Higher June 2024 Paper 5 Q20
20 The shape below is part of a circle, centre O and radius 12 cm.
Angle OAB = 30°.
Not to scale
Work out the perimeter of the shape.
Give your answer in its simplest terms in the form \(a\sqrt{b} + k\pi\).
You must show your working. [7]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(12\sqrt{3} + 16\pi\) final answer with correct working | 7 | Correct working requires evidence of at least M1M1M2 | |
| M1 for angle of major sector = 240° or \(\frac{240}{360}\) oe soi | For M1 accept e.g. \(\frac{2}{3}\) soi 240 may be on diagram | ||
| M1 for \(\frac{\textit{their } 240}{360} \times 2 \times \pi \times 12\) A1 for \(16\pi\) AND | For M1 \(120 \leqslant \textit{their } 240 \leqslant 300\) but not 180 \(24\pi \div 3 \times 2\) oe eg done in stages implies M1M1 | ||
| M2 for [ 2 ×] 12 cos30° oe | M2 oe accept e.g. [\(\text{AB}^2 =\) ] \(12^2 + 12^2 - 2.12.12.\cos 120\) or \(\dfrac{12\sin 120}{\sin 30}\), [2×] \(\sqrt{12^2 - (12\sin 30)^2}\) oe | ||
| or M1 for \(\frac{x}{12} = \cos 30\) oe | where \(x\) is ½ AB M1 for any correct implicit method for finding AB or ½ AB | ||
| B1 for \(\cos 30 = \frac{\sqrt{3}}{2}\) soi A1 for \(12\sqrt{3}\) | In this method and other methods used, B1 is awarded for the correct trig value[s] associated with their method for find [½]AB, even if their method is incorrect, not just seen in a table of trig values unless selected Award maximum of 6 marks if answer incorrect | ||
| If 0, 1 or 2, scored instead award SC3 for answer \(12\sqrt{3} + 16\pi\) If 0 or 1 scored, instead award SC2 for \(12\sqrt{3}\) or \(16\pi\) in answer | SEE AG | ||
Additional guidance: Question 20
The length AB or \(\frac{1}{2}\) AB (\(12\sqrt{3}\) oe or \(6\sqrt{3}\) oe may be seen unsimplified e.g. \(2\sqrt{108}\) or \(\sqrt{108}\)) can be obtained by a number of different methods.
M2 is for the explicit method leading to AB or \(\frac{1}{2}\)AB and examples of this are in the guidance column of the scheme
M1 is for a correct implicit method
The B mark is awarded for a correct trig value[s] for their chosen method
Implicit methods may include
| \(\text{AB}^2 + (12\sin 30)^2 = 12^2\) oe or better | \(\cos 120 = \dfrac{12^2 + 12^2 - \text{AB}^2}{2 \times 12 \times 12}\) oe or better | \(\dfrac{\sin 120}{AB} = \dfrac{\sin 30}{12}\) oe or better |
| Award B1 for \(\sin 30 = \frac{1}{2}\) soi | Award B1 for \(\cos 120 = -\frac{1}{2}\) soi | Award B1 for \(\sin 120 = \frac{\sqrt{3}}{2}\) and \(\sin 30 = \frac{1}{2}\) soi |