Let z be a complex number such that the imaginary part of z is non-zero and a = z 2 + z + 1 is real. Then a cannot take the value [2012]

Question

Let $z$ be a complex number such that the imaginary part of $z$ is non-zero and $a = z^{2} + z + 1$ is real. Then a cannot take the value [2012]

Smart Achievers · Accepted Answer

(4)

$∵$ $a = z^{2} + z + 1$

$\Rightarrow z is non-real$

$and equation z^{2} + z + (1 - a) = 0 will have non-real roots, if D < 0$

$\Rightarrow 1 - 4 (1 - a) < 0 \Rightarrow 4 a < 3 \Rightarrow a < \frac{3}{4}$

$∴$ $a cannot take the value \frac{3}{4} .$

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