Question 55

1. Sign Bit ($S$):
   The number is negative ($-14.25$), so sign bit $S = 1$.

2. Binary Conversion of Magnitude $14.25$:

   • Integer part: $14_{10} = 1110_2$.
   • Fractional part: $0.25_{10} = 0.01_2$.
   • Magnitude: $14.25_{10} = 1110.01_2$.

3. Normalized Scientific Notation:

   $$1110.01_2 = 1.11001_2 \times 2^3.$$

   Exponent $e = 3$.

4. Biased Exponent ($E$):

   $$E = e + \text{Bias} = 3 + 127 = 130_{10} = 10000010_2.$$

5. Mantissa ($M$):
   Drop the leading implicit 1 and pad to 23 bits:

   $$M = 11001000000000000000000_2.$$

6. Hexadecimal Encoding:
   Combine fields $(S \mid E \mid M)$:

   $$\underbrace{1}_{S} \ \underbrace{1000001}_{E[7..1]} \ \underbrace{0}_{E[0]} \ \underbrace{11001000000000000000000}_{M}$$

   Group into 4-bit nibbles:

   • 1100 = C
   • 0001 = 1
   • 0110 = 6
   • 0100 = 4
   • 0000 = 0
   • 0000 = 0
   • 0000 = 0
   • 0000 = 0

Result: C1640000H (Option A).