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Give the answer WITHOUT using calculator, paper, pencil. Only using ‘wetware’:
A
Simplify $\,\,\displaystyle{\frac13+\frac12 -\frac{1}{12}}\,.$
A
Given the numbers $\,\,z=3(i-10)-5(7-2i)-i(3i-5)+3i(i-5)\,.\,$ Find the rectangular form for $z\,.$
B
Given the number
$$a=5-i(3-i)+6i\,\,\, \mathrm{and} \,\,\,b=-5-4(-2i+1)\,.$$
Write the number $\,z=a+ib\,$ in rectangular form.
A
Determine the real part and the imaginary part of
$$\frac{-2+3i}i$$
and write the number in rectangular form.
Is easily done by multiplying the numerator and the denominator by the same number.
B
Reduce the following expression and write it in rectangular form.
$\displaystyle{\frac{3}{5}- \frac{3-2i}{2+i}}$
It is probably easiest to write the last fraction in rectangular form.
$$-\frac{1}{5} + \frac{7}{5} i$$
Let $b,c$ and $d$ be the following real numbers:
$$b=5 \,,\, \, c=\frac{6}{7} \,,\, \, d=\frac{2}{3} \,$$
C
Find the following numbers:
$$c+d\,,\, \, d \cdot b\,,\, \,\frac{b}{d}, \, \, \, \frac{d}{c}$$
Let $k,n,m$ og $s$ be the following complex numbers:
$$k=1+i \cdot \sqrt{3} \,,\, \, n=5 \cdot i \,,\, \, m=1+i \,,\, \, s=i \cdot 4 +3$$
D
Write the following complex numbers in rectangular form:
$$\frac{m}{n} \,,\, \, \frac{k}{s} \,,\, \, \frac{1}{m} + s$$
Use the square theorems in the two following questions, where $u$ and $v$ denotes two complex numbers
A
Reduce
$$(u+v)^2+(u-v)^2$$
B
Reduce
$$\frac{u^2-v^2}{u+v}+ \frac{v^2-u^2}{v-u}$$
C
Find by use of elementary computations the rectangular form of the following complex numbers.
-
$(3+5i)(3+5i)$
-
$(3i+5)(3i-5)$
-
$\displaystyle{\frac{3-4i}{3+4i}}$
$\,1) -16+30i \, \, , \, \, 2) -34 \, \, , \, \,3) -\frac{7}{25} - \frac{24}{25} i \, \,.$
D
Prove the formula
$$z\cdot \overline{z}=\left|z\right|^2\,.$$
A
Solve the Equation
$$(1-i)z+1=2+i\,.$$
The solution is wanted in rectangular form.
B
Determine all solutions to
$$(x+2i)(x-2i)(x-5)=0\,.$$
What does the zero rule say?
$$-2i \, \, , \, \, 2i \, \, , \, \, 5$$
C
Show that
$$x^4-x^3+4x^2-4x=0\,$$
has the roots $\,0,\,1,\,2i\,$ og $\,-2i\,.$
Here you do not have to compute the roots!
You only have to verify the solution.
Find the real solutions to the following equations:
C
What are the complex solutions to the equations above?
Let $A$ and $B$ be finite sets, given on the list forms:
$$A = \{n \in \Bbb{N}\, | \, n=m^2 \,\,\,\mathrm{where} \,\,\, m \in \{1,2,3,4,5\}\}$$
$$B = \{n \in \Bbb{N} \, |\, n=2m-1 \,\,\,\mathrm{where} \,\,\, m \in \{1,2,3,4,5\}\}$$
A
Which elements are members of the sets $A \cap B$ and $A \cup B\,$?
Let $C$ and $D$ be sets, given on the list forms:
$$C = \{n \in \Bbb{N}\, | \, n=2m \,\,\,\mathrm{hvor} \,\,\, m \in \Bbb{N}\}$$
$$D = \{n \in \Bbb{N}\, |\, n=3m \,\,\,\mathrm{hvor} \,\,\, m \in \Bbb{N}\}$$
B
Which elements are members of the sets $C \cap D$ and $C \cup D\,$?
C
Describe in you own words the sets $\Bbb{R} \setminus \Bbb{Q}$ and $\Bbb{C} \setminus \Bbb{R}\,.$
