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  4. If ((1+i)2/2-i)=x+iy, then x+y is equal to

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Q. If $ \frac{{{(1+i)}^{2}}}{2-i}=x+iy, $ then $ x+y $ is equal to

J & K CETJ & K CET 2007Complex Numbers and Quadratic Equations

Solution:

Given that, $ \frac{{{(1+i)}^{2}}}{2-i}=x+iy $
$ \Rightarrow $ $ \frac{1+{{i}^{2}}+2i}{2-i}=x+iy $
$ \Rightarrow $ $ \frac{1-1+2i}{2-i}=x+iy $
$ \Rightarrow $ $ \frac{2i}{2-i}\times \frac{2+i}{2+i}=x+iy $
$ \Rightarrow $ $ \frac{2i(2+i)}{{{(2)}^{2}}+{{(i)}^{2}}}=x+iy $
$ \Rightarrow $ $ \frac{4i-2}{5}=x+iy $
$ \Rightarrow $ $ x+iy=-\frac{2}{5}+\frac{4}{5}i $
Now, $ x+y=-\frac{2}{5}+\frac{4}{5}=\frac{2}{5} $