Equations and inequalities with parameters

2000019104

Level:

Consider the following equation with a parameter

a

5 x - a = a x + 4

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

a

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

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

\begin{array}{cc} Parameter & Solution set \\ a = 5 & R \\ a \neq 5 & \emptyset \end{array}

2000019103

Level:

Determine the set of all values of the parameter

a \in R ∖ {3}

for which the given equation has no solution.

\frac{5 x - 2}{a - 3} = 4 + \frac{2 x}{3}

{\frac{21}{2}}

{\frac{2}{5}}

{- 3}

{0}

2000019102

Level:

Determine the set of all values of the real parameter

a

for which the given equation has no solution.

a^{2} x = x + a

{- 1; 1}

{- 1; 0; 1}

{1}

{0; 1}

2000019101

Level:

Determine the set of all values of the parameter

a \in R ∖ {0}

for which the given equation has no solution.

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

{\frac{1}{2}}

{\frac{1}{2}; 2}

{1}

{\frac{1}{2}; 1}

2010008408

Level:

Solve the following equation with unknown

x

and a real parameter

a \in R ∖ {1}

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

\begin{array}{cc} Parameter & Solution set \\ a = 2 & \emptyset \\ a \notin {1; 2} & \frac{a - a^{2}}{2 - a} \end{array}

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

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

2010008407

Level:

Determine the set of all values of the real parameter

a

for which the equation will have exactly one solution.

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

R ∖ {0; - 2}

{0; \frac{1}{3}}

R ∖ {0; - 2}

R

2010008406

Level:

Consider equation

q (3 - q) x = 6 - 2 q

with a real parameter

q

. Solve the equation for

q = 3

R

\emptyset

R ∖ {0}

{\frac{6 - 2 q}{q (3 - q)}}

2010008405

Level:

Consider the equation

x^{2} (1 - q) + 2 x + 1 + q = 0

with the real parameter

q

. Solve the equation for

q = - 1

{- 1; 0}

{- 1; 1}

{0; 1}

\emptyset

2010008404

Level:

Identify the set of the values of the real parameter

t

for which the following equation has no solution in

R

x^{2} + (t + 2) x + 1 = 0

(- 4; 0)

(- \infty; - 4) \cup (0; \infty)

(- \infty; - 4)

(0; \infty)

2010008403

Level:

Identify the set of the values of the real parameter

d

for which the following equation has two different real roots.

x^{2} - 2 d x + 2 d^{2} - 9 = 0

(- 3; 3)

(- \infty; - 3) \cup (3; \infty)

(- \infty; - 3)

(3; \infty)

Equations and inequalities with parameters

2000019104

2000019103

2000019102

2000019101

2010008408

2010008407

2010008406

2010008405

2010008404

2010008403

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