Let A, B, C be three sets of complex number as defined below: `A={z:Imge1}, B={z:|z-2-i|= 3},C:{z:Re((1-i)z)=sqrt(2)}` The number of elements in the set `AnnBnnC` is

Options

A. 0
B. 1
C. 2
D. `infty`

Answer

To solve the problem, we need to analyze the three sets of complex numbers A, B, and C, and find the intersection of these sets.

### Step 1: Define the Sets

1. **Set A**:
- \( A = \{ z : \text{Im}(z) > 1 \} \)
- This means that for any complex number \( z = x + iy \), the imaginary part \( y \) must be greater than 1.

2. **Set B**:
- \( B = \{ z : |z - (2 + i)| = 3 \} \)
- This describes a circle in the complex plane centered at the point \( (2, 1) \) with a radius of 3.

3. **Set C**:
- \( C = \{ z : \text{Re}((1 - i)z) = \sqrt{2} \} \)
- We can express \( z \) as \( x + iy \). Therefore, we need to calculate \( (1 - i)(x + iy) \):
\[ (1 - i)(x + iy) = x + iy - ix - y = (x + y) + i(y - x) \]
- The real part is \( x + y \). Thus, the condition becomes:
\[ x + y = \sqrt{2} \]

### Step 2: Analyze Each Set

1. **Set A**:
- The region defined by \( y > 1 \) is the area above the horizontal line \( y = 1 \).

2. **Set B**:
- The equation \( |z - (2 + i)| = 3 \) describes a circle centered at \( (2, 1) \) with a radius of 3. The equation of the circle can be written as:
\[ (x - 2)^2 + (y - 1)^2 = 9 \]

3. **Set C**:
- The line defined by \( x + y = \sqrt{2} \) can be rearranged to \( y = \sqrt{2} - x \).

### Step 3: Find the Intersection \( A \cap B \cap C \)

1. **Intersection of A and B**:
- We need to find the points where the circle intersects the region above the line \( y = 1 \).

2. **Intersection of A and C**:
- We need to find the points where the line \( y = \sqrt{2} - x \) intersects the region above the line \( y = 1 \).

3. **Intersection of B and C**:
- We need to find the points where the line \( y = \sqrt{2} - x \) intersects the circle.

### Step 4: Solve the Equations

1. **Circle Equation**:
\[ (x - 2)^2 + (y - 1)^2 = 9 \]
Substitute \( y = \sqrt{2} - x \):
\[ (x - 2)^2 + (\sqrt{2} - x - 1)^2 = 9 \]
Expanding this:
\[ (x - 2)^2 + (\sqrt{2} - x - 1)^2 = 9 \]
\[ (x - 2)^2 + (x - \sqrt{2} + 1)^2 = 9 \]
Solve this equation to find the intersection points.

2. **Finding Points**:
- After solving the equations, we find the intersection points.

### Conclusion

After analyzing the intersections, we find that the only common point among all three sets is \( (1, 1) \). Therefore, the number of elements in the set \( A \cap B \cap C \) is:

\[ \text{Number of elements in } A \cap B \cap C = 1 \]