Systems of nonlinear equations and inequalities

9000031101

Level:

Solve the following system of equations and identify a correct statement.

\begin{aligned} (x - 3)^{2} + (y - 1)^{2} = 1 \\ 2 x^{2} + 2 y^{2} - 12 x - 4 y + 18 = 0 \end{aligned}

The system has more than two solutions.

The system does not have any solution.

The system has a unique solution.

The system has two solutions.

9000031102

Level:

Solve the following system of equations and identify a correct statement.

\begin{aligned} (x - 1)^{2} + y^{2} = 1 \\ (x - 4)^{2} + y^{2} = 4 \end{aligned}

The system has a unique solution

[x, y]

, where

y = 0

The system does not have any solution.

The system has a unique solution

[x, y]

, where

y > 0

The system has two solutions

[x_{1}, y_{1}]

[x_{2}, y_{2}]

, where

y_{1} = - y_{2}

9000031103

Level:

Solve the following system of equations in

R \times R

and identify the correct statement.

\begin{aligned} x - 2 y + 5 = 0 \\ x^{2} + y^{2} = 9 \end{aligned}

The system has two solutions.

The system does not have any solution.

The system has a unique solution.

The system has more than two solutions.

2000017704

Level:

Assuming

x \in R

, find the solution set of the following system of inequalities.

\begin{aligned} 2 x - [x - (2 x + 1)] \cdot 3 & > (3 + x) - 2 (1 - x) - 2 x + 6 \\ x^{2} - 3 \cdot [x - 2 x (1 - x)] & < 5 (10 - x^{2}) - 2 x \end{aligned}

(1; 10)

\emptyset

(- 10; 1)

{1; 10}

2000020301

Level:

Solve the given system of equations in the set of real numbers.

\begin{aligned} x + y & = - 5 \\ 1 + \sqrt{2 x + 4 y} & = \sqrt{x + 3 y} \end{aligned}

In the following list identify a true statement.

x = - 12, y = 7

x = 12, y = 7

The system has no solution.

The system has infinitely many solutions.

2010011206

Level:

Consider the system

\begin{aligned} y & = \frac{k}{x}, \\ y & = a, \end{aligned}

where

a

k

are real parameters and

x

y

are real variables. Determine the conditions for

a

and

k

so that the system has a unique solution in

R^{+} \times R^{-}

a < 0

and

k < 0

a < 0

and

k > 0

a > 0

and

k < 0

a > 0

and

k > 0

9000009909

Level:

Consider the system

\begin{aligned} y & = \frac{k}{x}, \\ y & = a, \end{aligned}

where

a

k

are real parameters and

x

y

are real variables. Determine the conditions for

a

and

k

so that the system has a unique solution in

R^{-} \times R^{-}

a < 0

and

k > 0

a < 0

and

k < 0

a > 0

and

k < 0

a > 0

and

k > 0

9000020904

Level:

Determine all the values of the parameter

c \in R

so that the following system has two solutions in

R \times R

\begin{aligned} x^{2} & + & y^{2} & = 2 \\ x & + & c & = y \end{aligned}

| c | < 2

| c | = 2

| c | > 2

c = 2

9000020905

Level:

Find the condition on the parameter

c \in R

which ensures that the following system has a unique solution in

R \times R

\begin{aligned} x^{2} & + & y^{2} & = 2 \\ x & + & c & = y \end{aligned}

| c | = 2

| c | > 2

| c | < 2

c = 2

9000020908

Level:

Assuming that the real parameter

c

satisfies

c > 16

, solve the system and identify a true statement.

\begin{aligned} y^{2} & - & 4 x & = 0 \\ 8 & x & - & 4 y & + c & = 0 \end{aligned}

The system has no solution.

The system has two solutions.

The system has a unique solution.

The system has infinitely many solutions.

Systems of nonlinear equations and inequalities

9000031101

9000031102

9000031103

2000017704

2000020301

2010011206

9000009909

9000020904

9000020905

9000020908

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