UGC NET Questions (Paper – 1)

A three-digit octal number has the form abc where each digit ranges from 0 to 7. The largest such number is 777 in base eight. Its decimal value is 7×8^2 + 7×8 + 7 = 7×64 + 56 + 7 = 448 + 63 = 511. Therefore, 511 is the maximum decimal value representable by a three-digit octal number.

Option A:
Option A, 255, is the maximum decimal value for an 8-bit unsigned binary pattern, not for three-digit octal. An octal number 377 corresponds to 255, which is smaller than the maximum 777. Hence, 255 does not represent the full range of three-digit octal numbers.

Option B:
Option B, 343, is just an intermediate decimal value and has no special significance as a maximum here. When converting 777 from octal, we obtain 511, which is clearly larger than 343. Thus, 343 cannot be the maximum decimal value for three-digit octal numbers.

Option C:
Option C, 512, is 1 greater than 511 and equals 8^3. It represents the total count of distinct three-digit octal combinations starting from 000. However, since numbering starts at 0, the highest representable value is 512 - 1 = 511, not 512.

Option D:
Option D, 511, results from converting 777₈ using place values 64, 8, and 1. The calculation 7×64 + 7×8 + 7 = 511 confirms it is the largest decimal value achievable. Hence, it is the correct maximum for three-digit octal numbers.

To convert binary to octal, we group the bits in triples from the right. The number 111011110 becomes 111 011 110. Each group is then converted: 111₂ = 7₈, 011₂ = 3₈, and 110₂ = 6₈. Combining these octal digits gives 736₈ as the equivalent of 111011110₂.

Option A:
Option A, 756, would correspond to binary groups 111 101 110, which is different from the original 111 011 110.

Option B:
Option B, 7360, implies an extra 3-bit group (or trailing zeros) that are not present in the given binary numeral.

Option C:
Option C, 734, would correspond to 111 011 100; the last group does not match 110.

Option D:
Option D, 736, comes directly from converting 111, 011, and 110 into 7, 3, and 6.

To convert 1023 to octal, we repeatedly divide by 8 and record remainders. Dividing 1023 by 8 gives a quotient of 127 and remainder 7. Dividing 127 by 8 gives 15 with remainder 7, and 15 divided by 8 gives 1 with remainder 7. Reading the remainders from last to first, we get 1777₈, so 1023₁₀ equals 1777₈.

Option A:
Option A, 1770, corresponds to a decimal value slightly less than 1023. If we convert 1770₈ back to decimal, we do not obtain 1023. Therefore, it cannot be the correct octal representation of 1023.

Option B:
Option B, 1775, differs in the last digit from the correct octal representation. This change in the unit place affects the decimal value. Converting 1775₈ gives a number slightly smaller than 1023, so it does not match the given decimal.

Option C:
Option C, 1777, is formed by the sequence of remainders 7, 7, 7 from the division process. Its decimal value is 1×8^3 + 7×8^2 + 7×8 + 7 = 512 + 448 + 56 + 7 = 1023. Thus, it is the correct octal equivalent.

Option D:
Option D, 2000, is equal to 2×8^3 in decimal, which is 2×512 = 1024. This is one more than 1023, so it cannot represent the given decimal value. Therefore, 2000₈ is not the correct octal representation.