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3.1 Übungen

Aus Online Mathematik Brückenkurs 2

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Exercise 3.1:1

Write in the form \displaystyle \,a+bi\,, where \displaystyle \,a\, and \displaystyle \,b\, are real numbers

a)	\displaystyle (5-2i)+(3+5i)	b)	\displaystyle 3i -(2-i)
c)	\displaystyle i(2+3i)	d)	\displaystyle (3-2i)(7+5i)
e)	\displaystyle (1+i)(2-i)^2	f)	\displaystyle i^{\,20} + i^{\,11}

Svar

Lösning a

Lösning b

Lösning c

Lösning d

Lösning e

Lösning f

Exercise 3.1:2

Write in the form \displaystyle \,a+bi\,, where \displaystyle \,a\, and \displaystyle \,b\, where b are real numbers,

a)	\displaystyle \displaystyle\frac{3-2i}{1+i}	b)	\displaystyle \displaystyle\frac{3i}{4-6i} - \displaystyle\frac{1+i}{3+2i}
c)	\displaystyle \displaystyle\frac{(2-i\sqrt{3}\,)^2}{1+i\sqrt{3}}	d)	\displaystyle \displaystyle\frac{5-\displaystyle\frac{1}{1+i}}{3i + \displaystyle\frac{i}{2-3i}}

Svar

Lösning a

Lösning b

Lösning c

Lösning d

Exercise 3.1:3

Determine the real number \displaystyle \,a\, such that the expression \displaystyle \ \displaystyle\frac{3+i}{2+ai}\ becomes purely imaginary (i.e. the real part equals zero).

Svar

Svar 3.1:3

Lösning

Lösning 3.1:3

Exercise 3.1:4

Solve the equations

a)	\displaystyle z+3i=2z-2	b)	\displaystyle (2-i) z= 3+2i
c)	\displaystyle iz+2= 2z-3	d)	\displaystyle (2+i) \overline{z} = 1+i
e)	\displaystyle \displaystyle\frac{iz+1}{z+i} = 3+i	f)	\displaystyle (1+i)\overline{z}+iz = 3+5i