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

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Warm-up-exercises

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, which is .


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


b) Solve this system of equations.


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?


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?


Find a polynomial

with , such that the following conditions hold.


Find a polynomial

with , such that the following conditions hold.


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


Determine an equation for the secant of the function

to the points and .


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

lie.


Given a complex number

find its inverse complex number with the help of a real system of linear equations, with two equations in two variables.


Solve, over the complex numbers, the linear system of equations


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


Show by 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.



Exercise (3 marks)

Consider in the two planes

Determine the intersecting line .



Exercise (3 marks)

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

lie.



Exercise (3 marks)

Find a polynomial

with , such that the following conditions hold.



Exercise (4 marks)

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?