Complex Numbers — Easy JEE math MCQ
If $z = \frac{1 + i}{1 - i}$, then $|z|$ and $\arg(z)$ are respectively:
- A. $1$ and $\frac{\pi}{2}$
- B. $\sqrt{2}$ and $\frac{\pi}{4}$
- C. $1$ and $\frac{\pi}{4}$
- D. $\sqrt{2}$ and $\frac{\pi}{2}$
Solution
Simplify $z$ by multiplying numerator and denominator by the conjugate of denominator:
$$z = \frac{1 + i}{1 - i} \times \frac{1 + i}{1 + i} = \frac{(1 + i)^2}{1 - i^2}$$
$$= \frac{1 + 2i + i^2}{1 - (-1)} = \frac{1 + 2i - 1}{2} = \frac{2i}{2} = i$$
Therefore:
- $|z| = |i| = 1$
- $\arg(z) = \arg(i) = \frac{\pi}{2}$