If ( 1-i 1+i )^ 100 =a+ib, find (a, b). - Vidyadip

Question

If $\Big(\frac{1-\text{i}}{1+\text{i}}\Big)^{100}=\text{a}+\text{ib},$ find (a, b).

✓

Answer

$\Big(\frac{1-\text{i}}{1+\text{i}}\Big)^{100}=\text{a}+\text{ib}$ $\Rightarrow\Big(\frac{(1-\text{i})(1-\text{i})}{(1+\text{i})(1-\text{i})}\Big)^{100}=\text{a}+\text{ib}$ [Rationalizing the denominator] $\Rightarrow\Big(\frac{(1-2\text{i}-1)}{(1+1)}\Big)^{100}=\text{a}+\text{ib}$ $\Rightarrow\Big(\frac{-2\text{i}}{2}\Big)^{100}=\text{a}+\text{ib}$ $\Rightarrow(-\text{i})^{100}=\text{a}+\text{ib}$ $\Rightarrow1=\text{a}+\text{ib}$ Comparing, we get $(\text{a},\text{b})=(1,0)$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 4 marks)" from Complex Numbers→Complex Numbers — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let A = {1, 2, 3, ... ,14}. Define a relation R from A to A by R = {(x, y) : 3x – y = 0, where x, y ∈ A}. Write down its domain, codomain and range.

Sketch the graphs of the following trigonometric functions: $\text{u(x)}=\cos^2\frac{\text{x}}{2}$

Find the equation of the hyperbola whose Focus is at $(4, 2)$ centre at $(6, 2)$ and $e = 2$.

Determine the points xy-plane equidistant from the points $A(1, -1, 0), B(2, 1, 2)$ and $C(3, 2, -1)$.

Solve the system of inequality graphically: x + y $\le$ 6, x + y $\ge$ 4

If a, b, c, are in G.P., prove that: $\big(\text{a}^2-\text{b}^2\big),\big(\text{b}^2-\text{c}^2\big),\big(\text{c}^2-\text{d}^2\big)\text{ are in G.P.}$

Show that the product of perpendiculars on the line $\frac{\text{x}}{\text{a}}\cos\theta+\frac{\text{y}}{\text{b}}\sin\theta=1$ from the points $\Big(\sqrt{\text{a}^2-\text{b}^2},0\Big)$ is $\text{b}^2.$

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{\sqrt{2}}}\frac{\sqrt{5+2\text{x}}-\big(\sqrt{2}+1\big)}{\text{x}^2-2}$

If a, b, c, d are in G.P., prove that: $\big(\text{a}^2+\text{b}^2+\text{c}^2\big),\big(\text{ab}+\text{bc}+\text{cd}\big),\big(\text{b}^2+\text{c}^2+\text{d}^2\big)\text{ are in G.P.}$

If $\text{f}(\text{x})=\begin{cases}|\text{x}|+1,& \text{x}< 0\\0 & \text{x} = 0\\|\text{x}|-1,&\text{x}>1\end{cases}$. For what value (s) of a does $\lim\limits_{\text{x}\rightarrow\text{a}}\text{f}(\text{x})$ exists?