9000031003
Level:
B
Assuming \(x\in \mathbb{R}\),
solve the following algebraic equation.
\[
x^{4} + 4x^{2} - 5 = 0
\]
\( \{ - 1,1\}\)
\( \{1\}\)
\( \{ -\sqrt{5},-1,1,\sqrt{5}\}\)
\( \emptyset \)
B
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 \(\mathbb{R}\).
9000031005
Level:
B
Assuming \(x\in \mathbb{R}\),
solve the following algebraic equation.
\[
(x + 1)^{4} - 5(x + 1)^{2} + 4 = 0
\]
\( \{ - 3,-2,0,1\}\)
\( \{1,4\}\)
\( \{ - 2,-1,1,2\}\)
\( \{ - 1,3\}\)
9000031207
Level:
B
Find the algebraic form of the complex number
\(z = 2\left (\cos \frac{3\pi }
{4} + \mathrm{i}\sin \frac{3\pi }
{4}\right )\).
\(-\sqrt{2} + \mathrm{i}\sqrt{2}\)
\(\sqrt{2} + \mathrm{i}\sqrt{2}\)
\(\sqrt{2} -\mathrm{i}\sqrt{2}\)
\(-\sqrt{2} -\mathrm{i}\sqrt{2}\)
9000031008
Level:
B
Assuming \(x\in \mathbb{R}\),
solve the following equation.
\[
4x^{3} - 3x^{2} - x = 0
\]
\( \left \{-\frac{1}
{4},0,1\right \}\)
\(\{0,1,4\}\)
\( \{1,4\}\)
\( \{0\}\)
9000031208
Level:
B
Find the polar form of the complex number
\(z = -3 + 3\mathrm{i}\).
\(3\sqrt{2}\left (\cos \frac{3\pi }
{4} + \mathrm{i}\sin \frac{3\pi }
{4}\right )\)
\(3\left (\cos \frac{\pi }{4} + \mathrm{i}\sin \frac{\pi }{4}\right )\)
\(3\left (\cos \frac{5\pi }
{4} + \mathrm{i}\sin \frac{5\pi }
{4}\right )\)
\(3\sqrt{2}\left (\cos \frac{7\pi }
{4} + \mathrm{i}\sin \frac{7\pi }
{4}\right )\)
9000031009
Level:
B
Find the sum of the solutions of the following equation.
\[
6(3x + 1)(2x^{2} + 3x - 2) = 0
\]
\(-\frac{11}
{6} \)
\(-\frac{7}
{6}\)
\(-\frac{1}
{2}\)
\(\frac{11}
{6} \)
9000031101
Level:
B
Solve the following system of equations and identify a correct statement.
\[\begin{aligned}
(x - 3)^{2} + (y - 1)^{2} = 1 & &
\\2x^{2} + 2y^{2} - 12x - 4y + 18 = 0 & &
\end{aligned}\]
The system has more than two solutions.
The system does not have any solution.
The system has a unique solution.
The system has two solutions.
9000031103
Level:
B
Solve the following system of equations in \( \mathbb{R} \times \mathbb{R}\) and identify the correct statement.\[\begin{aligned}
x - 2y + 5 = 0 & &
\\x^{2} + y^{2} = 9 & &
\end{aligned}\]
The system has two solutions.
The system does not have any solution.
The system has a unique solution.
The system has more than two solutions.
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
\(y_{1} = -y_{2}\).