3.2 Exercises
From Förberedande kurs i matematik 2
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| - | {{ | + | {{Not selected tab|[[3.2 Polar form|Theory]]}} |
| - | {{ | + | {{Selected tab|[[3.2 Exercises|Exercises]]}} |
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|a) | |a) | ||
| - | |width="50%"|<math>z\,</math> | + | |width="50%"|<math>z\,</math> and <math>\,w</math> |
|b) | |b) | ||
| - | |width="50%"| <math>z+u\,</math> | + | |width="50%"| <math>z+u\,</math> and <math>\,z-u</math> |
|- | |- | ||
|c) | |c) | ||
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|width="50%"| <math>z-\overline{w}+u</math> | |width="50%"| <math>z-\overline{w}+u</math> | ||
|} | |} | ||
| - | </div>{{#NAVCONTENT:Answer|Answer 3.2:1|Solution a| | + | </div>{{#NAVCONTENT:Answer|Answer 3.2:1|Solution a|Solution 3.2:1a|Solution b|Solution 3.2:1b|Solution c|Solution 3.2:1c|Solution d|Solution 3.2:1d}} |
===Exercise 3.2:2=== | ===Exercise 3.2:2=== | ||
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|width="50%"| <math>2<|z-i|\le3</math> | |width="50%"| <math>2<|z-i|\le3</math> | ||
|} | |} | ||
| - | </div>{{#NAVCONTENT:Answer|Answer 3.2:2|Solution a| | + | </div>{{#NAVCONTENT:Answer|Answer 3.2:2|Solution a|Solution 3.2:2a|Solution b|Solution 3.2:2b|Solution c|Solution 3.2:2c|Solution d|Solution 3.2:2d|Solution e|Solution 3.2:2e|Solution f|Solution 3.2:2f}} |
===Exercise 3.2:3=== | ===Exercise 3.2:3=== | ||
<div class="ovning"> | <div class="ovning"> | ||
The complex numbers <math>\,1+i\,</math>, <math>\,3+2i\,</math> and <math>\,3i\,</math> constitute three corners of a square in the complex number plane. Determine the square's fourth corner. | The complex numbers <math>\,1+i\,</math>, <math>\,3+2i\,</math> and <math>\,3i\,</math> constitute three corners of a square in the complex number plane. Determine the square's fourth corner. | ||
| - | </div>{{#NAVCONTENT:Answer|Answer 3.2:3|Solution| | + | </div>{{#NAVCONTENT:Answer|Answer 3.2:3|Solution|Solution 3.2:3}} |
===Exercise 3.2:4=== | ===Exercise 3.2:4=== | ||
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|width="50%"| <math>\displaystyle\frac{3-4i}{3+2i}</math> | |width="50%"| <math>\displaystyle\frac{3-4i}{3+2i}</math> | ||
|} | |} | ||
| - | </div>{{#NAVCONTENT:Answer|Answer 3.2:4|Solution a| | + | </div>{{#NAVCONTENT:Answer|Answer 3.2:4|Solution a|Solution 3.2:4a|Solution b|Solution 3.2:4b|Solution c|Solution 3.2:4c|Solution d|Solution 3.2:4d}} |
===Exercise 3.2:5=== | ===Exercise 3.2:5=== | ||
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|width="50%"| <math>\displaystyle\frac{i}{1+i}</math> | |width="50%"| <math>\displaystyle\frac{i}{1+i}</math> | ||
|} | |} | ||
| - | </div>{{#NAVCONTENT:Answer|Answer 3.2:5|Solution a| | + | </div>{{#NAVCONTENT:Answer|Answer 3.2:5|Solution a|Solution 3.2:5a|Solution b|Solution 3.2:5b|Solution c|Solution 3.2:5c|Solution d|Solution 3.2:5d}} |
===Exercise 3.2:6=== | ===Exercise 3.2:6=== | ||
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|width="50%"| <math>\displaystyle\frac{(2+2i)(1+i\sqrt{3}\,)}{3i(\sqrt{12} -2i)}</math> | |width="50%"| <math>\displaystyle\frac{(2+2i)(1+i\sqrt{3}\,)}{3i(\sqrt{12} -2i)}</math> | ||
|} | |} | ||
| - | </div>{{#NAVCONTENT:Answer|Answer 3.2:6|Solution a| | + | </div>{{#NAVCONTENT:Answer|Answer 3.2:6|Solution a|Solution 3.2:6a|Solution b|Solution 3.2:6b|Solution c|Solution 3.2:6c|Solution d|Solution 3.2:6d|Solution e|Solution 3.2:6e|Solution f|Solution 3.2:6f}} |
Current revision
| Theory | Exercises |
Exercise 3.2:1
Given the complex numbers \displaystyle \,z=2+i\,, \displaystyle \,w=2+3i\, and \displaystyle \,u=-1-2i\,. Mark the following numbers on the complex plane:
| a) | \displaystyle z\, and \displaystyle \,w | b) | \displaystyle z+u\, and \displaystyle \,z-u |
| c) | \displaystyle 2z+w | d) | \displaystyle z-\overline{w}+u |
Answer
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Solution a
Solution b
Solution c
Solution d
Exercise 3.2:2
Draw the following sets in the complex number plane
| a) | \displaystyle 0\le \mbox{Im}\, z \le 3 | b) | \displaystyle 0 \le \mbox{Re} \, z \le \mbox{Im}\, z \le 3 |
| c) | \displaystyle |z|=2 | d) | \displaystyle |z-1-i|=3 |
| e) | \displaystyle \mbox{Re}\, z = i + \bar z | f) | \displaystyle 2<|z-i|\le3 |
Answer
Solution a
Solution b
Solution c
Solution d
Solution e
Solution f
Exercise 3.2:3
The complex numbers \displaystyle \,1+i\,, \displaystyle \,3+2i\, and \displaystyle \,3i\, constitute three corners of a square in the complex number plane. Determine the square's fourth corner.
Answer
Solution
Exercise 3.2:4
Determine the magnitude of
| a) | \displaystyle 3+4i | b) | \displaystyle (2-i) + (5+3i) |
| c) | \displaystyle (3-4i)(3+2i) | d) | \displaystyle \displaystyle\frac{3-4i}{3+2i} |
Answer
Solution a
Solution b
Solution c
Solution d
Exercise 3.2:5
Determine the argument of
| a) | \displaystyle -10 | b) | \displaystyle -2+2i |
| c) | \displaystyle (\sqrt{3} +i)(1-i) | d) | \displaystyle \displaystyle\frac{i}{1+i} |
Answer
Solution a
Solution b
Solution c
Solution d
Exercise 3.2:6
Write the following numbers in polar form
| a) | \displaystyle 3 | b) | \displaystyle -11i |
| c) | \displaystyle -4-4i | d) | \displaystyle \sqrt{10} + \sqrt{30}\,i |
| e) | \displaystyle \displaystyle\frac{1+i\sqrt{3}}{1+i} | f) | \displaystyle \displaystyle\frac{(2+2i)(1+i\sqrt{3}\,)}{3i(\sqrt{12} -2i)} |
Answer
Solution a
Solution b
Solution c
Solution d
Solution e
Solution f
