February/March 2026 Paper 32

By sketching a suitable pair of graphs, show that the equation $\ln x=\mathrm{cosec}\frac{1}{2}x$ has exactly one root in the interval $0<x<\pi$.

Verify by calculation that this root lies between 2.6 and 2.9.

Use the iterative formula $x_{n+1}=\exp \left(\mathrm{cosec}\frac{1}{2}{x}_n\right)$ to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

[$\exp (x)$ is an alternative notation for ${\mathrm{e}}^x$.]