May/June 2026 Paper 32

By sketching a suitable pair of graphs, show that the equation $\mathrm{cosec}\frac{1}{2}x=\frac{1}{2}(x+3)$ has exactly one root in the interval $0\le x\le \pi$.

Show by calculation that this root lies between $1$ and $1.2$.

Show that, if a sequence of real values given by the iterative formula converges, then it converges to the root of the equation in part (a).

Use the iterative formula in part (c) to calculate this root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.