a) (i) For \(\lambda = -1\), eigenvector \(\binom{3}{1}\): motion toward origin along \(y = \frac{1}{3}x\).
For \(\lambda = 2\), eigenvector \(\binom{1}{3}\): motion away from origin along \(y = 3x\). [2]

(ii) General solution: \( \begin{pmatrix} x(t) \\ y(t) \end{pmatrix} = c_1 e^{-t} \begin{pmatrix} 3 \\ 1 \end{pmatrix} + c_2 e^{2t} \begin{pmatrix} 1 \\ 3 \end{pmatrix} \).
Region I (\(y > 3x\)): Trajectory from third quadrant, crosses first quadrant, diverges along \(y = 3x\).
Region II (\(3x > y > \frac{1}{3}x\)): Trajectory approaches origin along \(y = \frac{1}{3}x\), diverges along \(y = 3x\).
Region III (\(y < \frac{1}{3}x\)): Trajectory approaches origin along \(y = \frac{1}{3}x\), diverges to third quadrant along \(y = 3x\).
[3]

b) Initial: \( (252, 60 + \delta) \). Solve: \( 3c_1 + c_2 = 252 \), \( c_1 + 3c_2 = 60 + \delta \).
Gives: \( c_1 = \frac{696 – \delta}{8} \), \( c_2 = \frac{3\delta – 72}{8} \).
\( y(t) = \frac{696 – \delta}{8} e^{-t} + \frac{9\delta – 216}{8} e^{2t} \).
For \( y(t) \geq 0 \) as \( t \to \infty \), need \( 9\delta – 216 > 0 \implies \delta > 24 \). Minimum integer: \(\delta = 25\). [4]