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

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 (A) -1 (B) (1/3) (C) (1/2) (D) (3/4)

Tardigrade Admin · Accepted Answer

Given equation is z2+z+1-a=0 Clearly this equation do not have real roots if D <0 ⇒ 1-4(1-a)<0 ⇒ 4 a <3 a <(3/4)

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

$- 1$

$\frac{1}{3}$

$\frac{1}{2}$

$\frac{3}{4}$

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

Q. 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

−1

31​

21​

43​