Equations and inequalities with parameters

2000019106

Level: 
C
Consider the following equation with a parameter \( a\). \[ \frac{x-a}{x-3}=2a \] Choose the table that summarizes solutions of the equation according to the value of \(a\).
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a \in \left\{\frac12;3\right\} & \emptyset \\ a \in \mathbb{R} \setminus \left\{\frac12;3\right\}& \left\lbrace\frac{5a}{2a-1}\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a =3 & \emptyset \\ a \neq 3& \left\lbrace\frac{5a}{2a-1}\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=\frac12 & \emptyset \\ a \neq \frac12 & \left\lbrace\frac{5a}{2a-1}\right\rbrace \\\hline \end{array}\)

2000019105

Level: 
C
Consider the following equation with a parameter \( a\). \[ \frac{2x-a}{x-5}=a \] Choose the table that summarizes solutions of the equation according to the value of \(a\).
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a \in \{2;10\} & \emptyset \\ a \in \mathbb{R} \setminus \{2;10\}& \left\lbrace\frac{4a}{a-2}\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=5 & \emptyset \\ a \neq 5 & \left\lbrace\frac{4a}{a-2}\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=2 & \emptyset \\ a \neq 5 & \left\lbrace\frac{4a}{a-2}\right\rbrace \\\hline \end{array}\)

2000019104

Level: 
A
Consider the following equation with a parameter \( a\). \[ 5x-a=ax+4 \] Choose the table that summarizes solutions of the equation according to the value of \(a\).
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=5 & \emptyset \\ a \neq 5 & \left\lbrace\frac{a+4}{5-a}\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=5 & \mathbb{R} \\ a \neq 5 & \left\lbrace\frac{a+4}{5-a}\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=5 & \mathbb{R}\\ a \neq 5 & \emptyset \\\hline \end{array}\)

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 \(a\in\mathbb{R}\setminus\{1\}\). \[\frac{x} {1-a} = a-x\]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=2 & \emptyset \\ a\notin\{1;2\} & \frac{a-a^2}{2-a} \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=2 & \mathbb{R} \\ a\notin\{1;2\} & \frac{a-a^2}{2-a} \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=2 & \mathbb{R} \\ a\notin\{1;2\} & \emptyset \\\hline \end{array}\)