Key Concepts and Formulas

• Complex Number Representation: A complex number $z$ can be represented as $z = x + yi$, where $x = Re(z)$ is the real part and $y = Im(z)$ is the imaginary part.
• Equality of Complex Numbers: Two complex numbers $a + bi$ and $c + di$ are equal if and only if $a = c$ and $b = d$.
• Complex Number Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$.

Step-by-Step Solution

Step 1: Represent the complex number $z$.

We are given that $Im(z) = 10$. Let $Re(z) = x$. Therefore, we can write $z$ as: $z = x + 10i$ Explanation: Representing $z$ in terms of its real and imaginary components is essential for manipulating the complex equation and separating its real and imaginary parts later.*

Step 2: Substitute $z$ into the given equation.

The given equation is: $\frac{2z - n}{2z + n} = 2i - 1$ Substitute $z = x + 10i$ into the equation: $\frac{2(x + 10i) - n}{2(x + 10i) + n} = -1 + 2i$ $\frac{2x + 20i - n}{2x + 20i + n} = -1 + 2i$ $\frac{(2x - n) + 20i}{(2x + n) + 20i} = -1 + 2i$ Explanation: Substituting the expression for $z$ allows us to rewrite the equation in terms of real variables ($x$ and $n$) and the imaginary unit $i$. This sets the stage for isolating the complex number and equating real and imaginary parts.*

Step 3: Eliminate the denominator by cross-multiplication.

Multiply both sides by $(2x + n) + 20i$: $(2x - n) + 20i = (-1 + 2i)((2x + n) + 20i)$ Explanation: Cross-multiplication is used to get rid of the fraction, transforming the division into a multiplication, which is easier to handle.*

Step 4: Expand the right-hand side of the equation.

Expand the product of the complex numbers on the right-hand side: $(-1 + 2i)((2x + n) + 20i) = (-1)(2x + n) + (-1)(20i) + (2i)(2x + n) + (2i)(20i)$ $= -2x - n - 20i + 4xi + 2ni + 40i^2$ Since $i^2 = -1$, we have: $= -2x - n - 20i + 4xi + 2ni - 40$ $= (-2x - n - 40) + (4x + 2n - 20)i$ Therefore, the equation becomes: $(2x - n) + 20i = (-2x - n - 40) + (4x + 2n - 20)i$ Explanation: Expanding the product carefully, remembering that $i^2 = -1$, is crucial. Errors in this step will propagate through the rest of the solution.*

Step 5: Equate real and imaginary parts.

Equate the real parts: $2x - n = -2x - n - 40$ Equate the imaginary parts: $20 = 4x + 2n - 20$ Explanation: This step applies the principle of complex number equality, breaking down the complex equation into two simpler equations involving only real numbers.*

Step 6: Solve the system of linear equations.

From the real parts equation: $2x - n = -2x - n - 40$ $4x = -40$ $x = -10$ So, $Re(z) = -10$.

From the imaginary parts equation: $20 = 4x + 2n - 20$ Substitute $x = -10$: $20 = 4(-10) + 2n - 20$ $20 = -40 + 2n - 20$ $20 = -60 + 2n$ $80 = 2n$ $n = 40$ We have $n = 40$, which is a natural number.

Explanation: Solving the system of equations gives us the values of $x$ and $n$. The fact that $n$ is a natural number confirms our solution.*

Common Mistakes & Tips

• Sign Errors: Be extremely careful with signs during expansion and simplification.
• Forgetting $i^2 = -1$: Always remember to substitute $i^2$ with $-1$.
• Checking Constraints: Make sure the solution satisfies any given conditions, like $n$ being a natural number.

Summary

By representing the complex number $z$ in terms of its real and imaginary parts, substituting it into the given equation, and then equating the real and imaginary components, we obtained a system of two linear equations. Solving this system yielded $Re(z) = -10$ and $n = 40$.

Final Answer

The final answer is \boxed{n = 40 and Re(z) = -10}, which corresponds to option (C). The correct answer is (C).