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

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Warm-up-exercises

Exercise

and are the members of one family. In this case is three times as old as and together, is older than and is older than , moreover the age difference between and is twice as large as the difference between and . Furthermore is seven times as old as and the sum of the ages of all family members is equal to the paternal grandmother's age, that is .

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

b) Solve this system of equations.


Exercise

Kevin pays € for a winter bunch of flowers with snowdrops and mistletoes and Jennifer pays € for a bunch with snowdrops and mistletoes. How much does a bunch with one snowdrop and 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

with , such that the following conditions hold.


Exercise

Find a polynomial

with , such that the following conditions hold.


Exercise

Exhibit a linear equation for the straight line in , which runs through the two points and .


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

On the subset it is given a function

and two points , the straight line through and is called secant of to and .


Exercise

Determine an equation of the secant of the function

to the points  and .


Exercise

Determine a linear equation for the plane in , where the three points

lie.


Exercise

Given a complex number , 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


Exercise

Let be the field with two elements of Example 2.3. Solve in the inhomogeneous linear system


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.


Exercise (3 points)

Consider in the two planes

Determine the intersection line .


Exercise (3 points)

Determine a linear equation for the plane in , where the three points

lie.


Exercise (3 points)

Find a polynomial

with , such that the following conditions hold.


Exercise (4 points)

We consider the linear system

over the real numbers, depending on the parameter . For which does the system of equations have no solution, one solution or infinitely many solutions?




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