| Date | May 2011 | Marks available | 2 | Reference code | 11M.1.hl.TZ2.7 |
| Level | HL only | Paper | 1 | Time zone | TZ2 |
| Command term | Show that | Question number | 7 | Adapted from | N/A |
Question
The diagram shows a tangent, (TP) , to the circle with centre O and radius r . The size of \({\rm{P\hat OA}}\) is \(\theta \) radians.
Find the area of triangle AOP in terms of r and \(\theta \) .
Find the area of triangle POT in terms of r and \(\theta \) .
Using your results from part (a) and part (b), show that \(\sin \theta < \theta < \tan \theta \) .
Markscheme
area of \({\text{AOP}} = \frac{1}{2}{r^2}\sin \theta \) A1
[1 mark]
\({\text{TP}} = r\tan \theta \) (M1)
area of POT \( = \frac{1}{2}r(r\tan \theta )\)
\( = \frac{1}{2}{r^2}\tan \theta \) A1
[2 marks]
area of sector OAP \( = \frac{1}{2}{r^2}\theta \) A1
area of triangle OAP < area of sector OAP < area of triangle POT R1
\(\frac{1}{2}{r^2}\sin \theta < \frac{1}{2}{r^2}\theta < \frac{1}{2}{r^2}\tan \theta \)
\(\sin \theta < \theta < \tan \theta \) AG
[2 marks]
Examiners report
The majority of candidates were able to find the area of Triangle AOP correctly. Most were then able to get an expression for the other triangle. In the final section, few saw the connection between the area of the sector and the relationship.
The majority of candidates were able to find the area of Triangle AOP correctly. Most were then able to get an expression for the other triangle. In the final section, few saw the connection between the area of the sector and the relationship.
The majority of candidates were able to find the area of Triangle AOP correctly. Most were then able to get an expression for the other triangle. In the final section, few saw the connection between the area of the sector and the relationship.
