ALGEBRA OF COMPLEX NUMBERS (Excersice)

\[\LARGE{\color {purple} {PART- A}}\]
\[1. \ Find\ the\ value\ of\ (2i\ +\ i)(i\ +\ 3i)\ \hspace{15cm}\]
\[\color {black}{Solution:}\ (2i\ +\ i)(i\ +\ 3i)\ \hspace{18cm}\]
\[ =\ (3\ i)\ (4\ i)\ \hspace{15cm}\]
\[ =\ 12\ i^2\ \hspace{15cm}\]
\[ =\ 12\ (-\ 1)\ \hspace{15cm}\]
\[ =\ -12\ \hspace{15cm}\]
\[2. \ If\ Z_1 = 1 + i,\ Z_2 = 3 + 2i,\ find\ 3Z_1 + 4Z_2\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Z_1 = 1 +\ i,\ Z_2 =\ 3 +\ 2i\ \hspace{18cm}\]
\[3Z_1 + 4Z_2 = 3(1 + i) +\ 4( 3 +\ 2i)\ \hspace{10cm}\]
\[ = 3 + i +\ 12 + 8i\ \hspace{10cm}\]
\[= 3 +\ 12 +\ i (1 + 12)\ \hspace{10cm}\]
\[=\ 15\ +\ 13i\ \hspace{10cm}\]
\[3. \ If\ Z_1 = 2 + 3i,\ Z_2 = 4 – 5i,\ find\ Z_1 – Z_2\ \hspace{15cm}\]
\[\color {black}{Solution:}\ Z_1 = 2 + 3i,\ Z_2 = 4 – 5i,\ \hspace{18cm}\]
\[Z_1 – Z_2 = 2 +\ 3i -\ (4 -\ 5i)\ \hspace{10cm}\]
\[Z_1 – Z_2 =\ 2 +\ 3i -\ 4 +\ 5i\ \hspace{10cm}\]
\[Z_1 – Z_2 = 2 – 4 +\ 3i +\ 5i\ \hspace{10cm}\]
\[Z_1 – Z_2 = – 2 +\ 8i\ \hspace{10cm}\]
\[4. \ Find\ the\ Real\ and\ Imaginary\ parts\ of\ \frac{1}{2\ +\ 3i} \hspace{18cm}\]
\[\color {black}{Solution:}\ z = \frac{1}{2\ +\ 3i}\ ×\ \frac{2\ -\ 3i}{2\ -\ 3i}\ \hspace{18cm}\]
\[ = \frac{2\ -\ 3i}{(2)^2 + (3)^2}\ \hspace{3cm}\ \because [(a +ib)(a – ib)\ =\ (a)^2 + (b)^2]\]
\[ = \frac{2\ -\ 3i}{4\ +\ 9}\ \hspace{15cm}\]
\[ = \frac{2\ -\ 3i}{13}\ \hspace{15cm}\]
\[ = \frac{2}{13}\ +\ i\ \frac{-3}{13}\ \hspace{15cm}\]
\[Re(Z)\ =\ \frac{2}{13};\ Im(Z)\ =\ \frac{-3}{13}\ \hspace{15cm}\]
\[5. \ Find\ the\ modulus\ and\ amplitude\ of\ 1 + \ i\ \hspace{18cm}\]
\[\color {black}{Solution:} \hspace{20cm}\]
\[Let\ z = 1 + \ i\ = a\ + ib\ \hspace{15cm}\]
\[a = 1,\ b\ =\ 1\ \hspace{15cm}\]
\[\color {brown} {To\ find\ modulus}:\ \hspace{18cm}\]
\[|z| = \sqrt{a^2 + b^2}\ \hspace{12cm}\]
\[ = \sqrt{(1)^2 + (1)^2}\ \hspace{12cm}\]
\[ = \sqrt{1 + 1}\ =\ 2\ \hspace{12cm}\]
\[|z| = 2\ \hspace{15cm}\]
\[\color {brown} {To\ find\ amplitude}:\ \hspace{18cm}\]
\[θ = tan^{-1} (\frac{b}{a})\ =\ tan^{-1} \frac{1}{1}\ =\ tan^{-1} \ (1)\]
\[θ = 45^0\ \hspace{15cm}\]
\[\LARGE{\color {purple} {PART- B}}\]
\[6.\ Find\ the\ Real\ and\ Imaginary\ parts\ of\ \frac{1\ -\ i}{1\ +\ i}\ \hspace{18cm}\]
\[\color {black}{Solution:}\ z = \frac{1\ -\ i}{1\ +\ i}\ ×\ \frac{1\ -\ i}{1\ -\ i}\ \hspace{18cm}\]
\[ = \frac{1 – i – i – 1}{(1)^2 + (1)^2}\ \hspace{3cm}\ \because [(a +ib)(a – ib)\ =\ (a)^2 + (b)^2]\]
\[ = \frac{1 – i – i – 1}{1\ +\ 1}\ \hspace{15cm}\]
\[ = \frac{-\ 2i}{2}\ \hspace{15cm}\]
\[ = \frac{0}{2}\ +\ i\ \frac{-2}{2}\ \hspace{15cm}\]
\[Re(Z)\ =\ 0\ Im(Z)\ =\ -1\ \hspace{15cm}\]
\[7.\ Find\ the\ Real\ and\ Imaginary\ parts\ of\ \frac{2\ +\ 5i}{2\ +\ 3i}\ \hspace{18cm}\]
\[\color {black}{Solution:}\ z = \frac{2+ 5i}{2+ 3i}\ ×\ \frac{2\ – 3i}{2\ – 3i}\ \hspace{18cm}\]
\[ = \frac{4 – 6i + 10i + 15}{(2)^2 + (3)^2}\ \hspace{3cm}\ \because [(a +ib)(a – ib)\ =\ (a)^2 + (b)^2]\]
\[ = \frac{8 – 10i + 12i + 15}{4\ +\ 9}\ \hspace{15cm}\]
\[ = \frac{19\ +\ 4i}{13}\ \hspace{15cm}\]
\[ = \frac{19}{13}\ +\ i\ \frac{4}{13}\ \hspace{15cm}\]
\[Re(Z)\ =\ \frac{19}{13};\ Im(Z)\ =\ \frac{4}{13}\ \hspace{15cm}\]
\[8.\ Find\ the\ Real\ and\ Imaginary\ parts\ of\ \frac{2\ +\ 5i}{4\ -\ 3i}\ \hspace{18cm}\]
\[\color {black}{Solution:}\ z = \frac{2+ 5i}{4 – 3i}\ ×\ \frac{4\ + 3i}{4\ + 3i}\ \hspace{18cm}\]
\[ = \frac{8 + 6i + 20i – 15\ i^2}{(4)^2 + (3)^2}\ \hspace{3cm}\ \because [(a +ib)(a – ib)\ =\ (a)^2 + (b)^2]\]
\[ = \frac{8 + 6i + 20i + 15}{16\ +\ 9}\ \hspace{15cm}\]
\[ = \frac{23\ +\ 26i}{25}\ \hspace{15cm}\]
\[ = \frac{23}{25}\ +\ i\ \frac{26}{25}\ \hspace{15cm}\]
\[Re(Z)\ =\ \frac{23}{25};\ Im(Z)\ =\ \frac{26}{25}\ \hspace{15cm}\]
\[Ex\ 9:\ Find\ the\ modulus\ and\ amplitude\ of\ \sqrt{3} – \ i\ \hspace{18cm}\]
\[\color {black}{Solution:} \hspace{20cm}\]
\[Let\ z = \sqrt{3} – \ i\ = a\ + ib\ \hspace{15cm}\]
\[a = \sqrt{3},\ b\ = – 1\ \hspace{15cm}\]
\[\color {brown} {T0\ find\ modulus}:\ \hspace{18cm}\]
\[|z| = \sqrt{a^2 + b^2}\ \hspace{12cm}\]
\[ = \sqrt{(\sqrt{3})^2 + (-1)^2}\ \hspace{12cm}\]
\[ = \sqrt{3 + 1}\ =\ \sqrt{4}\ =\ 2\ \hspace{12cm}\]
\[|z| = 2\ \hspace{15cm}\]
\[\color {brown} {To\ find\ amplitude}:\ \hspace{18cm}\]
\[θ = tan^{-1} (\frac{b}{a})\ =\ tan^{-1}( \frac{- 1}{\sqrt{3}})\]
\[θ =\ -\ 30^0\ \hspace{15cm}\]
\[\LARGE{\color {purple} {PART- C}}\]
\[1.\ Prove\ that\ the\ complex\ numbers\ 3\ +\ 4\ i,\ 9\ +\ 8\ i,\ 5\ +\ 2\ i\ and\ -\ 1\ -\ 2\ i\ form\ a\ rhombus\ \hspace{10cm}\]\[in\ the\ argand\ diagram\ \hspace{10cm}\]
\[\color {black}{Solution:}\ Part\ -\ C\ of Question\ 2\ \hspace{20cm}\]

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