GATE CS 2026 Set 1 Q26 — Number Representations (Fixed and Floating Point)

To find $Z = X + Y$, we first decode the IEEE 754 single-precision representations of $X$ and $Y$. The format consists of 1 sign bit ($S$), 8 exponent bits ($E$), and 23 fraction bits ($F$).

1. Decoding $X = 35\text{C}00000_{16}$:

• Binary: $0011\ 0101\ 1100\ 0000\ 0000\ 0000\ 0000\ 0000_2$
• $S = 0$ (positive)
• $E = 01101011_2 = 107_{10} \implies \text{Actual Exponent} = 107 - 127 = -20$
• $F = 1000\dots_2 \implies \text{Mantissa} = 1.1_2$
• $X = 1.1_2 \times 2^{-20}$

2. Decoding $Y = 34\text{A}00000_{16}$:

• Binary: $0011\ 0100\ 1010\ 0000\ 0000\ 0000\ 0000\ 0000_2$
• $S = 0$ (positive)
• $E = 01101001_2 = 105_{10} \implies \text{Actual Exponent} = 105 - 127 = -22$
• $F = 0100\dots_2 \implies \text{Mantissa} = 1.01_2$
• $Y = 1.01_2 \times 2^{-22}$

3. Aligning Exponents and Addition:

Align $Y$ to the larger exponent of $-20$:

4. Encoding $Z$ back to IEEE 754:

• $S = 0$
• $\text{Actual Exponent} = -20 \implies E = -20 + 127 = 107 = 01101011_2$
• $\text{Mantissa} = 1.1101_2 \implies F = 11010000\dots_2$

Combining these fields into a 32-bit binary representation:

Grouping into 4-bit hexadecimal nibbles:

Thus, the correct option is C.