Question 7, Exercise 1.2

Solutions of Question 7 of Exercise 1.2 of Unit 01: Complex Numbers. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

Separate into real and imaginary parts $\dfrac{2+3i}{5-2i}$.

$$\begin{align}&\dfrac{2+3i}{5-2i} \\ =&\dfrac{2+3i}{5-2i}\times \dfrac{5+2i}{5+2i} \quad \text{by rationalizing} \\ =&\dfrac{10-6+15i+4i}{25+4}\\ =&\dfrac{4+19i}{29}\\ =&\dfrac{4}{29}+\dfrac{19}{29}i \end{align}$$ Real part $=\dfrac{4}{29}$
Imaginary part $=\dfrac{19}{29}$

Separate into real and imaginary parts $\dfrac{{{\left( 1+2i \right)}^{2}}}{1-3i}$.

$$\begin{align}&\dfrac{(1+2i)^2}{1-3i}\\ =&\dfrac{1-4+4i}{1-3i}\\ =&\dfrac{-3+4i}{1-3i}\\ =&\dfrac{-3+4i}{1-3i}\times \dfrac{1+3i}{1+3i} \quad \text{by rationalizing}\\ =&\dfrac{-3-12+4i-9i}{1+9}\\ =&\dfrac{-15-5i}{10}\\ =&\dfrac{-3}{2}-\dfrac{1}{2}i\end{align}$$ Real part $=\dfrac{-3}{2}$
Imaginary part $=-\dfrac{1}{2}$

Separate into real and imaginary parts $\dfrac{1-i}{{{\left( 1+i \right)}^{2}}}$.

$$\begin{align}&\dfrac{1-i}{{{\left( 1+i \right)}^{2}}}\\ =&\dfrac{1-i}{1-1+2i}\\ =&\dfrac{1-i}{2i}\times \dfrac{-2i}{-2i}\\ =&\dfrac{1-i}{2i}\times \dfrac{-2i}{-2i}\\ =&\dfrac{-2-2i}{4}\\ =&-\dfrac{1}{2}-\dfrac{1}{2}i\end{align}$$ Real part $=-\dfrac{1}{2}$
Imaginary part $=-\dfrac{1}{2}$

Separate into real and imaginary parts ${{\left( 2a-bi \right)}^{-2}}$.

$$\begin{align}&{{\left( 2a-bi \right)}^{-2}}\\ =&\dfrac{1}{{{\left( 2a-bi \right)}^{2}}}\\ =&\dfrac{1}{\left( 4{{a}^{2}}-{{b}^{2}} \right)-4abi}\\ =&\dfrac{1}{\left( 4{{a}^{2}}-{{b}^{2}} \right)-4abi}\times \dfrac{\left( 4{{a}^{2}}-{{b}^{2}} \right)+4abi}{\left( 4{{a}^{2}}-{{b}^{2}} \right)+4abi}\\ =&\dfrac{\left( 4{{a}^{2}}-{{b}^{2}} \right)+4abi}{{{\left( 4{{a}^{2}}-{{b}^{2}} \right)}^{2}}+16{{a}^{2}}{{b}^{2}}}\\ =&\dfrac{\left( 4{{a}^{2}}-{{b}^{2}} \right)+4abi}{{{\left( 4{{a}^{2}}-{{b}^{2}} \right)}^{2}}+16{{a}^{2}}{{b}^{2}}}\\ =&\dfrac{\left( 4{{a}^{2}}-{{b}^{2}} \right)+4abi}{16{{a}^{4}}+{{b}^{4}}-8{{a}^{2}}{{b}^{2}}+16{{a}^{2}}{{b}^{2}}}\\ =&\dfrac{\left( 4{{a}^{2}}-{{b}^{2}} \right)+4abi}{{{\left( 4{{a}^{2}}+{{b}^{2}} \right)}^{2}}}\\ =&\dfrac{4{{a}^{2}}-{{b}^{2}}}{{{\left( 4{{a}^{2}}+{{b}^{2}} \right)}^{2}}}+\dfrac{4abi}{{{\left( 4{{a}^{2}}+{{b}^{2}} \right)}^{2}}}\end{align}$$ Real part $=\dfrac{4{{a}^{2}}-{{b}^{2}}}{{{4{{a}^{2}}+{{b}^{2}}}^{2}}}$
Imaginary part $=\dfrac{4ab}{{{\left( 4{{a}^{2}}+{{b}^{2}} \right)}^{2}}}$

Separate into real and imaginary parts ${{\left( \dfrac{3-4i}{4-3i} \right)}^{-2}}$.

$$\begin{align}{{\left( \dfrac{3-4i}{4-3i} \right)}^{-2}}\\ =&{{\left( \dfrac{4-3i}{3-4i} \right)}^{2}}\\ =&\dfrac{16-9-24i}{9-16-24i}\\ =&\dfrac{7-24i}{-7-24i}\\ =&\dfrac{7-24i}{-7-24i}\times \dfrac{-7+24i}{-7+24i}\\ =&\dfrac{-\left( 49+576-336i \right)}{49+576}\\ =&\dfrac{-\left( 625-336i \right)}{625}\\ =&\dfrac{-625}{625}+\frac{336}{625}i\\ =&-1+\dfrac{336}{625}i\end{align}$$ Real part $=-1$
Imaginary part $=\dfrac{336}{625}$

Separate into real and imaginary parts ${{\left( \dfrac{4-5i}{2+3i} \right)}^{2}}$.

$$\begin{align}&{{\left( \dfrac{4-5i}{2+3i} \right)}^{2}}\\ =&\dfrac{16-25-40i}{4-9+12i}\\ =&\dfrac{-9-40i}{-5+12i}\\ =&\dfrac{-9-40i}{-5+12i}\times \dfrac{-5-12i}{-5-12i}\\ =&\dfrac{45-480+200i+108i}{25+144}\\ =&\dfrac{-435+308i}{169}\\ =&\dfrac{435}{169}+\dfrac{308i}{169}\end{align}$$ Real part $=\dfrac{435}{169}$
Imaginary part $=\dfrac{308}{169}$

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