NIMCET 2026 — Computer PYQ

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Question

NIMCET 2026 — Computer PYQ

NIMCET | Computer | 2026

Consider the following statements about the range of numbers in a 9-bit 1's complement and 2's complement system.

I. In 9-bit 1's complement, the range is -255 to +255, and there exist two representations of zero.

II. In 9-bit 2's complement, the range is -256 to +255, and both 1's complement and 2's complement can represent exactly 512 unique values.

III. The maximum positive number representable is +255 in both 1's complement and 2's complement 9-bit systems.

Identify the CORRECT option.

Choose the correct answer:

Explanation

Complete Solution

To evaluate the correctness of each statement, let's use the standard formulas for the range of numbers representable using an $n$ -bit system where $n = 9$ .

Analysis of Statement I

Formula for 1's Complement Range:
$[- (2^{n - 1} - 1) to + (2^{n - 1} - 1)]$
Substituting $n = 9$ :
$Range = [- (2^{9 - 1} - 1) to + (2^{9 - 1} - 1)]$
$Range = [- (2^{8} - 1) to + (2^{8} - 1)] = [- (256 - 1) to + (256 - 1)] = - 255 to + 255$
Representation of Zero: In a 1's complement system, there are two distinct representations for zero: positive zero ( $+ 0$ , represented as 000000000) and negative zero ( $- 0$ , represented as 111111111).
Conclusion: Statement I is CORRECT.

Analysis of Statement II

Formula for 2's Complement Range:
$[- 2^{n - 1} to + (2^{n - 1} - 1)]$
Substituting $n = 9$ :
$Range = [- 2^{9 - 1} to + (2^{9 - 1} - 1)]$
$Range = [- 2^{8} to + (2^{8} - 1)] = - 256 to + 255$
Unique Values Represented:
- With $9$ bits, there are a total of $2^{9} = 512$ possible binary combinations.
- In 2's complement, because there is only one unique representation for zero, all $512$ combinations correspond to 512 unique values.
- In 1's complement, because zero has two representations ( $+ 0$ and $- 0$ ), two different bit patterns represent the exact same value. Therefore, it can only represent $512 - 1 = 511$ unique values.
Conclusion: Statement II is INCORRECT because 1's complement does not represent 512 unique values.

Analysis of Statement III

Maximum Positive Number in 1's Complement:
$+ (2^{n - 1} - 1) = + (2^{8} - 1) = + 255$
Maximum Positive Number in 2's Complement:
$+ (2^{n - 1} - 1) = + (2^{8} - 1) = + 255$
Conclusion: Statement III is CORRECT as both systems share the exact same maximum positive limit for a 9-bit configuration.

Summary Table

Number System (n=9)	Minimum Value	Maximum Value	Total Unique Values	Representations of Zero
1's Complement	$- 255$	$+ 255$	$511$	Two ( $+ 0$ and $- 0$ )
2's Complement	$- 256$	$+ 255$	$512$	One ( $0$ )

Since only statements I and III are true, the correct choice is option 3.