B

9000021801

Level: 
B
Solve the following system of inequalities. \[\begin{aligned} \frac{1} {3}(2x + 5) &\geq 0.5\left (\frac{2 + 3x} {2} + 2\right ) & & \\0.2(3 - 2x) &\leq \frac{1} {3}\left (\frac{4 - 2x} {5} + 2\right ) & & \end{aligned}\]
\(x\in \left [ -\frac{5} {4};2\right ] \)
\(x\in [ 2;\infty )\)
\(x\in \left (-\infty ;-\frac{5} {4}\right ] \)
\(x\in \emptyset \)

9000020907

Level: 
B
Identify a true statement related to the solution of the following system in \(\mathbb{R}\times \mathbb{R}\). \[ \begin{alignedat}{80} &2x^{2} & - & &y^{2} & - &2 &x & - 5 & = 0 & & & & & & & & & & \\ & & & &3x & - & &y & - 5 & = 0 & & & & & & & & & & \\\end{alignedat}\]
The system has no solution.
The system has two solutions.
The system has a unique solution.
None of the above conclusions can be obtained.

9000019808

Level: 
B
Assuming \(x\in \mathbb{C}\), find the solution set of the following equation. \[ x\left (x + 1\right )\left (x^{2} + 1\right ) = 0 \]
\(\left \{-1;0;-\mathrm{i};\mathrm{i}\right \}\)
\(\left \{-1;0;1;-\mathrm{i};\mathrm{i}\right \}\)
\(\left \{-1;1;-\mathrm{i};\mathrm{i}\right \}\)
\(\left \{-1;0;-\mathrm{i}\right \}\)

9000019809

Level: 
B
Find the factorization of the following equation. \[ x^{3} + 3x^{2} - x - 3 = 0 \]
\(\left (x + 3\right )\left (x + 1\right )\left (x - 1\right ) = 0\)
\(\left (x - 3\right )\left (x + 1\right )\left (x - 1\right ) = 0\)
\(\left (x + 3\right )\left (x - 3\right )\left (x - 1\right ) = 0\)
\(\left (x + 3\right )\left (x - 3\right )\left (x + 1\right ) = 0\)

9000019904

Level: 
B
The coefficient matrix of a \(3\times 3\) linear system is \(A\) and the augmented matrix \(A'\). Find \(\mathop{\mathrm{rank}}(A)\) and \(\mathop{\mathrm{rank}}(A')\). \[ A = \begin{pmatrix} -1 & 3 & 2 \\ 0 & 4 & -5 \\ 0 & 0 & 2 \end{pmatrix} \qquad A' = \left(\begin{array}{ccc|c} -1 & 3 & 2 & 5 \\ 0 & 4 & -5 & 10\\ 0 & 0 & 2 & 0 \end{array}\right) \]
\(\mathop{\mathrm{rank}}(A) = 3,\ \mathop{\mathrm{rank}}(A') = 3\)
\(\mathop{\mathrm{rank}}(A) = 2,\ \mathop{\mathrm{rank}}(A') = 3\)
\(\mathop{\mathrm{rank}}(A) = 3,\ \mathop{\mathrm{rank}}(A') = 2\)
\(\mathop{\mathrm{rank}}(A) = 2,\ \mathop{\mathrm{rank}}(A') = 2\)

9000019810

Level: 
B
Find the factorization of the following equation. \[ 5x^{4} - 30x^{2} + 40 = 0 \]
\(5\left (x -\sqrt{2}\right )\left (x + \sqrt{2}\right )\left (x - 2\right )\left (x + 2\right ) = 0\)
\(\left (x -\sqrt{2}\right )\left (x + \sqrt{2}\right )\left (x - 2\right )\left (x + 2\right ) = 0\)
\(5x\left (x -\sqrt{2}\right )\left (x + \sqrt{2}\right )\left (x - 2\right ) = 0\)
\(5x\left (x -\sqrt{2}\right )\left (x + \sqrt{2}\right )\left (x + 2\right ) = 0\)

9000019905

Level: 
B
Let \(A\) and \(A'\) be the coefficient matrix and the augmented matrix of the following linear system, respectively. Find the ranks of these matrices. \[ \begin{array}{cl} \phantom{ -} 3x + 5y +\phantom{ 2}z =\phantom{ -}10& \\ - 2x - 3y + 2z = -10& \\ \phantom{ - 2}x +\phantom{ 2}y - 5z =\phantom{ -}10& \end{array} \]
\(\mathop{\mathrm{rank}}(A) = 2,\ \mathop{\mathrm{rank}}(A') = 2\)
\(\mathop{\mathrm{rank}}(A) = 3,\ \mathop{\mathrm{rank}}(A') = 3\)
\(\mathop{\mathrm{rank}}(A) = 3,\ \mathop{\mathrm{rank}}(A') = 2\)
\(\mathop{\mathrm{rank}}(A) = 2,\ \mathop{\mathrm{rank}}(A') = 3\)