Key Concepts and Formulas

• A complex number $z$ can be expressed as $z = x + iy$, where $x = \operatorname{Re}(z)$ and $y = \operatorname{Im}(z)$. The conjugate of $z$ is $\bar{z} = x - iy$.
• $z\bar{z} = (x+iy)(x-iy) = x^2 + y^2$.
• To find the real part of a complex number in the form $\frac{a+bi}{c+di}$, multiply the numerator and denominator by the conjugate of the denominator, i.e., $\frac{a+bi}{c+di} \cdot \frac{c-di}{c-di}$.

Step-by-Step Solution

1. Express $z$ and $\bar{z}$ in terms of $y$

We are given that $\operatorname{Re}(z) = 3$. Let $z = x + iy$, where $x, y \in \mathbb{R}$. Since $x = 3$, we have $z = 3 + iy$. Therefore, $\bar{z} = 3 - iy$.

Explanation: We represent the complex number $z$ using its real and imaginary parts, incorporating the given condition to express $z$ in terms of a single real variable $y$. This simplifies the algebraic manipulations in subsequent steps.

2. Substitute $z$ and $\bar{z}$ into the numerator

The numerator is $z - \bar{z} + z\bar{z}$. Substituting $z = 3 + iy$ and $\bar{z} = 3 - iy$: $z - \bar{z} + z\bar{z} = (3 + iy) - (3 - iy) + (3 + iy)(3 - iy)$ $= 3 + iy - 3 + iy + (9 - (iy)^2)$ $= 2iy + (9 - (-y^2))$ $= 2iy + 9 + y^2$ $= (9 + y^2) + 2iy$

Explanation: Here, we substitute the expressions for $z$ and $\bar{z}$ into the numerator. We expand the product $z\bar{z}$ using the difference of squares formula, $(a+b)(a-b) = a^2 - b^2$, and simplify the expression to obtain a complex number in the form $a+bi$.

3. Substitute $z$ and $\bar{z}$ into the denominator

The denominator is $2 - 3z + 5\bar{z}$. Substituting $z = 3 + iy$ and $\bar{z} = 3 - iy$: $2 - 3z + 5\bar{z} = 2 - 3(3 + iy) + 5(3 - iy)$ $= 2 - 9 - 3iy + 15 - 5iy$ $= (2 - 9 + 15) + (-3iy - 5iy)$ $= 8 - 8iy$

Explanation: We substitute the expressions for $z$ and $\bar{z}$ into the denominator, distribute the constants, and combine the real and imaginary terms to express the denominator as a complex number in the form $a+bi$.

4. Form the complex fraction and find its real part

The expression becomes: $\frac{z - \bar{z} + z\bar{z}}{2 - 3z + 5\bar{z}} = \frac{(9 + y^2) + 2iy}{8 - 8iy}$ To find the real part, multiply the numerator and denominator by the conjugate of the denominator, which is $8 + 8iy$: $\frac{(9 + y^2) + 2iy}{8 - 8iy} \cdot \frac{8 + 8iy}{8 + 8iy} = \frac{((9 + y^2) + 2iy)(8 + 8iy)}{(8 - 8iy)(8 + 8iy)}$ $= \frac{8(9 + y^2) + 8i(9 + y^2) + 16iy + 16i^2y^2}{64 - (64i^2y^2)}$ $= \frac{72 + 8y^2 + 72i + 8iy^2 + 16iy - 16y^2}{64 + 64y^2}$ $= \frac{(72 + 8y^2 - 16y^2) + i(72y + 8y^3 + 16y)}{64(1 + y^2)}$ $= \frac{(72 - 8y^2) + i(88y + 8y^3)}{64(1 + y^2)}$ The real part is: $\operatorname{Re}\left(\frac{z - \bar{z} + z\bar{z}}{2 - 3z + 5\bar{z}}\right) = \frac{72 - 8y^2}{64(1 + y^2)} = \frac{8(9 - y^2)}{64(1 + y^2)} = \frac{9 - y^2}{8(1 + y^2)}$

Explanation: We multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary term from the denominator. Then we expand the product in the numerator, simplify by using $i^2=-1$, and separate the real and imaginary parts. Finally, we isolate the real part of the complex expression.

5. Determine the range of the real part expression

Let $K = \frac{9 - y^2}{8(1 + y^2)}$. We can rewrite this as: $K = \frac{1}{8} \left(\frac{9 - y^2}{1 + y^2}\right) = \frac{1}{8} \left(\frac{-1(y^2+1) + 10}{y^2+1}\right) = \frac{1}{8} \left(-1 + \frac{10}{y^2+1}\right)$ Since $y \in \mathbb{R}$, $y^2 \ge 0$, so $y^2 + 1 \ge 1$. Thus, $\frac{1}{y^2 + 1} \le 1$, and $\frac{1}{y^2+1} > 0$. Therefore, $0 < \frac{1}{y^2 + 1} \le 1$. Multiplying by 10, we get $0 < \frac{10}{y^2 + 1} \le 10$. Subtracting 1, we get $-1 < \frac{10}{y^2 + 1} - 1 \le 9$. Dividing by 8, we get $-\frac{1}{8} < \frac{1}{8}\left(\frac{10}{y^2 + 1} - 1\right) \le \frac{9}{8}$. Therefore, $K \in \left(-\frac{1}{8}, \frac{9}{8}\right]$.

Explanation: We rewrite the expression to make it easier to analyze. Since $y$ can be any real number, we determine the range of $y^2$, then $y^2+1$, and systematically build up the range of the entire expression. We pay attention to whether the endpoints of the intervals are included or excluded.

6. Identify $\alpha$ and $\beta$ and calculate the final value

We have the interval $(\alpha, \beta] = \left(-\frac{1}{8}, \frac{9}{8}\right]$. Therefore, $\alpha = -\frac{1}{8}$ and $\beta = \frac{9}{8}$. We need to find $24(\beta - \alpha)$: $24(\beta - \alpha) = 24\left(\frac{9}{8} - \left(-\frac{1}{8}\right)\right) = 24\left(\frac{9}{8} + \frac{1}{8}\right) = 24\left(\frac{10}{8}\right) = 24\left(\frac{5}{4}\right) = 6 \cdot 5 = 30$

Explanation: We identify $\alpha$ and $\beta$ by comparing the derived range with the given interval format. We then substitute these values into the expression $24(\beta-\alpha)$ and perform the arithmetic to obtain the final result.

Common Mistakes & Tips

• When finding the real part of a complex fraction, remember to multiply both the numerator and denominator by the conjugate of the denominator.
• Pay close attention to sign errors, especially when expanding expressions and dealing with $i^2 = -1$.
• When finding the range of an expression involving $y^2$, remember that $y^2 \ge 0$ for all real $y$.

Summary

The problem requires us to find the range of the real part of a complex expression, given that the real part of $z$ is 3. We express $z$ as $3 + iy$, substitute this into the expression, simplify by multiplying by the conjugate, and then analyze the range of the resulting real-valued function of $y$. Finally, we calculate $24(\beta - \alpha)$. The final answer is 30.

Final Answer

The final answer is $\boxed{30}$, which corresponds to option (D).