Equations and Inequalities with Parameters

9000375401

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

9000140001

Level: 
C
Consider the equation \[ \frac{4a} {x} - \frac{1} {ax} + \frac{2} {a} = 4 \] with unknown \(x\) and a parameter \(a\in \mathbb{R}\setminus \{0\}\). Identify a true statement.
If \(a = \frac{1} {2}\), then the solution is \(x\in \mathbb{R}\setminus \{0\}\).
If \(a = \frac{1} {2}\), then the equation has no solution.
If \(a = \frac{1} {2}\), then the solution is \(x\in \mathbb{R}\).

9000140002

Level: 
A
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\setminus\{0\}\). \[ \frac{x+a} {a} = ax - 1\]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a\in\{-1,1\} & \emptyset \\ a\notin\{-1,0,1\} & \left\{\frac{2a}{(a-1)(a+1)}\right\} \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=-1 & \emptyset \\ a\notin\{-1,0\} & \left\{\frac{2a}{(a-1)(a+1)}\right\} \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a\in\{-1,1\} & \mathbb{R} \\ a\notin\{-1,0,1\} & \left\{\frac{2a}{(a-1)(a+1)}\right\} \\\hline \end{array}\)

9000140003

Level: 
A
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\setminus\{0\}\). \[ax - \frac{2} {a^{2}} = \frac{4x+1} {a} \]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=-2 & \mathbb{R} \\ a=2 & \emptyset \\ a\notin\{-2,0,2\} & \left\{\frac1{a(a-2)}\right\} \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=-2 & \mathbb{R}\setminus\{1\} \\ a=2 & \emptyset \\ a\notin\{-2,0,2\} & \left\{\frac1{a(a-2)}\right\} \\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=-2 & \emptyset \\ a=2 & \mathbb{R} \\ a\notin\{-2,0,2\} & \left\{\frac1{a(a-2)}\right\} \\\hline \end{array}\)

9000140004

Level: 
C
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\). \[ \frac{a^{2}(x-1)} {ax-3} = 3 \]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \emptyset \\ a=3 & \mathbb{R}\setminus\{1\} \\ a\notin\{0,3\} & \left\{\frac{a+3}a\right\} \\ \hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \emptyset \\ a=3 & \{1\} \\ a\notin\{0,3\} & \left\{\frac{a+3}a\right\} \\ \hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a\in\{0,3\} & \emptyset \\ a\notin\{0,3\} & \left\{\frac{a+3}a\right\} \\ \hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \emptyset \\ a=3 & \mathbb{R} \\ a\notin\{0,3\} & \left\{\frac{a+3}a\right\} \\ \hline \end{array}\)

9000140005

Level: 
C
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\setminus\{0\}\). \[\frac ax-\frac4{ax}=1-\frac2a\]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=-2 & \emptyset \\ a=2 & \mathbb{R}\setminus\{0\} \\ a\notin\{-2,0,2\} & \left\{a+2\right\} \\ \hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=2 & \mathbb{R}\setminus\{0\} \\ a\notin\{0,2\} & \left\{a+2\right\} \\ \hline \end{array}\)
\( \begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=2 & \mathbb{R} \\ a\notin\{0,2\} & \left\{a+2\right\} \\ \hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=2 & \mathbb{R}\setminus\{1\} \\ a\notin\{0,2\} & \left\{a+2\right\} \\ \hline \end{array}\)

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.