Express (1-2 i)^ -3 in the form of (a+i b). - Vidyadip

Question

Express $(1-2 i)^{-3}$ in the form of $(a+i b)$.

✓

Answer

Let $z=(1-2 i)^{-3}$
$=\frac{1}{(1-2 i)^3}=\frac{1}{1-8 i^3-6 i+12 i^2}\left[\because(a-b)^3=a^3-b^3-3 a^2 b+3 a b^2\right]$
$=\frac{1}{1-8 i^2 \cdot i-6 i+12(-1)}$
$=\frac{1}{1+8 i-6 i-12}\left[\because i^2=-1\right]$
$=\frac{1}{-11+2 i}=\frac{1}{-11+2 i} \times \frac{-11-2 i}{-11-2 i}[\text { multiplying numerator and denominator by }-11-2 i]$
$=\frac{-11-2 i}{(-11)^2-(2 i)^2}=\frac{-11-2 i}{121+4}\left[\because(a-b)(a+b)=a^2-b^2\right]$
$=\frac{-11-2 i}{125}=\frac{-11}{125}-\frac{2 i}{125}=a+i b[\text { say }]$
where, $a=\frac{-11}{125}$ and $b =\frac{-2}{125}$

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 "3 Marks Question" from Model Paper 9→Model Paper 9 — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Which term of the G.P.:$\sqrt{3},3,3\sqrt{3},\ \dots\text{is}729?$

A bag contains six white marbles and five red marbles. Find the number of ways in which four marbles can be drawn from the bag if:

They can be of any colour.
Two must be white and two red and.
They must all be of the same colour.

A farmer buys a used tractor for $₹\ 12000.$ He pays $₹\ 6000$ cash and agrees to pay the balance in annual installments of $₹\ 500$ plus $12\ %$ interest on the unpaid amount. How much will the tractor cost him$?$

If $\text{a}\in[-1,2,3,4]$ and $\text{b}\in [0, 3, 6],$ write the set of all ordered pairs (a, b) such that a + b= 5.

Define the function $f: R \rightarrow R$ by $y = f(x) = x^2, x \in R$. Complete the Table given below by using this definition. What is the domain and range of this function? Draw the graph of f.

$x$	$– 4$	$–3$	$–2$	$–1$	$0$	$1$	$2$	$3$	$4$
$y = f(x) = x^2$

Find the equation of the right bisector of the line segment joining the points A(1, 0) and B(2, 3).

If $\sin\text{A}=\frac{1}{2},$ $\cos\text{B}=\frac{\sqrt{3}}{2},$ where $\frac{\pi}{2}<\text{A}<\pi$ and $0<\text{B}<\frac{\pi}{2},$find the following:$\tan{\text{(A+B)}}$

Solve the following linear inequations in R:
$\text{x}-2\leq\frac{5\text{x}+8}{3}$

$\text{b}\cos\text{B + c}\cos\text{C}=\text{a}\cos(\text{B}-\text{C})$

Solve the following system of equations in R.
$|3-4\text{x}|\geq9$