2110016202
Level:
B
Solve the following inequality and choose the diagram that shows the correct solution set marked in red on the number line.
\[ 4.5 - 0.7x < 1 - 2.1x\]
B
2010016203
Level:
B
Find the minimal integer which satisfies the following inequality.
\[
3x - 1 \leq 7x-8
\]
\( 2 \)
\( 0 \)
\(3\)
\( 1 \)
2010016201
Level:
B
Find the linear inequality of which the solution set is shown in red in the graph below.
\( \frac{x+4}2 > 6-x \)
\( \frac{x-4}2 > 6-x \)
\( x+2 \geq 6-x \)
\( \frac{x+4}2 < 6-x \)
2000017413
Level:
B
Consider the matrices
\[
A=\left (\array{
1& 0 & 0\cr
-1 & 1 & 1 \cr
2& 1 & 0} \right ),~
B=\left (\array{
-1& 1 & 0\cr
0 & 1 & 1 \cr
1 &1& -1 } \right ).
\]
For which matrix \(X\) holds \(A \cdot B - X= B^2\)?
\(
\left (\array{
-2& 1 & -1\cr
1 & -1& 0\cr
0 & 2& -1 } \right )
\)
\(
\left (\array{
-2& 1 & -1\cr
1 & 1& 0\cr
0 & 2& -1 } \right )
\)
\(
\left (\array{
-2& 1 & 1\cr
1 & -1& 0\cr
0 & 2& -1 } \right )
\)
\(
\left (\array{
2& 1 & -1\cr
1 & -1& 0\cr
0 & 2& -1 } \right )
\)
2000017412
Level:
B
For which matrices \(X\) and \(Y\) are both following equations true?
\[
3\cdot X - Y=
\left (\array{
2& -2 & -5\cr
8 & 3 & 2\cr
} \right )
\]
\[
X + 2 \cdot Y=
\left (\array{
3& 4& -4\cr
5 & 1 & 3\cr
} \right )
\]
\(X=
\left (\array{
1& 0& -2\cr
3 & 1 & 1\cr
} \right )
\),
\(Y=
\left (\array{
1& 2& -1\cr
1 & 0 & 1\cr
} \right )
\)
\(X=
\left (\array{
1& 0& 2\cr
3 & 1 & 1\cr
} \right )
\),
\(Y=
\left (\array{
1& 2& -1\cr
1 & 0 & 1\cr
} \right )
\)
\(X=
\left (\array{
1& 0& -2\cr
3 & 1 & 1\cr
} \right )
\),
\(Y=
\left (\array{
1& 2& 1\cr
1 & 0 & 1\cr
} \right )
\)
\(X=
\left (\array{
1& 0& -2\cr
3 & -1 & 1\cr
} \right )
\),
\(Y=
\left (\array{
1& 2& -1\cr
-1 & 0 & 1\cr
} \right )
\)
2000017411
Level:
B
For which matrix \(M\) the following equality holds?
\[
2 \cdot \left (\array{
5& -4\cr
7 & -3 \cr
} \right ) - 4\cdot M =
\left (\array{
2& -20 \cr
18 & -22 \cr
} \right )
\]
\(
\left (\array{
2& 3\cr
-1 & 4 \cr
} \right )
\)
\(
\left (\array{
2& 3\cr
1 & 4 \cr
} \right )
\)
\(
\left (\array{
2& -3\cr
-1 & 4 \cr
} \right )
\)
\(
\left (\array{
-2& 3\cr
-1 & 4 \cr
} \right )
\)
2000017410
Level:
B
Find the matrix \(M\) for which the following equality is true:
\[
3 \cdot \left (\array{
4& -1\cr
2 & 5 \cr
} \right ) - 2\cdot M =
\left (\array{
14& -7 \cr
4 & 7 \cr
} \right )
\]
\(
\left (\array{
-1& 2\cr
1 & 4 \cr
} \right )
\)
\(
\left (\array{
1& 2\cr
1 & 4 \cr
} \right )
\)
\(
\left (\array{
-1& 2\cr
-1 & 4 \cr
} \right )
\)
\(
\left (\array{
-1& -2\cr
1 & 4 \cr
} \right )
\)
2000017210
Level:
B
In the following matrix, what is the position of the entry with the largest value?
\[
\left (\array{
1+2& \sin \frac{\pi}4& \mathrm{tg}\,\frac{3\pi}4\cr
3+\cos \frac{\pi}2 & 2^{\sqrt3} & 5+\mathrm{tg} (-\pi)\cr
\sqrt{23}& 3-4 & \sin 2\pi -2
} \right )
\]
It lays above the main diagonal.
It lays on the main diagonal.
It lays under the main diagonal.
It lays on the counterdiagonal.
2000017209
Level:
B
For what real value of \(a\) is the following matrix an upper triangular matrix?
\[
\left (\array{
2^{a-3}& 3+a& a-2\cr
3-a & (\sqrt3)^{a-1} & 1+a\cr
\sin (\pi a) & a-\sqrt9 & 2+\sqrt3
} \right )
\]
\(a=3\)
\(a=2\)
\(a=-3\)
\(a=-2\)
2000017208
Level:
B
Let \(x \in \mathbb{R}\). What is the sum of all the entries on the main diagonal of the following matrix?
\[
\left (\array{
\sin^2 x& 0& 1\cr
5 & \cos^2 x & 4\cr
\sin^2 x & 3 & -1
} \right )
\]
\(0\)
\(1\)
\(2\)
\(4\)