 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-3/31b/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For

k = 1

, the interval in part (b) contains one exactly when

a > 1

. With

V = x^{2} + y^{2} = r^{2}

\dot{V} = - 2 a r^{2} + 4 x y + 2 r^{4} \leq 2 r^{2} (r^{2} - a + 1),

because

2 x y \leq r^{2}

. Hence

\dot{V} < 0

throughout

0 < r^{2} < a - 1.

Every trajectory starting in the disc

x^{2} + y^{2} < a - 1

remains there and approaches the origin. Thus this disc is contained in the basin of attraction of the origin.

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