The conjugate of `( 2- i)/((1-2i)^(2) ` is

Options

A. `( 2)/( 25) - (i11)/( 25)`
B. `(- 2)/( 25) - (i11)/( 25)`
C. `(- 2)/( 25) + (i11)/( 25)`
D. `(2)/( 25) + (i11)/( 25)`

Answer

To find the conjugate of the expression \(\frac{2 - i}{(1 - 2i)^2}\), we will follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ \frac{2 - i}{(1 - 2i)^2} \]

### Step 2: Calculate the denominator
First, we need to calculate \((1 - 2i)^2\):
\[ (1 - 2i)^2 = 1^2 - 2 \cdot 1 \cdot 2i + (2i)^2 = 1 - 4i - 4(-1) = 1 - 4i + 4 = 5 - 4i \]

### Step 3: Substitute back into the expression
Now we substitute back into the expression:
\[ \frac{2 - i}{5 - 4i} \]

### Step 4: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(5 + 4i\):
\[ \frac{(2 - i)(5 + 4i)}{(5 - 4i)(5 + 4i)} \]

### Step 5: Calculate the denominator
Calculating the denominator:
\[ (5 - 4i)(5 + 4i) = 5^2 - (4i)^2 = 25 - 16(-1) = 25 + 16 = 41 \]

### Step 6: Calculate the numerator
Now we calculate the numerator:
\[ (2 - i)(5 + 4i) = 2 \cdot 5 + 2 \cdot 4i - i \cdot 5 - i \cdot 4i = 10 + 8i - 5i - 4(-1) = 10 + 8i - 5i + 4 = 14 + 3i \]

### Step 7: Combine the results
Now we combine the results:
\[ \frac{14 + 3i}{41} \]

### Step 8: Write the final expression
Thus, the expression simplifies to:
\[ \frac{14}{41} + \frac{3}{41}i \]

### Step 9: Find the conjugate
The conjugate of a complex number \(a + bi\) is \(a - bi\). Therefore, the conjugate of \(\frac{14}{41} + \frac{3}{41}i\) is:
\[ \frac{14}{41} - \frac{3}{41}i \]

### Final Answer
The conjugate of \(\frac{2 - i}{(1 - 2i)^2}\) is:
\[ \frac{14}{41} - \frac{3}{41}i \]