GATE 2023 IN Q.28: solution and answer

Official answer: -3Source: Official answer key, GATE 2023 IN (IIT Kanpur)

1. Time-reversed, shifted step. $u[-n+5]=1$ when $-n+5\ge 0$, that is for $n\le 5$, and $u[-n+5]=0$ for $n>5$ (i.e. $n\ge 6$).

2. Shifted step. $u[n+3]=1$ when $n+3\ge 0$, that is for $n\ge -3$, and $u[n+3]=0$ for $n<-3$ (i.e. $n\le -4$).

3. Region $n\le -4$: $u[-n+5]=1$ and $u[n+3]=0$, so $x[n]=1-0=1$.

4. Region $-3\le n\le 5$: both steps equal 1, so $x[n]=1-1=0$.

5. Region $n\ge 6$: $u[-n+5]=0$ and $u[n+3]=1$, so $x[n]=0-1=-1$.

6. Putting the three regions together,
   $$x[n]=\{\begin{array}{ll}1,& n\le -4\\ 0,& -3\le n\le 5\\ -1,& n\ge 6\end{array}$$
   $x[n]$ is non-zero for every $n\le -4$ and first becomes zero at $n=-3$, so the smallest $n$ with $x[n]=0$ is $-3$.