UGC NET Questions (Paper – 1)

Decimal 8 can be expressed in octal as one group of eight, which is 10₈. In octal, the place values are 1, 8, 64 and so on, similar to how decimal uses 1, 10 and 100. Thus a single 8 corresponds to 1 in the eights place and 0 in the ones place, giving 10₈.

Option A:
Option A, 7₈, equals seven in decimal, which is one less than 8. It does not include the necessary value in the eights place.

Option B:
Option B, 11₈, equals 1×8 + 1 = 9 in decimal, which is greater than 8. It overshoots the correct value and therefore cannot be the right representation.

Option C:
Option C is correct because 10₈ means 1×8 + 0×1 = 8. This exactly matches the decimal value and follows the octal place value structure.

Option D:
Option D, 8, is not a valid single octal digit since octal digits range from 0 to 7. Writing 8 as a digit in an octal number would violate the base-8 rules.

Dividing 2024 repeatedly by 8 yields remainders that form the octal digits 3, 7, 5 and 0, leading to 3750₈. Expanding 3750₈ as 3×8³ + 7×8² + 5×8¹ + 0×8⁰ equals 1536 + 448 + 40 + 0 = 2024. Hence 3750 is the correct octal representation of 2024.

Option A:
3750₈ corresponds exactly to 3×512 + 7×64 + 5×8 + 0 = 2024. This confirms it as the correct octal form of the given decimal number. Its digits reflect the sequence of remainders from the division process.

Option B:
3740₈ represents 3×512 + 7×64 + 4×8 + 0 = 1536 + 448 + 32 = 2016. Since 2016 is less than 2024, this option is not correct.

Option C:
3570₈ equals 3×512 + 5×64 + 7×8 + 0 = 1536 + 320 + 56 = 1912. This decimal value is far from 2024, so the digit pattern is not appropriate.

Option D:
3650₈ gives 3×512 + 6×64 + 5×8 + 0 = 1536 + 384 + 40 = 1960. As 1960 is below 2024, this cannot be the correct conversion.

The decimal 4095 is 2¹² − 1, which is also one less than 8⁴, so in octal it is represented by four 7s. 7777₈ expands to 7×8³ + 7×8² + 7×8¹ + 7×8⁰. This equals 3584 + 448 + 56 + 7 = 4095. Thus 7777 is the correct octal representation of 4095.

Option A:
7770₈ equals 7×512 + 7×64 + 7×8 + 0 = 3584 + 448 + 56 + 0 = 4088. Since this value is slightly less than 4095, it cannot represent the given decimal. It is a close distractor but not exact.

Option B:
7707₈ converts to 7×512 + 7×64 + 0×8 + 7 = 3584 + 448 + 0 + 7 = 4039. This value is again less than 4095, making it an incorrect conversion. The zero in the 8¹ place reduces the total.

Option C:
7077₈ stands for 7×512 + 0×64 + 7×8 + 7 = 3584 + 0 + 56 + 7 = 3647. This is much smaller than 4095, so it cannot be correct. Missing 7 in the 8² position significantly decreases the value.

Option D:
7777₈, when expanded, yields 3584 + 448 + 56 + 7 = 4095. Because this matches the specified decimal exactly, it is the correct answer. It also illustrates that a sequence of maximal digits represents the largest value for a fixed number of positions.

Using repeated division of 1729 by 8 gives remainders that form the octal digits 3, 3, 0 and 1 from most significant to least significant. Thus the correct octal form is 3301₈. Expanding 3301₈ as 3×8³ + 3×8² + 0×8¹ + 1×8⁰ yields 1536 + 192 + 0 + 1 = 1729. This confirms that 3301 is the correct octal representation.

Option A:
3231₈ expands to 3×512 + 2×64 + 3×8 + 1 = 1536 + 128 + 24 + 1 = 1689. Since 1689 is less than 1729, this octal number cannot represent the given decimal value. It is a reasonable distractor but numerically wrong.

Option B:
3301₈ evaluated as 3×512 + 3×64 + 0×8 + 1 gives 1536 + 192 + 0 + 1 = 1729. This matches the target decimal exactly, demonstrating a correct conversion. The pattern of digits and place values reflects proper repeated division by eight.

Option C:
3401₈ gives 3×512 + 4×64 + 0×8 + 1 = 1536 + 256 + 0 + 1 = 1793. Because 1793 is greater than 1729, it overshoots the required value. Thus, it cannot be the correct conversion.

Option D:
3310₈ represents 3×512 + 3×64 + 1×8 + 0 = 1536 + 192 + 8 + 0 = 1736. As 1736 is slightly above 1729, this octal number is again not the exact representation. It shows how sensitive the last digit is in positional systems.

The decimal 511 is equal to 2⁹ − 1, which is one less than 512 and corresponds to nine ones in binary. In octal, this boundary value is represented as 777. 777₈ expands to 7×8² + 7×8¹ + 7×8⁰ = 448 + 56 + 7 = 511. Thus 777 is the correct octal representation of 511.

Option A:
757₈ equals 7×64 + 5×8 + 7 = 448 + 40 + 7 = 495. This is less than 511, so it does not match the given decimal number.

Option B:
767₈ represents 7×64 + 6×8 + 7 = 448 + 48 + 7 = 503. Because 503 differs from 511, this octal number is not correct.

Option C:
775₈ stands for 7×64 + 7×8 + 5 = 448 + 56 + 5 = 509. This value is close to but still less than 511, so it cannot be the exact representation.

Option D:
777₈ expands to 448 + 56 + 7 = 511, exactly equal to the decimal number specified. Its all-7 pattern reflects the maximum value for three octal digits, analogous to all ones in binary.