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

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Warm-up-exercises

### Exercise

${\displaystyle {}M,P,S}$ and ${\displaystyle {}T}$ are the members of one family. In this case ${\displaystyle {}M}$ is three times as old as ${\displaystyle {}S}$ and ${\displaystyle {}T}$ together, ${\displaystyle {}M}$ is older than ${\displaystyle {}P}$ and ${\displaystyle {}S}$ is older than ${\displaystyle {}T}$, moreover the age difference between ${\displaystyle {}S}$ and ${\displaystyle {}T}$ is twice as large as the difference between ${\displaystyle {}M}$ and ${\displaystyle {}P}$. Furthermore ${\displaystyle {}P}$ is seven times as old as ${\displaystyle {}T}$ and the sum of the ages of all family members is equal to the paternal grandmother's age, that is ${\displaystyle {}83}$.

a) Set up a linear system of equations that expresses the conditions described.

b) Solve this system of equations.

### Exercise

Kevin pays ${\displaystyle {}2,50}$€ for a winter bunch of flowers with ${\displaystyle {}3}$ snowdrops and ${\displaystyle {}4}$ mistletoes and Jennifer pays ${\displaystyle {}2,30}$€ for a bunch with ${\displaystyle {}5}$ snowdrops and ${\displaystyle {}2}$ mistletoes. How much does a bunch with one snowdrop and ${\displaystyle {}11}$ mistletoes cost?

### Exercise

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?

### Exercise

Find a polynomial

${\displaystyle f=a+bX+cX^{2}}$
with ${\displaystyle {}a,b,c\in \mathbb {R} }$, such that the following conditions hold.
${\displaystyle f(-1)=2,\,f(1)=0,\,f(3)=5.}$

### Exercise

Find a polynomial

${\displaystyle f=a+bX+cX^{2}+dX^{3}}$
with ${\displaystyle {}a,b,c,d\in \mathbb {R} }$, such that the following conditions hold.
${\displaystyle f(0)=1,\,f(1)=2,\,f(2)=0,\,f(-1)=1.}$

### Exercise

Exhibit a linear equation for the straight line in ${\displaystyle {}\mathbb {R} ^{2}}$, which runs through the two points ${\displaystyle {}(2,3)}$ and ${\displaystyle {}(5,-7)}$.

Before the next tasks, we recall the concept of secant.

On the subset ${\displaystyle {}T\subseteq \mathbb {R} }$ it is given a function

${\displaystyle f\colon T\longrightarrow \mathbb {R} }$

and two points ${\displaystyle {}a,b\in T}$, the straight line through ${\displaystyle {}(a,f(a))}$ and ${\displaystyle {}(b,f(b))}$ is called secant of ${\displaystyle {}f}$ to ${\displaystyle {}a}$ and ${\displaystyle {}b}$.

### Exercise

Determine an equation of the secant of the function

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,\,x\longmapsto -x^{3}+x^{2}+2,}$

to the points ${\displaystyle {}3}$ and ${\displaystyle {}4}$.

### Exercise

Determine a linear equation for the plane in ${\displaystyle {}\mathbb {R} ^{3}}$, where the three points

${\displaystyle (1,0,0),\,\,(0,1,2)\,{\text{ and }}\,(2,3,4)}$

lie.

### Exercise

Given a complex number ${\displaystyle {}z=a+bi\neq 0}$, find its inverse complex number with the help of a real system of linear equations with two variables and two equations.

### Exercise

Solve over the complex numbers the linear system of equations

${\displaystyle {\begin{matrix}ix&+y&+(2-i)z&=&2\\&7y&+2iz&=&-1+3i\\&&(2-5i)z&=&1\,.\end{matrix}}}$

### Exercise

Let ${\displaystyle {}K}$ be the field with two elements of Example 2.3. Solve in ${\displaystyle {}K}$ the inhomogeneous linear system

${\displaystyle {\begin{matrix}x&+y&&=&1\\&y&+z&=&0\\x&+y&+z&=&0\,.\end{matrix}}}$

### Exercise

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.

Hand-in-exercises

### Exercise (4 points)

Solve the following system of inhomogeneous linear equations.

${\displaystyle {\begin{matrix}x&+2y&+3z&+4w&=&1\\2x&+3y&+4z&+5w&=&7\\x&\,\,\,\,\,\,\,\,&+z&\,\,\,\,\,\,\,\,&=&9\\x&+5y&+5z&+w&=&0\,.\end{matrix}}}$

### Exercise (3 points)

Consider in ${\displaystyle {}\mathbb {R} ^{3}}$ the two planes

${\displaystyle E={\left\{(x,y,z)\in \mathbb {R} ^{3}\mid 3x+4y+5z=2\right\}}{\text{ and }}F={\left\{(x,y,z)\in \mathbb {R} ^{3}\mid 2x-y+3z=-1\right\}}.}$

Determine the intersection line ${\displaystyle {}E\cap F}$.

### Exercise (3 points)

Determine a linear equation for the plane in ${\displaystyle {}\mathbb {R} ^{3}}$, where the three points

${\displaystyle (1,0,2),\,\,(4,-3,2)\,{\text{ and }}\,(2,1,-1)}$

lie.

### Exercise (3 points)

Find a polynomial

${\displaystyle f=a+bX+cX^{2}}$
with ${\displaystyle {}a,b,c\in \mathbb {C} }$, such that the following conditions hold.
${\displaystyle f(i)=1,\,f(1)=1+i,\,f(1-2i)=-i.}$

### Exercise (4 points)

We consider the linear system

${\displaystyle {\begin{matrix}2x&-ay&&=&-2\\ax&&+3z&=&3\\-{\frac {1}{3}}x&+y&+z&=&2\end{matrix}}}$

over the real numbers, depending on the parameter ${\displaystyle {}a\in \mathbb {R} }$. For which ${\displaystyle {}a}$ does the system of equations have no solution, one solution or infinitely many solutions?