UGC NET Questions (Paper – 1)

In a positional system with radix r and k digits, the smallest value is all zeros and the largest is all digits equal to r−1. The numeric value of k digits all set to r−1 is (r−1)(1 + r + r² + … + r^(k−1)), which sums to (r−1)(r^k − 1)/(r−1) = r^k − 1. This is the highest value attainable without increasing the number of digits.

Option A:
Option A precisely captures the algebraic result of a geometric series of place values. It recognises that with k digits, the largest magnitude is one less than the next power of the base, r^k. Therefore the formula r^k − 1 is logically consistent with positional representation.

Option B:
Option B, r^(k−1) − 1, only considers up to the (k−1)-th power and would correspond to the maximum value of a (k−1)-digit number instead. It underestimates the representable range for k digits and thus is incorrect.

Option C:
Option C, r^k, corresponds to the value requiring k+1 digits in general, because r^k is written as 1 followed by k zeros in base r. It therefore exceeds the maximum value that fits in exactly k digits.

Option D:
Option D, r^k + 1, extends the overestimation even further and represents a number that is not just larger but two units beyond the next power of r. Such a value cannot be expressed using only k digits in that base.