9000019806
Level:
B
Find the smallest integer solution of the following equation.
\[
x^{4} - 2x^{3} - x^{2} + 2x = 0
\]
\(- 1\)
\(0\)
\(1\)
\(2\)
B
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.
\[
\begin{array}{cl}
\phantom{ -} 3x + 5y +\phantom{ 2}z =\phantom{ -}10&
\\
- 2x - 3y + 2z = -10&
\\
\phantom{ - 2}x +\phantom{ 2}y - 5z =\phantom{ -}10& \end{array}
\]
\(\mathop{\mathrm{rank}}(A) = 2,\ \mathop{\mathrm{rank}}(A') = 2\)
\(\mathop{\mathrm{rank}}(A) = 3,\ \mathop{\mathrm{rank}}(A') = 3\)
\(\mathop{\mathrm{rank}}(A) = 3,\ \mathop{\mathrm{rank}}(A') = 2\)
\(\mathop{\mathrm{rank}}(A) = 2,\ \mathop{\mathrm{rank}}(A') = 3\)
9000018105
Level:
B
Find all the values of \(x\)
which ensure that the following fraction is positive.
\[
- \frac{3}
{5 - 2x}
\]
\(x > \frac{5}
{2}\)
\(x < \frac{5}
{2}\)
\(x < -\frac{5}
{2}\)
\(x > -\frac{5}
{2}\)
9000019906
Level:
B
Consider a linear system of four equations with four unknowns. The rank of the coefficient
matrix \(A\) is
\(\mathop{\mathrm{rank}}(A) = 3\). The rank of the
augmented matrix \(A'\)
is \(\mathop{\mathrm{rank}}(A') = 4\).
Identify a true statement on this system.
The system does not have any solution.
The system has infinitely many solutions.
The system has a unique solution.
It is not possible to draw any conclusion from this information.
9000019803
Level:
B
Find the solution set of the following equation.
\[
x^{4} - 5x^{2} + 4 = 0
\]
\(\left \{-2;-1;1;2\right \}\)
\(\left \{-1;1\right \}\)
\(\left \{-2;2\right \}\)
\(\left \{1;2\right \}\)
9000019907
Level:
B
The augmented matrix of a system of three equations with
three unknowns is row equivalent with the following matrix
\(A'\). Find
the solution of the system.
\[
A' = \left(\begin{array}{ccc|c}
1 & 2 & 4 & 0\\
0 & 2 & 7 & 7\\
0 & 0 & 7 & 35
\end{array}\right)
\]
\([8;-14;5]\)
\([-62;21;5]\)
\([8;14;-5]\)
\([-22;-21;5]\)