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2000020301

Level:

Solve the given system of equations in the set of real numbers.

\begin{aligned} x + y & = - 5 \\ 1 + \sqrt{2 x + 4 y} & = \sqrt{x + 3 y} \end{aligned}

In the following list identify a true statement.

x = - 12, y = 7

x = 12, y = 7

The system has no solution.

The system has infinitely many solutions.

2010020012

Level:

How many of the following functions have exactly two asymptotes?

f (x) = \frac{- x^{3} + 1}{(x - 3)^{2}},

h (x) = {(\frac{x + 2}{x - 2})}^{4},

i (x) = - \frac{x^{2}}{x^{2} - 2},

g (x) = \sqrt{6 x + 2}

None of these functions has exactly two asymptotes.

2010020011

Level:

How many of the following functions have exactly two asymptotes?

f (x) = {(\frac{4 + x}{4 - x})}^{4},

g (x) = \frac{x^{3}}{(x - 2)^{2}},

h (x) = \sqrt{6 - 4 x},

i (x) = \frac{x^{2}}{4 - x^{2}}

None of these functions has exactly two asymptotes.

2010020010

Level:

A function

f

is given by the formula

f (x) = \frac{1 + \ln x}{x^{2} - 3}

. For the function

f

identify all vertical asymptotes.

x = \sqrt{3}, x = 0

x = \sqrt{3}, x = - \sqrt{3}, x = 0

x = \sqrt{3}, x = - \sqrt{3}

This function has no vertical asymptote.

2010020009

Level:

A function

f

is given by the formula

f (x) = \frac{\ln x}{2 - x^{2}}

. For the function

f

identify all vertical asymptotes.

x = \sqrt{2}, x = 0

x = \sqrt{2}, x = - \sqrt{2}, x = 0

x = \sqrt{2}, x = - \sqrt{2}

This function has no vertical asymptote.

2010020008

Level:

A function

f

is given by the formula

f (x) = \frac{3 x^{2} - 1}{x - 2}

. For the function

f

identify a slant or horizontal asymptote.

y = 3 x + 6

y = 3

y = 3 x - 3

This function has neither horizontal nor slant asymptote.

2010020007

Level:

A function

f

is given by the formula

f (x) = \frac{2 x^{2} + 3}{x + 1}

. For the function

f

identify a slant or horizontal asymptote.

y = 2 x - 2

y = 2

y = - 2 x

This function has neither horizontal nor slant asymptote.

2010020006

Level:

The picture shows the part of the graph of a function

f

. Choose the formula of the function

f

whose graph corresponds to this picture.

f (x) = - \frac{x^{2}}{4 - x^{2}} - 1

f (x) = \frac{x^{2} + 4}{x^{2} - 4}

f (x) = \frac{x}{x^{2} - 4} + 1

f (x) = \frac{x^{2}}{4 - x^{2}}

2010020005

Level:

The picture shows the part of the graph of a function

f

. Choose the formula of the function

f

whose graph corresponds to this picture.

f (x) = \frac{x^{2} + 4}{x^{2} - 4}

f (x) = \frac{x}{x^{2} - 4} + 1

f (x) = - \frac{x^{2}}{4 - x^{2}} - 1

f (x) = \frac{x^{2}}{4 - x^{2}}

2000019110

Level:

Determine the set of all values of the real parameter

a

for which the equation has a unique solution.

\frac{a (x + 2) - 3 (x - 1)}{x + 1} = 1

R ∖ {- 6; 4}

R ∖ {- 1; - 2; 1}

R ∖ {0; - 1}

R ∖ {4}

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