A-LevelMathematicsNumerical methods9709/32 · May/June 2018 · Q68 Marks

The diagram shows a triangle ABC in which AB = AC = a and angle BAC = θ radians. Semicircles are drawn outside the triangle with AB and AC as diameters. A circular arc with centre A joins B and C. The area of the shaded segment is equal to the sum of the areas of the semicircles. [Figure X.X]

(i)Show that θ = 1/2 π + sin θ.[3]
(ii)Verify by calculation that θ lies between 2.2 and 2.4.[2]
(iii)Use an iterative formula based on the equation in part (i) to determine θ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[3]

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Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

[M1] Use correct method for finding the area of a segment and area of semicircle and form an equation in θ
[A1] State a correct equation in any form
[A1] Obtain the given answer correctly
[M1] Calculate values of a relevant expression or pair of expressions at θ = 2.2 and θ = 2.4
[A1] Complete the argument correctly with correct calculated values
[M1] Use θ_(n+1) = 1/2 π + sin θ_n correctly at least once
[A1] Obtain final answer 2.31
[A1] Show sufficient iterations to 4 d.p. to justify 2.31 to 2 d.p. or show there is a sign change in the interval (2.305, 2.315)

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