Complex Numbers and Quadratic Equations — MATHS STD 11 Science — Question

1. 1

Solution:

Let $\text{z}=\frac{1+\text{i}}{1-\text{i}}$

Rationalising the denominator:

$\text{z}=\frac{1+\text{i}}{1-\text{i}}\times\frac{1+\text{i}}{1+\text{i}}$

$\Rightarrow\text{z}=\frac{1+\text{i}^2+2\text{i}}{1-\text{i}^2}$

$\Rightarrow\text{z}=\frac{1-1+2\text{i}}{1+1}$

$\Rightarrow\text{z}=\frac{2\text{i}}{2}$

$\Rightarrow\text{z}=\text{i}$

$\Rightarrow\text{z}^4=\text{i}^4$

Since $\text{i}^2=-1,$ we have:

$\Rightarrow\text{z}^4=\text{i}^2\times\text{i}^2$

$\Rightarrow\text{z}^4=1$