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

Differentiate the following function with respect to $(\text{x})$:
$\Big(\text{x}+\frac{1}{\text{x}}\Big)\Big(\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}}\Big)$

If the 5th and 12th terms of an A.P. are 30 and 65 respectively, what is the sum of first 20 terms?

If $a, b, c$ are in G.P. prove that:
(i) $\log a^n, \log b^n, \log c^n$ are in G.P.
(ii) $\log a, \log b, \log c$ are in G.P.
(iii) $\left(a^2+b^2\right)\left(a^2+c^2\right)=(a b+b c)^2$
(iv) $a\left(b^2+c^2\right)=c\left(a^2+b^2\right)$

Let $S$ be the sum, $P$ the product and $R$ the sum of reciprocals of n terms in a G.P. Prove that $P^2R^n = S^n.$

Let A and B are sets. If $A \cap X = B \cap X = \phi $ and $A \cup X = B \cup X$ for some set X. Show that A = B.
[Hints A = A $\cap$ ( A $\cup$ X ) , B = B $\cap$ ( B $\cup$ X ) and use Distributive law ]

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

If $\text{a}^2,\ \text{b}^2,\ \text{c}^2$ are in A.P., prove that $\frac{\text{a}}{\text{b}+\text{c}},\frac{\text{b}}{\text{c}+\text{a}},\frac{\text{c}}{\text{a}+\text{b}}$ are in A.P.

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$?$

Using binomial theorem, write down the expansions of the following:
$\Big(\text{x}-\frac{1}{\text{x}}\Big)^6$

From a well shuffled deck of 52 cards, 4 cards are drawn at random. What is the probability that all the drawn cards are of the same colour.