UGC NET Questions (Paper – 1)

Four hexadecimal digits can represent values from 0000₁₆ to FFFF₁₆, and FFFF is the largest of these. FFFF₁₆ means 15×16³ + 15×16² + 15×16¹ + 15×16⁰. Calculating this gives 61440 + 3840 + 240 + 15 = 65535. Therefore FFFF is the exact hexadecimal representation of 65535. None of the other options expand to this decimal value.

Option A:
FFFF₁₆ expands directly to 65535 because each F contributes 15 times the appropriate power of 16. This makes it the maximum value for four hex digits and matches the decimal given. Therefore this choice is correct.

Option B:
FFFE₁₆ gives 65534 since it differs from FFFF only in the last digit being 14 instead of 15. This is one less than 65535, so it is a near miss but not correct.

Option C:
FFF0₁₆ calculates as 15×4096 + 15×256 + 15×16 + 0, which is 61440 + 3840 + 240 + 0 = 65520. This is slightly smaller than 65535, so it cannot be the correct answer.

Option D:
F0FF₁₆ corresponds to 15×4096 + 0×256 + 15×16 + 15, yielding 61440 + 0 + 240 + 15 = 61695. Because 61695 differs from 65535, this hexadecimal string does not represent the required decimal value.

Four hexadecimal digits range from 0000₁₆ to FFFF₁₆, and FFFF corresponds to 16⁴ − 1. Evaluating 16⁴ gives 65536, so subtracting 1 yields 65535. Therefore, 65535 is the largest decimal value representable with exactly four hex digits. This follows the general pattern that an n-digit base-b number has a maximum value of bⁿ − 1.

Option A:
16383 is the maximum value for 14 bits in binary, but it does not correspond to four hex digits. It is associated with the range up to 2¹⁴ − 1, not with 16⁴ − 1. Hence this option is not appropriate for four hexadecimal positions.

Option B:
32767 is the maximum value for 15 bits in binary (2¹⁵ − 1) and is often seen for signed 16-bit integers. However, it does not match the range of four hexadecimal digits, which is governed by powers of 16, not 2¹⁵.

Option C:
65534 is one less than 65535, so it would correspond to FFFE₁₆ rather than FFFF₁₆. While it is extremely close, it is not the maximum possible value for four hex digits.

Option D:
65535 equals 16⁴ − 1 and is represented as FFFF₁₆. This is the largest value achievable when all four hexadecimal positions are set to their maximum digit F, satisfying the condition of the question.

To convert 1023 to base five, we divide repeatedly by 5 and collect the remainders, obtaining digits 1, 3, 0, 4 and 3 which give 13043₅. Expanding 13043₅ as 1×5⁴ + 3×5³ + 0×5² + 4×5¹ + 3×5⁰ gives 625 + 375 + 0 + 20 + 3 = 1023. Thus 13043 is the correct base-five representation of 1023.

Option A:
13034₅ expands to 1×625 + 3×125 + 0×25 + 3×5 + 4 = 625 + 375 + 0 + 15 + 4 = 1019. This is slightly less than 1023 and therefore cannot be the correct representation.

Option B:
12403₅ corresponds to 1×625 + 2×125 + 4×25 + 0×5 + 3 which is 625 + 250 + 100 + 0 + 3 = 978. This value is far smaller than 1023. Hence this option is incorrect.

Option C:
13103₅ stands for 1×625 + 3×125 + 1×25 + 0×5 + 3 = 625 + 375 + 25 + 0 + 3 = 1028. This is greater than 1023 by five, showing that the digit configuration is not correct.

Option D:
13043₅ gives 625 + 375 + 0 + 20 + 3 = 1023, which exactly matches the decimal number. The pattern of digits reflects accurate repeated division by 5 and correct positional weighting.

To convert 111.011₂ to decimal, we sum the weighted values of each bit. The integer part 111₂ equals 4 + 2 + 1 = 7. The fractional part .011₂ equals 0/2 + 1/4 + 1/8 = 0.375. Adding 7 and 0.375 gives 7.375. Therefore, the correct decimal equivalent is 7.375.

Option A:
If we interpreted the value as 7.25, the fractional part would need to be .01₂, which equals 0.25. However, 111.011₂ contains two fractional ones at positions 1/4 and 1/8, so the fraction is 0.375, not 0.25. Thus this option is not correct.

Option B:
7.375 matches the sum of 7 from the integer part and 0.375 from the fractional part .011₂. Each bit contributes correctly according to its negative power of two. This makes 7.375 the correct conversion from binary to decimal.

Option C:
A value of 7.5 would require the fractional part to be .1₂, corresponding to 0.5. Since the given binary fraction has bits in the 1/4 and 1/8 places instead, the value does not reach 7.5. Hence this option is incorrect.

Option D:
7.625 would correspond to .101₂ as the fractional portion, giving 0.625. The actual fraction .011₂ is 0.375, so 7.625 does not match the given binary fraction either.

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.

In hexadecimal, 3E7 means 3×16² + 14×16¹ + 7×16⁰. Calculating this gives 3×256 + 14×16 + 7 = 768 + 224 + 7 = 999. This matches the target decimal value exactly. Hence 3E7 is the correct hexadecimal representation of 999.

Option A:
3D7 corresponds to 3×256 + 13×16 + 7 which equals 768 + 208 + 7 = 983. As 983 is less than 999, this option does not match the given decimal number. It shows how changing a single digit alters the decimal result.

Option B:
3E6 converts as 3×256 + 14×16 + 6 = 768 + 224 + 6 = 998. This value is very close to 999 but still one less, so it is not acceptable as the exact representation. Small differences in the least significant digit matter in conversions.

Option C:
3E7, with digits 3, E (14) and 7, expands to 768 + 224 + 7 = 999. Because this equals the required decimal, this hexadecimal code is exactly correct. Its structure shows how three hex digits can represent a three-digit decimal number just under 1000.

Option D:
3F7 stands for 3×256 + 15×16 + 7 = 768 + 240 + 7 = 1015. Since 1015 is greater than 999, the value overshoots the target. Therefore it cannot be considered the correct conversion for 999.

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.