 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/10/solution

(a, a + 1) \cdot (a + 1, a) = 2 a (a + 1) = 2 a^{2} + 2 a .

a^{2} + (a + 1)^{2} = 2 a^{2} + 2 a + 1

. Under the given condition

a^{2} + a = 1/2

, their dot product is

1

and each squared Euclidean norm is

2

. Hence

cos θ = \frac{1}{2 2} = \frac{1}{2}, θ = π /3 = 6 0^{\circ} .

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