Q113 / 180 · KCET 2013 · Mathematics

The modulus and amplitude of $\frac{1+2 i}{1-(1-i)^{2}}$ are

  • A
    $\sqrt{2}$ and $\frac{\pi}{6}$
  • B
    1 and $\frac{\pi}{4}$
  • Correct answer
    C
    1 and 0
  • D
    1 and $\frac{\pi}{3}$

Solution

Let $z=\frac{1+2 i}{1-(1-i)^{2}}$ $\Rightarrow z=\frac{1+2 i}{1-\left(1+i^{2}-2 i\right)}=\frac{1+2 i}{-i^{2}+2 i}$ $=\frac{1+2 i}{1+2 i} \quad\left(\because i^{2}=-1\right)$ $\Rightarrow \quad z=1+0 i$ $\therefore$ Modulus of $z=|z|=|1+0 \cdot i|$ $=\sqrt{1^{2}+0^{2}}=1$ and amplitude of $z \equiv \tan ^{-1}\left(\frac{0}{1}\right)$ $=\tan ^{-1}(0)=\tan ^{-1}(\tan 0)$ $=0$

Hence, the correct answer is option: 'C'.

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