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Question

Express in the form of $$a+ib,a,b, \in R$$ and $$i=\sqrt{-1}$$ then find conjugate modulus and amplitude of the complex number
$$\dfrac{1+3i}{2-i}+\dfrac{1+2i}{2+i}$$


Solution

Given,

$$\dfrac{1+3i}{2-i}+\dfrac{1+2i}{2+i}$$

$$=\dfrac{(1+3i)(2+i)+(1+2i)(2-i)}{(2+i)(2-i)}$$

$$=\dfrac{3+10i}{5}$$

$$\therefore z=\dfrac{3}{5}+i2$$

$$|z|=\sqrt{\left ( \dfrac{3}{5} \right )^2+2^2}$$

$$=\dfrac {\sqrt{109}}{5}$$

$$amp=\theta =\tan^{-1}\left (\dfrac{2}{\frac{3}{5}}  \right )\Rightarrow \theta =73.30^{\circ}$$

Mathematics

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