User interface language: English | Español

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]

Examiners report

[N/A]

Syllabus sections

Topic 1 - Core: Algebra » 1.5 » Complex numbers: the number \({\text{i}} = \sqrt { - 1} \) ; the terms real part, imaginary part, conjugate, modulus and argument.
Show 33 related questions

View options