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?