\((i) Start with the equation xⁿy = C. Taking natural logarithms on both sides gives:\)

\(n \ln x + \ln y = \ln C\)

Substitute the given values:

\(For x = 1.10, y = 5.20:\)

\(n \ln(1.10) + \ln(5.20) = \ln C\)

\(For x = 3.20, y = 1.05:\)

\(n \ln(3.20) + \ln(1.05) = \ln C\)

Solving these two equations simultaneously, we find:

\(n = 1.50\)

Substitute back to find \(C\):

\(C = 6.00\)

(ii) The equation \(n \ln x + \ln y = \ln C\) is linear in \(\ln y\) and \(\ln x\), which means the graph of \(\ln y\) against \(\ln x\) is a straight line.