Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 5/en/latex

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\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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