9000031001
Level:
B
Find the sum of all real roots of the following equation.
\[
(3x - 1)(2x + 1)(4x^{2} + 3x - 1) = 0
\]
\(-\frac{11}
{12}\)
\(- \frac{1}
{12}\)
\(-\frac{1}
{6}\)
\(\frac{1}
{6}\)
B
9000031102
Level:
B
Solve the following system of equations and identify a correct statement.
\[\begin{aligned}
(x - 1)^{2} + y^{2} = 1 & &
\\(x - 4)^{2} + y^{2} = 4 & &
\end{aligned}\]
The system has a unique solution \(\left [x,y\right ]\),
where \(y = 0\).
The system does not have any solution.
The system has a unique solution \(\left [x,y\right ]\),
where \(y > 0\).
The system has two solutions \(\left [x_{1},y_{1}\right ]\),
\(\left [x_{2},y_{2}\right ]\), where
\(y_{1} = -y_{2}\).
9000031004
Level:
B
Assuming \(y\in \mathbb{R}\),
find the number of the solutions of the following algebraic equation.
\[
y^{4} + 5y^{2} + 6 = 0
\]
\(0\)
\(4\)
\(3\)
\(2\)
9000031005
Level:
B
Assuming \(x\in \mathbb{R}\),
solve the following algebraic equation.
\[
(x + 1)^{4} - 5(x + 1)^{2} + 4 = 0
\]
\( \{ - 3;-2;0;1\}\)
\( \{1;4\}\)
\( \{ - 2;-1;1;2\}\)
\( \{ - 1;3\}\)
9000031008
Level:
B
Assuming \(x\in \mathbb{R}\),
solve the following equation.
\[
4x^{3} - 3x^{2} - x = 0
\]
\( \left \{-\frac{1}
{4};0;1\right \}\)
\(\{0;1;4\}\)
\( \{1;4\}\)
\( \{0\}\)
9000031010
Level:
B
Identify a true statement on the following equation.
\[
x^{5} - x^{3} - 6x = 0
\]
The equation has three solutions in \(\mathbb{R}\).
The equation does not have solution in
\(\mathbb{R}\).
The equation has five solutions in \(\mathbb{R}\).
The equation has one solution in \(\mathbb{R}\).
9000031002
Level:
B
One of the solutions of the following equation is
\(x = 2\). Find
the set of all solutions.
\[
x^{3} + 2x^{2} - 5x - 6 = 0
\]
\(\{ - 3;-1;2\}\)
\( \{ - 3;-1\}\)
\( \{ - 3;0;2\}\)
\(\{ - 1;2;3\}\)
9000031207
Level:
B
Find the algebraic form of the complex number
\(z = 2\left (\cos \frac{3\pi }
{4} + \mathrm{i}\sin \frac{3\pi }
{4}\right )\).
\(-\sqrt{2} + \mathrm{i}\sqrt{2}\)
\(\sqrt{2} + \mathrm{i}\sqrt{2}\)
\(\sqrt{2} -\mathrm{i}\sqrt{2}\)
\(-\sqrt{2} -\mathrm{i}\sqrt{2}\)
9000031208
Level:
B
Find the polar form of the complex number
\(z = -3 + 3\mathrm{i}\).
\(3\sqrt{2}\left (\cos \frac{3\pi }
{4} + \mathrm{i}\sin \frac{3\pi }
{4}\right )\)
\(3\left (\cos \frac{\pi }{4} + \mathrm{i}\sin \frac{\pi }{4}\right )\)
\(3\left (\cos \frac{5\pi }
{4} + \mathrm{i}\sin \frac{5\pi }
{4}\right )\)
\(3\sqrt{2}\left (\cos \frac{7\pi }
{4} + \mathrm{i}\sin \frac{7\pi }
{4}\right )\)
9000028307
Level:
B
Solve the following equation.
\[
x^{3} + 6x^{2} - 8x = 0
\]
\(0\),
\(- 3 -\sqrt{17}\),
\(- 3 + \sqrt{17}\)
\(0\),
\(3 -\sqrt{17}\),
\(3 + \sqrt{17}\)
\(0\),
\(- 3\),
\(\sqrt{
17}\)
\(0\),
\(3\),
\(-\sqrt{17}\)