If z is a complex number satisfying `|z|^(2)-|z|-2 lt 0`, then the value of `|z^(2)+zsintheta|`, for all values of `theta`, is

Options

A. equal to 4
B. equal to 6
C. more than 6
D. less than 6

Explanation

To solve the problem, we start with the inequality given for the complex number \( z \):

### Step 1: Analyze the inequality
We are given:
\[ |z|^2 - |z| - 2 < 0 \]
This can be rearranged to:
\[ |z|^2 - |z| - 2 < 0 \]

### Step 2: Factor the quadratic expression
We can factor the left-hand side:
\[ (|z| - 2)(|z| + 1) < 0 \]

### Step 3: Determine the intervals
To find the values of \( |z| \) that satisfy this inequality, we analyze the factors:
- The expression \( |z| - 2 < 0 \) gives \( |z| < 2 \).
- The expression \( |z| + 1 > 0 \) is always true since \( |z| \) is non-negative.

Thus, the only condition we need to satisfy is:
\[ |z| < 2 \]

### Step 4: Find the expression we need to evaluate
Next, we need to evaluate:
\[ |z^2 + z \sin \theta| \]
Using the property of modulus, we can express this as:
\[ |z^2 + z \sin \theta| = |z^2| + |z \sin \theta| \]
This can be rewritten as:
\[ |z^2| + |z| |\sin \theta| \]

### Step 5: Substitute the modulus of \( z \)
Since we know \( |z| < 2 \), we can substitute:
\[ |z^2| = |z|^2 < 2^2 = 4 \]
Thus, we have:
\[ |z^2 + z \sin \theta| < 4 + |z| |\sin \theta| \]

### Step 6: Determine the maximum value of \( |z| |\sin \theta| \)
Since \( |z| < 2 \) and \( |\sin \theta| \) can take values from -1 to 1, the maximum value of \( |z| |\sin \theta| \) occurs when both \( |z| \) and \( |\sin \theta| \) are at their maximum:
\[ |z| |\sin \theta| < 2 \cdot 1 = 2 \]

### Step 7: Combine the results
Now we combine our results:
\[ |z^2 + z \sin \theta| < 4 + 2 = 6 \]

### Conclusion
Thus, the value of \( |z^2 + z \sin \theta| \) for all values of \( \theta \) is:
\[ |z^2 + z \sin \theta| < 6 \]