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GATE CSE 2023 Q43: State never reached by a T–D–T flip-flop circuit

Question

Consider a sequential digital circuit consisting of T flip-flops and D flip-flops as shown in the figure. CLKIN is the clock input to the circuit. At the beginning, Q1, Q2 and Q3 have values 0, 1 and 1, respectively.

Three flip-flops sharing clock CLKIN: a T flip-flop with output Q1, a D flip-flop with input Q1 and output Q2, and a T flip-flop with input Q2 and output Q3. Q3 passes through an inverter back to the T input of the first flip-flop
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Which one of the given values of (Q1, Q2, Q3) can NEVER be obtained with this digital circuit?

Try it: multiple choice: exactly one correct option

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Wrong answer costs 2/3 mark in GATE.
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Answer: A (official GATE 2023 final answer key)

Concept used: Build the state sequence from the next-state equations Q1+=Q1⊕Q3‾Q_1^+ = Q_1 \oplus \overline{Q_3}, Q2+=Q1Q_2^+ = Q_1, Q3+=Q3⊕Q2Q_3^+ = Q_3 \oplus Q_2.

First step: Start from (0, 1, 1) and apply the three equations clock by clock until the state repeats.

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Source: official GATE 2023 CS question paper and final answer key, organised by IIT Kanpur. Question paper · Answer key

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