GATE 2025 IN Q.17: solution and answer

Official answer: ASource: Official answer key, GATE 2025 IN (IIT Roorkee)

1. First MUX (4:1). Its output is $F=\sum m_i{I}_i$, where $m_i$ is the minterm of the select word. The select word is $S_0{S}_1=PQ$, and the data inputs are $I_{00}=0,I_{01}=1,I_{10}=1,I_{11}=0$.

2. $F$ is 1 only for $PQ=01$ and $PQ=10$, so $F=\overline{P}Q+P\overline{Q}=P\oplus Q$. Here $\overline{}$ means complement.

3. The result does not depend on which select line is the MSB. The two 1-inputs (01 and 10) are mirror images of each other, so swapping $S_0$ and $S_1$ still gives $P\oplus Q$.

4. Second MUX (2:1). Its output is $Y=\overline{S}{I}_0+S{I}_1$. With $S=R$, $I_0=F$ and $I_1=1$: $$Y=\overline{R}(P\oplus Q)+R\cdot 1$$

5. Simplify with $A+\overline{A}B=A+B$: $$Y=R+\overline{R}(P\oplus Q)=R+(P\oplus Q)$$

6. Check by cases. When $R=1$, the second MUX passes the constant 1, so $Y=1$. When $R=0$, it passes the first MUX's output, so $Y=P\oplus Q$. Option (A) gives the same value in both cases. (B) gives $Y=0$ when $R=0$. (C) uses $\overline{P\oplus Q}$ when $R=0$. (D) gives $Y=\overline{P\oplus Q}$ when $R=1$. The answer is (A).