GATE CS 2022 Q.18: Number Representations and Computer Arithmetic

Let $R_1$ and $R_2$ be two 4-bit registers that store numbers in 2's complement form. For the operation $R_1 + R_2$ , which one of the following values of $R_1$ and $R_2$ gives an arithmetic overflow?

$R_1 = 1011$ and $R_2 = 1110$

$R_1 = 1100$ and $R_2 = 1010$

$R_1 = 0011$ and $R_2 = 0100$

$R_1 = 1001$ and $R_2 = 1111$

Official answer and solution

Official answer: B
From: Official GATE 2022 answer key

Solution

Evaluate the 4-bit two's complement values and detect overflow when the sum of two numbers of the same sign produces a result of the opposite sign.

In 4-bit 2's complement representation, the range of representable integers is $[-2^3, 2^3 - 1] = [-8, +7]$ .

Let us evaluate each option:

(A) $R_1 = 1011_2 = -5$ , $R_2 = 1110_2 = -2$ . Sum: $-5 + (-2) = -7$ . Since $-7 \in [-8, +7]$ , no overflow occurs.
(B) $R_1 = 1100_2 = -4$ , $R_2 = 1010_2 = -6$ . Sum: $-4 + (-6) = -10$ . Since $-10 < -8$ , the true sum cannot be represented in 4 bits. In binary addition: $1100_2 + 1010_2 = (1)0110_2$ . The result has sign bit 0 (representing $+6$ ), indicating negative overflow.
(C) $R_1 = 0011_2 = +3$ , $R_2 = 0100_2 = +4$ . Sum: $+3 + (+4) = +7 \in [-8, +7]$ , no overflow.
(D) $R_1 = 1001_2 = -7$ , $R_2 = 1111_2 = -1$ . Sum: $-7 + (-1) = -8 \in [-8, +7]$ , no overflow.

Thus, only Option (B) results in arithmetic overflow.

Final answer

Option (B)

Common trap

Confusing the end carry out of the MSB with an arithmetic overflow condition.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2022 Computer Science and Information Technology (CS)
Question: 18 of 65
Section: Computer Science and Information Technology

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