Higher June 2023 Paper 1 Q22
22 The diagram shows two circles with centre \(O\) and a regular pentagon \(ABCDE\)
Diagram NOT accurately drawn
\(A\), \(B\), \(C\), \(D\) and \(E\) are points on the larger circle.
The pentagon has sides of length 8 cm.
The diagram is shaded such that
shaded area = unshaded area
Work out the radius of the smaller circle.
Give your answer correct to 3 significant figures.
(6)
| Scheme | Marks |
|---|---|
(radius of large circle =) \(\dfrac{4}{\cos 54}\) or \(\dfrac{4}{\sin 36}\) or \(\dfrac{8\sin 54}{\sin 72}\) or \(\sqrt{\dfrac{8^2}{2 - 2\cos 72}}\) (= 6.805...) or (height of 1 triangle within pentagon =) 4tan54 (=5.505...) oe | M1 |
(area of large circle =) \(\pi \times \left(\text{``}{6.805...}\text{''}\right)^2\) (= 145.489...) oe or (area of sector =) \(\dfrac{72}{360} \times \pi \times \left(\text{``}{6.805...}\text{''}\right)^2\) (= 29.097...) oe | M1 |
| (area of pentagon =) 5 × ½ × 8 × “5.505...” (= 80tan54 = 110.11...) or 10 × ½ × 4 × “5.505...” (= 80tan54 = 110.11...) or 5 × ½ × “6.805...” × “6.805...” × sin72 (= 110.11...) oe OR (area of one triangle =) ½ × 8 × “5.505...” (= 22.022...) or ½ × “6.805...” × “6.805...” × sin72 (= 22.022...) or ½ × “6.805...” × 8 × sin54 (= 22.022...) oe | M1 |
\(\text{``}{145.489...}\text{''} - \text{``}{110.11...}\text{''} + \pi r^2 = \text{``}{110.11...}\text{''} - \pi r^2\) oe or \(5 \times \left(\text{``}{29.097...}\text{''} - \text{``}{22.022...}\text{''}\right) + \pi r^2 = 5 \times \text{``}{22.022...}\text{''} - \pi r^2\) oe | M1 |
| \(2\pi r^2 = 2 \times \text{``}{110.11...}\text{''} - \text{``}{145.489...}\text{''}\) (= 74.731...) oe | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: 3.45 | A1 |
| (6) | |
| (6 marks) |
Notes
M1: for a complete method to find the radius of the large circle or the perpendicular height of one triangle within the pentagon
M1: for a complete method to find the area of the large circle or the area of a sector of the large circle
M1: for a complete method to find the area of the pentagon OR the area of one triangle eg \(OED\) or equivalent
M1: for a correct equation for the radius of the smaller circle
M1: for a correct rearranged equation with the area of the circle the subject or better
A1: accept 3.43 – 3.45