Key Concepts and Formulas

• Complex Number Representation: A complex number $z$ can be expressed as $z = x + iy$, where $x$ is the real part (Re(z)) and $y$ is the imaginary part (Im(z)), and $i = \sqrt{-1}$. $x$ and $y$ are real numbers.
• Modulus of a Complex Number: The modulus of a complex number $z = x + iy$ is denoted by $|z|$ and is defined as $|z| = \sqrt{x^2 + y^2}$.
• Equality of Complex Numbers: Two complex numbers are equal if and only if their real and imaginary parts are equal. If $a + ib = c + id$, then $a = c$ and $b = d$.

Step-by-Step Solution

Step 1: Represent the complex number and its modulus

We are given $|z| + z = 3 + i$. Let $z = x + iy$, where $x$ and $y$ are real numbers. Then $|z| = \sqrt{x^2 + y^2}$. Substitute these into the given equation: $\sqrt{x^2 + y^2} + (x + iy) = 3 + i$ Group the real and imaginary terms on the left side: $(\sqrt{x^2 + y^2} + x) + iy = 3 + i$ Rationale: Expressing $z$ in standard form allows us to separate the real and imaginary components of the equation, a common strategy for solving complex number problems.

Step 2: Equate real and imaginary parts

Using the equality of complex numbers, we equate the real and imaginary parts from both sides of the equation. Equating the real parts: $\sqrt{x^2 + y^2} + x = 3 \quad (*)$ Equating the imaginary parts: $y = 1 \quad (**)$ Rationale: This step transforms the complex equation into a system of two real equations, which simplifies the problem. The equation for the imaginary parts directly gives us the value of $y$.

Step 3: Express $x$ in terms of $|z|$ and use the value of $y$

Notice that $\sqrt{x^2 + y^2} = |z|$. Equation $(*)$ can be rewritten as: $|z| + x = 3$ From this, we express $x$ in terms of $|z|$: $x = 3 - |z| \quad (***)$ We know from equation $(**)$ that $y = 1$. Rationale: Expressing $x$ in terms of $|z|$ brings us closer to an equation with only $|z|$ as the unknown.

Step 4: Use the modulus definition to solve for $|z|$

Now we have expressions for $x$ and $y$. We substitute these into the definition of the modulus: $|z| = \sqrt{x^2 + y^2}$ Substitute $x = 3 - |z|$ and $y = 1$: $|z| = \sqrt{(3 - |z|)^2 + (1)^2}$ To eliminate the square root, we square both sides of the equation. Let $r = |z|$ for simpler calculation: $r^2 = (3 - r)^2 + 1^2$ Expand the squared term: $r^2 = (9 - 6r + r^2) + 1$ Combine constant terms: $r^2 = 10 - 6r + r^2$ Subtract $r^2$ from both sides: $0 = 10 - 6r$ Solve for $r$: $6r = 10$ $r = \frac{10}{6}$ Simplify the fraction: $r = \frac{5}{3}$ Since $r = |z|$, we conclude that: $|z| = \frac{5}{3}$ Rationale: Substituting back into the modulus definition creates a single equation with $|z|$ as the unknown, which we then solve algebraically.

Step 5: Verification (Optional)

It's good to verify the solution. If $|z| = \frac{5}{3}$: From $x = 3 - |z|$, we get $x = 3 - \frac{5}{3} = \frac{4}{3}$. We found $y = 1$. So, $z = \frac{4}{3} + i$. Check the original equation: $|z| + z = 3 + i$. Left-Hand Side (LHS): $|z| + z = \frac{5}{3} + \left( \frac{4}{3} + i \right)$ LHS $= \left( \frac{5}{3} + \frac{4}{3} \right) + i$ LHS $= \frac{9}{3} + i$ LHS $= 3 + i$ Since LHS = RHS, our solution is correct.

Common Mistakes & Tips

• Modulus is non-negative: Remember that $|z|$ is a real, non-negative number.
• Equating parts is crucial: Separating complex equations into real and imaginary components is a key first step.
• Check for extraneous solutions: Squaring both sides of an equation can introduce extraneous solutions. Always verify your solutions in the original equation.

Summary

This problem demonstrates a standard technique for solving equations involving complex numbers and their moduli. The strategy involves substituting $z = x + iy$, separating the equation into real and imaginary parts, and solving the resulting system of real equations. The solution demonstrates that $|z|$ is $\frac{5}{3}$.

Final Answer

The final answer is \boxed{\frac{5}{3}}, which corresponds to option (B).