6 (a) By sketching a suitable pair of graphs, show that the equation cot x = 2 − cos x has one root in the interval 0 < x < ½π. (b) Show by calculation that this root lies between 0.6 and 0.8. (c) Use the iterative formula xn+1 = tan⁻¹(1 / (2 − cos xn)) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. B1 Sketch a relevant graph. e.g. y = cot x: x intercept should be correct. Not touching the y-axis. No incorrect curvature. Ignore anything outside 0 < x ≤ ½π.
2. B1 Sketch a second relevant graph and justify the given statement e.g. y = 2-cosx : Condone if looks almost straight, but not if drawn with a ruler and not incorrect curvature. Correct y intercept. Needs to be drawn for 0 < x ≤ ½π. Ignore outside this. 2nd B1 requires a mark at the point of intersection or a suitable comment for the justification.
3. M1 Calculate the value of a relevant expression or values of a pair of expressions at x = 0.6 and x = 0.8 Must be working in radians. Values correct to at least 2 significant figures. Need all relevant values but only one (pair) needs to be correct to award M1. Complete set of values for their expression. If not comparing with 0 or 1 then the pairing must be clear, not just embedded values.
4. A1 Complete the argument correctly with correct calculated values (awrt 2 s.f.). Clear comparison for their expression. Allow work on a smaller interval.
5. M1 Use the iterative process correctly at least once. Must be working in radians
6. A1 Show sufficient iterations to at least 4 d.p. to justify 0.68 to 2 d.p. or show there is a sign change in the interval (0.675, 0.685). Allow recovery. Allow truncation. Allow small differences in the 4th s.f.
7. A1 Obtain final answer 0.68 Must be a clear conclusion.