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Question

If |u|=|v|=1, uv1, and z=uv1+uv, then

A
|z|=1
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B
Re(z)=0
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C
Im(z)=0
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D
Re(z)=Im(z)
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Solution

The correct option is B Re(z)=0
Given that |u|=|v|=1
Let u=cosA+isinA and u=cosA+isinAv=cosB+isinB
z=uv1+uv
z=cosA+isinAcosBisinB1+(cosA+isinA)(cosB+isinB)
z=cosAcosB+i(sinAsinB)1+cos(A+B)+isin(A+B)
z=2sin(A+B2)sin(AB2)+2cos(A+B2)isin(AB2)2cos2(A+B2)+2isin(A+B2)cos(A+B2)
z=isin(AB2)cos(A+B2)⎢ ⎢ ⎢ ⎢isin(A+B2)+cos(A+B2)cos(A+B2)+isin(A+B2)⎥ ⎥ ⎥ ⎥
z=isin(AB2)cos(A+B2)
Therefore, Re(z)=0
Ans: B

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