Key Concepts and Formulas

• Purely Imaginary Numbers: A complex number $w$ is purely imaginary if $w = -\bar{w}$, which is equivalent to $w + \bar{w} = 0$.
• Complex Conjugate Properties:
  • $\overline{z_1 \pm z_2} = \bar{z_1} \pm \bar{z_2}$
  • $\overline{z_1 / z_2} = \bar{z_1} / \bar{z_2}$
  • $\overline{ki} = -ki$ for a real constant $k$.
• Modulus of a Complex Number: If $z = x + iy$, then $|z|^2 = z\bar{z} = x^2 + y^2$.

Step-by-Step Solution

1. Define the complex expression and apply the purely imaginary condition. Let $w = \frac{2z - 3i}{2z + i}$. We are given that $w$ is purely imaginary. Why: By definition, a complex number $w$ is purely imaginary if and only if $w = -\bar{w}$, which can be rewritten as $w + \bar{w} = 0$. This allows us to convert the problem into an algebraic equation involving $z$ and its conjugate $\bar{z}$. $\frac{2z - 3i}{2z + i} + \overline{\left(\frac{2z - 3i}{2z + i}\right)} = 0$

2. Apply properties of complex conjugates. Using the properties of complex conjugates: $\overline{\left(\frac{2z - 3i}{2z + i}\right)} = \frac{\overline{2z - 3i}}{\overline{2z + i}} = \frac{2\bar{z} - \overline{3i}}{2\bar{z} + \overline{i}} = \frac{2\bar{z} + 3i}{2\bar{z} - i}$ Why: Taking the conjugate of the entire fraction simplifies the expression by applying the conjugate operation to the numerator and denominator separately. It's crucial to correctly find the conjugate of terms like $-3i$ (which is $+3i$) and $+i$ (which is $-i$). Substituting this back into our condition: $\frac{2z - 3i}{2z + i} + \frac{2\bar{z} + 3i}{2\bar{z} - i} = 0$

3. Clear the denominators and expand the terms. Multiply both sides by $(2z + i)(2\bar{z} - i)$ to eliminate the fractions. Why: This is a standard algebraic step to simplify equations with fractions, allowing us to work with polynomials. $(2z - 3i)(2\bar{z} - i) + (2\bar{z} + 3i)(2z + i) = 0$ Now, expand both products:

• First product: $(2z - 3i)(2\bar{z} - i) = 4z\bar{z} - 2zi - 6i\bar{z} + 3i^2 = 4|z|^2 - 2iz - 6i\bar{z} - 3$
• Second product: $(2\bar{z} + 3i)(2z + i) = 4\bar{z}z + 2i\bar{z} + 6iz + 3i^2 = 4|z|^2 + 2i\bar{z} + 6iz - 3$ Why: Careful expansion of products is essential to avoid algebraic errors. Recognizing $z\bar{z} = |z|^2$ simplifies the expression by converting complex number products into real magnitudes.

4. Combine and simplify the expanded terms. Add the results from the two products: $(4|z|^2 - 2iz - 6i\bar{z} - 3) + (4|z|^2 + 2i\bar{z} + 6iz - 3) = 0$ Combine like terms: $8|z|^2 + 4iz - 4i\bar{z} - 6 = 0$ Why: Grouping and combining like terms makes the expression more manageable and moves us closer to an equation in terms of $x$ and $y$.

5. Substitute $z = x + iy$ and $|z|^2 = x^2 + y^2$. We are given $z = x + iy$. Therefore, $\bar{z} = x - iy$ and $|z|^2 = x^2 + y^2$. Substitute these into the simplified equation: $8(x^2 + y^2) + 4i(x + iy) - 4i(x - iy) - 6 = 0$ Expand the terms: $8(x^2 + y^2) + 4ix - 4y - 4ix - 4y - 6 = 0$ The terms $4ix$ and $-4ix$ cancel out: $8(x^2 + y^2) - 8y - 6 = 0$ Divide the entire equation by 2 to simplify: $4(x^2 + y^2) - 4y - 3 = 0$ Why: By substituting $z=x+iy$ and $\bar{z}=x-iy$, we convert the complex number equation into an equation solely involving real variables $x$ and $y$, which is easier to work with. This also utilizes the definition of the modulus squared, $|z|^2 = x^2+y^2$.

6. Utilize the second given condition. We are given the condition $x + y^2 = 0$, so $x = -y^2$. Why: This condition provides a link between $x$ and $y$, allowing us to reduce the number of variables in our equation and solve for a specific expression involving only $y$. Substitute $x = -y^2$ into the equation $4(x^2 + y^2) - 4y - 3 = 0$: $4((-y^2)^2 + y^2) - 4y - 3 = 0$ $4(y^4 + y^2) - 4y - 3 = 0$ $4y^4 + 4y^2 - 4y - 3 = 0$

7. Isolate the required expression. We need to find the value of $y^4 + y^2 - y$. Notice that $4y^4 + 4y^2 - 4y$ is $4$ times the expression we are looking for. Rearrange the equation: $4(y^4 + y^2 - y) = 3$ Divide by 4: $y^4 + y^2 - y = \frac{3}{4}$ Why: By carefully observing the algebraic form of the equation and the desired expression, we can directly solve for it without needing to find the specific value of $y$. This often simplifies calculations and avoids dealing with roots.

Common Mistakes & Tips:

• Conjugate Errors: Mistakes in taking the conjugate, especially with terms involving 'i', are common. Remember $\overline{i} = -i$.
• Algebraic Simplification: Be careful with signs when expanding and simplifying.
• Using $w + \bar{w} = 0$: This is the key to solving purely imaginary problems.

Summary This problem uses the properties of complex conjugates and the definition of purely imaginary numbers to find the value of an expression. Substituting $x=-y^2$ allowed us to directly compute the target expression $y^4 + y^2 - y$. The final answer is $\frac{3}{4}$.

Final Answer The final answer is $\boxed{\frac{3}{4}}$, which corresponds to option (C).