GATE CS 2024 Set 1 Q.64: Combinational Circuits

Consider a digital logic circuit consisting of three 2-to-1 multiplexers M1, M2, and M3 as shown below. X1 and X2 are inputs of M1. X3 and X4 are inputs of M2. A, B, and C are select lines of M1, M2, and M3, respectively.

Three 2-to-1 multiplexers M1, M2, and M3. M1 has inputs 0=X1, 1=X2, select S1=A, and output Q1 feeding input 0 of M3. M2 has inputs 0=X3, 1=X4, select S2=B, and output Q2 feeding input 1 of M3. M3 has select S3=C and output Y.

For an instance of inputs X1=1, X2=1, X3=0, and X4=0, the number of combinations of A, B, C that give the output Y=1 is _________

Your answer

Official answer and solution

Official answer: 4
From: Official GATE 2024 answer key

Solution

Evaluate the outputs $Q_1$ and $Q_2$ of multiplexers M1 and M2 for the given constant data inputs, then find the select values $(A, B, C)$ that make $Y = 1$ .

For the given input instance $X_1 = 1, X_2 = 1, X_3 = 0, X_4 = 0$ :

Output of M1: $Q_1 = \overline{A}X_1 + AX_2 = \overline{A}(1) + A(1) = 1 \quad (\text{for both } A = 0 \text{ and } A = 1)$
Output of M2: $Q_2 = \overline{B}X_3 + BX_4 = \overline{B}(0) + B(0) = 0 \quad (\text{for both } B = 0 \text{ and } B = 1)$
Output of M3: $Y = \overline{C}Q_1 + CQ_2 = \overline{C}(1) + C(0) = \overline{C}$

Thus $Y = 1$ if and only if $C = 0$ , while $A \in \{0, 1\}$ and $B \in \{0, 1\}$ can take any values. This gives $2 \times 2 \times 1 = 4$ combinations of $(A, B, C)$ : namely $(0, 0, 0), (0, 1, 0), (1, 0, 0),$ and $(1, 1, 0)$ .

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2024 Computer Science and Information Technology - Set 1 (CS1)
Question: 64 of 65
Section: Computer Science and Information Technology

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