UGC NET Questions (Paper – 1)

To convert 999 to base seven, we perform repeated division by 7 and collect the remainders, which yields the digits 2, 6, 2 and 5 in base 7. Reading them from most significant to least significant gives 2625₇. Expanding 2625₇ as 2×7³ + 6×7² + 2×7¹ + 5 gives 686 + 294 + 14 + 5 = 999. Hence 2625 is the correct base-seven representation of 999.

Option A:
2624₇ expands as 2×343 + 6×49 + 2×7 + 4 = 686 + 294 + 14 + 4 = 998. Because it is one less than 999, it cannot be the correct base-seven form. This is a typical near-miss distractor in conversion questions.

Option B:
2665₇ equals 2×343 + 6×49 + 6×7 + 5 = 686 + 294 + 42 + 5 = 1027. This value is larger than 999, so it does not match the required decimal. The higher middle digits inflate the base-seven value.

Option C:
2635₇ represents 2×343 + 6×49 + 3×7 + 5 = 686 + 294 + 21 + 5 = 1006. Since 1006 is slightly above 999, this cannot be the correct representation. It again illustrates sensitivity to changes in the lower-order digits.

Option D:
2625₇ when expanded gives 686 + 294 + 14 + 5 = 999. That matches the decimal value exactly, confirming its correctness. This option reflects accurate execution of repeated division and positional weighting in base-seven conversions.

Converting 345 to base seven requires dividing by 7 and tracking remainders, which produces digits 1, 0, 0 and 2, giving 1002₇. Expanding 1002₇ as 1×7³ + 0×7² + 0×7¹ + 2×7⁰ gives 343 + 0 + 0 + 2 = 345. Hence, 1002 is the correct base-seven representation of 345.

Option A:
1003₇ expands as 1×343 + 0×49 + 0×7 + 3 = 343 + 0 + 0 + 3 = 346. This value is one more than 345, so this option is incorrect.

Option B:
1052₇ corresponds to 1×343 + 0×49 + 5×7 + 2 = 343 + 0 + 35 + 2 = 380. As this is significantly larger than 345, it cannot be the correct conversion.

Option C:
1002₇ represents 343 + 0 + 0 + 2 = 345, which exactly equals the decimal given. Its structure shows that only the highest and lowest positions carry nonzero digits in this base-seven form.

Option D:
1012₇ gives 1×343 + 0×49 + 1×7 + 2 = 343 + 0 + 7 + 2 = 352. Since 352 is larger than 345, this pattern is not the right representation.