2010006603
Level:
B
Assuming \(m\neq 0\), simplify the expression: \[m+n-\frac{m^2-n^2}{m}\]
\( \frac{n(m+n)}m\)
\( \frac{n(m-n)}m\)
\( \frac{mn-n^2}m\)
\( \frac{n^2-mn}m\)
B
2010006506
Level:
B
Solve the following system of equations in \( \mathbb{R} \times \mathbb{R}\) and identify the correct statement.
\[\begin{aligned}
x^2 - y^2 = 5 & &
\\2x + y = 1 & &
\end{aligned}\]
The system does not have any solution.
The system has a unique solution.
The system has more than two solutions.
The system has two solutions.
2010006307
Level:
B
The equation of a parabola is given by \( 4x^2+16x-y+18=0 \). Find the equation of its directrix.
\( x=\frac{31}{16} \)
\( x=-\frac{33}{16} \)
\( x=\frac{33}{16} \)
\( x=-\frac{31}{16} \)
\( x=\frac{15}{8} \)
2010006306
Level:
B
The equation of a hyperbola with the center \( S=[1;-3] \), the focus \( F=[1;2] \), and the vertex \( A=[1;0] \) is given by:
\( \frac{(y+3)^2}{9}-\frac{(x-1)^2}{16} =1 \)
\( \frac{(x-1)^2}{16}-\frac{(y+3)^2}{9} =1 \)
\( \frac{(x-1)^2}{9}-\frac{(y+3)^2}{16} =1 \)
\( \frac{(y+3)^2}{16}-\frac{(x-1)^2}{9} =1 \)
\( \frac{(x+1)^2}{16}-\frac{(y-3)^2}{9} =1 \)
2010006305
Level:
B
Decide whether the equation \( 9x^2+4y^2-18x+8y+14=0 \) (in \( x \)-\( y \) plane) describes:
an empty set
an ellipse
a parabola
a hyperbola
a point
2010006304
Level:
B
The equation of a parabola is given by \( y^2-x-6y+10=0 \). What are the coordinates of its vertex?
\( [1;3] \)
\( [3;1] \)
\( [3;-1] \)
\( [-1;-3] \)
\( [-3;-1] \)
2010006303
Level:
B
A parabola passes through the points \( A=[5;0] \) and \( B=[-6;2] \) and it is symmetric with respect to \( x\)-axis. What are the coordinates of its vertex?
\( [5;0] \)
\( [6;-2] \)
\( [0;5] \)
\( [-6;2] \)
\( [-1;1] \)
2010006302
Level:
B
The equation of a hyperbola is given by \( -16x^2+9y^2-96x+108y+36=0\). The length of its semi-minor axis is:
\( 3 \)
\( 9\)
\( 4 \)
\( 16 \)
\( 5 \)
2010006209
Level:
B
The angle \( \beta \) is by \( 20\% \) smaller than the angle \( \gamma \) and by \( 100\% \) greater than the angle \( \alpha \). The angle \( \alpha \) is
by \( 60\% \) smaller than \( \gamma \).
by \( 40\% \) smaller than \( \gamma \).
by \( 20\% \) smaller than \( \gamma \).
by \( 80\% \) smaller than \( \gamma \).
2010006208
Level:
B
There are \(14\) right-handed and \(18\) left-handed children among the children attending the ceramic group. Find the difference between percentages of right-handers and left-handers in the group. (Round the result to one decimal place.)
differ by \(27.3\, \%\)
differ by \(63.6\, \%\)
differ by \(36.4\, \%\)
differ by \(57.1\, \%\)