Equations and inequalities with parameters

9000104404

Level:

Find a set of the values of the real parameter

a

which ensure that the following equation has infinitely many solutions.

a^{2} x + 1 = a^{2} + a x

{1}

{- 1; 1}

{0}

{- 1}

9000104405

Level:

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 = 3 a^{2} x + a

R ∖ {0; 3}

{0}

{0; 3}

R ∖ {3}

9000104501

Level:

Consider equation

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

with an unknown

x \in R

and a real parameter

a \in R ∖ {0}

. Identify a statement which is not true.

For

a \in {- 3; 0}

we have

x = \frac{1}{a + 3}

For

a \notin {- 3; 0}

we have

x = a + 3

a = - 3

, then the equation has infinitely many solutions.

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}

9000104301

Level:

Assuming

a < 0

, solve the following inequality.

3 x + 2 a \geq 0

[- \frac{2 a}{3}; \infty)

(- \infty; - \frac{2 a}{3}]

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

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

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}

9000104302

Level:

Assuming

a = 0

, solve the following inequality.

2 a x + 4 a < 1

R

\emptyset

(\frac{1 - 4 a}{2 a}; \infty)

(- \infty; \frac{1 - 4 a}{2 a})

9000104505

Level:

Solve the following equation with unknown

x

and a real parameter

a \in R ∖ {- 3; 3}

\frac{a - x}{a - 3} - \frac{6 a}{a^{2} - 9} = \frac{x - 3}{a + 3}

\begin{array}{cc} Parameter & Solution set \\ a = 0 & \emptyset \\ a \notin {- 3; 0; 3} & {\frac{a^{2} - 9}{2 a}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 0 & R \\ a \notin {- 3; 0; 3} & {\frac{a^{2} - 9}{2 a}} \end{array}

\begin{array}{cc} Parameter & Solution set \\ a = 0 & R ∖ {0} \\ a \notin {- 3; 0; 3} & {\frac{a^{2} - 9}{2 a}} \end{array}

9000104303

Level:

Assuming

a < 3

, solve the following inequality.

a x - 3 \geq 3 x - a

(- \infty; - 1]

(- \infty; - 1)

(- 1; \infty)

R

9000104401

Level:

Find a set of the values of the real parameter

a

which ensure that the following equation has no solution.

a^{2} x + 2 a x - 3 a = 0

{- 2}

{2}

{0}

{- 3; 1}

Equations and inequalities with parameters

9000104404

9000104405

9000104501

9000104503

9000104301

9000104504

9000104302

9000104505

9000104303

9000104401

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