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Let's go through the problems step-by-step.

### Problem 1(a)
We need to find the values of \( m \) and \( n \) such that:

\[
\frac{m}{3 + 6i} + \frac{n}{3 - 6i} = 1 + \frac{2}{5}i
\]

1. **Rationalize** both denominators by multiplying by the conjugate of each. This will help simplify the expression.
2. Equate the real and imaginary parts separately to find \( m \) and \( n \).

### Problem 1(b)
Given:

\[
z = \frac{1}{2} \left( 1 + \sqrt{3}i \right)
\]

satisfies the equation:

\[
\frac{1}{\overline{z}} = kz^2
\]

Determine \( k \). This involves calculating \( \overline{z} \) and then solving for \( k \).

### Problem 2
#### Part (a)
Solve:

\[
3 \left( x^{\frac{1}{2}} - x - \frac{1}{2} \right) = x^{\frac{1}{2}} + x^{\frac{1}{2}}
\]

Simplify and solve for \( x \).

#### Part (b)
Solve the equation:

\[
x^4 e^{2 \ln x} = 18 - 3x
\]

Use logarithmic properties and algebraic manipulation.

#### Part (c)
Solve:

\[
10 \log_3 3 - 3 \log_2 8 = \log_3 x
\]

Simplify using properties of logarithms.

### Problem 3
#### Part (a)
Given:

\[
f(x) = \ln(x - 3)
\]

Find the inverse function \( f^{-1}(x) \) and state its domain and range.

#### Part (b)
Sketch the graphs of \( f \) and \( f^{-1} \).

#### Part (c)
Find \( (g \circ f^{-1})(x) \) where \( g(x) = 4x + 9 \) and sketch the graph.

Let's start solving each part in detail. Do you want to begin with a specific problem, or shall I proceed with the first one?

Solve a series of algebraic and logarithmic problems involving complex numbers, exponentials, and functions. Problems include finding values of unknowns in complex equations, solving algebraic and logarithmic equations, and working with inverse functions.

This page provides detailed solutions to a series of algebraic problems including solving complex equations, exponential and logarithmic functions, and finding inverse functions. Suitable for advanced high school mathematics students (Grades 11-12).

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