Equations and Inequalities with Parameters
7400320164
Submitted by michaela.bailova on Fri, 10/18/2024 - 13:387400220164
Submitted by michaela.bailova on Fri, 10/18/2024 - 13:347400120164
Submitted by michaela.bailova on Fri, 10/18/2024 - 13:322000019110
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.
\[
\frac{x-1}{x} = \frac{2-a}{3a}
\]
\(\mathbb{R} \setminus \left\{\frac12,0\right\}\)
\(\mathbb{R} \setminus \left\{0,2,\frac12\right\}\)
\(\mathbb{R} \setminus \{0\}\)
\(\mathbb{R} \setminus \left\{\frac13,0,2,1\right\}\)
2000019108
Level:
A
Determine the set of all values of the parameter \( a \in \mathbb{R} \setminus \{-2,2\}\) for which the equation has an infinite number of solutions.
\[
\frac{x-a}{2-a} = \frac{x+a}{2+a}
\]
\( \{0\}\)
\( \{-1\}\)
\( \{1\}\)
\(\emptyset\)
2000019107
Level:
A
Determine the set of all values of the real parameter \(a\) for which the equation has an infinite number of solutions.
\[
a^2x+1=x+a
\]
\( \{1\}\)
\( \{-1\}\)
\( \{0\}\)
\(\emptyset\)