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?