Q110 / 160 · WBJEE 2009 · Mathematics
The modulus of $\frac{1-i}{3+i}+\frac{4 i}{5}$ is
- A$\sqrt{5}$ unit
- B$\frac{\sqrt{11}}{5}$ unit
- Correct answerC$\frac{\sqrt{5}}{5}$ unit
- D$\frac{\sqrt{12}}{5}$ unit
Solution
$$
\text { Hints: } \frac{1-i}{3+i}+\frac{4 i}{5}=\frac{5-5 i+4 i(3+i)}{5(3+i)}=\frac{5-5 i+12 i-4}{5(3+i)}=\frac{1+7 i}{5(3+i)}=\frac{(1+7 i)(3-i)}{5(9+1)}
$$
$$
=\frac{3+21 i-i+7}{5 \times 10}=\frac{10+20 i}{5 \times 10}=\frac{1+2 i}{5}
$$
$$
\therefore \text { Modulus }=\sqrt{\left(\frac{1}{5}\right)^2+\left(\frac{2}{5}\right)^2}=\sqrt{\frac{1}{25}+\frac{4}{25}}=\sqrt{\frac{1}{5}}=\frac{\sqrt{5}}{5} \text { unit }
$$
Hence, the correct answer is option: 'C'.