Equations and inequalities with parameters

2000019110

Level: 
C
Determine the set of all values of the real parameter \( a \) for which the equation has a unique solution. \[ \frac{a(x+2)-3(x-1)}{x+1} = 1 \]
\(\mathbb{R} \setminus \{-6;4\}\)
\(\mathbb{R} \setminus \{-1;-2;1\}\)
\(\mathbb{R} \setminus \{0;-1\}\)
\(\mathbb{R} \setminus \{4\}\)

2000019109

Level: 
C
Determine the set of all values of the parameter \( a \in \mathbb{R} \setminus \{0\}\) for which the equation has a unique solution. \[ \frac{x-1}{x} = \frac{2-a}{3a} \]
\(\mathbb{R} \setminus \left\{\frac12;0\right\}\)
\(\mathbb{R} \setminus \left\{0;2;\frac12\right\}\)
\(\mathbb{R} \setminus \{0\}\)
\(\mathbb{R} \setminus \left\{\frac13;0;2;1\right\}\)

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 \(a\).
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=5 & \emptyset \\ a \neq 5 & \left\lbrace\frac{a+4}{5-a}\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=5 & \mathbb{R} \\ a \neq 5 & \left\lbrace\frac{a+4}{5-a}\right\rbrace \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=5 & \mathbb{R}\\ a \neq 5 & \emptyset \\\hline \end{array}\)