# Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 5

    Warm-up-exercises

### Exercise

$M,P,S$ and $T$ are the members of one family. In this case, $M$ is three times as old as $S$ 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.

### Exercise *

Kevin pays $2.50$€ for a winter bunch of flowers with $3$ snowdrops and $4$ mistletoes, and Jennifer pays $2.30$€ for a bunch with $5$ snowdrops and $2$ mistletoes. How much does a bunch with one snowdrop and $11$ 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
    $f=a+bX+cX^{2}\,$
with $a,b,c\in \mathbb {R}$, such that the following conditions hold.
    $f(-1)=2,\,f(1)=0,\,f(3)=5.$

### Exercise

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

### Exercise *

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

### Exercise

Determine an equation for the secant of the function
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto -x^{3}+x^{2}+2,$
to the points $3$ and $4$.

### Exercise

Determine a linear equation for the plane in $\mathbb {R} ^{3}$, where the three points
    $(1,0,0),\,(0,1,2){\text{ and }}(2,3,4)$
lie.

### Exercise

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

### Exercise

Solve, over the complex numbers, the linear system of equations
    ${\begin{matrix}{\mathrm {i} }x&+y&+(2-{\mathrm {i} })z&=&2\\&7y&+2{\mathrm {i} }z&=&-1+3{\mathrm {i} }\\&&(2-5{\mathrm {i} })z&=&1\,.\end{matrix}}$

### Exercise

Let $K$ be the field with two elements. Solve in $K$ the inhomogeneous linear system
    ${\begin{matrix}x&+y&&=&1\\&y&+z&=&0\\x&+y&+z&=&0\,.\end{matrix}}$

### 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 marks)

Solve the following system of inhomogeneous linear equations.
    ${\begin{matrix}x&+2y&+3z&+4w&=&1\\2x&+3y&+4z&+5w&=&7\\x&\,\,\,\,\,\,\,\,&+z&\,\,\,\,\,\,\,\,&=&9\\x&+5y&+5z&+w&=&0\,.\end{matrix}}$

### Exercise (3 marks)

Consider in $\mathbb {R} ^{3}$ the two planes
    $E={\left\{(x,y,z)\in \mathbb {R} ^{3}\mid 3x+4y+5z=2\right\}}{\text{ and }}F={\left\{(x,y,z)\in \mathbb {R} ^{3}\mid 2x-y+3z=-1\right\}}.$
Determine the intersecting line $E\cap F$.

### Exercise (3 marks)

Determine a linear equation for the plane in $\mathbb {R} ^{3}$, where the three points
    $(1,0,2),\,\,(4,-3,2)\,{\text{ and }}\,(2,1,-1)$
lie.

### Exercise (3 marks)

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

### Exercise (4 marks)

We consider the linear system
    ${\begin{matrix}2x&-ay&&=&-2\\ax&&+3z&=&3\\-{\frac {1}{3}}x&+y&+z&=&2\end{matrix}}$
over the real numbers, depending on the parameter $a\in \mathbb {R}$. For which $a$ does the system of equations have no solution, one solution or infinitely many solutions?