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.
Question 7(i)
Separate into real and imaginary parts $\dfrac{2+3i}{5-2i}$.
Solution
\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 $=\dfrac{19}{29}$
Question 7(ii)
Separate into real and imaginary parts $\dfrac{{{\left( 1+2i \right)}^{2}}}{1-3i}$.
Solution
\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 $=-\dfrac{1}{2}$
Question 7(iii)
Separate into real and imaginary parts $\dfrac{1-i}{{{\left( 1+i \right)}^{2}}}$.
Solution
\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 $=-\dfrac{1}{2}$
Question 7(iv)
Separate into real and imaginary parts ${{\left( 2a-bi \right)}^{-2}}$.
Solution
\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 $=\dfrac{4ab}{{{\left( 4{{a}^{2}}+{{b}^{2}} \right)}^{2}}}$
Question 7(v)
Separate into real and imaginary parts ${{\left( \dfrac{3-4i}{4-3i} \right)}^{-2}}$.
Solution
\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 $=\dfrac{336}{625}$
Question 7(vi)
Separate into real and imaginary parts ${{\left( \dfrac{4-5i}{2+3i} \right)}^{2}}$.
Solution
\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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