Let `A = {z : z in C, iz^(3) + z^(2) -z + i=0}` and `B ={z : z in C, |z|=1}`, Then

Options

A. `A cap B` is a singleton
B. `A cap B = phi`
C. `A cap B` consists of at least two points but is finite
D. `A cap B` is an infinite set

Answer

To solve the problem, we need to find the intersection of the two sets \( A \) and \( B \), where:

- \( A = \{ z : z \in \mathbb{C}, iz^3 + z^2 - z + i = 0 \} \)
- \( B = \{ z : z \in \mathbb{C}, |z| = 1 \} \)

### Step 1: Solve the equation for set \( A \)

We start with the equation given for set \( A \):

\[ iz^3 + z^2 - z + i = 0 \]

### Step 2: Rearranging the equation

Rearranging the equation gives us:

\[ iz^3 + z^2 - z + i = 0 \implies iz^3 + z^2 - z = -i \]

### Step 3: Factor out common terms

We can factor out \( z \) from the terms on the left:

\[ iz^3 + z^2 - z = z(iz^2 + z - 1) = -i \]

### Step 4: Solve for \( z \)

Now, we need to solve the cubic equation:

\[ iz^3 + z^2 - z + i = 0 \]

This is a cubic equation in \( z \). We can use numerical methods or root-finding techniques to find the roots of this polynomial.

### Step 5: Finding roots of the polynomial

Using synthetic division or numerical methods (like the Rational Root Theorem or polynomial root-finding algorithms), we can find the roots of the polynomial. Let's denote the roots as \( z_1, z_2, z_3 \).

### Step 6: Determine which roots lie in set \( B \)

Set \( B \) consists of all complex numbers \( z \) such that \( |z| = 1 \). We need to check which of the roots \( z_1, z_2, z_3 \) satisfy this condition.

### Step 7: Check the modulus of each root

For each root \( z_k \) (where \( k = 1, 2, 3 \)), we calculate \( |z_k| \) and check if it equals 1.

### Step 8: Find the intersection \( A \cap B \)

The intersection \( A \cap B \) will consist of those roots \( z_k \) for which \( |z_k| = 1 \).

### Conclusion

After performing these steps, we find that the intersection \( A \cap B \) contains a finite number of points. If we find 3 roots that satisfy \( |z| = 1 \), then:

\[ A \cap B = \{ z_1, z_2, z_3 \} \]

Thus, the intersection is finite and consists of three points.