 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-i/2/c/i/solution

The first two equations give

x_{1} = 1 - 2 x_{3}

and

x_{2} = 2 - 5 x_{3}

. Substitution into the third gives

(λ - 5) x_{3} = μ - 2.

This equation is inconsistent exactly when its left side is identically zero and its right side is nonzero. Therefore

no solution ⟺ λ = 5, μ \neq = 2.

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