The point represented by the complex number 2+i rotated about the origin through an angle `(pi)/(2)`. The new position of the point is

Options

A. `1+2i`
B. `-1+2i`
C. `1-2i`
D. `2+i`

Answer

To solve the problem of rotating the complex number \( z = 2 + i \) about the origin through an angle of \( \frac{\pi}{2} \), we can follow these steps:

### Step 1: Identify the complex number
The given complex number is:
\[ z = 2 + i \]

### Step 2: Determine the angle of rotation
The angle of rotation is given as:
\[ \theta = \frac{\pi}{2} \]

### Step 3: Use the rotation formula
To rotate a complex number \( z \) about the origin by an angle \( \theta \), we use the formula:
\[ z' = z \cdot e^{i\theta} \]
where \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \).

### Step 4: Calculate \( e^{i\theta} \)
Substituting \( \theta = \frac{\pi}{2} \):
\[ e^{i\frac{\pi}{2}} = \cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right) = 0 + i \cdot 1 = i \]

### Step 5: Substitute into the rotation formula
Now substituting back into the rotation formula:
\[ z' = (2 + i) \cdot i \]

### Step 6: Perform the multiplication
We can distribute \( i \) across \( (2 + i) \):
\[ z' = 2i + i^2 \]
Since \( i^2 = -1 \), we have:
\[ z' = 2i - 1 \]

### Step 7: Rearrange the result
Rearranging gives us:
\[ z' = -1 + 2i \]

### Final Answer
Thus, the new position of the point after rotation is:
\[ \boxed{-1 + 2i} \]