UGC NET Questions (Paper – 1)

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.

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.