 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-2/3/a/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

A homogeneous linear system has a nonzero solution exactly when its coefficient matrix is singular. Expansion of the determinant gives

det M = 5 α^{2} + 3 α - 8 = (α - 1) (5 α + 8) .

Thus the allowed values are

α = 1 or - 8/5

. Solving the first two equations and checking the third gives

α = 1 : x = \frac{z}{5}, y = - \frac{7 z}{5}; α = - \frac{8}{5} : x = - 5 z, y = \frac{5 z}{2} .

In each case

z \neq = 0

gives a nonzero solution and spans the one-dimensional null space;

z = 0

gives only the zero vector.

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