to sys \to timedelay \to Z_{k}[n]=(k+1)^2 \delta [n-N-(k+1)] $
Since $ (k+1)^2 \delta [n-N-(k+1)] $ is equal to $ (k+1)^2 \delta [n-N-(k+1)] $, the system is time-invariant.
Part b
In order for $ Y[n]=u[n-1] $ to be true, $ X[n]=u[n] $ must also be true.
Proof:
$ u[n]=\delta[n]-\delta[n-N] $ where $ N=1 $
$ \delta[n] \to sys \to \delta[n-1] \to $ $ - \to \delta[n-1]-\delta[n-2]=u[n-1] $ $ \delta[n-N] \to sys \to \delta[n-N-1] \to $
