A

9000106610

Level: 
A
Determine whether two lines are identical, parallel, intersecting or skew. The first line is the line passes through the points \(A = [1;-4;2]\) and \(B = [3;0;0]\) and the second line is the line passes through the points \(C = [3;-5;5]\) and \(D = [-1;-3;-1]\).
intersecting lines
parallel lines, not identical
identical lines
skew lines

9000104501

Level: 
A
Consider equation \[ \frac{x - 3} {a} = \frac{a - x} {3} + 2 \] with an unknown \(x\in \mathbb{R}\) and a real parameter \(a\in \mathbb{R}\setminus \{0\}\). Identify a statement which is not true.
For \(a\mathrel{\in }\{ - 3;0\}\) we have \(x = \frac{1} {a+3}\).
For \(a\mathrel{\notin }\{ - 3;0\}\) we have \(x = a + 3\).
If \(a = -3\), then the equation has infinitely many solutions.

9000104505

Level: 
A
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\setminus\{-3;3\}\). \[\frac{a-x} {a-3} - \frac{6a} {a^{2}-9} = \frac{x-3} {a+3} \]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \emptyset \\ a\notin\{-3;0;3\} & \left\lbrace\frac{a^2-9}{2a}\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \mathbb{R} \\ a\notin\{-3;0;3\} & \left\lbrace\frac{a^2-9}{2a}\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \mathbb{R}\setminus\{0\} \\ a\notin\{-3;0;3\} & \left\lbrace\frac{a^2-9}{2a}\right\rbrace \\\hline \end{array}\)

9000101810

Level: 
A
Given points \(A = [1;2]\) and \(B = [4;4]\), find the point \(X\) on the \(x\)-axis such that the distance from \(X\) to \(B\) is a double of the distance from \(X\) to \(A\). Find all solutions of the problem.
\(X_{1} = [2;0],\ X_{2} = [-2;0]\)
\(X = [2;0]\)
\(X = [8;0]\)
\(X_{1} = [2;0],\ X_{2} = [-4;0]\)

9000104502

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

9000101804

Level: 
A
In the following list identify a valid relation involving the vectors \(\vec{a} = (2;-3)\), \(\vec{b} = (1;3)\) and \(\vec{c} = (5;-3)\).
\(\vec{c} = 2\vec{a} +\vec{ b}\)
\(\vec{b} = \frac{1} {2}\vec{a} +\vec{ c}\)
\(2\vec{a} +\vec{ b} +\vec{ c} =\vec{ o}\)
\(\vec{a} = \frac{1} {2}\vec{b} +\vec{ c}\)