Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 5/en/latex
\setcounter{section}{5}
\zwischenueberschrift{Warm-up-exercises}
\inputexercise
{}
{
\mathl{M,P,S}{} and $T$ are the members of one family. In this case $M$ is three times as old as
\mathkor {} {S} {and} {T} {}
together, $M$ is older than $P$ and $S$ is older than $T$, moreover the age difference between $S$ and $T$ is twice as large as the difference between $M$ and $P$. Furthermore $P$ is seven times as old as $T$ and the sum of the ages of all family members is equal to the paternal grandmother's age, that is $83$.
a) Set up a linear system of equations that expresses the conditions described.
b) Solve this system of equations.
}
{} {}
\inputexercise
{}
{
Kevin pays
\mathl{2,50}{}\euro\ for a winter bunch of flowers with $3$ snowdrops and $4$ mistletoes and Jennifer pays
\mathl{2,30}{}\euro\ for a bunch with $5$ snowdrops and $2$ mistletoes. How much does a bunch with one snowdrop and $11$ mistletoes cost?
}
{} {}
\inputexercise
{}
{
We look at a clock with hour and minute hands. Now it is 6 o'clock, so that both hands have opposite directions. When will the hands have opposite directions again?
}
{} {}
\inputexercise
{}
{
Find a
\definitionsverweis {polynomial}{}{}
\mathdisp {f=a+bX+cX^2} { }
with $a,b,c \in \R$, such that the following conditions hold.
\mathdisp {f(-1) =2,\, f(1) = 0,\, f(3) = 5} { . }
}
{} {}
\inputexercise
{}
{
Find a
\definitionsverweis {polynomial}{}{}
\mathdisp {f=a+bX+cX^2+dX^3} { }
with $a,b,c,d \in \R$, such that the following conditions hold.
\mathdisp {f(0) =1,\, f(1) = 2,\, f(2) = 0, \, f(-1) = 1} { . }
}
{} {}
\inputexercise
{}
{
Exhibit a linear equation for the straight line in $\R^2$, which runs through the two points \mathkor {} {(2,3)} {and} {(5,-7)} {.}
}
{} {}
Before the next tasks, we recall the concept of secant.
On the subset
\mathl{T \subseteq \R}{} it is given a function
\maabbdisp {f} {T} {\R
} {}
and two points
\mathl{a,b \in T}{,} the straight line through
\mathkor {} {(a,f(a))} {and} {(b,f(b))} {}
is called
\definitionswort {secant}{}
of $f$ to
\mathkor {} {a} {and} {b} {.}
\inputexercise
{}
{
Determine an equation of the secant of the function
\maabbeledisp {} {\R} {\R
} {x} {-x^3+x^2+2
} {,}
to the points
\mathbed {3} {and}
{4} {}
{} {} {} {.}
}
{} {}
\inputexercise
{}
{
Determine a linear equation for the plane in $\R^3$, where the three points \mathlistdisp {(1,0,0)} {} {(0,1,2)} {and} {(2,3,4)} {} lie.
}
{} {}
\inputexercise
{}
{
Given a
\definitionsverweis {complex number}{}{}
\mathl{z=a+bi \neq 0}{}, find its inverse complex number with the help of a real system of linear equations with two variables and two equations.
}
{} {}
\inputexercise
{}
{
Solve over the
\definitionsverweis {complex numbers}{}{} the
\definitionsverweis {linear system}{}{} of equations
\mathdisp {\begin{matrix}
i x &+y & +(2-i)z & = & 2 \\
& 7y& +2iz &=& -1+3i \\
& & (2-5i) z &=& 1 \, .
\end{matrix}} { }
}
{} {}
\inputexercise
{}
{
Let $K$ be the field with two elements of
Example 2.3. Solve in $K$ the
\definitionsverweis {inhomogeneous linear system}{}{}
\mathdisp {\begin{matrix}
x &+y & & = & 1 \\
& y& +z &=& 0 \\
x& +y & +z &=& 0 \, .
\end{matrix}} { }
}
{} {}
\inputexercise
{}
{
Show with an example that the linear system given by three equations I, II, III is not equivalent to the linear system given by the three equations I-II, I-III, II-III.
}
{} {}
\zwischenueberschrift{Hand-in-exercises}
\inputexercise
{4}
{
Solve the following system of inhomogeneous linear equations.
\mathdisp {\begin{matrix} x &
+2 y &
+3 z &
+4 w & = & 1 \\ 2 x &
+3 y &
+4 z &
+5 w & = & 7 \\ x &
\, \, \, \, \, \, \, \, &
+ z &
\, \, \, \, \, \, \, \, & = & 9 \\ x &
+5 y &
+5 z &
+ w & = & 0 \, . \end{matrix}} { }
}
{} {}
\inputexercise
{3}
{
Consider in $\R^3$ the two planes
\mathdisp {E = { \left\{ (x,y,z) \in \R^3 \mid 3x+4y+5z = 2 \right\} } \text{ and } F = { \left\{ (x,y,z) \in \R^3 \mid 2x-y+3z = -1 \right\} }} { . }
Determine the intersection line $E \cap F$.
}
{} {}
\inputexercise
{3}
{
Determine a linear equation for the plane in $\R^3$, where the three points \mathlistdisp {(1,0,2)} {} {(4,-3,2)} {and} {(2,1,-1)} {} lie.
}
{} {}
\inputexercise
{3}
{
Find a polynomial
\mathdisp {f=a+bX+cX^2} { }
with $a,b,c \in {\mathbb C}$, such that the following conditions hold.
\mathdisp {f(i) =1,\, f(1) = 1+i,\, f(1-2i) = -i} { . }
}
{} {}
\inputexercise
{4}
{
We consider the linear system
\mathdisp {\begin{matrix}
2 x &-ay & & = & -2 \\
ax & & +3 z &=& 3 \\
-{ \frac{ 1 }{ 3 } }x & +y & + z &=& 2
\end{matrix}} { }
over the real numbers, depending on the parameter
\mathl{a \in \R}{.} For which
\mathl{a}{} does the system of equations have no solution, one solution or infinitely many solutions?
}
{} {}
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