Equations and inequalities with parameters

2000019101

Level: 
B
Determine the set of all values of the parameter \( a \in \mathbb{R} \setminus \{0\}\) for which the given equation has no solution. \[ \frac{x-1}{x} = \frac{2-a}{3a} \]
\(\left\{ \frac12\right\}\)
\(\left\{ \frac12; 2\right\}\)
\( \{ 1 \}\)
\(\left\{ \frac12; 1\right\}\)

2010008408

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

9000375401

Level: 
A
Find a set of the values of the real parameter \(a\) which ensure that the following equation has a unique solution. \[ a^{3}x + 4a - 1 = a^{2}x + 3 \]
\(\mathbb{R}\setminus \{0;1\}\)
\(\mathbb{R}\setminus \{ - 1;1\}\)
\(\mathbb{R}\setminus \{0\}\)
\(\mathbb{R}\setminus \{ - 1;0\}\)