Suppose that 12-bit two's complement numbers are being employed. What decimal integer would give rise to the twelve-bit sequence 1111 1110 0010:

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First, restating the problem helps ground the analysis: we are using 12-bit two's complement numbers, and the bit pattern given is 1111 1110 0010. The multiple-choice options are: +226, −226, −30, −31, +158.

Option 1: +226. In two's complement with 12 bits, a positive number would have a leading 0. Since the given pattern starts with 1, it cannot represent a positive value in 12-bit two's complement. Therefore +226 is inconsistent with the sign bit and is incorrect.

Option 2: −226. To evaluate a negative value in two's complement, one can invert all bits and add 1, or recognize that the magnitude should correspond to the two's complement magnitude. Inverting 1111 1110 0010 yields 0000 0001 1101, which is 29 in decimal; adding 1 gives 30, so the magnitude is 30 with a negative sign. This indicates the value is −30, not −226, so this option is incorrect.

Option 3: −30. This matches the result obtained by standard two's complement conversion: since the MSB is 1, the number is negative; inverting the bits 1111 1110 0010 gives 0000 0001 1101 (which is 29), and adding 1 yields 30, so the value is −30. This aligns perfectly with the 12-bit two's complement interpretation, so this option is consistent with the given bit sequence.

Option 4: −31. If the value were −31, the process would yield a magnitude of 31 after conversion (invert bits to get 0000 0001 1110, which is 30, plus 1 would be 31). The actual inverted-and-added result is 30, not 31, so −31 does not fit the bit pattern provided.

Option 5: +158. As discussed earlier, the leading bit is 1, which designates a negative value in 12-bit two's complement. A positive interpretation like +158 would require a leading 0, which is not the case here. Hence this option is invalid for the given pattern.

In summary, the pattern 1111 1110 0010 in 12-bit two's complement corresponds to a negative value whose magnitude is 30, so the correct interpretation is −30, while all other options fail to match the sign bit rule or the magnitude derived from the two's complement conversion.

COMP10002_2025_SM2 Exam: Foundations of Algorithms (COMP10002_2025_SM2)- Requires Respondus LockDown Browser

Suppose that 12-bit two's complement numbers are being employed. What decimal integer would give rise to the twelve-bit sequence 1111 1110 0010:

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