| Date | May 2011 | Marks available | 3 | Reference code | 11M.1.hl.TZ2.4 |
| Level | HL only | Paper | 1 | Time zone | TZ2 |
| Command term | Calculate | Question number | 4 | Adapted from | N/A |
Question
The complex numbers \({z_1} = 2 - 2{\text{i}}\) and \({{\text{z}}_2} = 1 - \sqrt 3 {\text{i}}\) are represented by the points A and B respectively on an Argand diagram. Given that O is the origin,
Find AB, giving your answer in the form \(a\sqrt {b - \sqrt 3 } \) , where a , \(b \in {\mathbb{Z}^ + }\) .
Calculate \({\rm{A\hat OB}}\) in terms of \(\pi \).
Markscheme
\({\text{AB}} = \sqrt {{1^2} + {{(2 - \sqrt 3 )}^2}} \) M1
\( = \sqrt {8 - 4\sqrt 3 } \) A1
\( = 2\sqrt {2 - \sqrt 3 } \) A1
[3 marks]
METHOD 1
\(\arg {z_1} = - \frac{\pi }{4}{\text{ }}\arg {z_2} = - \frac{\pi }{3}\) A1A1
Note: Allow \(\frac{\pi }{4}\) and \(\frac{\pi }{3}\) .
Note: Allow degrees at this stage.
\({\rm{A\hat OB}} = \frac{\pi }{3} - \frac{\pi }{4}\)
\( = \frac{\pi }{{12}}{\text{ (accept }} - \frac{\pi }{{12}})\) A1
Note: Allow FT for final A1.
METHOD 2
attempt to use scalar product or cosine rule M1
\(\cos {\rm{A\hat OB}} = \frac{{1 + \sqrt 3 }}{{2\sqrt 2 }}\) A1
\({\rm{A\hat OB}} = \frac{\pi }{{12}}\) A1
[3 marks]
Examiners report
It was disappointing to note the lack of diagram in many solutions. Most importantly the lack of understanding of the notation AB was apparent. Teachers need to make sure that students are aware of correct notation as given in the outline. A number used the cosine rule but then confused the required angle or sides.
It was disappointing to note the lack of diagram in many solutions. Most importantly the lack of understanding of the notation AB was apparent. Teachers need to make sure that students are aware of correct notation as given in the outline. A number used the cosine rule but then confused the required angle or sides.
