Let z be a complex number such that imaginary part of z is non zero and a=z^2+z+1 is real. Then a ca

English

Gujarati

Hindi

4-1.Complex numbers

hard

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

A

$-1$

B

$\frac{1}{3}$

C

$\frac{1}{2}$

D

$\frac{3}{4}$

(IIT-2012)

Solution

Here $z^2+z+1-a=0$

$\Rightarrow \quad z=\frac{-1 \pm \sqrt{4 a-3}}{2}$

Here $\quad a \neq \frac{3}{4}$ otherwise $z$ will be purely real.

Standard 11

Mathematics

Share

Similar Questions

If $(1 + i)(1 + 2i)(1 + 3i)…..(1 + ni) = a + ib$, then $2.5.10….$$(1 + {n^2})$ is equal to

medium

Let ${z_1},{z_2}$ be two complex numbers such that ${z_1} + {z_2}$ and ${z_1}{z_2}$ both are real, then

medium

Let $\mathrm{n}$ denote the number of solutions of the equation $z^{2}+3 \bar{z}=0$, where $\mathrm{z}$ is a complex number. Then the value of $\sum_{k=0}^{\infty} \frac{1}{n^{k}}$ is equal to:

hard

(JEE MAIN-2021)

Let complex number $z$ be such that $|z- \frac{6}{z}|=5$ then maximum value of $|z|$ will be –

normal

If $z \neq 0$ be a complex number such that $\left| z -\frac{1}{ z }\right|=2$, then the maximum value of $|z|$ is.

hard

(JEE MAIN-2022)