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GATE CS 2026 Set 1 Q27 — Sequential Circuits

Consider a $2$ -bit saturating up/down counter that performs the saturating up count when the input $P$ is $0$ , and the saturating down count when $P$ is $1$ . The Next State table of the counter is as shown. The counter is built as a synchronous sequential circuit using D flip-flops.

Input	Cureent State	Next State
$\text{P}$	$\text{Q}_{1}$	$\text{Q}_{0}$	$\text{Q}_{1}^{+}$	$\text{Q}_{0}^{+}$
$0$	$0$	$0$	$0$	$1$
$0$	$0$	$1$	$1$	$0$
$0$	$1$	$0$	$1$	$1$
$0$	$1$	$1$	$1$	$1$
$1$	$0$	$0$	$0$	$0$
$1$	$0$	$1$	$0$	$0$
$1$	$1$	$0$	$0$	$1$
$1$	$1$	$1$	$1$	$0$

Which one of the following options corresponds to the expressions for the inputs of the D flip-flops, $D_{1}$ and $D_{0}$ ?

Choose one optionA

D_{1}=P Q_{1}+\bar{P} Q_{0}+Q_{1} Q_{0} \quad D_{0}=P Q_{0}+\bar{P} Q_{1}+Q_{1} \overline{Q_{0}}

D_{1}=\bar{P} Q_{1}+\bar{P} Q_{0}+Q_{1} Q_{0} \quad D_{0}=\bar{P} \overline{Q_{0}}+\bar{P} Q_{1}+Q_{1} \overline{Q_{0}}

D_{1}=\bar{P} \overline{Q_{1}}+\bar{P} Q_{0}+Q_{1} Q_{0} \quad D_{0}=\bar{P} Q_{0}+\bar{P} Q_{1}+Q_{1} \overline{Q_{0}}

D_{1}=P \overline{Q_{1}}+\bar{P} Q_{0}+Q_{1} Q_{0} \quad D_{0}=P \overline{Q_{0}}+\bar{P} Q_{1}+Q_{1} \overline{Q_{0}}

View worked solution

Answer: B

For a synchronous sequential circuit using D flip-flops, the excitation equations are directly equal to the next-state equations:

D_1 = Q_1^+ \quad \text{and} \quad D_0 = Q_0^+

1. Minimizing $D_1$ :

From the given next-state table, the minterms for $Q_1^+$ are:

D_1(P, Q_1, Q_0) = \sum m(1, 2, 3, 7)

Using a 3-variable K-map for $D_1$ :

Group 1: $m(1, 3) \implies \bar{P}Q_0$
Group 2: $m(2, 3) \implies \bar{P}Q_1$
Group 3: $m(3, 7) \implies Q_1Q_0$

D_1 = \bar{P}Q_1 + \bar{P}Q_0 + Q_1Q_0

2. Minimizing $D_0$ :

The minterms for $Q_0^+$ are:

D_0(P, Q_1, Q_0) = \sum m(0, 2, 3, 6)

Using a 3-variable K-map for $D_0$ :

Group 1: $m(0, 2) \implies \bar{P}\bar{Q_0}$
Group 2: $m(2, 3) \implies \bar{P}Q_1$
Group 3: $m(2, 6) \implies Q_1\bar{Q_0}$

D_0 = \bar{P}\bar{Q_0} + \bar{P}Q_1 + Q_1\bar{Q_0}

Thus, the correct option is B.

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Published 10 Oct 2026

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