Higher June 2017 Paper 2 Q24
24. A thin piece of card, which is a sector of a circle with centre \(O\), is shown below.
Diagram not drawn to scale
(a) Find an expression for the length of the arc \(DE\).
Give your answer, in terms of \(x\) and \(\pi\), in its simplest form. [2]
Give your answer, in terms of \(x\) and \(\pi\), in its simplest form. [2]
(b) The thin card is made into a cone by sticking edges \(OD\) and \(OE\) together without overlapping.
Diagram not drawn to scale
Show that the expression for the perpendicular height, \(h\), of the cone in terms of \(x\) is given by \(\dfrac{3\sqrt{7}x}{8}\). [5]| Scheme | Marks | Comments |
|---|---|---|
| (\(DE =\)) \(\tfrac14\pi x\) | B2 | B1 for \(\pi \times 2 \times x \times 45/360\) or equivalent unsimplified expression |
| Scheme | Marks | Comments |
|---|---|---|
| (Cone radius =) \(x/8\) or equivalent | B2 | FT ‘their derived DE’ B1 for \(\tfrac14\pi x = 2 \times \pi \times\) cone radius, or \(\tfrac14\pi x = \pi \times\) diameter |
| perpendicular height\(^2 = x^2 - (x/8)^2\) | M1 | FT ‘their derived cone radius’ provided it is in terms of \(x\) and equivalent level of difficulty and at least B1 previously awarded |
| \(= (64x^2 - x^2)/64\) or \(63x^2/64\) | A1 | FT correct stage towards simplifying as a single term |
| Perpendicular height = \(\dfrac{3\sqrt{7}x}{8}\) cm | A1 | CAO. Must be convincing from correct working, e.g. with sight of \(63 = 3^2 \times 7\) or \(9 \times 7\) or \(63x^2/64\) |
| (7) |