4400 Higher November 2010 Paper 4 Q19
19
Diagram NOT accurately drawn
\(AB\) is a diameter of a circle.
\(CD\) is a chord of the circle.
\(AB\) and \(CD\) intersect at \(E\).
\(BE = 4\) cm, \(CE = 16\) cm and \(DE = 5\) cm.
(a) Calculate the length of \(AE\). (2)
(b)
(i) Find the radius of the circle.
(ii) Calculate the size of angle \(AED\).
Give your answer correct to 1 decimal place.
(5)Give your answer correct to 1 decimal place.
| Scheme | Marks |
|---|---|
| \(AE \times 4 = 16 \times 5\) | M1 |
| 20 | A1 |
| (2) |
Notes
A1: cao
| Scheme | Marks |
|---|---|
| (i) Answer: 12 | B1 |
(ii) \((\cos x^\circ =)\ \dfrac{5^2 + 8^2 - 12^2}{2 \times 8 \times 5}\) or \(\dfrac{5^2 + OE^2 - \text{“}12\text{”}^2}{2 \times OE \times 5}\) \((\cos \angle OEC =)\ \dfrac{16^2 + 8^2 - 12^2}{2 \times 16 \times 8}\) or \(\dfrac{16^2 + OE^2 - \text{“}12\text{”}^2}{2 \times 16 \times OE}\) or, using the midpoint of \(CD\), \(\cos \angle OEC = \dfrac{5.5}{8}\) or \(\dfrac{5.5}{OE}\) or complete, correct method of finding \(\sin \angle OEC\) or \(\tan \angle OEC\) | M2 |
| 133.4 | A2 |
| (5) | |
| (7 marks) |
Notes
B1: cao
M2: M1 for
\(12^2 = 5^2 + 8^2 - 2 \times 8 \times 5 \cos x^\circ\)
or \(\text{“}12\text{”}^2 = 5^2 + OE^2 - 2 \times OE \times 5 \cos x^\circ\)
or \(12^2 = 16^2 + 8^2 - 2 \times 16 \times 8 \times \cos \angle OEC\)
or \(\text{“}12\text{”}^2 = 16^2 + OE^2 - 2 \times 16 \times OE \times \cos \angle OEC\)
A2: for answer rounding to 133.4 (133.4325...)
A1 for \(\dfrac{-55}{80}\) oe or −0.6875
If \(\angle OEC\) is used, award A1 for \(\dfrac{176}{256}\) oe or 0.6875 or value rounding to 46.6 seen.
If midpoint of \(CD\) is used, award A1 for \(\dfrac{5.5}{8}\) oe or 0.6875 or value rounding to 46.6 seen.