Equations and inequalities with parameters

9000104307

Level:

Assuming

a \in (0; 2)

, solve the following inequality.

a (a - 2) x > 1

(- \infty; \frac{1}{a (a - 2)})

(\frac{1}{a (a - 2)}; \infty)

\emptyset

{\frac{1}{a (a - 2)}}

9000104310

Level:

Assuming

a \in (0; 1)

, solve the following inequality.

2 a (1 - a) x > 3

(\frac{3}{2 a (1 - a)}; \infty)

(- \frac{3}{2 a (1 - a)}; \infty)

(- \frac{3}{2 a (1 - a)}; \frac{3}{2 a (1 - a)})

(- \infty; \frac{3}{2 a (1 - a)})

2000019105

Level:

Consider the following equation with a parameter

a

\frac{2 x - a}{x - 5} = a

Choose the table that summarizes solutions of the equation according to the value of

a

\begin{array}{cc} Parameter & Solution set \\ a \in {2; 10} & \emptyset \\ a \in R ∖ {2; 10} & {\frac{4 a}{a - 2}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 5 & \emptyset \\ a \neq 5 & {\frac{4 a}{a - 2}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 2 & \emptyset \\ a \neq 5 & {\frac{4 a}{a - 2}} \end{array}

2000019106

Level:

Consider the following equation with a parameter

a

\frac{x - a}{x - 3} = 2 a

Choose the table that summarizes solutions of the equation according to the value of

a

\begin{array}{cc} Parameter & Solution set \\ a \in {\frac{1}{2}; 3} & \emptyset \\ a \in R ∖ {\frac{1}{2}; 3} & {\frac{5 a}{2 a - 1}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 3 & \emptyset \\ a \neq 3 & {\frac{5 a}{2 a - 1}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = \frac{1}{2} & \emptyset \\ a \neq \frac{1}{2} & {\frac{5 a}{2 a - 1}} \end{array}

2000019109

Level:

Determine the set of all values of the parameter

a \in R ∖ {0}

for which the equation has a unique solution.

\frac{x - 1}{x} = \frac{2 - a}{3 a}

R ∖ {\frac{1}{2}; 0}

R ∖ {0; 2; \frac{1}{2}}

R ∖ {0}

R ∖ {\frac{1}{3}; 0; 2; 1}

2000019110

Level:

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

R ∖ {- 6; 4}

R ∖ {- 1; - 2; 1}

R ∖ {0; - 1}

R ∖ {4}

9000104503

Level:

Solve the following equation with unknown

x

and a real parameter

a \in R

\frac{a^{2} (x - 1)}{a x - 2} = 2

\begin{array}{cc} Parameter & Solution set \\ a = 0 & \emptyset \\ a = 2 & R ∖ {1} \\ a \notin {0; 2} & {\frac{a + 2}{a}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a \in {0; 2} & R \\ a \notin {0, 2} & {\frac{a + 2}{a}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 0 & \emptyset \\ a = 2 & R \\ a \notin {0; 2} & {\frac{a + 2}{a}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 0 & R ∖ {1} \\ a = 2 & \emptyset \\ a \notin {0; 2} & {\frac{a + 2}{a}} \end{array}

9000104504

Level:

Solve the following equation with unknown

x

and a real parameter

a \in R ∖ {0}

\frac{1}{x - a} + 1 = \frac{1}{a}

\begin{array}{cc} Parameter & Solution set \\ a = 1 & \emptyset \\ a \notin {0, 1} & {\frac{a (a - 2)}{a - 1}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 1 & R ∖ {1} \\ a \notin {0; 1} & {\frac{a (a - 2)}{a - 1}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 1 & R \\ a \notin {0, 1} & {\frac{a (a - 2)}{a - 1}} \end{array}

9000140001

Level:

Consider the equation

\frac{4 a}{x} - \frac{1}{a x} + \frac{2}{a} = 4

with unknown

x

and a parameter

a \in R ∖ {0}

. Identify a true statement.

a = \frac{1}{2}

, then the solution is

x \in R ∖ {0}

a = \frac{1}{2}

, then the equation has no solution.

a = \frac{1}{2}

, then the solution is

x \in R

9000140004

Level:

Solve the following equation with unknown

x

and a real parameter

a \in R

\frac{a^{2} (x - 1)}{a x - 3} = 3

\begin{array}{cc} Parameter & Solution set \\ a = 0 & \emptyset \\ a = 3 & R ∖ {1} \\ a \notin {0; 3} & {\frac{a + 3}{a}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 0 & \emptyset \\ a = 3 & {1} \\ a \notin {0; 3} & {\frac{a + 3}{a}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a \in {0; 3} & \emptyset \\ a \notin {0; 3} & {\frac{a + 3}{a}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 0 & \emptyset \\ a = 3 & R \\ a \notin {0; 3} & {\frac{a + 3}{a}} \end{array}

Equations and inequalities with parameters

9000104307

9000104310

2000019105

2000019106

2000019109

2000019110

9000104503

9000104504

9000140001

9000140004

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