C

9000106806

Level: 
C
Given points \(A = [0;5]\), \(B = [6;1]\), \(C = [7;9]\), find the direction vector of the line passing through the point \(A\) and perpendicular to the segment \(BC\) (i.e. the line which contains the altitude of the triangle \(ABC\) through the point \(A\)).
\((8;-1)\)
\((1;8)\)
\((1;9)\)
\((-9;1)\)

9000106807

Level: 
C
Consider the points \(A = [0;5]\), \(B = [6;1]\), \(C = [7;9]\) and the triangle \(ABC\). Find the direction vector of the line which is the perpendicular bisector of the side \(b\) (i.e. the line through the midpoint of the side \(AC\) which is perpendicular to the segment \(AC\)).
\((4;-7)\)
\((7;4)\)
\((7;9)\)
\((7;-9)\)

9000106808

Level: 
C
Consider the points \(A = [0;5]\), \(B = [6;1]\), \(C = [7;9]\) and the triangle \(ABC\). Find the direction vector of the line which is the bisector of the angle \(ACB\) (i.e. the line which splits the internal angle at the point \(C\) into two angles with equal measures).
\((2;3)\)
\((6;-4)\)
\((7;9)\)
\((7;8)\)

9000106307

Level: 
C
Given points \(A = [0;0;1]\), \(B = [2;0;-1]\) and \(S = [2;1;0]\), find the parametric equations of the image of the line \(AB\) in a point reflection about the point \(S\).
\(\begin{aligned}[t] x& =\phantom{ -}4 + t, & \\y& =\phantom{ -}2, \\z& = -1 - t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& = 2 + 2m, & \\y& = 2 +\phantom{ 2}m, \\z& = 1 -\phantom{ 2}m;\ m\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}4 + 2k, & \\y& =\phantom{ -}2 +\phantom{ 2}k, \\z& = -1 -\phantom{ 2}k;\ k\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& = -2 + 2u, & \\y& =\phantom{ -}2, \\z& =\phantom{ -}1 - 2u;\ u\in \mathbb{R} \\ \end{aligned}\)

9000104801

Level: 
C
Consider the hyperbola \[ xy = -1 \] and a line \(p\) parallel to one of the axes but not identical with this axis. Find the true statement.
The line \(p\) has a unique intersection with the hyperbola.
The line \(p\) has two intersections with the hyperbola.
The line \(p\) does not have any intersection with the hyperbola.
We cannot draw any conclusion on the number of intersections of the line \(p\) with the hyperbola.

9000104803

Level: 
C
Consider the hyperbola \[ \frac{x^{2}} {16} -\frac{y^{2}} {4} = 1 \] and a line \(p\) parallel to one of the axes. Find the true statement.
We cannot draw any conclusion on the number of intersections of the line \(p\) with the hyperbola.
The line \(p\) has two intersections with the hyperbola.
The line \(p\) has a unique intersection with the hyperbola.
The line \(p\) does not have any intersection with the hyperbola.

9000104805

Level: 
C
Find the slope of a line through the center of the hyperbola \[ \frac{(x - 2)^{2}} {4} -\frac{(y + 3)^{2}} {9} = 1 \] which has a unique intersection with this hyperbola.
There is no solution, the line with these properties does not exist.
\(\frac{3} {2}\)
\(-\frac{3} {2}\)
\(\frac{2} {3}\)
\(1\)
\(0\)

9000104809

Level: 
C
Among the following lines (which all pass through the point \([-1;3]\)) identify a line which is tangent to the following hyperbola. \[ (x + 2)\cdot (y - 2) = 1 \]
\(k\colon \ y = -x + 2\)
\(p\colon \ y = 3\)
\(q\colon \ x = -1\)
\(r\colon \ y = x + 4\)
None of the given answers is correct.

9000104503

Level: 
C
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\). \[\frac{a^{2}(x-1)} {ax-2} = 2\]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \emptyset \\ a=2 & \mathbb{R}\setminus\{1\} \\ a\notin\{0;2\} & \left\lbrace\frac{a+2}a\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a\in\{0;2\} & \mathbb{R} \\ a\notin\{0,2\} & \left\{\frac{a+2}a\right\} \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \emptyset \\ a=2 & \mathbb{R} \\ a\notin\{0;2\} & \left\lbrace\frac{a+2}a\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \mathbb{R}\setminus\{1\} \\ a=2 & \emptyset \\ a\notin\{0;2\} & \left\lbrace\frac{a+2}a\right\rbrace \\\hline \end{array}\)