(a) To find \(a\), consider the term in \(x^4\) from \(\left( 2x^2 + \frac{k^2}{x} \right)^5\). The term is given by:

\([10 \times x](2x^2)^3 \left( \frac{k^2}{x} \right)^2\)

\(80k^4 x^4 \Rightarrow a = 80k^4\).

To find \(b\), consider the term in \(x^2\) from \((2kx - 1)^4\). The term is given by:

\([6 \times 1](2kx)^2 \times 1 = 24k^2 x^2 \Rightarrow b = 24k^2\)

(b) Given \(a + b = 216\), substitute the expressions for \(a\) and \(b\):

\(80k^4 + 24k^2 = 216\)

\(10k^4 + 3k^2 - 27 = 0\)

Factorizing gives:

\((2k^2 - 3)(5k^2 + 9) = 0\)

\(k^2 = \frac{3}{2}\) or \(k^2 = -\frac{9}{5}\)

Since \(k^2\) must be non-negative, \(k^2 = \frac{3}{2}\).

Thus, \(k = \pm \frac{\sqrt{3}}{2}\).