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GATE CS 2004 Q61 — Combinational Circuits

Consider the partial implementation of a $2-bit$ counter using $T$ flip-flops following the sequence $0-2-3-1-0,$ as shown below.

To complete the circuit, the input $X$ should be

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Answer: D

To find the input $X$ , we analyze the state transitions of the 2-bit counter. The given sequence is $0 o 2 o 3 o 1 o 0$ . Let the state be represented by $Q_2 Q_1$ .

For a $T$ flip-flop, the excitation equation is $T = Q_{present} \oplus Q_{next}$ . We construct the transition table as follows:

Present State ( $Q_2 Q_1$ )	Next State ( $Q_2^+ Q_1^+$ )	$T_2 = Q_2 \oplus Q_2^+$	$T_1 = Q_1 \oplus Q_1^+$
$00$	$10$	$1$	$0$
$10$	$11$	$0$	$1$
$11$	$01$	$1$	$0$
$01$	$00$	$0$	$1$

From the table, we derive the simplified expressions for the flip-flop inputs:

$T_1$ is $1$ when $Q_2 Q_1 = 01$ or $10$ . Thus, $T_1 = Q_2 \oplus Q_1$ .
$T_2$ is $1$ when $Q_2 Q_1 = 00$ or $11$ . Thus, $T_2 = Q_2 \odot Q_1$ .

Comparing with the given options, the input $X$ corresponds to $T_1$ , which is $Q_1 \oplus Q_2$ .

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Published 10 Oct 2026

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