Systems of nonlinear equations and inequalities

2010006704

Level: 
B
Determine all the values of the parameter \(c\in \mathbb{R}\) so that the following system has two solutions in \(\mathbb{R}\times \mathbb{R}\). \[ \begin{alignedat}{80} &x^{2} & + &2y^{2} & = 6 & & & & & & \\ &x & + &y & = c & & & & & & \\\end{alignedat}\]
\(|c| < 3\)
\(|c| =3\)
\(|c| > 3\)
\(|c| \in \mathbb{R}\)

2010006703

Level: 
B
Identify a true statement related to the solution of the following system in \(\mathbb{R}\times \mathbb{R}\). \[ \begin{alignedat}{80} &x^{2} & + &2 & &y^{2} & - & &4x & = &0 & & & & & & & & & & & & \\ &x & + & & &y & = &4 & & & & & & & & & & & & \\\end{alignedat}\]
The system has two solutions.
The system has a unique solution.
The system does not have any solution.
The system has infinitely many solutions.

2010006506

Level: 
B
Solve the following system of equations in \( \mathbb{R} \times \mathbb{R}\) and identify the correct statement. \[\begin{aligned} x^2 - y^2 = 5 & & \\2x + y = 1 & & \end{aligned}\]
The system does not have any solution.
The system has a unique solution.
The system has more than two solutions.
The system has two solutions.

1003160803

Level: 
A
Use the substitution method to find the solution \( [x;y] \) of the following system of equations. \[ \begin{aligned} \frac{x+y}x+\frac1{x+y}=1 \\ \frac{2\cdot(x+y)}x-\frac1{x+y}=-7 \end{aligned} \]
\( \left[-\frac16;\frac12\right] \)
\( [-2;3] \)
\( \left[-\frac12;-\frac12\right] \)
\( \left[\frac12;\frac3{-2}\right] \)

1003160801

Level: 
A
Use the substitution method to find the solution \( [x;y] \) of the following system of equations. \[ \begin{aligned} \frac2{x+4}-\frac1{2-y}=-6 \\ \frac1{x+4}+\frac5{2-y}=8 \end{aligned} \]
\( \left[-\frac92;\frac32\right] \)
\( [-2;2] \)
\( [2;10] \)
\( \left[-\frac92;3\right] \)

1103085403

Level: 
A
Two resistors of unknown resistances \( R_1 \) and \( R_2 \), where \( R_1 < R_2 \) are connected in series (Figure A) and total resistance of the circuit is \( R_S=64\,\Omega \). If the resistors are connected in parallel (Figure B), the total resistance \( R_P=15\,\Omega \). Find \( R_1 \).
\( 24 \)
\( 22 \)
\( 12 \)
\( 15 \)

9000031110

Level: 
C
Solve the following system of equations and identify a correct statement. \[\begin{aligned} \left |x - 2\right | & = y & & \\\left |y + 2\right | & = x - 6 & & \end{aligned}\]
The system has no solution.
The system has a unique solution.
The system has two solutions.
The system has more than two solutions.

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.

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.