For the four-cycle, the quadratic form is
$$
q(x)=\sum_{j=1}^4(x_j-x_{j+1})^2,
\qquad x_5=x_1.
$$
It is nonnegative, but it vanishes on the nonzero vector $(1,1,1,1)$. Equivalently, the Gram eigenvalues are $0,2,2,4$. The form is positive semidefinite, not positive definite, and degenerate: the three answers are no, yes, and no. This is the affine Coxeter graph $\widetilde A_3$.

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Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 111 / 2 / a / 3 / Solution

 Solution

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