(a) [Figure 1]
In Fig. 1, OAB is a sector of a circle with centre O and radius r. AX is the tangent at A to the arc AB and angle BAX = α.
(b) [Figure 2]
In Fig. 2, ABC is an equilateral triangle of side 4 cm. The lines AX, BX and CX are tangents to the equal circular arcs AB, BC and CA. Use the results in part (a) to find the area of the shaded region, giving your answer in terms of π and √3.

Q: What topic does this question test?

This question tests Circular measure in A-Level Mathematics (syllabus code 9709). It is worth 10 marks.

Q: Which exam paper is this question from?

This question appeared in the Cambridge A-Level Mathematics Feb/Mar 2016 examination, Paper 1 Variant 2 (9709_m16_qp_12.pdf).

Question

(a) [Figure 1]
In Fig. 1, OAB is a sector of a circle with centre O and radius r. AX is the tangent at A to the arc AB and angle BAX = α.
(b) [Figure 2]
In Fig. 2, ABC is an equilateral triangle of side 4 cm. The lines AX, BX and CX are tangents to the equal circular arcs AB, BC and CA. Use the results in part (a) to find the area of the shaded region, giving your answer in terms of π and √3.

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