| Date | May 2014 | Marks available | 7 | Reference code | 14M.1.hl.TZ2.7 |
| Level | HL only | Paper | 1 | Time zone | TZ2 |
| Command term | Express and Find | Question number | 7 | Adapted from | N/A |
Question
Consider the complex numbers \(u = 2 + 3{\text{i}}\) and \(v = 3 + 2{\text{i}}\).
(a) Given that \(\frac{1}{u} + \frac{{1}}{v} = \frac{{10}}{w}\), express w in the form \(a + b{\text{i, }}a,{\text{ }}b \in \mathbb{R}\).
(b) Find \(w\)* and express it in the form \(r{e^{{\text{i}}\theta }}\).
Markscheme
(a) METHOD 1
\(\frac{1}{{2 + 3{\text{i}}}} + \frac{1}{{3 + 2{\text{i}}}} = \frac{{2 - 3{\text{i}}}}{{4 + 9}} + \frac{{3 - 2{\text{i}}}}{{9 + 4}}\) M1A1
\(\frac{{10}}{w} = \frac{{5 - 5{\text{i}}}}{{13}}\) A1
\(w = \frac{{130}}{{5 - 5{\text{i}}}}\)
\( = \frac{{130 \times 5 \times (1 + {\text{i}})}}{{50}}\)
\(w = 13 + 13{\text{i}}\) A1
[4 marks]
METHOD 2
\(\frac{1}{{2 + 3{\text{i}}}} + \frac{1}{{3 + 2{\text{i}}}} = \frac{{3 + 2{\text{i}} + 2 + 3{\text{i}}}}{{(2 + 3{\text{i}})(3 + 2{\text{i}})}}\) M1A1
\(\frac{{10}}{w} = \frac{{5 + 5{\text{i}}}}{{13{\text{i}}}}\) A1
\(\frac{w}{{10}} = \frac{{13{\text{i}}}}{{5 + 5{\text{i}}}}\)
\(w = \frac{{130{\text{i}}}}{{(5 + 5{\text{i}})}} \times \frac{{(5 - 5{\text{i}})}}{{(5 - 5{\text{i}})}}\)
\( = \frac{{650 + 650{\text{i}}}}{{50}}\)
\( = 13 + 13{\text{i}}\) A1
[4 marks]
(b) w* \( = 13 - 13{\text{i}}\) A1
\(z = \sqrt {338} {e^{ - \frac{\pi }{4}{\text{i}}}}{\text{ }}\left( { = 13\sqrt 2 {e^{ - \frac{\pi }{4}{\text{i}}}}} \right)\) A1A1
Note: Accept \(\theta = \frac{{7\pi }}{4}\).
Do not accept answers for \(\theta \) given in degrees.
[3 marks]
Total [7 marks]
