If `i z^3+z^2-z+i=0` , where `i=sqrt(-1)` , then `|z|` is equal to 1 (b) `1/2` (c) `1/4` (d) None of these

Answer

To solve the equation \( i z^3 + z^2 - z + i = 0 \), where \( i = \sqrt{-1} \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ i z^3 + z^2 - z + i = 0 \]
We can rearrange this equation to group the terms:
\[ i z^3 + z^2 - z + i = 0 \]

### Step 2: Factoring the Equation
Next, we can factor out common terms. We can take \( z - i \) as a common factor:
\[ i z^3 + z^2 - z + i = (z - i)(i z^2 + 1) \]
This gives us two factors:
1. \( z - i = 0 \)
2. \( i z^2 + 1 = 0 \)

### Step 3: Solving for \( z \)
From the first factor, we have:
\[ z - i = 0 \implies z = i \]

From the second factor:
\[ i z^2 + 1 = 0 \implies i z^2 = -1 \implies z^2 = -\frac{1}{i} \]
To simplify \( -\frac{1}{i} \), we multiply the numerator and denominator by \( i \):
\[ z^2 = -\frac{1}{i} \cdot \frac{i}{i} = \frac{-i}{-1} = i \]

### Step 4: Finding the Values of \( z \)
Now we have two values for \( z \):
1. \( z = i \)
2. \( z^2 = i \)

From \( z^2 = i \), we can find \( z \):
\[ z = \sqrt{i} \]
To find \( \sqrt{i} \), we express \( i \) in polar form:
\[ i = e^{i \frac{\pi}{2}} \]
Thus,
\[ \sqrt{i} = e^{i \frac{\pi}{4}} = \frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} \]

### Step 5: Calculating the Modulus
Now we need to find \( |z| \):
1. For \( z = i \):
\[ |z| = |i| = 1 \]
2. For \( z = \sqrt{i} \):
\[ |z| = \left| \frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} \right| = \sqrt{\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2} = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1 \]

### Conclusion
Thus, in all cases, we find that:
\[ |z| = 1 \]
The correct answer is (a) \( 1 \).

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