Range of 12-bit two’s complement – UGC NET Question

In an n-bit two's complement system, one bit is reserved for the sign and the remaining n-1 bits contribute to magnitude. The range of values is from -2ⁿ⁻¹ to 2ⁿ⁻¹ - 1. For 12 bits, this is -2¹¹ to 2¹¹ - 1, which equals -2048 to 2047. This asymmetric range is characteristic of two's complement representation. Hence, option D correctly states the representable range.

Option A:
Option A describes the range of an unsigned 12-bit representation, where all combinations are non-negative and go up to 4095. It does not apply to two's complement, which must allow negative numbers. Therefore, it is incorrect in this context.

Option B:
Option B suggests a symmetric range around zero from -4096 to 4095, which would require 13 bits rather than 12 in two's complement. It misuses the formula for the minimum and maximum values and thus is not accurate.

Option C:
Option C gives a symmetric range -2047 to 2047, which is appropriate for sign-magnitude but not for two's complement. In two's complement you get one extra negative value, making the minimum -2048. Hence this range is off by one on the negative side.

Option D:
Option D uses the correct two's complement formula for 12 bits, giving a minimum of -2048 and a maximum of 2047. This matches how the sign bit and magnitude bits interact in two's complement encoding. Therefore, it is the only option that correctly describes the 12-bit two's complement range.