9000028304
Level:
C
The following equation has solutions
\(x_1= 1\) and
\(x_2 = 3\). Find
the sum of the remaining real solutions.
\[
x^{4} - 12x^{3} + 47x^{2} - 72x + 36 = 0
\]
\(8\)
\(- 1\)
\(3\)
\(5\)
C
9000028305
Level:
C
The following equation has solution
\(x_1= 2\) and
\(x_2= 4\). Find
the sum of the remaining real solutions.
\[
x^{4} - 6x^{3} - x^{2} + 54x - 72 = 0
\]
\(0\)
\(- 1\)
\(1\)
\(2\)
9000024805
Level:
C
A falling body dropped at a velocity
\(60\, \mathrm{m}\mathrm{s}^{-1}\). Find the
initial height \(h\),
if the relation between the velocity and the initial height
\(h\) is
\(v = \sqrt{2hg}\). Use
\(g = 10\, \mathrm{m}\mathrm{s}^{-2}\) for
acceleration of gravity.
The initial height is between \(150\, \mathrm{m}\)
and \(200\, \mathrm{m}\).
The initial height is smaller than \(100\, \mathrm{m}\).
The initial height is between \(100\, \mathrm{m}\)
and \(150\, \mathrm{m}\).
The initial height is bigger than \(200\, \mathrm{m}\).
9000024810
Level:
C
Find the solution set of the following inequality.
\[
\sqrt{|x|} > x
\]
\(K = (-\infty ;0)\cup (0;1)\)
\(K = (-\infty ;1)\)
\(K =\mathbb{R} ^{+}\)
\(K = (0;1)\)
9000025802
Level:
C
Find all intersections of the graph of the following function with
\(x\)-axis.
\[
f(x) = \frac{x^{2} + x - 2}
{x + 1}
\]
\(X_{1} = [-2;0]\),
\(X_{2} = [1;0]\)
\(X = [0;0]\)
\(X_{1} = [-2;0]\),
\(X_{2} = [-1;0]\),
\(X_{3} = [1;0]\)
\(X = [-1;0]\)
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.