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

Options

A. 2
B. 3
C. 6
D. 5

Answer

To solve the equation \( z^3 + \frac{3(\bar{z})^2}{|z|} = 0 \), where \( z \) is a complex number, we will follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the equation in terms of \( z \) and its conjugate \( \bar{z} \). We know that \( |z| = z \bar{z} \). Therefore, we can rewrite the equation as:
\[ z^3 + \frac{3(\bar{z})^2}{|z|} = 0 \]
This can be rewritten as:
\[ z^3 + \frac{3(\bar{z})^2}{z \bar{z}} = 0 \]
Multiplying both sides by \( z \bar{z} \) gives:
\[ z^3 \bar{z} + 3 \bar{z}^2 = 0 \]

### Step 2: Factor out \( \bar{z} \)
Factoring out \( \bar{z} \) from the equation, we get:
\[ \bar{z}(z^3 + 3\bar{z}) = 0 \]
This implies two cases:
1. \( \bar{z} = 0 \)
2. \( z^3 + 3\bar{z} = 0 \)

### Step 3: Solve the first case
For the first case \( \bar{z} = 0 \), this means \( z = 0 \). So, we have one solution:
\[ z = 0 \]

### Step 4: Solve the second case
For the second case \( z^3 + 3\bar{z} = 0 \), we can express \( \bar{z} \) in terms of \( z \):
\[ \bar{z} = -\frac{z^3}{3} \]
Now, substituting \( z = re^{i\theta} \) (where \( r = |z| \) and \( \theta \) is the argument of \( z \)):
\[ \bar{z} = re^{-i\theta} \]
This gives us:
\[ re^{-i\theta} = -\frac{(re^{i\theta})^3}{3} \]
Simplifying this, we have:
\[ re^{-i\theta} = -\frac{r^3 e^{3i\theta}}{3} \]
Multiplying both sides by \( 3 \):
\[ 3re^{-i\theta} = -r^3 e^{3i\theta} \]
Dividing both sides by \( r \) (assuming \( r \neq 0 \)):
\[ 3e^{-i\theta} = -r^2 e^{3i\theta} \]

### Step 5: Equate magnitudes and arguments
Equating magnitudes:
\[ 3 = r^2 \implies r = \sqrt{3} \]

Equating arguments:
\[ -e^{-i\theta} = e^{3i\theta} \implies -1 = e^{4i\theta} \implies 4\theta = (2n + 1)\pi \text{ for } n \in \mathbb{Z} \]
Thus:
\[ \theta = \frac{(2n + 1)\pi}{4} \]
For \( n = 0, 1, 2, 3 \), we find:
- \( \theta = \frac{\pi}{4} \)
- \( \theta = \frac{3\pi}{4} \)
- \( \theta = \frac{5\pi}{4} \)
- \( \theta = \frac{7\pi}{4} \)

### Step 6: Find the solutions
The corresponding solutions for \( z \) are:
1. \( z = \sqrt{3} e^{i\frac{\pi}{4}} \)
2. \( z = \sqrt{3} e^{i\frac{3\pi}{4}} \)
3. \( z = \sqrt{3} e^{i\frac{5\pi}{4}} \)
4. \( z = \sqrt{3} e^{i\frac{7\pi}{4}} \)

### Conclusion
Thus, we have found a total of 5 solutions:
1. \( z = 0 \)
2. \( z = \sqrt{3} e^{i\frac{\pi}{4}} \)
3. \( z = \sqrt{3} e^{i\frac{3\pi}{4}} \)
4. \( z = \sqrt{3} e^{i\frac{5\pi}{4}} \)
5. \( z = \sqrt{3} e^{i\frac{7\pi}{4}} \)

The total number of solutions is **5**.