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Question

Polar form of z=(1+7i)(2i)2 is

A
2(cos3π4+isin3π4)
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B
2(cos3π4+isin3π4)
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C
2(cosπ4+isinπ4)
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D
2(cosπ4+isinπ4)
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Solution

The correct option is A 2(cos3π4+isin3π4)
Let z=(1+7i)(2i)2
z=1+7i44i+i2z=1+7i34iz=(1+7i34i)(3+4i3+4i)z=25+25i25z=1+ir=|z|=(1)2+(1)2=2
Let α be the acute angle given by
tanα=Im(z)Re(z)=11=1α=π4
Since the point (1,1) representing z lies in the second quadrant,
θ=arg(z)=πα=ππ4=3π4
Hence, z in the polar form is given by
z=2(cos3π4+isin3π4)

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