 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-1/14/b/solution

For

h = π /6

E [X] = \int_{- h}^{h} x (a x + cos x) d x = a \int_{- h}^{h} x^{2} d x = \frac{2 a h ^{3}}{3},

because

x cos x

is an odd function. Comparing with the specified expected value gives

a = 1

, which lies in the permitted parameter range.

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