 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-62/3/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

Use the standard form

C z \leq d

with unrestricted

z = (x, y)^{T}

C = - 1 11 0 - 1 1, d = 01 - 2 .

The nonnegative multiplier

w = 211

satisfies

C^{T} w = 0

and

d^{T} w = - 1 < 0

. This is a Farkas certificate for linear inequalities, so the system is infeasible by Farkas lemma. In scalar form, adding twice the first inequality to the other two produces

0 \leq - 1,

which directly exhibits the contradiction.

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