A

9000106606

Level: 
A
Determine whether the following two lines are identical, parallel, intersecting or skew. \[ \begin{aligned}[t] p\colon x& = 2t, & \\y & = 3 - t, \\z & = 4 - t;\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}[t] q\colon x& = 2 - 2s, & \\y & = -1 + s, \\z & = 6 + 3s;\ s\in \mathbb{R} \\ \end{aligned} \]
skew lines
parallel lines, not identical
intersecting lines
identical lines

9000106607

Level: 
A
Determine whether the following two lines are identical, parallel, intersecting or skew. \[\begin{aligned} p\colon &x = 2, &q\colon &x =\phantom{ -}1 -\phantom{ 3}s, & & & & \\ &y = 3 -\phantom{ 2}t, & &y =\phantom{ -}2 + 3s, & & & & \\ &z = 3 + 2t;\ t\in \mathbb{R}, & &z = -1 - 2s;\ s\in \mathbb{R} & & & & \end{aligned}\]
skew lines
parallel lines, not identical
intersecting lines
identical lines

9000104402

Level: 
A
Find a set of the values of the real parameter \(a\) which ensure that the following equation has no solution. \[ 2a^{2}x - ax - 2a = -1 \]
\(\left \{0\right \}\)
\(\left \{\frac{1} {2}\right \}\)
\(\left \{-\frac{1} {2}\right \}\)
\(\left \{-\frac{1} {2}; \frac{1} {2}\right \}\)

9000104403

Level: 
A
Find a set of the values of the real parameter \(a\) which ensure that the following equation has infinitely many solutions. \[ 3a^{2}x - 2ax + 4 = 6a \]
\(\left \{\frac{2} {3}\right \}\)
\(\left \{-\frac{2} {3}\right \}\)
\(\left \{0\right \}\)
\(\left \{0; \frac{2} {3}\right \}\)

9000104405

Level: 
A
Find a set of the values of the real parameter \(a\) which ensure that the following equation has a unique solution. \[ a^{3}x + 3 = 3a^{2}x + a \]
\(\mathbb{R}\setminus \left \{0;3\right \}\)
\(\left \{0\right \}\)
\(\left \{0;3\right \}\)
\(\mathbb{R}\setminus \left \{3\right \}\)

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.
\(X_{1} = [2;0],\ X_{2} = [-2;0]\)
\(X = [2;0]\)
\(X = [8;0]\)
\(X_{1} = [2;0],\ X_{2} = [-4;0]\)