(a) To find the square roots of \(-3 + 4i\), express \(x + iy\) and equate real and imaginary parts:

\(x^2 - y^2 = -3\) and \(2xy = 4\).

From \(2xy = 4\), we have \(xy = 2\).

Substitute \(y = \frac{2}{x}\) into \(x^2 - y^2 = -3\):

\(x^2 - \left(\frac{2}{x}\right)^2 = -3\).

Simplify to get \(x^4 + 3x^2 - 4 = 0\).

Solving gives \(x = 1, y = 2\) and \(x = -1, y = -2\).

Thus, the square roots are \(1 + 2i\) and \(-1 - 2i\).

(b)(i) Multiply numerator and denominator of \(\frac{-1 + 3i}{2 + i}\) by the conjugate \(2 - i\):

\(z = \frac{(-1 + 3i)(2 - i)}{(2 + i)(2 - i)} = \frac{-2 + i + 6i - 3}{4 + 1} = \frac{-5 + 7i}{5}\).

Simplify to get \(z = 0.2 + 1.4i\).

(b)(ii) Plot the points \(-1 + 3i\), \(2 + i\), and \(0.2 + 1.4i\) on an Argand diagram.

(b)(iii) The equation relating the lengths is \(OC = \frac{OA}{OB}\).