Higher June 2021 Paper 1 Q25
25 The diagram shows two circles such that the region R, shown shaded in the diagram, is the region common to both circles.
Diagram NOT accurately drawn
One of the circles has centre \(O\) and radius 5 cm.
The other circle has centre \(P\) and radius 4 cm.
Angle \(AOB = 50°\)
Calculate the area of region R.
Give your answer correct to 3 significant figures.
(6)
| Scheme | Marks |
|---|---|
[chord \(AB\) =] \(\sqrt{5^2 + 5^2 - 2 \times 5 \times 5 \times \cos 50}\) or \(2 \times 5 \times \sin 25\) (= 10sin25 or 4.226...) | M1 |
\([\angle APB =]\ \cos^{-1}\left(\dfrac{4^2 + 4^2 - \text{“4.226…”}^2}{2 \times 4 \times 4}\right) (= 63.77\ldots)\) or \([\angle OPA =]\ \sin^{-1}\left(\dfrac{0.5 \times \text{“4.226…”}}{4}\right) (= 31.88\ldots)\) | M1 |
| [Area sector \(AOB\) =] \(\dfrac{50}{360} \times \pi \times 5^2 \left(= \dfrac{125}{36}\pi \text{ or } 10.9\ldots\right)\) | M1 |
| [Area sector \(APB\) =] \(\dfrac{\text{“63.77…”}}{360} \times \pi \times 4^2\ (= 8.90\ldots)\) | M1 |
| \(\left(\dfrac{50}{360}\pi \times 5^2 - \dfrac{1}{2} \times 5^2 \times \sin 50\right) + \left(\dfrac{\text{“63.77…”}}{360} \times \pi \times 4^2 - \dfrac{1}{2} \times 4^2 \times \sin \text{“63.77…”}\right)\) | M1 |
| Working not required, so correct answer scores full marks (unless from obvious incorrect working) Answer: 3.06 | A1 |
| (6) | |
| (6 marks) |
Notes
M1: oe
M1: oe may use other methods but must be a complete method for \(\angle APB\) or \(\angle OPA\) (see below for sine rule)
M1: oe independent
M1: oe NB: 2 × “31.88…” = “63.77…”
M1: oe (10.9…– 9.57…) + (8.90… – 7.17…)
A1: allow 3 – 3.1
| Scheme | Marks |
|---|---|
| \([\angle OPA] = \sin^{-1}\left(\dfrac{5\sin 25}{4}\right) (= 31.88\ldots)\) | M1 |
| [Area sector \(APB\) =] \(\dfrac{2 \times \text{“31.88…”}}{360} \times \pi \times 4^2\ (= 8.90\ldots)\) | M1 |
| [Area \(OAPB\) =] 2 × ½ × 5 × 4 × sin(180 – “31.88..” – 25) (=16.75...) | M1 |
| [Area sector \(AOB\) =] \(\dfrac{50}{360} \times \pi \times 5^2 \left(= \dfrac{125}{36}\pi = 10.9\ldots\right)\) | M1 |
| [Area R =] “10.9...” + “8.90...” – “16.75...” | M1 |
| Working not required, so correct answer scores full marks (unless from obvious incorrect working) Answer: 3.06 | A1 |
Notes
M1: oe (see above for cosine rule & trig)
M1: oe
M1: oe
M1: oe independent
M1: oe
A1: allow 3 – 3.1