A-LevelMathematicsNumerical methods9709/32 · Oct/Nov 2022 · Q99 Marks

The diagram shows a semicircle with diameter AB, centre O and radius r. The shaded region is the minor segment on the chord AC and its area is one third of the area of the semicircle. The angle CAB is θ radians. [Figure 9.1]

(a)Show that θ = (⅓)(π – 1.5 sin 2θ).[4]
(b)Verify by calculation that 0.5 < θ < 0.7.[2]
(c)Use an iterative formula based on the equation in part (a) to determine θ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.[3]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[B1] (a) State or imply angle AOC = π−2θ
[M1] (a) Use correct formulae for the area of a sector and triangle, or of a segment, and find the area of the shaded region
[A1] (a) Equate to (1/6)πr² and obtain a correct equation in any form
[A1] (a) Obtain θ = (1/3)(π – 1.5sin2θ) correctly
[M1] (b) Evaluate a relevant expression or pair of expressions at θ=0.5 and θ=0.7
[A1] (b) Complete the argument correctly with correct calculated values
[M1] (c) Use the iterative process θn+1 = (1/3)(π – 1.5sin2θn) correctly at least once
[A1] (c) Obtain final answer 0.586
[A1] (c) Show sufficient iterations to 5 d.p. to justify 0.586 to 3 d.p., or show there is a sign change in the interval (0.5855, 0.5865).

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