The number of distinct complex number(s) z, such that `|z|=1 and z^(3)` is purely imagninary, is/are equal to

Answer

To solve the problem of finding the number of distinct complex numbers \( z \) such that \( |z| = 1 \) and \( z^3 \) is purely imaginary, we can follow these steps:

### Step 1: Represent \( z \)
Let \( z \) be represented in polar form, where \( z = e^{i\theta} \) for some angle \( \theta \). Since \( |z| = 1 \), this representation is valid.

### Step 2: Find \( z^3 \)
Calculate \( z^3 \):
\[ z^3 = (e^{i\theta})^3 = e^{3i\theta} \]

### Step 3: Condition for \( z^3 \) to be purely imaginary
A complex number is purely imaginary if its real part is zero. The real part of \( e^{3i\theta} \) is given by:
\[ \text{Re}(z^3) = \cos(3\theta) \]
For \( z^3 \) to be purely imaginary, we need:
\[ \cos(3\theta) = 0 \]

### Step 4: Solve for \( \theta \)
The cosine function is zero at odd multiples of \( \frac{\pi}{2} \):
\[ 3\theta = \frac{\pi}{2} + k\pi \quad \text{for } k \in \mathbb{Z} \]
This simplifies to:
\[ \theta = \frac{\pi}{6} + \frac{k\pi}{3} \]

### Step 5: Determine distinct values of \( z \)
Now we need to find distinct values of \( z \) for different integer values of \( k \):
- For \( k = 0 \): \( \theta = \frac{\pi}{6} \) → \( z = e^{i\frac{\pi}{6}} \)
- For \( k = 1 \): \( \theta = \frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{2} \) → \( z = e^{i\frac{\pi}{2}} \)
- For \( k = 2 \): \( \theta = \frac{\pi}{6} + \frac{2\pi}{3} = \frac{5\pi}{6} \) → \( z = e^{i\frac{5\pi}{6}} \)
- For \( k = 3 \): \( \theta = \frac{\pi}{6} + \pi = \frac{7\pi}{6} \) → \( z = e^{i\frac{7\pi}{6}} \)
- For \( k = 4 \): \( \theta = \frac{\pi}{6} + \frac{4\pi}{3} = \frac{3\pi}{2} \) → \( z = e^{i\frac{3\pi}{2}} \)
- For \( k = 5 \): \( \theta = \frac{\pi}{6} + \frac{5\pi}{3} = \frac{11\pi}{6} \) → \( z = e^{i\frac{11\pi}{6}} \)

### Step 6: Identify distinct values
The angles \( \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{3\pi}{2}, \frac{11\pi}{6} \) correspond to distinct points on the unit circle.

### Conclusion
Thus, the number of distinct complex numbers \( z \) such that \( |z| = 1 \) and \( z^3 \) is purely imaginary is:
\[ \boxed{6} \]