O-LevelAdditional MathematicsCircular measure4037/12 · Oct/Nov 2016 · Q89 Marks

The diagram shows a sector AOB of the circle, centre O, radius 12cm, together with points C and D such that ABCD is a rectangle. The angle AOB is θ radians and the perimeter of the sector AOB is 47 cm. [Figure 8.1]

(i)Show that θ = 1.92 radians correct to 2 decimal places.[2]
(ii)Find the length of CD.[2]
(iii)Given that the total area of the shape is 425 cm², find the length of AD.[5]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2016). The mark or mark type is shown in brackets.

[1] (i) 47 - 24 = 12θ M1 for complete correct method to get θ =
[1] (i) θ = 23/12 so θ=1.917 or better A1 must have evidence of working to more than 2 dp, allow if 1.916 seen (truncated) θ=1.92 to 2dp
[1] (ii) sin(θ/2) = (CD/2)/12 M1 for a complete method, may use cosine rule to get CD
[1] (ii) CD = awrt 19.6 or 19.7 A1
[1] (iii) Area of sector = awrt 138 B1 for sector area, allow unsimplified
[1] (iii) Area of triangle AOB = awrt 67 or 68 M1 for a correct attempt at area
[1] (iii) Area of segment = awrt 70 or 71 M1 for segment area (their sector area - their triangle area)
[1] (iii) AD × AB + segment area = 425 M1 for complete method to find AD
[1] (iii) leading to AD = awrt 18.1 or 18.0 A1 Allow A1 for 18

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