Key Concepts and Formulas

• Modulus of a Complex Number: For a complex number $z = x + iy$, where $x$ and $y$ are real numbers, the modulus is $|z| = \sqrt{x^2 + y^2}$.
• Equality of Complex Numbers: If $a + bi = c + di$, where $a, b, c, d$ are real numbers, then $a = c$ and $b = d$.

Step-by-Step Solution

Step 1: Calculate $z + 1$

We are given $z = \frac{1}{2} - 2i$. We need to find $z + 1$. $z + 1 = \left(\frac{1}{2} - 2i\right) + 1$ $z + 1 = \frac{1}{2} + 1 - 2i$ $z + 1 = \frac{3}{2} - 2i$ This simplifies the expression inside the modulus.

Step 2: Calculate $|z + 1|$

Now we find the modulus of $z + 1 = \frac{3}{2} - 2i$. Using the formula $|z| = \sqrt{x^2 + y^2}$: $|z + 1| = \left|\frac{3}{2} - 2i\right| = \sqrt{\left(\frac{3}{2}\right)^2 + (-2)^2}$ $|z + 1| = \sqrt{\frac{9}{4} + 4}$ $|z + 1| = \sqrt{\frac{9}{4} + \frac{16}{4}}$ $|z + 1| = \sqrt{\frac{25}{4}}$ $|z + 1| = \frac{5}{2}$ So, the left-hand side (LHS) of the equation is $\frac{5}{2}$.

Step 3: Simplify the Right-Hand Side (RHS)

The right-hand side (RHS) of the equation is $\alpha z + \beta(1 + i)$. We substitute $z = \frac{1}{2} - 2i$ into this expression. $\alpha z + \beta(1 + i) = \alpha\left(\frac{1}{2} - 2i\right) + \beta(1 + i)$ $= \frac{\alpha}{2} - 2\alpha i + \beta + \beta i$ $= \left(\frac{\alpha}{2} + \beta\right) + (-2\alpha + \beta)i$ This expresses the RHS in the form $A + Bi$.

Step 4: Equate LHS and RHS

We are given $|z + 1| = \alpha z + \beta(1 + i)$. We have calculated $|z + 1| = \frac{5}{2}$ and $\alpha z + \beta(1 + i) = \left(\frac{\alpha}{2} + \beta\right) + (-2\alpha + \beta)i$. Therefore: $\frac{5}{2} = \left(\frac{\alpha}{2} + \beta\right) + (-2\alpha + \beta)i$ Now, we equate the real and imaginary parts:

• Real parts: $\frac{\alpha}{2} + \beta = \frac{5}{2}$
• Imaginary parts: $-2\alpha + \beta = 0$ This leads to a system of two linear equations.

Step 5: Solve for $\alpha$ and $\beta$

We have the system of equations:

1. $\frac{\alpha}{2} + \beta = \frac{5}{2}$
2. $-2\alpha + \beta = 0$

From equation (2), we have $\beta = 2\alpha$. Substituting this into equation (1): $\frac{\alpha}{2} + 2\alpha = \frac{5}{2}$ $\frac{\alpha + 4\alpha}{2} = \frac{5}{2}$ $\frac{5\alpha}{2} = \frac{5}{2}$ $5\alpha = 5$ $\alpha = 1$ Now, substitute $\alpha = 1$ into $\beta = 2\alpha$: $\beta = 2(1) = 2$ Thus, $\alpha = 1$ and $\beta = 2$.

Step 6: Calculate $\alpha + \beta$

Finally, we calculate $\alpha + \beta$: $\alpha + \beta = 1 + 2 = 3$

Common Mistakes & Tips:

• Remember to square both the real and imaginary parts when calculating the modulus of a complex number.
• Be careful with signs when solving simultaneous equations.
• Double-check your arithmetic throughout the process.

Summary

We simplified the given complex equation by calculating the modulus and separating the real and imaginary parts. This led to a system of two linear equations, which we solved to find $\alpha = 1$ and $\beta = 2$. Therefore, $\alpha + \beta = 3$.

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