O-LevelAdditional MathematicsCircular measure4037/22 · Oct/Nov 2014 · Q1110 Marks

The diagram shows a sector OPQ of a circle with centre O and radius x cm. Angle POQ is 0.8 radians. The point S lies on OQ such that OS = 5cm. The point R lies on OP such that angle ORS is a right angle. Given that the area of triangle ORS is one-fifth of the area of sector OPQ, find [Figure 11.1]

(i)the area of sector OPQ in terms of x and hence show that the value of x is 8.837 correct to 4 significant figures,[5]
(ii)the perimeter of PQSR,[3]
(iii)the area of PQSR.[2]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2014). The mark or mark type is shown in brackets.

[B1] (i) Area of sector = 1/2 x² × 0.8 (= 0.4x² cm²). anywhere
[B1] (i) SR = 5 sin 0.8 (=3.59) or OR = 5 cos 0.8 (= 3.48). SR may be seen in stated 1/2ab sin C
[M1] (i) Area of triangle = 1/2 5cos 0.8×5 sin 0.8 = 6.247 cm². insert correct terms into correct area formulae
[A1] (i) 0.08x² = 6.247
[A1] (i) x = 8.837 cm AG
[B1] (ii) SQ=8.84-5(=3.84cm). PR = 8.84-5cos 0.8 (= 5.35 or 5.36cm). two lengths from SQ, PR, PQ awrt
[B1] (ii) PQ = 8.84×0.8 (= 7.07 cm). third length awrt
[B1] (ii) Perimeter = 19.84 to 19.86 cm or rounded to 19.8 or 19.9. sum
[M1] (iii) Area PQSR = 4 × 6.247
[A1] (iii) = 25 cm². 24.95 to 25

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.