Step-by-step Solution:

1. Understand the definitions of the sets A, B, and C.

Let a complex number be represented as z = x + iy, where x and y are real numbers.

• Set A: $A = \left\{ {z:\,{\mathop{\rm Im}\nolimits} \,z\, \ge \,1} \right\}$ The imaginary part of z is y. So, this condition is $y \ge 1$. Geometrically, this represents the upper half-plane including the line y = 1.

• Set B: $B = \left\{ {z:\,\,\left| {z - 2 - i} \right| = 3} \right\}$ This can be written as |z - (2 + i)| = 3. This represents all points z in the complex plane whose distance from the point (2 + i) is 3. Geometrically, this is a circle with center $C_B = (2, 1)$ and radius r = 3. The equation in Cartesian coordinates is $(x - 2)^2 + (y - 1)^2 = 3^2 = 9$.

• Set C: $C = \left\{ {z:\,{\mathop{\rm Re}\nolimits} (1 - i)z) = \sqrt 2 \,} \right\}$ Let's compute (1 - i)z: $(1 - i)(x + iy) = x + iy - ix - i^2y = x + y + i(y - x)$ The real part is Re((1 - i)z) = x + y. So, the condition is $x + y = \sqrt 2$. Geometrically, this is a straight line.

2. Analyze the expression to be evaluated.

The expression is $E = |z + 1 - i|^2 + |z - 5 - i|^2$. Let's express this in terms of x and y: $z + 1 - i = (x + 1) + i(y - 1)$ $|z + 1 - i|^2 = (x + 1)^2 + (y - 1)^2$

$z - 5 - i = (x - 5) + i(y - 1)$ $|z - 5 - i|^2 = (x - 5)^2 + (y - 1)^2$

Now, sum these two parts: $E = [(x + 1)^2 + (y - 1)^2] + [(x - 5)^2 + (y - 1)^2]$ $E = (x^2 + 2x + 1) + (y - 1)^2 + (x^2 - 10x + 25) + (y - 1)^2$ $E = 2x^2 - 8x + 26 + 2(y - 1)^2$

Let's complete the square for the terms involving x: $E = 2(x^2 - 4x) + 26 + 2(y - 1)^2$ $E = 2(x^2 - 4x + 4 - 4) + 26 + 2(y - 1)^2$ $E = 2((x - 2)^2 - 4) + 26 + 2(y - 1)^2$ $E = 2(x - 2)^2 - 8 + 26 + 2(y - 1)^2$ $E = 2(x - 2)^2 + 2(y - 1)^2 + 18$ $E = 2 \left[ (x - 2)^2 + (y - 1)^2 \right] + 18$

3. Use the condition from Set B.

Any point z in the intersection $A \cap B \cap C$ must belong to set B. The condition for z to be in B is $(x - 2)^2 + (y - 1)^2 = 9$.

Substituting this into our simplified expression for E: $E = 2(9) + 18 = 18 + 18 = 36$

This means that for any point z on the circle defined by set B, the value of the expression is exactly 36.

4. Verify that the intersection $A \cap B \cap C$ is not empty.

We need to ensure there exists at least one point z that satisfies the conditions for all three sets. We need to find if the line $x + y = \sqrt 2$ (from C) intersects the circle $(x-2)^2 + (y-1)^2 = 9$ (from B) in the region $y \ge 1$ (from A).

• Intersection of B and C: The distance d from the center of the circle (2, 1) to the line $x + y - \sqrt 2 = 0$ is: $d = \frac{|2 + 1 - \sqrt 2|}{\sqrt{1^2 + 1^2}} = \frac{|3 - \sqrt 2|}{\sqrt 2} = \frac{3 - \sqrt 2}{\sqrt 2}$ Since $\sqrt 2 \approx 1.414$, $d \approx (3 - 1.414) / 1.414 \approx 1.586 / 1.414 \approx 1.12$. The radius of the circle is r = 3. Since d < r, the line intersects the circle at two distinct points.

• Checking condition A: We need to see if at least one of these intersection points has a y-coordinate greater than or equal to 1. The center of the circle is at y=1. The line $x+y=\sqrt 2$ passes through the point $(\sqrt 2 - 1, 1)$. Since $\sqrt 2 - 1 \approx 0.414 < 2$, and the line has a slope of -1, one part of the line segment chord inside the circle will be above y=1 and the other below. Thus, one of the intersection points will have y > 1. Therefore, the intersection $A \cap B \cap C$ is non-empty.

5. Final Conclusion.

Since a point z exists in $A \cap B \cap C$, and for any such point (as it must be in B), the expression evaluates to 36. We need to find which of the given intervals contains the value 36.

• A: 25 and 29 (25 < 36 < 29 is false)
• B: 30 and 34 (30 < 36 < 34 is false)
• C: 35 and 39 (35 < 36 < 39 is true)
• D: 40 and 44 (40 < 36 < 44 is false)

The value 36 lies between 35 and 39.