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Question

Mathematics Question on Quadratic Equations

The argument of the complex number (i2−2i)\left( \frac{i}{2}-\frac{2}{i} \right) is equal to

A

π4\frac{\pi }{4}

B

3π4\frac{3\pi }{4}

C

π12\frac{\pi }{12}

D

π2\frac{\pi }{2}

Answer

π2\frac{\pi }{2}

Explanation

Solution

Let z=i2−2i=i2−2ii2z=\frac{i}{2}-\frac{2}{i}=\frac{i}{2}-\frac{2i}{{{i}^{2}}} Z=i2+2i=52i=0+52iZ=\frac{i}{2}+2i=\frac{5}{2}i=0+\frac{5}{2}i arg⁡(z)=tan⁡−1(Im⁡(z)Re⁡(z))\arg (z)={{\tan }^{-1}}\left( \frac{\operatorname{Im}(z)}{\operatorname{Re}(z)} \right)
=tan⁡−1(5/20)=tan⁡−1(∞)={{\tan }^{-1}}\left( \frac{5/2}{0} \right)={{\tan }^{-1}}(\infty )
=tan⁡−1(tan⁡π2)=π2={{\tan }^{-1}}\left( \tan \frac{\pi }{2} \right)=\frac{\pi }{2}