All the roots of the equation `1 lz^(10) + 10iz^(9) + 10iz -11=0` lie

Options

A. inside `|z|=1`
B. one `|z|=1`
C. outside `|z|=1`
D. cannot say

Explanation

To solve the equation \( 11z^{10} + 10iz^{9} + 10iz - 11 = 0 \) and determine where all the roots lie, we can follow these steps:

### Step 1: Rewrite the Equation
We start with the given equation:
\[ 11z^{10} + 10iz^{9} + 10iz - 11 = 0 \]

### Step 2: Factor Out Common Terms
We can factor out \( z^9 \) from the first two terms:
\[ z^9(11z + 10i) + 10iz - 11 = 0 \]

### Step 3: Rearrange the Equation
Rearranging gives us:
\[ z^9(11z + 10i) = 11 - 10iz \]

### Step 4: Divide Both Sides
Now, we can divide both sides by \( 11z + 10i \) (assuming \( 11z + 10i \neq 0 \)):
\[ z^9 = \frac{11 - 10iz}{11z + 10i} \]

### Step 5: Analyze the Modulus
To find the modulus, we take the modulus of both sides:
\[ |z^9| = \left| \frac{11 - 10iz}{11z + 10i} \right| \]
This simplifies to:
\[ |z|^9 = \frac{|11 - 10iz|}{|11z + 10i|} \]

### Step 6: Find the Modulus of the Numerator and Denominator
Calculating the modulus of the numerator:
\[ |11 - 10iz| = \sqrt{11^2 + (-10y)^2} = \sqrt{121 + 100y^2} \]
Calculating the modulus of the denominator:
\[ |11z + 10i| = \sqrt{(11x)^2 + (10)^2} = \sqrt{121x^2 + 100} \]

### Step 7: Set Up the Inequality
We need to analyze the inequality:
\[ |z|^9 = \frac{\sqrt{121 + 100y^2}}{\sqrt{121x^2 + 100}} \]

### Step 8: Consider Cases for Modulus
Assuming \( |z| < 1 \):
\[ |z|^9 < 1 \implies \frac{\sqrt{121 + 100y^2}}{\sqrt{121x^2 + 100}} < 1 \]
This leads to a contradiction.

Assuming \( |z| > 1 \):
\[ |z|^9 > 1 \implies \frac{\sqrt{121 + 100y^2}}{\sqrt{121x^2 + 100}} > 1 \]
This also leads to a contradiction.

### Step 9: Conclusion
The only consistent solution is when:
\[ |z| = 1 \]
Thus, all the roots of the equation lie on the unit circle.

### Final Answer
All the roots of the equation lie on the circle defined by \( |z| = 1 \).

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