If `8i z^3+12 z^2-18 z+27 i=0,t h e n` `|z|=3/2` b. `|z|=2/3` c.`|z|=1` d. `|z|=3/4`

Options

A. `|z| = (3)/(2)`
B. `|z| = (3)/(4)`
C. `|z|=1`
D. `|z| = (3)/(4)`

Explanation

To solve the equation \(8i z^3 + 12 z^2 - 18 z + 27 i = 0\) and find the modulus of \(z\), we can follow these steps:

### Step 1: Rearranging the equation
The given equation is:
\[ 8i z^3 + 12 z^2 - 18 z + 27 i = 0 \]

### Step 2: Factoring out common terms
We can factor the equation by grouping terms. Notice that we can take \(4i\) as a common factor from the first and last terms:
\[ 4i(2z^3 + \frac{12}{4i} z^2 - \frac{18}{4i} z + \frac{27}{4i}) = 0 \]
This simplifies to:
\[ 4i z^2(2z - \frac{3}{2}i) + 9 = 0 \]

### Step 3: Setting up the factors
We can separate the equation into two parts:
1. \(2z - 3i = 0\)
2. \(4iz^2 - 9 = 0\)

### Step 4: Solving the first equation
From the first equation:
\[ 2z - 3i = 0 \implies 2z = 3i \implies z = \frac{3i}{2} \]
Now, we find the modulus of \(z\):
\[ |z| = \left|\frac{3i}{2}\right| = \frac{3}{2} \]

### Step 5: Solving the second equation
From the second equation:
\[ 4iz^2 - 9 = 0 \implies 4iz^2 = 9 \implies z^2 = \frac{9}{4i} \]
To find \(z\), we can multiply the numerator and denominator by the conjugate of the denominator:
\[ z^2 = \frac{9}{4i} \cdot \frac{-i}{-i} = \frac{-9i}{-4} = \frac{9i}{4} \]
Now, taking the modulus:
\[ |z|^2 = \left| \frac{9i}{4} \right| = \frac{9}{16} \]
Thus,
\[ |z| = \sqrt{\frac{9}{16}} = \frac{3}{4} \]

### Conclusion
The possible values of \(|z|\) from the equations are \(\frac{3}{2}\) and \(\frac{3}{4}\). The correct answer from the options provided is:
- **Option a: \(|z| = \frac{3}{2}\)**