 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ia/paper-1/11e/a/solution

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

Choose

z_{+}, z_{-} \in C

with

f (z_{+}) > 0, f (z_{-}) < 0.

Restrict

f

to the line segment between them:

h (t) = f ((1 - t) z_{-} + t z_{+}), 0 \leq t \leq 1.

This is continuous, with

h (0) < 0 < h (1)

t_{0}

with

h (t_{0}) = 0

. Therefore

z = (1 - t_{0}) z_{-} + t_{0} z_{+}

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