3.2 Exercises

From Förberedande kurs i matematik 2

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{{Ej vald flik|[[3.2 Polär form|Teori]]}}
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{{Not selected tab|[[3.2 Polar form|Theory]]}}
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{{Vald flik|[[3.2 Övningar|Övningar]]}}
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{{Selected tab|[[3.2 Exercises|Exercises]]}}
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===Övning 3.2:1===
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===Exercise 3.2:1===
<div class="ovning">
<div class="ovning">
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Givet de komplexa talen <math>\,z=2+i\,</math>, <math>\,w=2+3i\,</math> och <math>\,u=-1-2i\,</math>. Markera följande tal i det komplexa talplanet
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Given the complex numbers <math>\,z=2+i\,</math>, <math>\,w=2+3i\,</math> and <math>\,u=-1-2i\,</math>. Mark the following numbers on the complex plane:
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="50%"|<math>z\,</math> och <math>\,w</math>
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|width="50%"|<math>z\,</math> and <math>\,w</math>
|b)
|b)
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|width="50%"| <math>z+u\,</math> och <math>\,z-u</math>
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|width="50%"| <math>z+u\,</math> and <math>\,z-u</math>
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|-
|c)
|c)
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|width="50%"| <math>z-\overline{w}+u</math>
|width="50%"| <math>z-\overline{w}+u</math>
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</div>{{#NAVCONTENT:Svar|Svar 3.2:1|Lösning a|Lösning 3.2:1a|Lösning b|Lösning 3.2:1b|Lösning c|Lösning 3.2:1c|Lösning d|Lösning 3.2:1d}}
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</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}}
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===Övning 3.2:2===
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===Exercise 3.2:2===
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<div class="ovning">
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Rita in följande mängder i det komplexa talplanet
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Draw the following sets in the complex number plane
{| width="100%" cellspacing="10px"
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|a)
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|width="50%"| <math>2<|z-i|\le3</math>
|width="50%"| <math>2<|z-i|\le3</math>
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|}
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</div>{{#NAVCONTENT:Svar|Svar 3.2:2|Lösning a|Lösning 3.2:2a|Lösning b|Lösning 3.2:2b|Lösning c|Lösning 3.2:2c|Lösning d|Lösning 3.2:2d|Lösning e|Lösning 3.2:2e|Lösning f|Lösning 3.2:2f}}
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</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}}
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===Övning 3.2:3===
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===Exercise 3.2:3===
<div class="ovning">
<div class="ovning">
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De komplexa talen <math>\,1+i\,</math>, <math>\,3+2i\,</math> och <math>\,3i\,</math> bildar i det komplexa talplanet tre hörn i en kvadrat. Bestäm kvadratens fjärde hörn.
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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.
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</div>{{#NAVCONTENT:Svar|Svar 3.2:3|Lösning |Lösning 3.2:3}}
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</div>{{#NAVCONTENT:Answer|Answer 3.2:3|Solution|Solution 3.2:3}}
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===Övning 3.2:4===
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===Exercise 3.2:4===
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Bestäm beloppet av
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Determine the magnitude of
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{| width="100%" cellspacing="10px"
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|a)
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|width="50%"| <math>\displaystyle\frac{3-4i}{3+2i}</math>
|width="50%"| <math>\displaystyle\frac{3-4i}{3+2i}</math>
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|}
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</div>{{#NAVCONTENT:Svar|Svar 3.2:4|Lösning a|Lösning 3.2:4a|Lösning b|Lösning 3.2:4b|Lösning c|Lösning 3.2:4c|Lösning d|Lösning 3.2:4d}}
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</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}}
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===Övning 3.2:5===
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===Exercise 3.2:5===
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Bestäm argumentet av
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Determine the argument of
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
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|a)
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|width="50%"| <math>\displaystyle\frac{i}{1+i}</math>
|width="50%"| <math>\displaystyle\frac{i}{1+i}</math>
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</div>{{#NAVCONTENT:Svar|Svar 3.2:5|Lösning a|Lösning 3.2:5a|Lösning b|Lösning 3.2:5b|Lösning c|Lösning 3.2:5c|Lösning d|Lösning 3.2:5d}}
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</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}}
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===Övning 3.2:6===
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===Exercise 3.2:6===
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Skriv följande tal i polär form
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Write the following numbers in polar form
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|a)
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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>
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|}
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</div>{{#NAVCONTENT:Svar|Svar 3.2:6|Lösning a|Lösning 3.2:6a|Lösning b|Lösning 3.2:6b|Lösning c|Lösning 3.2:6c|Lösning d|Lösning 3.2:6d|Lösning e|Lösning 3.2:6e|Lösning f|Lösning 3.2:6f}}
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</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
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

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
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

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
Answer 3.2:3
Solution
Solution 3.2:3

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
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

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
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

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
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
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