4400 Higher June 2007 Paper 4 Q19

EdexcelHigherOld spec5 marksAdvanced Trigonometry Circles & Sectors

(a) Calculate the area of an equilateral triangle of side 5 cm.
Give your answer correct to 3 significant figures.

Diagram NOT accurately drawn

(2)

(b) The diagram shows two overlapping circles.
The centre of each circle lies on the circumference of the other circle.
The radius of each circle is 5 cm.
The distance between the centres is 5 cm.

Two overlapping circles of radius 5 cm whose centres are 5 cm apart; the equilateral triangle formed by the two centres and an intersection point, together with the two circular segments on its slanted sides, is shaded

Diagram NOT accurately drawn

Calculate the area of the shaded region.
Give your answer correct to 3 significant figures. (3)

Mark scheme (a)

Scheme	Marks
\(\frac{1}{2} \times 5 \times 5 \times \sin 60\)	M1
10.8...	A1
	(2)

Notes

M1: \(\frac{1}{2} \times 5 \times \sqrt{5^2 - (\frac{5}{2})^2}\) or \(\frac{1}{2} \times 5 \times 4.33\)

A1: \((25\sqrt{3})/4\) M1A0

Mark scheme (b)

Scheme	Marks
sect \(= \frac{1}{6} \times \pi \times 5^2\) or 13.1	M1
“10.8” \(+ 2(\frac{1}{6} \times \pi \times 5^2 - \text{“}10.8\text{”})\) or “10.8” + 2×2.26 or \(2 \times \frac{1}{6} \times \pi \times 5^2 - \text{“}10.8\text{”}\)	M1
15.4 cm²	A1
	(3)
	(5 marks)

Notes

M1: \(\triangle + 2(\text{sect} - \triangle)\) or \(2 \times \text{sect} - \triangle\) Allow eg \(\triangle = \frac{1}{2} \times 5 \times 5\)