C

9000028410

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 solutions and one of the solutions is a reciprocal value of the second solution.
\(b^{2} - 4ac > 0\text{ and }\frac{c} {a} = 1\)
\(b^{2} - 4ac > 0\text{ and }a = c\)
\(b^{2} - 4ac > 0\text{ and }\frac{c} {a} = -1\)
\(b^{2} - 4ac > 0\text{ and }a = -c\)

9000028409

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 \(a\), \(b\), \(c\) does not have a real solution.
\((b^{2} - 4ac < 0\text{ and }a\not = 0)\text{ or }(a = b = 0\text{ and }c\not = 0)\)
\(b^{2} - 4ac < 0\)
\(b^{2} - 4ac < 0\text{ and }a\not = 0\)
\((b^{2} - 4ac < 0\text{ and }a\not = 0)\text{ or }(ab = 0\text{ and }c\not = 0)\)

9000028408

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 and one of the solutions is bigger than the other one.
\(b^{2} - 4ac > 0\text{ and }a\not = 0\)
\(b^{2} - 4ac\not = 0\text{ and }a\not = 0\)
\(- \frac{b} {2a} > \frac{\sqrt{b^{2 } -4ac}} {2a} \)
\(- \frac{b} {2a} < \frac{\sqrt{b^{2 } -4ac}} {2a} \)

9000028407

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 unique positive and a unique negative real solution.
\(b^{2} - 4ac > 0\text{ and }\frac{c} {a} < 0\)
\(b^{2} - 4ac > 0\text{ and } - \frac{b} {2a} < 0\)
\(\left (\frac{c} {a} < 0\right )\text{ and }\left (\frac{b} {a} > 0\right )\)
\(\left (\frac{c} {a} < 0\right )\text{ and }\left (\frac{b} {a} < 0\right )\)

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.
\(\frac{c} {a} < 0\text{ and }b = 0\)
\(- \frac{b} {2a} = 0\)
\(b^{2} = 4ac\text{ and }a\not = 0\)
\(b^{2} = 4ac\text{ and }a\not = 0\text{ and }c\not = 0\)

9000026006

Level: 
C
In the picture, the shaded region corresponds to the set of points that is the solution to one of the given systems of inequalities. Which of the systems is it?
\(\begin{aligned}x +\phantom{ 2}y&\geq 3 & \\y - 2x& < -1 \\ \end{aligned}\)
\(\begin{aligned}x +\phantom{ 2}y& > 3 & \\y - 2x& < -1 \\ \end{aligned}\)
\(\begin{aligned}x +\phantom{ 2}y&\leq 3 & \\y - 2x& < -1 \\ \end{aligned}\)
\(\begin{aligned}x +\phantom{ 2}y& < 3 & \\y - 2x& > -1 \\ \end{aligned}\)

9000026007

Level: 
C
In the picture, the shaded region corresponds to the set of points that is the solution to one of the given systems of inequalities. Which of the systems is it?
\(\begin{aligned}y & < 2 & \\y + 1&\geq x + 1 \\ \end{aligned}\)
\(\begin{aligned}y &\geq 2 & \\y + 1& < x + 1 \\ \end{aligned}\)
\(\begin{aligned}y & > 2 & \\y + 1&\leq x + 1 \\ \end{aligned}\)
\(\begin{aligned}y&\leq 2 & \\y& > x \\ \end{aligned}\)

9000026008

Level: 
C
In the picture, the shaded region corresponds to the set of points that is the solution to one of the given systems of inequalities. Which of the systems is it?
\(\begin{aligned}2x - y&\leq 2 & \\2x + y&\geq - 2 \\ \end{aligned}\)
\(\begin{aligned}2x - y&\geq 2 & \\2x + y&\geq - 2 \\ \end{aligned}\)
\(\begin{aligned}2x - y&\leq 2 & \\2x + y&\leq - 2 \\ \end{aligned}\)
\(\begin{aligned}2x - y&\geq 2 & \\2x + y&\leq - 2 \\ \end{aligned}\)