C

9000106805

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 the midpoint of the segment \(BC\) (i.e. the median of the triangle \(ABC\) through the vertex \(A\)).
\((1;0)\)
\((1;8)\)
\((1;9)\)
\((6.5;5)\)

9000106903

Level: 
C
The motion with a constant acceleration is described by the relation \(s = \frac{1} {2}at^{2}\). Consequently, the graph which shows the distance as a function of time is part of a parabola. Find the directrix of this parabola, if \(a = 4\, \mathrm{m}/\mathrm{s}^{2}\).
\(s = -\frac{1} {8}\)
\(s = -1\)
\(s = \frac{1} {8}\)
\(s = 1\)

9000106901

Level: 
C
A body is thrown at the initial angle \(\alpha = 45^{\circ }\) and the initial velocity \(v_{0} = 10\, \mathrm{m}/\mathrm{s}\). The trajectory of the body is a parabola. Find the equation of this parabola. Hint: The coordinates of the moving body as functions of time are \[ \begin{aligned}x& = v_{0}t\cdot \cos \alpha , & \\y& = v_{0}t\cdot \sin \alpha -\frac{1} {2}gt^{2}. \\ \end{aligned} \] Consider the standard acceleration due to gravity \(g = 10\, \mathrm{m}/\mathrm{s}^{2}\).
\((x - 5)^{2} = -10\cdot (y - 2.5)\)
\((x - 5)^{2} = 10\cdot (y + 2.5)\)
\(x^{2} = -10\cdot (y - 5)\)
\((x - 5)^{2} = -10\cdot (y + 2.5)\)

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 \(S\).
\(\begin{aligned}[t] x& =\phantom{ -}4 + t, & \\y& =\phantom{ -}2, \\z& = -1 - t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& = 2 + 2m, & \\y& = 2 +\phantom{ 2}m, \\z& = 1 -\phantom{ 2}m;\ m\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}4 + 2k, & \\y& =\phantom{ -}2 +\phantom{ 2}k, \\z& = -1 -\phantom{ 2}k;\ k\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& = -2 + 2u, & \\y& =\phantom{ -}2, \\z& =\phantom{ -}1 - 2u;\ u\in \mathbb{R} \\ \end{aligned}\)

9000104503

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

9000104504

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

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.