(i) Integrate by parts to reach \(ax^{\frac{3}{2}} \ln x + b \int x^{\frac{3}{2}} \frac{1}{x} \, dx\).

Obtain \(\frac{2}{3} x^{\frac{3}{2}} \ln x - \frac{2}{3} \int x^{\frac{1}{2}} \, dx\).

Obtain integral \(\frac{2}{3} x^{\frac{3}{2}} \ln x - \frac{4}{9} x^{\frac{3}{2}}\).

Substitute limits correctly and equate to 2 to obtain the given answer.

(ii) Evaluate the expression at \(x = 2\) and \(x = 4\) to show \(a\) lies between these values.

(iii) Use the iterative formula:

1. Start with an initial guess, say \(a_1 = 3\).

2. Calculate \(a_2 = \left( \frac{7 + 2 \times 3^{\frac{3}{2}}}{3 \ln 3} \right)^{\frac{2}{3}}\).

3. Continue iterations until the value stabilizes to 5 decimal places.

4. Final answer is \(a = 3.031\) to 3 decimal places.