GATE CS 2017 Set 1 Q7 — Number Representations (Fixed and Floating Point)

For an unsigned $n$-bit fixed-point representation with $i$ integer bits and $f$ fractional bits ($n = i + f$):

• Minimum value: Since the representation is unsigned, the minimum value occurs when all $n$ bits are $0$, which represents $0$.
• Maximum value: The maximum value occurs when all $n$ bits are $1$. The weight of the least significant bit (LSB) in the fractional part is $2^{-f}$. The value represented by all $n$ bits being $1$ is:
  $$\sum_{k=-f}^{i-1} 2^k = 2^i - 2^{-f}$$

Thus, the range of decimal values for $X$ is $0$ to $\left(2^i - 2^{-f}\right)$, which corresponds to Option D.