C

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

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 \(x_{1}\neq x_{2}\), \(x_{1} > 0\), \(x_{2} > 0\).
\(b^{2} - 4ac > 0\text{ and }\frac{c} {a} > 0\text{ and }\frac{b} {a} < 0\)
\(a\not = 0\text{ and }c > 0\)
\(a > 0\text{ and }b < 0\text{ and }c > 0\text{ and }b^{2} - 4ac > 0\)
\(a\not = 0\text{ and }c > 0\text{ and }b^{2} - 4ac > 0\)

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?
\(\begin{aligned}2x - y&\leq 2 & \\2x + y&\geq - 2 \\ \end{aligned}\)
\(\begin{aligned}2x - y&\geq 2 & \\2x + y&\geq - 2 \\ \end{aligned}\)
\(\begin{aligned}2x - y&\leq 2 & \\2x + y&\leq - 2 \\ \end{aligned}\)
\(\begin{aligned}2x - y&\geq 2 & \\2x + y&\leq - 2 \\ \end{aligned}\)

9000026009

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?
\(\begin{aligned}2y -\phantom{ 2}x& < 4& \\x - 2y & < 2 \\ \end{aligned}\)
\(\begin{aligned}2y -\phantom{ 2}x& < 4& \\x - 2y & > 2 \\ \end{aligned}\)
\(\begin{aligned}2y - x& > 4 & \\2y - x& < -2 \\ \end{aligned}\)
\(\begin{aligned}2y - x& > 4 & \\2y - x& > -2 \\ \end{aligned}\)

9000026010

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?
\(\begin{aligned}x &\leq 3 & \\5x& < 9 - 3y \\ \end{aligned}\)
\(\begin{aligned}x & < 3 & \\5x& < 9 - 3y \\ \end{aligned}\)
\(\begin{aligned}x & > 3 & \\5x& < 9 - 3y \\ \end{aligned}\)
\(\begin{aligned}x &\leq 3 & \\5x& > 9 - 3y \\ \end{aligned}\)

9000025808

Level: 
C
In the following list identify a true statement on the function \(f\). \[ f(x) = \frac{(x - 1)(x + 2)} {(2x + 1)(3 - 2x)} \]
\(f(x) > 0 \iff x\in \left (-2;-\frac{1} {2}\right )\cup \left (1; \frac{3} {2}\right )\)
\(f(x) > 0 \iff x\in (-\infty ;-2)\cup \left (-\frac{1} {2};1\right )\cup \left (\frac{3} {2};\infty \right )\)
\(f(x) > 0 \iff x\in (-\infty ;-2)\cup (1;\infty )\)
\(f(x) > 0 \iff x\in \left (-2; \frac{3} {2}\right )\)