GATE 2024 EE Q.26: solution and answer

GATE 2024 · EEQ.26 · 1 mark · MCQ

To obtain the Boolean function $F (X, Y) = X \overline{Y} + \overline{X}$ , the inputs $P QR S$ in the figure should be

GATE 2024 EE question 26: A 4:1 multiplexer drawn as a box labelled '4:1 Multiplexer'. — Figure from the GATE 2024 EE paper. It is part of the question.

(A)
1010
(B)
1110
(C)
0110
(D)
0001

Answer and solution

ANSWER

$P QR S = 1110$ , option (B). The function simplifies to $F = \overline{X} + \overline{Y} = \overline{X Y}$ , which is 0 only when $X = Y = 1$ , so only input 3 ( $S$ ) is tied to 0.

Official answer: BSource: Official answer key, GATE 2024 EE (IISc Bengaluru)

How a 4:1 MUX works here: $S_{1} = X$ is the MSB and $S_{0} = Y$ is the LSB, so the output is $F = \overline{X} \overline{Y} P + \overline{X} Y Q + X \overline{Y} R + X Y S$ . This means data input $I_{k}$ must equal the value of $F$ at the minterm $X Y = k$ . (An overline means complement.)
Simplify first with the absorption rule $A + \overline{A} B = A + B$ , taking $A = \overline{X}$ : $F = \overline{X} + X \overline{Y} = \overline{X} + \overline{Y} = \overline{X Y}$ . That is a two-input NAND.
Work out $F$ for each select combination. $X Y = 00$ : $\overline{X} = 1$ , so $F = 1$ and $P = 1$ . $X Y = 01$ : $\overline{X} = 1$ , so $F = 1$ and $Q = 1$ .
$X Y = 10$ : $F = X \overline{Y} + \overline{X} = 1 \cdot 1 + 0 = 1$ , so $R = 1$ . $X Y = 11$ : $F = 1 \cdot 0 + 0 = 0$ , so $S = 0$ .
So $P QR S = 1110$ . This is the NAND truth table read in the order $I_{0} I_{1} I_{2} I_{3}$ , which is option (B).

KEY IDEA

In an $n$ -select MUX used as a function generator, data input $I_{k}$ is simply the value of $F$ at the minterm whose binary code, with the MSB select line as the leading bit, equals $k$ . Here $X$ drives the MSB, so the input order $P, Q, R, S$ matches $X Y = 00, 01, 10, 11$ .

WATCH OUT FOR

Mixing up which select line is the MSB. If you take $Y$ as the MSB, inputs 1 and 2 swap places. That gives the same pattern here only because $F$ is symmetric in $X$ and $Y$ , so it will not work in general. A second slip is to read only the $X \overline{Y}$ term, mark just input 2 as 1, and forget that $\overline{X}$ also covers inputs 0 and 1.

From the GATE 2024 EE paper. Every question of this paper · Download the full paper · All 12 Combinational questions, with notes

GATE 2024 EE Q.26: solution and answer

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