 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ia/paper-2/5a/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Take

q_{1} (x) = 1, φ_{1} (x) = sin (x + 1) .

Its consecutive zeros are

- 1

and

π - 1

. For the stated equation,

q_{2} (x) = 2 + cos 3 x \geq 1 = q_{1} (x),

and

q_{2}

is not identically equal to

1

on the interval. The Sturm comparison theorem therefore shows that every nontrivial solution has at least one zero in

[- 1, π - 1] .

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