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  4. Let z be a complex number such that the imaginary part of z is nonzero and a=z2+z+1 is real. Then a cannot take the value

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Q. Let $z$ be a complex number such that the imaginary part of $z$ is nonzero and $a=z^{2}+z+1$ is real. Then a cannot take the value

JEE AdvancedJEE Advanced 2012

Solution:

Given equation is $z^{2}+z+1-a=0$
Clearly this equation do not have real roots if
$D <0$
$\Rightarrow 1-4(1-a)<0 $
$\Rightarrow 4 a <3$
$ a <\frac{3}{4}$