(i) To find the values of \(k\) for which the line and curve meet at two distinct points, equate the equations:

\(3kx - 2k = x^2 - kx + 2\)

Rearrange to form a quadratic equation:

\(x^2 - 4kx + 2k + 2 = 0\)

For two distinct points, the discriminant \(b^2 - 4ac > 0\).

Here, \(a = 1\), \(b = -4k\), \(c = 2k + 2\).

Calculate the discriminant:

\((-4k)^2 - 4 \times 1 \times (2k + 2) > 0\)

\(16k^2 - 8k - 8 > 0\)

Factorize or solve the inequality:

\((k - 1)(k + \frac{1}{2}) > 0\)

Thus, \(k > 1\) or \(k < -\frac{1}{2}\).

(ii) For the line to be tangent to the curve, the discriminant must be zero:

\(16k^2 - 8k - 8 = 0\)

Solving gives \(k = 1\) and \(k = -\frac{1}{2}\).

For \(k = 1\), the line is \(y = 3x - 2\) and the curve is \(y = \frac{3}{2}x + 1\).

Equate the tangents:

\(3x - 2 = \frac{3}{2}x + 1\)

Solve for \(x\):

\(x = \frac{2}{3}\)

Thus, the tangents meet on the x-axis at \(x = \frac{2}{3}\).