\((i)(a) Substitute x = 1 + 2i into the equation 2x^3 - x^2 + 4x + k = 0.\)

\(Calculate (1 + 2i)^2 = 1 + 4i^2 + 4i = -3 + 4i.\)

\(Calculate (1 + 2i)^3 = (1 + 2i)(-3 + 4i) = -3 + 4i - 6i + 8i^2 = -11 - 2i.\)

\(Substitute into the equation: 2(-11 - 2i) - (-3 + 4i) + 4(1 + 2i) + k = 0.\)

\(Simplify: -22 - 4i + 3 - 4i + 4 + 8i + k = 0.\)

\(Combine: -15 + k = 0, so k = 15.\)

(i)(b) The conjugate of 1 + 2i is 1 - 2i, which is also a root.

\(Use the quadratic factor (x - (1 + 2i))(x - (1 - 2i)) = x^2 - 2x + 5.\)

\(Divide the cubic by this quadratic to find the third root: -\frac{3}{2}.\)

\((ii) The equation |z - (1 + 2i)| = 1 represents a circle with center 1 + 2i and radius 1.\)

The least value of arg z occurs at the point on the circle closest to the origin.

\(Calculate the angle using trigonometry: arg z = \tan^{-1}\left(\frac{2}{1}\right) - \sin^{-1}\left(\frac{1}{\sqrt{5}}\right).\)

Compute: arg z \approx 0.64 \, \text{radians}.