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Question

If complex numbers z1 and z2 both satisfy z+¯z=2|z1| and arg(z1z2)=π3, then find the value of Im(z1+z2). (where Im(z) denotes the imaginary part of z)

A
sinπ3
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B
cosec π3
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C
tanπ3
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D
cotπ3
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Solution

The correct option is B cosec π3
Let z=x+iy
z+¯z2=|z1|
x2=(x1)2+y2
y2=2x1
Let z1(t21+12,t1) and z2(t22+12,t2)
arg(z1z2)=π3
t2t1(t22t212)=3
2t1+t2=3
Im(z1+z2)=t1+t2=23

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