Kerala KEAM 5 June 2024 Solved Paper

We are given the expression: $$\left( \frac{1+i}{1-i} \right) \left( \frac{2+i}{2-i} \right)$$First, we simplify each fraction by multiplying the numerator and denominator by the conjugate of the denominator.Step 1: Simplifying $\frac{1+i}{1-i}$:Multiply the numerator and denominator by $1+i$ (the conjugate of $1-i$): $$\frac{1+i}{1-i} \times \frac{1+i}{1+i} = \frac{(1+i)^2}{(1-i)(1+i)}$$The denominator becomes: $$(1-i)(1+i) = 1^2 - i^2 = 1 - (-1) = 2$$The numerator becomes: $$(1+i)^2 = 1^2 + 2i + i^2 = 1 + 2i - 1 = 2i$$Thus: $$\frac{1+i}{1-i} = \frac{2i}{2} = i$$Step 2: Simplifying $\frac{2+i}{2-i}$:Multiply the numerator and denominator by $2+i$ (the conjugate of $2-i$): $$\frac{2+i}{2-i} \times \frac{2+i}{2+i} = \frac{(2+i)^2}{(2-i)(2+i)}$$The denominator becomes: $$(2-i)(2+i) = 2^2 - i^2 = 4 - (-1) = 5$$The numerator becomes: $$(2+i)^2 = 2^2 + 2 \cdot 2 \cdot i + i^2 = 4 + 4i - 1 = 3 + 4i$$Thus: $$\frac{2+i}{2-i} = \frac{3+4i}{5} = \frac{3}{5} + \frac{4i}{5}$$Step 3: Now, multiply the two simplified expressions: $$\left( i \right) \left( \frac{3}{5} + \frac{4i}{5} \right)$$Distribute $i$ across the terms: $$i \times \frac{3}{5} = \frac{3i}{5}, \quad i \times \frac{4i}{5} = \frac{4i^2}{5} = \frac{-4}{5}$$Thus, the result is: $$\frac{3i}{5} - \frac{4}{5}$$The real part of the complex number is $-\frac{4}{5}$.Thus, the correct answer is option (D).Quick Tip: When simplifying expressions with complex numbers, remember to multiply by the conjugate of the denominator. This will eliminate the imaginary part from the denominator, making the expression easier to handle.