2000019102
Level:
A
Determine the set of all values of the real parameter \( a \) for which the given equation has no solution.
\[
a^2x=x+a
\]
\(\{ -1;1\}\)
\(\{ -1;0;1\}\)
\( \{ 1 \}\)
\( \{ 0;1\}\)
Equations and inequalities with parameters
2000019103
Level:
A
Determine the set of all values of the parameter \( a \in \mathbb{R} \setminus \{3\} \) for which the given equation has no solution.
\[
\frac{5x-2}{a-3} = 4+ \frac{2x}3
\]
\(\left\{\frac{21}2\right\}\)
\(\left\{ \frac25\right\}\)
\( \{ -3 \}\)
\( \{0\}\)
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 \(a\).
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=5 & \emptyset \\
a \neq 5 & \left\lbrace\frac{a+4}{5-a}\right\rbrace
\\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=5 & \mathbb{R} \\
a \neq 5 & \left\lbrace\frac{a+4}{5-a}\right\rbrace
\\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=5 & \mathbb{R}\\
a \neq 5 & \emptyset
\\\hline \end{array}\)
2000019107
Level:
A
Determine the set of all values of the real parameter \(a\) for which the equation has an infinite number of solutions.
\[
a^2x+1=x+a
\]
\( \{1\}\)
\( \{-1\}\)
\( \{0\}\)
\(\emptyset\)
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.
\[
\frac{x-a}{2-a} = \frac{x+a}{2+a}
\]
\( \{0\}\)
\( \{-1\}\)
\( \{1\}\)
\(\emptyset\)
2010008401
Level:
A
Find all the values of the parameter \(k\) so that the solution of the following equation is bigger than \(6\).
\[
2x - 9 = \frac{7x - 3k}
{3}
\]
\(k\in (-\infty;11 )\)
\(k\in \{11\}\)
\(k\in (11;\infty )\)
\(k\in (-\infty;13 )\)
2010008402
Level:
A
Find all the values of the parameter \(k\) so that the solution of the following equation is positive.
\[
kx -2= 3x -k
\]
\(k\in (2;3)\)
\(k\in (0;\infty )\)
\(k\in (3;\infty )\)
\(k\in (-\infty;2 )\)
2010008405
Level:
A
Consider the equation
\[
x^{2}(1 - q) + 2x + 1 + q = 0
\]
with the real parameter \(q\).
Solve the equation for \(q = -1\).
\(\left \{-1;0\right \}\)
\(\left \{-1;1\right \}\)
\(\left \{0;1\right \}\)
\(\emptyset \)
2010008406
Level:
A
Consider equation
\[
q(3 - q)x = 6-2q
\]
with a real parameter \(q\).
Solve the equation for \(q = 3\).
\( \mathbb{R}\)
\( \emptyset \)
\( \mathbb{R}\setminus \{0\}\)
\( \left\{\frac{6-2q}{q(3-q)} \right\} \)
2010008407
Level:
A
Determine the set of all values of the real parameter \(a\) for which the equation will have exactly one solution.
\[
a^{2}x + 2ax - 3a = 0
\]
\( \mathbb{R}\setminus \{0;-2\}\)
\(\left\{0;\frac13\right\}\)
\( \mathbb{R}\setminus \{0;-2\}\)
\( \mathbb{R}\)