If the equation `z^(3)+(3+i)z^(2)-3z-(m+i)=0, " where " i=sqrt(-1) " and " m in R`, has atleast one real root, value of m is

Options

A. 1
B. 2
C. 3
D. 5

Answer

To solve the equation \( z^3 + (3+i)z^2 - 3z - (m+i) = 0 \) where \( i = \sqrt{-1} \) and \( m \in \mathbb{R} \), and to find the value of \( m \) such that the equation has at least one real root, we can follow these steps:

### Step 1: Substitute a Real Root
Let \( A \) be a real root of the equation. Then substituting \( z = A \) into the equation gives us:
\[ A^3 + (3+i)A^2 - 3A - (m+i) = 0 \]
This can be rearranged to:
\[ A^3 + 3A^2 - 3A - m + i(3A^2 - 1) = 0 \]
This means the real part and the imaginary part must both equal zero.

### Step 2: Set Up the Real and Imaginary Parts
From the equation above, we can separate the real and imaginary parts:
1. Real part: \( A^3 + 3A^2 - 3A - m = 0 \) (Equation 1)
2. Imaginary part: \( 3A^2 - 1 = 0 \) (Equation 2)

### Step 3: Solve for \( A \) from the Imaginary Part
From Equation 2:
\[ 3A^2 - 1 = 0 \implies 3A^2 = 1 \implies A^2 = \frac{1}{3} \implies A = \pm \frac{1}{\sqrt{3}} \]

### Step 4: Substitute \( A \) Back to Find \( m \)
Now we substitute \( A = \frac{1}{\sqrt{3}} \) into Equation 1:
\[ \left(\frac{1}{\sqrt{3}}\right)^3 + 3\left(\frac{1}{\sqrt{3}}\right)^2 - 3\left(\frac{1}{\sqrt{3}}\right) - m = 0 \]
Calculating each term:
\[ \frac{1}{3\sqrt{3}} + 3 \cdot \frac{1}{3} - \frac{3}{\sqrt{3}} - m = 0 \]
This simplifies to:
\[ \frac{1}{3\sqrt{3}} + 1 - \sqrt{3} - m = 0 \]
Rearranging gives:
\[ m = 1 + \frac{1}{3\sqrt{3}} - \sqrt{3} \]

Now substituting \( A = -\frac{1}{\sqrt{3}} \):
\[ \left(-\frac{1}{\sqrt{3}}\right)^3 + 3\left(-\frac{1}{\sqrt{3}}\right)^2 - 3\left(-\frac{1}{\sqrt{3}}\right) - m = 0 \]
Calculating gives:
\[ -\frac{1}{3\sqrt{3}} + 1 + \sqrt{3} - m = 0 \]
Rearranging gives:
\[ m = 1 - \frac{1}{3\sqrt{3}} + \sqrt{3} \]

### Step 5: Determine Values of \( m \)
Thus, we have two values for \( m \):
1. \( m = 1 + \frac{1}{3\sqrt{3}} - \sqrt{3} \)
2. \( m = 1 - \frac{1}{3\sqrt{3}} + \sqrt{3} \)

### Final Answer
The values of \( m \) that allow the equation to have at least one real root are:
- \( m = 1 + \frac{1}{3\sqrt{3}} - \sqrt{3} \)
- \( m = 1 - \frac{1}{3\sqrt{3}} + \sqrt{3} \)