 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-1/3/a/solution

At a point of tangency, the derivative of the curve equals the slope one of the straight line. The chain rule gives

2 cos 2 x = 1

. Since

0 \leq 2 x \leq π

, the unique solution is

x = π /6

. The corresponding height is

sin (π /3) = 3 /2

. Substitution in the tangent line gives

c = \frac{3}{2} - \frac{π}{6} .

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