The number of solutions of the equation z^2 + | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let $z = x + iy$,so that $\bar{z} = x - iy$. Therefore,the equation becomes:$z^2 + \bar{z} = 0 \iff (x + iy)^2 + (x - iy) = 0$$(x^2 - y^2 + x) + i

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(D) Let $z = x + iy$,so that $\bar{z} = x - iy$. Therefore,the equation becomes:$z^2 + \bar{z} = 0 \iff (x + iy)^2 + (x - iy) = 0$$(x^2 - y^2 + x) + i(2xy - y) = 0$Equating real and imaginary parts to zero,we get:$x^2 - y^2 + x = 0$ .....$(i)$$2xy - y = 0$ .....$(ii)$From $(ii)$,$y(2x - 1) = 0$,which implies $y = 0$ or $x = \frac{1}{2}$.Case $1$: If $y = 0$,then $(i)$ gives $x^2 + x = 0$,so $x(x + 1) = 0$,which gives $x = 0$ or $x = -1$. This provides two solutions: $z = 0$ and $z = -1$.Case $2$: If $x = \frac{1}{2}$,then $(i)$ gives $(\frac{1}{2})^2 - y^2 + \frac{1}{2} = 0$,so $\frac{1}{4} - y^2 + \frac{1}{2} = 0$,which means $y^2 = \frac{3}{4}$,so $y = \pm \frac{\sqrt{3}}{2}$. This provides two solutions: $z = \frac{1}{2} + i\frac{\sqrt{3}}{2}$ and $z = \frac{1}{2} - i\frac{\sqrt{3}}{2}$.Thus,there are $4$ solutions in total.

The number of solutions of the equation $z^2 + \bar{z} = 0$ is

Similar Questions

If ${z_r} = \cos \frac{{r\alpha }}{{{n^2}}} + i\sin \frac{{r\alpha }}{{{n^2}}}$,where $r = 1, 2, 3, \dots, n$,then $\mathop {\lim }\limits_{n \to \infty } {z_1}{z_2}{z_3} \dots {z_n}$ is equal to

If $(x-iy)^{1/3} = a-ib$,then the value of $\frac{x}{a} + \frac{y}{b}$ is

Let $A = \{\theta \in [0, 2\pi] : 1 + 10 \operatorname{Re}\left(\frac{2 \cos \theta + i \sin \theta}{\cos \theta - 3i \sin \theta}\right) = 0\}$. Then $\sum_{\theta \in A} \theta^2$ is equal to

Let $z = 1 + ai$ be a complex number,$a > 0$,such that $z^3$ is a real number. Then the sum $1 + z + z^2 + .... + z^{11}$ is equal to

If $z$ is a point on the circle $|z|=1$ with $\operatorname{Arg}(z)=\frac{\pi}{6}$,then $\frac{z^{12}+1-z^6}{z^{12}+i z^6-1}=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module