Question 8, Exercise 1.1

Solutions of Question 8 of Exercise 1.1 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.

Express the $\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}$ in the standard form of $a+ib.$

$$\begin{align}&\dfrac{1-2i}{2+i}+\dfrac{4-i}{3+2i}\\ &=\dfrac{\left( 3+2i \right)\left( 1-2i \right)+\left( 2+i \right)\left( 4-i \right)}{\left( 2+i \right)\left( 3+2i \right)}\\ &=\dfrac{\left( 3+4+2i-6i \right)+\left( 8+1+4i-2i \right)}{\left( 6-2+3i+4i \right)}\\ &=\dfrac{\left( 7-4i \right)+\left( 9+2i \right)}{4+7i}\\ &=\dfrac{16-2i}{4+7i}\\ &=\dfrac{16-2i}{4+7i}\times \dfrac{4-7i}{4-7i}\\ &=\dfrac{\left( 64-14 \right)-\left( 112+8 \right)i}{16+49}\\ &=\dfrac{50-120i}{65}\\ &=\dfrac{10-24i}{13}\\ &=\dfrac{10}{13}-\dfrac{24i}{13}\end{align}$$

Express the $\dfrac{2+\sqrt{-9}}{-5-\sqrt{-16}}$ in the standard form of $a+ib.$

$$\begin{align}\dfrac{2+\sqrt{-9}}{-5-\sqrt{-16}}&=\dfrac{2+3i}{-5-4i}\\ &=\dfrac{2+3i}{-5-4i}\times \dfrac{-5+4i}{-5+4i}\\ &=\dfrac{\left( -10-12 \right)+\left( 8-15 \right)i}{25+16}\\ &=\dfrac{-22-7i}{41}\\ &=\dfrac{-22}{41}-\dfrac{7i}{41}\end{align}$$

Express the $\dfrac{{{\left( 1+i \right)}^{2}}}{4+3i}$ in the standard form of $a+ib.$

$$\begin{align}\dfrac{\left( 1+i \right)\left( 1+i \right)}{4+3i}&=\dfrac{1-1+i+i}{4+3i}\\ &=\dfrac{2i}{4+3i}\\ &=\dfrac{2i}{4+3i}\times \dfrac{4-3i}{4-3i}\\ &=\dfrac{6+8i}{16+9}\\ &=\dfrac{6+8i}{25}\\ &=\dfrac{6}{25}+\dfrac{8i}{25}\end{align}$$

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