A-LevelMathematicsCircular measure9709/12 · Oct/Nov 2012 · Q1110 Marks

The diagram shows a sector of a circle with centre O and radius 20 cm. A circle with centre C and radius x cm lies within the sector and touches it at P, Q and R. Angle POR = 1.2 radians. [Figure 11]

(i)Show that x = 7.218, correct to 3 decimal places.[4]
(ii)Find the total area of the three parts of the sector lying outside the circle with centre C.[2]
(iii)Find the perimeter of the region OPSR bounded by the arc PSR and the lines OP and OR.[4]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2012). The mark or mark type is shown in brackets.

[B1] OQ = x + OC = 20 (used somewhere, needs "20".)
[M1] sin 0.6 = x/OC → OC = x/sin 0.6 (Use of trig in 90° triangle formed by O, C and midpoint of PR)
[M1] x + x/sin 0.6 = 20 (soln of linear equation for x)
[A1] → x = 7.218 (AG, ensure correct method)
[M1] Area = ½ r²θ - πx² = ½ 20² × 1.2 – π × 7.218² (Use of ½r²θ - needs r=20 and θ = 1.2)
[A1] = 76.3
[B1] Angle PCR = π – 1.2
[M1] Arc PR = x(π – 1.2) = 7.218(π – 1.2) (Use of s=rθ with r=7.218)
[M1] OP = OR = x/tan 0.6 (Correct use of trig)
[A1] → Perimeter of 35.1 cm

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