4400 Higher June 2008 Paper 3 Q20
20
Diagram NOT accurately drawn
\(P\), \(Q\) and \(R\) are points on a circle, centre \(O\).
Diagram NOT accurately drawn
\(A\), \(B\) and \(C\) are points on a circle, centre \(O\).\(BCD\) is a straight line.
Find the value of \(x\). (5)
| Scheme | Marks |
|---|---|
| (i) Answer: 59 | B1 |
| (ii) Answer: angle at the centre = twice angle at the circumference or angle at the circumference = half the angle at the centre | B1 |
| (2) |
Notes
B1: cao
B1: Three key points must be mentioned
1. angle at centre/middle/\(O\)/origin
2. twice/double/2× or half/\(\frac{1}{2}\) as appropriate
3. angle at circumference/edge/perimeter (NOT e.g. angle \(R\), angle \(PRQ\), angle at top, angle at outside)
| Scheme | Marks |
|---|---|
| \(180 - (x + 36)\) oe seen (possibly marked on diagram as size of \(\angle ACB\)) | B1 |
\(x = 2(180 - (x + 36))\) or \(x = 2(180 - x - 36)\) or \(180 - (x + 36) = \dfrac{x}{2}\) or \(180 - x - 36 = \frac{1}{2}x\) | M1 |
| \(x = 360 - 2x - 72\) or \(x + \frac{1}{2}x = 180 - 36\) | M1 |
| \(3x = 360 - 72\) or \(3x = 288\) or \(\frac{3}{2}x = 180 - 36\) or \(\frac{3}{2}x = 144\) | M1 |
| 96 | A1 |
| (5) | |
| (7 marks) |
Notes
B1: for \(180 - (x + 36)\) oe seen, either on its own or as part of an equation
(This mark may still be scored, even if brackets are later removed incorrectly.)
A1: cao
SC
(Max of 2 M marks) for omission of brackets in \(-(x + 36)\) or their incorrect removal:
\(x = 2(180 - x + 36)\) or \(180 - x + 36 = \frac{1}{2}x\) or \(180 - 36 + x = \frac{1}{2}x\) M1
\(x = 360 - 2x + 72\) or \(x + \frac{1}{2}x = 180 + 36\) M1
(Note - incorrect simplification results in an answer of \(x = 144\))
Alternative
\(\dfrac{x}{2}\) oe seen (possibly marked on diagram as size of \(\angle ACB\)) B1
\(x + 36 + \dfrac{x}{2} = 180\) M1
The rest of this alternative method is cut off in the source.