(i) Differentiate both equations: \(y = x \cos x\) gives \(\frac{dy}{dx} = \cos x - x \sin x\) and \(y = \frac{k}{x}\) gives \(\frac{dy}{dx} = -\frac{k}{x^2}\). Equate the derivatives at \(x = a\):

\(\cos a - a \sin a = -\frac{k}{a^2}\).

Since the curves touch, \(a \cos a = \frac{k}{a}\). Substitute \(k = a^2 \cos a\) into the derivative equation:

\(\cos a - a \sin a = -\frac{a^2 \cos a}{a^2} = -\cos a\).

Rearrange to get \(\tan a = \frac{2}{a}\).

(ii) Use the iterative formula \(a_{n+1} = \arctan \left( \frac{2}{a_n} \right)\) starting with an initial guess, say \(a_0 = 1\):

\(a_1 = \arctan \left( \frac{2}{1} \right) = 1.10715\)

\(a_2 = \arctan \left( \frac{2}{1.10715} \right) = 1.07989\)

\(a_3 = \arctan \left( \frac{2}{1.07989} \right) = 1.07735\)

\(a_4 = \arctan \left( \frac{2}{1.07735} \right) = 1.07708\)

\(a_5 = \arctan \left( \frac{2}{1.07708} \right) = 1.07703\)

Thus, \(a \approx 1.077\) to 3 decimal places.

(iii) Use \(a \cos a = \frac{k}{a}\) with \(a = 1.077\):

\(k = a^2 \cos a = (1.077)^2 \cos(1.077) \approx 0.55\).