GATE 2025 IN Q.18: solution and answer

GATE 2025 · INQ.18 · 1 mark · MCQ

The 4-bit signed 2's complement form of $(5)_{10} + (5)_{10}$ is

(A)
$(- 6)_{10}$
(B)
$(- 7)_{10}$
(C)
$(- 5)_{10}$
(D)
$(- 8)_{10}$

See the question in the paper

Answer and solution

ANSWER

In 4 bits, $0101 + 0101 = 1010$ . Read as a signed 2's complement number, $1010$ is $(- 6)_{10}$ , so the answer is (A). The true sum $+ 10$ overflows the range $- 8$ to $+ 7$ .

Official answer: ASource: Official answer key, GATE 2025 IN (IIT Roorkee)

Write each operand in 4-bit 2's complement: $(5)_{10} = (0101)_{2}$ . A 4-bit 2's complement number ranges from $- 2^{3}$ to $2^{3} - 1$ , that is, from $- 8$ to $+ 7$ .
Add in binary and keep only 4 bits: $+ 010101011010$ There is no carry out of the MSB, so the 4-bit result is $(1010)_{2}$ .
Overflow check: both operands are positive (sign bit 0), but the result has sign bit 1, so the addition has overflowed. The true sum, $+ 10$ , lies outside the range $- 8$ to $+ 7$ .
Read $(1010)_{2}$ as a signed 2's complement number. The MSB has weight $- 2^{3}$ : $- 8 + 0 \cdot 4 + 1 \cdot 2 + 0 \cdot 1 = - 6$
Cross-check by taking the 2's complement of $1010$ : inverting every bit gives $0101$ , and adding 1 gives $0110 = 6$ . So $1010$ stands for $- 6$ .
The 4-bit register therefore holds $(- 6)_{10}$ , which is option (A).

KEY IDEA

An $n$ -bit 2's complement register can hold only values from $- 2^{n - 1}$ to $2^{n - 1} - 1$ . When two positive numbers add up to more than $2^{n - 1} - 1$ , the sum wraps around. The stored pattern is the true sum minus $2^{n}$ , here $10 - 16 = - 6$ . You can spot the overflow because two operands with sign bit 0 give a result with sign bit 1.

WATCH OUT FOR

Reading $1010$ as unsigned and answering $+ 10$ , which is not an option, or treating it as sign-magnitude and getting $- 2$ . A related slip is choosing $- 8$ because it is the most negative 4-bit value. Some students also flip the bits and forget to add 1, getting $0101 = 5$ and choosing $- 5$ (C). The MSB carries weight $- 8$ , so $1010 = - 8 + 2 = - 6$ .

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GATE 2025 IN Q.18: solution and answer

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