3.3 Övningar
Aus Förberedande kurs i matematik 2
(Unterschied zwischen Versionen)
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| - | {{ | + | {{Ej vald flik|[[3.3 Potenser och rötter|Theory]]}} |
| - | {{ | + | {{Vald flik|[[3.3 Övningar|Exercises]]}} |
| style="border-bottom:1px solid #000" width="100%"| | | style="border-bottom:1px solid #000" width="100%"| | ||
|} | |} | ||
| - | === | + | ===Exercise 3.3:1=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | Write the following number in the form <math>\,a+ib\,</math>, where <math>\,a\,</math> and <math>\,b\,</math> are real numbers: | |
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
|a) | |a) | ||
| Zeile 25: | Zeile 25: | ||
|} | |} | ||
</div>{{#NAVCONTENT:Svar|Svar 3.3:1|Lösning a|Lösning 3.3:1a|Lösning b|Lösning 3.3:1b|Lösning c|Lösning 3.3:1c|Lösning d|Lösning 3.3:1d|Lösning e|Lösning 3.3:1e}} | </div>{{#NAVCONTENT:Svar|Svar 3.3:1|Lösning a|Lösning 3.3:1a|Lösning b|Lösning 3.3:1b|Lösning c|Lösning 3.3:1c|Lösning d|Lösning 3.3:1d|Lösning e|Lösning 3.3:1e}} | ||
| + | |||
| + | ===Exercise 3.3:2=== | ||
| + | <div class="ovning"> | ||
| + | Solve the equations | ||
| + | {| width="100%" cellspacing="10px" | ||
| + | |a) | ||
| + | |width="33%"|<math>z^4=1</math> | ||
| + | |b) | ||
| + | |width="33%"| <math>z^3=-1</math> | ||
| + | |c) | ||
| + | |width="33%"| <math> z^5=-1-i</math> | ||
| + | |- | ||
| + | |d) | ||
| + | |width="33%"| <math>(z-1)^4+4=0</math> | ||
| + | |e) | ||
| + | |width="33%"| <math>\displaystyle\Bigl(\frac{z+i}{z-i}\Bigr)^2 = -1</math> | ||
| + | |} | ||
| + | </div>{{#NAVCONTENT:Svar|Svar 3.3:2|Lösning a|Lösning 3.3:2a|Lösning b|Lösning 3.3:2b|Lösning c|Lösning 3.3:2c|Lösning d|Lösning 3.3:2d|Lösning e|Lösning 3.3:2e}} | ||
| + | |||
| + | ===Exercise 3.3:3=== | ||
| + | <div class="ovning"> | ||
| + | Complete the square of the following expressions | ||
| + | {| width="100%" cellspacing="10px" | ||
| + | |a) | ||
| + | |width="50%"|<math>z^2 +2z+3</math> | ||
| + | |b) | ||
| + | |width="50%"| <math>z^2 +3iz-\frac{1}{4}</math> | ||
| + | |- | ||
| + | |c) | ||
| + | |width="50%"| <math>-z^2-2iz +4z+1</math> | ||
| + | |d) | ||
| + | |width="50%"| <math>iz^2+(2+3i)z-1</math> | ||
| + | |} | ||
| + | </div>{{#NAVCONTENT:Svar|Svar 3.3:3|Lösning a|Lösning 3.3:3a|Lösning b|Lösning 3.3:3b|Lösning c|Lösning 3.3:3c|Lösning d|Lösning 3.3:3d}} | ||
| + | |||
| + | ===Exercise 3.3:4=== | ||
| + | <div class="ovning"> | ||
| + | Solve the equations | ||
| + | {| width="100%" cellspacing="10px" | ||
| + | |a) | ||
| + | |width="50%"|<math>z^2=i</math> | ||
| + | |b) | ||
| + | |width="50%"| <math>z^2-4z+5=0</math> | ||
| + | |- | ||
| + | |c) | ||
| + | |width="50%"| <math>-z^2+2z+3=0</math> | ||
| + | |d) | ||
| + | |width="50%"| <math>\displaystyle\frac{1}{z} + z = \frac{1}{2}</math> | ||
| + | |} | ||
| + | </div>{{#NAVCONTENT:Svar|Svar 3.3:4|Lösning a|Lösning 3.3:4a|Lösning b|Lösning 3.3:4b|Lösning c|Lösning 3.3:4c|Lösning d|Lösning 3.3:4d}} | ||
| + | |||
| + | ===Exercise 3.3:5=== | ||
| + | <div class="ovning"> | ||
| + | Solve the equations | ||
| + | {| width="100%" cellspacing="10px" | ||
| + | |a) | ||
| + | |width="50%"|<math>z^2-2(1+i)z+2i-1=0</math> | ||
| + | |b) | ||
| + | |width="50%"| <math>z^2-(2-i)z+(3-i)=0</math> | ||
| + | |- | ||
| + | |c) | ||
| + | |width="50%"| <math>z^2-(1+3i)z-4+3i=0</math> | ||
| + | |d) | ||
| + | |width="50%"| <math>(4+i)z^2+(1-21i)z=17</math> | ||
| + | |} | ||
| + | </div>{{#NAVCONTENT:Svar|Svar 3.3:5|Lösning a|Lösning 3.3:5a|Lösning b|Lösning 3.3:5b|Lösning c|Lösning 3.3:5c|Lösning d|Lösning 3.3:5d}} | ||
| + | |||
| + | ===Exercise 3.3:6=== | ||
| + | <div class="ovning"> | ||
| + | Determine the solution to <math>\,z^2=1+i\,</math> both in polar form and in the form <math>\,a+ib\,</math>, where <math>\,a\,</math> and <math>\,b\,</math> are real numbers. Use the result to calculate <math>\; \tan \frac{\pi}{8}\,</math>. | ||
| + | </div>{{#NAVCONTENT:Svar|Svar 3.3:6|Lösning |Lösning 3.3:6}} | ||
Aktuelle Version
| Theory | Exercises |
Exercise 3.3:1
Write the following number in the form \displaystyle \,a+ib\,, where \displaystyle \,a\, and \displaystyle \,b\, are real numbers:
| a) | \displaystyle (i+1)^{12} | b) | \displaystyle \displaystyle\Bigl(\frac{1+i\sqrt{3}}{2}\,\Bigr)^{12} |
| c) | \displaystyle (4\sqrt{3} -4i)^{22} | d) | \displaystyle \Bigl(\displaystyle\frac{1+i\sqrt{3}}{1+i}\,\Bigr)^{12} |
| e) | \displaystyle \displaystyle\frac{(1+i\sqrt{3}\,)(1-i)^8}{(\sqrt{3}-i)^9} |
Svar
Laden...
Lösning a
Lösning b
Lösning c
Lösning d
Lösning e
Exercise 3.3:2
Solve the equations
| a) | \displaystyle z^4=1 | b) | \displaystyle z^3=-1 | c) | \displaystyle z^5=-1-i |
| d) | \displaystyle (z-1)^4+4=0 | e) | \displaystyle \displaystyle\Bigl(\frac{z+i}{z-i}\Bigr)^2 = -1 |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
Lösning e
Exercise 3.3:3
Complete the square of the following expressions
| a) | \displaystyle z^2 +2z+3 | b) | \displaystyle z^2 +3iz-\frac{1}{4} |
| c) | \displaystyle -z^2-2iz +4z+1 | d) | \displaystyle iz^2+(2+3i)z-1 |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
Exercise 3.3:4
Solve the equations
| a) | \displaystyle z^2=i | b) | \displaystyle z^2-4z+5=0 |
| c) | \displaystyle -z^2+2z+3=0 | d) | \displaystyle \displaystyle\frac{1}{z} + z = \frac{1}{2} |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
Exercise 3.3:5
Solve the equations
| a) | \displaystyle z^2-2(1+i)z+2i-1=0 | b) | \displaystyle z^2-(2-i)z+(3-i)=0 |
| c) | \displaystyle z^2-(1+3i)z-4+3i=0 | d) | \displaystyle (4+i)z^2+(1-21i)z=17 |
Svar
Lösning a
Lösning b
Lösning c
Lösning d
Exercise 3.3:6
Determine the solution to \displaystyle \,z^2=1+i\, both in polar form and in the form \displaystyle \,a+ib\,, where \displaystyle \,a\, and \displaystyle \,b\, are real numbers. Use the result to calculate \displaystyle \; \tan \frac{\pi}{8}\,.
Svar
Lösning
