1. **Problem Statement:** (a) Define analogue and digital systems and explain their differences. (b) Explain two's complement and perform the operations: (i) $ (1010)_2 + (1101)_2 $ (ii) $ 48 - 21 $ 2. **Analogue vs Digital Systems:** - An analogue system processes continuous signals that vary smoothly over time. - A digital system processes discrete signals, typically represented by binary values (0s and 1s). **Differences:** - Analogue signals are continuous; digital signals are discrete. - Analogue systems are more susceptible to noise; digital systems are more noise-resistant. - Digital systems allow easier data storage and processing. 3. **Two's Complement:** - Two's complement is a method to represent signed integers in binary. - To find the two's complement of a number, invert all bits and add 1. - It allows easy binary addition and subtraction. 4. **Operation (i):** $ (1010)_2 + (1101)_2 $ - Convert to decimal: $1010_2 = 10$, $1101_2 = 13$ - Add: $10 + 13 = 23$ - Binary sum: Add directly using binary addition: \begin{align*} &\quad 1010 \\ +&\quad 1101 \\ \hline & 10111 \end{align*} - Result is $10111_2$ which is 23 in decimal. 5. **Operation (ii):** $48 - 21$ - Convert to binary: $48 = 00110000_2$, $21 = 00010101_2$ - Find two's complement of 21: - Invert bits: $11101010$ - Add 1: $11101011$ - Add to 48: \begin{align*} &00110000 \\ +&11101011 \\ \hline &00011011 \quad (ignore carry) \end{align*} - Result: $00011011_2 = 27$ decimal. **Final answers:** (i) $10111_2$ (ii) $27$