Systems of nonlinear equations and inequalities

9000031101

Level: 
B
Solve the following system of equations and identify a correct statement. \[\begin{aligned} (x - 3)^{2} + (y - 1)^{2} = 1 & & \\2x^{2} + 2y^{2} - 12x - 4y + 18 = 0 & & \end{aligned}\]
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: 
B
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 \(\left [x,y\right ]\), where \(y = 0\).
The system does not have any solution.
The system has a unique solution \(\left [x,y\right ]\), where \(y > 0\).
The system has two solutions \(\left [x_{1},y_{1}\right ]\), \(\left [x_{2},y_{2}\right ]\), where \(y_{1} = -y_{2}\).

9000031103

Level: 
B
Solve the following system of equations in \( \mathbb{R} \times \mathbb{R}\) and identify the correct statement.\[\begin{aligned} x - 2y + 5 = 0 & & \\x^{2} + y^{2} = 9 & & \end{aligned}\]
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: 
C
Assuming \( x \in \mathbb{R}\), find the solution set of the following system of inequalities. \[\begin{aligned} 2x- [x-(2x+1)]\cdot 3 &> (3+x)-2(1-x)-2x+6 \\ x^2-3\cdot [x-2x(1-x)] &< 5(10-x^2)-2x \end{aligned}\]
\( (1;10)\)
\( \emptyset \)
\( (-10;1)\)
\( \{1;10\}\)

2000020301

Level: 
C
Solve the given system of equations in the set of real numbers. \[ \begin{aligned} x+y&=-5\\ 1+\sqrt{2x+4y}&=\sqrt{x+3y}\\ \end{aligned}\] 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: 
C
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 \(\mathbb{R}^{+}\times \mathbb{R}^{-}\).
\(a < 0\) and \(k < 0\)
\(a < 0\) and \(k > 0\)
\(a > 0\) and \(k < 0\)
\(a > 0\) and \(k > 0\)

9000009909

Level: 
C
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 \(\mathbb{R}^{-}\times \mathbb{R}^{-}\).
\(a < 0\) and \(k > 0\)
\(a < 0\) and \(k < 0\)
\(a > 0\) and \(k < 0\)
\(a > 0\) and \(k > 0\)

9000020904

Level: 
C
Determine all the values of the parameter \(c\in \mathbb{R}\) so that the following system has two solutions in \(\mathbb{R}\times \mathbb{R}\). \[ \begin{alignedat}{80} &x^{2} & + &y^{2} & = 2 & & & & & & \\ &x & + &c & = y & & & & & & \\\end{alignedat}\]
\(|c| < 2\)
\(|c| = 2\)
\(|c| > 2\)
\(c = 2\)

9000020905

Level: 
C
Find the condition on the parameter \(c\in \mathbb{R}\) which ensures that the following system has a unique solution in \(\mathbb{R}\times \mathbb{R}\). \[ \begin{alignedat}{80} &x^{2} & + &y^{2} & = 2 & & & & & & \\ &x & + &c & = y & & & & & & \\\end{alignedat}\]
\(|c| = 2\)
\(|c| > 2\)
\(|c| < 2\)
\(c = 2\)

9000020908

Level: 
C
Assuming that the real parameter \(c\) satisfies \(c > 16\), solve the system and identify a true statement. \[ \begin{alignedat}{80} &y^{2} & - &4x & & = 0 & & & & & & \\8 &x & - &4y & + c & = 0 & & & & & & \\\end{alignedat}\]
The system has no solution.
The system has two solutions.
The system has a unique solution.
The system has infinitely many solutions.