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

Engineering

Mathematics

Introduction to Complex Number

Question

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

$\frac{1}{2}$

$\frac{3}{4}$

$\frac{1}{3}$

–1

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Let z = x + iy (y $\neq$ 0) then a = x² – y² + 2ixy + x + iy + 1 or a = (x² – y² + x + 1) + i (2xy + y)

Since a is real $\Rightarrow$ Im(a) = 0 $\Rightarrow$ 2xy + y = 0 $\Rightarrow$ 2x = – 1 (as y $\neq$ 0)

Now, $a = {(\frac{- 1}{2})}^{2} - y^{2} - \frac{1}{2} + 1 = \frac{1}{4} - y^{2} - \frac{1}{2} + 1 = \frac{3}{4} - y^{2}$

$\Rightarrow a < \frac{3}{4}$