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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