A

2000018801

Level: 
A
Consider a triangle with area of \(5\, \mathrm{cm}^{2}\). Find the formula which relates the length of its side \(a\) to the length of the height \(v_a\) , where \(v_a\) is the height to the side \(a\).
\(v_a = \frac{10} {a}\)
\(v_a = \frac{5} {a}\)
\(v_a =5 {a}\)
\(v_a = \frac{5} {2a}\)

2010018502

Level: 
A
One of the angles \( \alpha \), \( \beta \), \( \gamma \), \( \delta \) takes the same position on the unit circle as the angle \( ASB \). Which of the angles \( \alpha \), \( \beta \), \( \gamma \), \( \delta \) it is?
\( \alpha = 135^{\circ} \)
\( \beta = -100^{\circ} \)
\( \gamma= -315^{\circ} \)
\( \delta= 210^{\circ} \)

2000018305

Level: 
A
Consider three matrices \[ A = \left (\array{ 3 &4\cr 1 & 2\cr } \right ),~ B = \left (\array{ 1 &1\cr 0&1\cr } \right ),~ C = \left (\array{ 1 &0\cr 1&1\cr } \right ). \] Let \(E\) denote the identity matrix of order \(2\). Then, find \(X\), which is the solution to the following equation. \[ C \cdot (A+X)\cdot B=E\]
\( X = \left (\array{ -2 &-5\cr -2& 0\cr } \right ) \)
\( X = \left (\array{ -2 &-5\cr 2& 0\cr } \right ) \)
\( X = \left (\array{ -2 &5\cr -2& 0\cr } \right ) \)
\( X = \left (\array{ -2 &5\cr 2& 0\cr } \right ) \)

2000018303

Level: 
A
Let \(E\) denote an identity matrix of order \(2\) and let matrix \[ M = \left (\array{ m &0\cr 0 & 2\cr } \right ) . \] Find all the values of \(m\) so that the equality below holds. \[ M^2-\frac52M+E=0 \]
\(m=2\) or \(m=\frac12\)
\(m=\frac12\)
\(m=2\)
\(m=2\) or \(m=-\frac12\)

2000018302

Level: 
A
Find the matrix \(M\) so that the equality given below is true. \[ 2 \cdot \left (\array{ -1&4\cr 3&-5\cr } \right ) - M = \left (\array{ -3 &6\cr 9 & -14\cr } \right ) \]
\[ M=\left (\array{ 1 &2\cr -3 & 4\cr } \right ) \]
\[ M=\left (\array{ -1 &2\cr -3 & 4\cr } \right ) \]
\[ M=\left (\array{ -1 &-2\cr 3 & -4\cr } \right ) \]
\[ M=\left (\array{ 1 &2\cr 3 & -4\cr } \right ) \]

2000018301

Level: 
A
Find the matrix \(B\), the solution to the equation given below. \[ \left (\array{ 3&-1 &5\cr 1 &0&3 } \right ) + B = \left (\array{ 5 & 0 & 4 \cr 3 & 2 & 1\cr } \right ) \]
\[ B= \left (\array{ 2 & 1 & -1\cr 2 & 2 & -2 } \right ) \]
\[ B= \left (\array{ 2 & -1 & -1\cr 2 & 2 & -2 } \right ) \]
\[ B= \left (\array{ 2 & 1 & -1\cr 2 & -2 & -2 } \right ) \]
\[ B= \left (\array{ 2 & 1 & -1\cr 2 & 2 & 2 } \right ) \]

2010013606

Level: 
A
In a set of \( 200 \) items, \( 20 \) are defective. We pick randomly \( 10 \) items from this set. First nine items were not defective. Find the probability that the tenth item selected is not defective too. Results are rounded to three decimal places.
\( \frac{171}{191}\doteq 0.895 \)
\( \frac{180}{191}\doteq 0.942 \)
\( \frac{180}{200}\doteq 0.9\)
\( \frac{1}{171}\doteq 0.006 \)

2010013605

Level: 
A
The wooden cube with the edges of length \( 4\,\mathrm{cm} \) has faces painted in blue. Suppose we cut the cube into small unit cubes (the edge length is \( 1\,\mathrm{cm}\)) and select one of the unit cubes at random. What is the probability that the selected cube has at most one face painted in blue?
\( 0.5 \)
\( 0.375 \)
\( 0.438 \)
\( 0.75 \)