C

9000028406

Level: 
C
Find the condition which is equivalent to the fact that the equation \(ax^{2} + bx + c = 0\) with \(x\in \mathbb{R}\) and real coefficients \(a\), \(b\), \(c\) has a solution in a form of a pair of two opposite real nonzero numbers.
\(\frac{c} {a} < 0\text{ and }b = 0\)
\(- \frac{b} {2a} = 0\)
\(b^{2} = 4ac\text{ and }a\not = 0\)
\(b^{2} = 4ac\text{ and }a\not = 0\text{ and }c\not = 0\)

9000028403

Level: 
C
Find the condition which is equivalent to the fact that the equation \(ax^{2} + bx + c = 0\) with \(x\in \mathbb{R}\) and real coefficients \(a\), \(b\), \(c\) has two real solutions \(x_{1}\neq x_{2}\), \(x_{1} > 0\), \(x_{2} > 0\).
\(b^{2} - 4ac > 0\text{ and }\frac{c} {a} > 0\text{ and }\frac{b} {a} < 0\)
\(a\not = 0\text{ and }c > 0\)
\(a > 0\text{ and }b < 0\text{ and }c > 0\text{ and }b^{2} - 4ac > 0\)
\(a\not = 0\text{ and }c > 0\text{ and }b^{2} - 4ac > 0\)

9000028402

Level: 
C
Find the condition which is equivalent to the fact that the equation \(ax^{2} + bx + c = 0\) with \(x\in \mathbb{R}\) and real coefficients \(a\), \(b\), \(c\) has two real solutions satisfying \(x_{1} = 0\) and \(x_{2}\neq 0\).
\(c = 0\text{ and }a\not = 0\text{ and }b\not = 0\)
\((a = b = 0)\text{ and }c\not = 0\)
\(a\not = 0\text{ and }c = 0\)
\(b\not = 0\text{ and }c = 0\)

9000028401

Level: 
C
Find the condition which is equivalent to the fact that the equation \(ax^{2} + bx + c = 0\) with \(x\in \mathbb{R}\) and real coefficients \(a\), \(b\), \(c\) has at least one real solution.
\((b^{2} - 4ac\geq 0\text{ and }a\not = 0)\text{ or }(a = 0\text{ and }b\not = 0)\text{ or }(a = b = c = 0)\)
\(a\not = 0\text{ and }b^{2} - 4ac\geq 0\)
\(b^{2} - 4ac\leq 0\)
\((b^{2} - 4ac\geq 0\text{ and }a\not = 0)\text{ or }(a = 0)\text{ or }(b = 0)\)

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.

9000031108

Level: 
C
Solve the following system of equations and identify a correct statement. \[\begin{aligned} 2x^{2} - y = 2 & & \\\left |x\right | + y = 1 & & \end{aligned}\]
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.

9000031109

Level: 
C
Solve the following system of equations and identify a correct statement. \[\begin{aligned} \left |x\right | = x + y & & \\\left |y\right | = 1 + x & & \end{aligned}\]
The system has a unique solution.
The system does not have any solution.
The system has two solutions.
The system has more than two solutions.

9000031106

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

9000031107

Level: 
C
Solve the following system of equations and identify a correct statement. \[\begin{aligned} \sqrt{x} = y & & \\x^{2} + y^{2} = 6 & & \end{aligned}\]
The system has a unique solution.
The system does not have any solution.
The system has two solutions.
The system has more than two solutions.