C

9000025803

Level: 
C
Find all intersections of the graph of the following function with \(x\)-axis. \[ f(x) = \frac{2x + 1} {x^{2} - x - 6} \]
\(X = \left [-\frac{1} {2};0\right ]\)
\(X = \left [-\frac{1} {6};0\right ]\)
\(X_{1} = [-2;0]\), \(X_{2} = [3;0]\)
\(X_{1} = [-2;0]\), \(X_{2} = \left [-\frac{1} {2};0\right ]\), \(X_{3} = [3;0]\)

9000025806

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

9000024807

Level: 
C
A body hangs on a string of the length \(l_{1}\). The length \(l\) of the spring defines the period \(T\) of motion by the relation \[ T = 2\pi \sqrt{ \frac{l} {g}}, \] where \(g\) is a standard acceleration of gravity. We have to adjust the length of the string such that the period doubles. Find the new length of the string.
We elongate the string by \(3\cdot l_{1}\), i.e. \(l_{2} = l_{1} + 3l_{1}\).
The length doubles, i.e. \(l_{2} = 2l_{1}\).
The new length will be half of the original length, i.e. \(l_{2} = \frac{1} {2}l_1\).
We shorten the string by \(3\cdot l_{1}\), i.e. \(l_{2} = l_{1} - 3l_{1}\).

9000024808

Level: 
C
In the following list identify a true statement referring to the following equation. \[ \sqrt{4x^{2 } - \sqrt{8x + 5}} = 2x + 1 \]
The equation has a unique solution, this solution is a negative number.
The equation has two solutions, both solutions have an opposite sign.
The equation has a unique solution, this solution is a positive number.
The equation does not have a solution.

9000022907

Level: 
C
Solve the following system of equations and identify a true statement. \[ \begin{alignedat}{80} |x - 2| & + &y & = &2 & & & & & & \\ - 2|5 + x| &- 3 &y & = - &5 & & & & & & \\\end{alignedat}\]
The system has two solutions. Both solutions satisfy \(y < 0\).
The system has a unique solution. This solution satisfies \(y > 0\).
The system has two solutions. Both solutions satisfy \(y > 0\).
The system has more than two solutions.
The system does not have any solution.

9000022901

Level: 
C
An arrow has been shot at the angle \(60^{\circ }\) at the speed \(10\, \mathrm{m}\, \mathrm{s}^{-1}\). Find the time when the height equals to the horizontal distance from the take-off point. Hint: The position is given by the equations \(x = v_{0}t\cdot \cos \alpha \), \(y = v_{0}t\cdot \sin \alpha -\frac{1} {2}gt^{2}\). Use \(g = 10\, \mathrm{m}\, \mathrm{s}^{-2}\) as an acceleration of gravity.
\(\left (\sqrt{3} - 1\right )\, \mathrm{s}\)
\(\left (\sqrt{3} + 1\right )\, \mathrm{s}\)
\(\sqrt{3}\, \mathrm{s}\)
\(\left (\sqrt{2} - 1\right )\, \mathrm{s}\)
\(\left (\sqrt{2} + 1\right )\, \mathrm{s}\)

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.