Onlinebrückenkurs Mathematik Abschnitt 1.1.1 Restart

Please read the instructions on the test procedure.
The sample part of the test is optional and will not be graded. It serves to introduce you to the use of the input boxes.
Important notice: please solve the exercises on a sheet of paper without using a calculator. Enter your solutions into the input boxes, and they will be checked. As soon as you are familiar with the use of the input boxes, you can click "next" in the upper right-hand corner and take the graded entrance test.
Many exercises are followed by blue info texts that give hints for the use of the interactive input boxes. For more complex terms, a blue info box appears where the entered mathematical term is displayed as you type in your entry. Do not confirm your entry until the displayed term agrees completely with the solution you intend to enter.

Exercise 1.1.1

Simplify the following complex fractions so that only a simple fraction remains:

$\frac{2 + \frac{3}{2}}{1 - \frac{1}{3}}$ is equivalent to .
Enter, for example, $\frac{11}{12}$ as 11/12.
$\frac{4 x^{2} + y^{2}}{3 x - y} - \frac{5 x^{2} - 2 y^{2}}{y - 3 x}$ is equivalent to .
Do not enter any brackets.

Exercise 1.1.2

Expand the brackets completely and collect like terms together:

(x - 2) (x + 1) \cdot x

= .
Enter, for example,

(x + 1) (x + 2)

= x^2+3*x+2 or also x*x+3*x+2.

Exercise 1.1.3

Apply one of the binomial formulas to transform each of the following terms:

$(- x - 3) (- x + 3)$ = .
$(s + 2 r + t)^{2}$ = .

First apply the binomial formula to $(s + (2 r + t {))}^{2}$ .

Enter, for example,

(x + 1)^{2}

= x^2+2*x+1 or also x*x+2*x+1, however, your input must not contain any products of bracketed terms.

Exercise 1.1.4

Rewrite this power and radical expression as a simple power with a rational number as its exponent, without using the root sign:

\sqrt{\sqrt{x} \cdot x}

=

.
Enter, for example,

\sqrt{x} \cdot x^{2}

= x^(5/2) or also x^(2.5),
remember the brackets around the fraction in the exponent.

Exercise 1.1.5

Transform the fractions such that the denominator disappears:

$\frac{2 x}{\sqrt{2} - \sqrt{3}}$ = .
$\frac{1}{\sqrt{n + 1} + \sqrt{n}}$ = .

Your input must not contain any fractions or powers. However, radicals are allowed. For example, enter the square root

\sqrt{x y z}

as sqrt(x*y*z).

Exercise 1.1.6

Solve the equation

\frac{t - 2}{t + 1} = 2

for

t

.
Answer:

t

=
.

Exercise 1.1.7

Specify the solution sets of the following quadratic equations:

$x^{2} + 3 x - 10 = 0$ has the solution set .
$x^{2} + 2 x + 3 = 0$ has the solution set .
$(x - 1)^{2} - (x + 1)^{2} = 0$ has the solution set .

Enter the solutions as sets. On a German keyboard, the curly brackets are accessed by holding the AltGr key and tapping 7 and 0, respectively. Enter the sets in the form {1;2;3}. The empty set is entered as {}.

Exercise 1.1.8

Specify the solution sets of the following equations:

$\frac{1}{x} + 1 = x - 1$ has the solution set .
$\frac{1}{x} + \frac{2}{x} = \frac{4}{x}$ has the solution set .
$\sqrt{2 x^{2} + 1} = 3 x$ has the solution set .

The solution sets are allowed to contain radical terms. However, different choices of sign have to be entered explicitly. For example, enter

{1 \pm \frac{1}{2} \sqrt{3}}

as {1+1/2*sqrt(3);1-1/2*sqrt(3)}.
How many solutions does the equation

(b - a)^{100} + (b + a)^{100} = 0

have if

a

and

b

are independent variables?
No solution
Exactly one solution for both

a

and

b

One solution for

a

and an infinite number of solutions for

b

An infinite number of solutions for both variables

Exercise 1.1.9

Rewrite the absolute value expression

| 2 x - 1 | - 3 x

as a case analysis with two expressions that do not contain any absolute values.
Answer:

| 2 x - 1 | - 3 x

= .
A short version of a case analysis is entered in the form if(condition,value1,value2).

In a mathematical case analysis, absolute values are defined by different cases. For example, the absolute value expression

| x - 1 |

can be written as:

| x - 1 | = {\begin{matrix} x - 1 & if x \geq 1 \\ - x + 1 & if x < 1 \end{matrix}

For example, this expression can be entered in the input box as if(x>=1;x-1;-x+1). Please note that there are several correct ways to write these expressions.

Exercise 1.1.10

Find all solutions of the absolute value equation

| 2 x - 7 | = x - 2

.
Answer: The solution set is

L

=

.
Enter the solutions as sets. On a German keyboard, the curly brackets are accessed by holding the AltGr key and tapping 7 and 0, respectively.

Exercise 1.1.11

Specify the solution set of the inequality

x^{2} + 4 x < 5

as an interval.
Answer:

L

=

.
Typical inputs for intervals are (-3;2), or [5;infty), and also (-infty;infty). Enter

\infty

as infinity or infty for short. Do not use the notation

] a; b [

to enter open intervals but

(a; b)

Exercise 1.1.12

Specify the domains and the solution sets of the following inequalities as intervals:

The inequality $\sqrt{x} > x$ has the domain
and the solution set .
The inequality $\sqrt{\sqrt{x - 1} + 2} > \sqrt{x + 1}$ has the domain
and the solution set .

As soon as you are familiar with the use of the input boxes, you can click "next" in the upper right corner and take the graded entrance test.