The number of solutions of the equation z^3+(3z¯^2)/z=0 (where, z is a complex number) are

Correct answer: (D) 5. Let, z=re^(iθ) ⇒r^3e^(i3θ)+3re^(-i2θ)=0 ⇒r^2e^(i5θ)=-3 ⇒r^2=3 and e^(i5θ)=-1 ⇒r=√3 and θ=(π)/5+2k/5 where, k=0,1,2,3,4 ⇒5 solutions

Q10 / 108 · NTA Abhyas (JEE Main) 2020 · Mathematics · Complex Number

The number of solutions of the equation $z^{3} + \frac{3 {(\bar{z})}^{2}}{|z|} = 0$ (where, $z$ is a complex number) are

A
$2$
B
$3$
C
$6$
Correct answer
D
$5$

Solution

Let,

z = r e^{i θ}

\Rightarrow r^{3} e^{i 3 θ} + 3 r e^{- i 2 θ} = 0

\Rightarrow r^{2} e^{i 5 θ} = - 3

\Rightarrow r^{2} = 3

and

e^{i 5 θ} = - 1

\Rightarrow r = \sqrt{3}

and

θ = \frac{π}{5} + \frac{2 k}{5}

where,

k = 0, 1, 2, 3, 4

\Rightarrow 5

solutions

Hence, the correct answer is option: 'D'.

The number of solutions of the equation z3+3z¯2z=0 (where, z is a complex number) are

Solution

More Complex Number Questions

The number of solutions of the equation $z^{3} + \frac{3 {(\bar{z})}^{2}}{|z|} = 0$ (where, $z$ is a complex number) are