Key Concepts and Formulas

• Complex Number Representation: A complex number $z$ can be written as $z = x + iy$, where $x = \text{Re}(z)$ is the real part and $y = \text{Im}(z)$ is the imaginary part.
• Modulus of a Complex Number: The modulus of $z = x + iy$ is given by $|z| = \sqrt{x^2 + y^2}$, which represents the distance from the origin to the point $(x, y)$ in the complex plane.
• Perpendicular Distance from a Point to a Line: The perpendicular distance $d$ from a point $(x_0, y_0)$ to a line $Ax + By + C = 0$ is given by $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$.

Step-by-Step Solution

Step 1: Represent the Complex Number and Interpret the Condition

Let $z = x + iy$, where $x$ and $y$ are real numbers. We are given the condition: $|\text{Re}(z)| + |\text{Im}(z)| = 4$ Substituting $x$ and $y$, we have: $|x| + |y| = 4$ This equation represents a square centered at the origin with vertices at $(4, 0)$, $(0, 4)$, $(-4, 0)$, and $(0, -4)$. This can be seen by considering each quadrant:

• Quadrant I ($x \ge 0, y \ge 0$): $x + y = 4$
• Quadrant II ($x < 0, y \ge 0$): $-x + y = 4$
• Quadrant III ($x < 0, y < 0$): $-x - y = 4$
• Quadrant IV ($x \ge 0, y < 0$): $x - y = 4$

Step 2: Determine the Range of $|z|$

The modulus $|z|$ is the distance from the origin to a point $(x, y)$ satisfying $|x| + |y| = 4$. We need to find the minimum and maximum possible values of $|z|$.

• Maximum Value of $|z|$: The maximum distance from the origin occurs at the vertices of the square. For example, at the vertex $(4, 0)$: $|z|_{\text{max}} = \sqrt{4^2 + 0^2} = \sqrt{16} = 4$ Similarly, for $(0, 4)$, $|z| = \sqrt{0^2 + 4^2} = 4$. Thus, the maximum value of $|z|$ is $4$.

• Minimum Value of $|z|$: The minimum distance occurs along the line segments forming the sides. Consider the side in the first quadrant, $x + y = 4$. The shortest distance from the origin to this line is the perpendicular distance. Using the perpendicular distance formula, where the line is $x + y - 4 = 0$ and the point is $(0, 0)$: $|z|_{\text{min}} = \frac{|1(0) + 1(0) - 4|}{\sqrt{1^2 + 1^2}} = \frac{|-4|}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}$ We can express this as $\sqrt{(2\sqrt{2})^2} = \sqrt{4 \times 2} = \sqrt{8}$.

Therefore, the possible range for $|z|$ is: $|z| \in [2\sqrt{2}, 4] = [\sqrt{8}, \sqrt{16}]$

Step 3: Evaluate the Given Options

We are looking for the option that lies outside the range $[\sqrt{8}, \sqrt{16}]$. To simplify comparison, we will compare the squares of the values with the squares of the range boundaries, i.e., $[8, 16]$.

• (A) $\sqrt{10}$: Its square is $10$. Since $8 \le 10 \le 16$, $\sqrt{10}$ is within the allowed range for $|z|$.
• (B) $\sqrt{7}$: Its square is $7$. Since $7 < 8$, $\sqrt{7}$ is not within the allowed range for $|z|$.
• (C) $\sqrt{\frac{17}{2}}$: Its square is $\frac{17}{2} = 8.5$. Since $8 \le 8.5 \le 16$, $\sqrt{\frac{17}{2}}$ is within the allowed range for $|z|$.
• (D) $\sqrt{8}$: Its square is $8$. Since $8 \le 8 \le 16$, $\sqrt{8}$ is within the allowed range for $|z|$.

Therefore, $\sqrt{7}$ is the only value that falls outside the possible range for $|z|$.

Common Mistakes & Tips

• Geometric Visualization: Always try to visualize the given condition in the complex plane. This simplifies the problem and helps in finding minimum/maximum distances.
• Perpendicular Distance Formula: Remember the formula for the perpendicular distance from a point to a line. This is essential for finding the minimum distance.
• Range vs. Specific Value: Understand whether the question asks for a value within the range or outside the range.

Summary

The condition $|\text{Re}(z)| + |\text{Im}(z)| = 4$ defines a square in the complex plane. The modulus $|z|$ represents the distance from the origin to any point on this square. The range of possible values for $|z|$ is $[\sqrt{8}, \sqrt{16}]$. Among the given options, $\sqrt{7}$ is the only value outside this range, so $|z|$ cannot be $\sqrt{7}$.

Final Answer

The final answer is \boxed{\sqrt{7}}, which corresponds to option (B).