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Q. Let z be a complex number with non-zero imaginary part. If
2+3z+4z223z+4z2
is a real number, then the value of |z|2 is ___

JEE AdvancedJEE Advanced 2022

Solution:

Given that
z¯z
Let α=2+3z+4z223z+4z2=(23z+4z2)+6z23z+4z2
α=1+6z23z+4z2
If α is a real number, then
α=ˉα
z23z+4z2=ˉz23ˉz+4ˉz2
2(zˉz)=4zˉz(zˉz)
(zˉz)(24zˉz)=0
As z¯z (Given)
zˉz=24=12
|z|2=0.50