B

2010008702

Level: 
B
We are given the point \( P=[3;-4;-5] \) and planes \( \alpha \) by \( 2x-y-3z-5=0 \) and \( \beta \) by \( 3x-2y-4z+3=0 \). Find the general form of the equation of the plane \( \sigma \) which passes through the point \( P \) and is perpendicular to both planes \(\alpha\) and \(\beta\) (see the picture).
\( \sigma\colon 2x+y+z+3=0 \)
\( \sigma\colon 2x-y-z+15=0 \)
\( \sigma\colon 2x-y+z-5=0 \)
\( \sigma\colon 2x+y-z-7=0 \)

2010008701

Level: 
B
We are given the points \(K = [ 1; −2; 1]\), \(L = [2; 0; −3]\) and the plane \(\rho\) by \(x-2z+3=0\). Find the general form of the equation of the plane \(\sigma\) in which the line \(KL\) is located and is perpendicular to the plane \(\rho\) (see the picture).
\( \sigma\colon 2x+y+z-1=0 \)
\( \sigma\colon 2x+3y+2z+2=0 \)
\( \sigma\colon 2y+z+3=0 \)
\( \sigma\colon 2x+y-4=0 \)

200001604

Level: 
B
Let \( A= \left\{ x \in \mathbb{R}\colon \left(\frac{\sqrt{2}}2\right)^{5x} < 8 \cdot 4^{3-2x}\right\}\) and \( B=\{x \in \mathbb{R}\colon 2^x-4\cdot 2^{-x}>3\}\). Find \(A \cap B\).
\(A \cap B=(2;6)\)
\(A \cap B=(-\infty;-1)\cup(4;6)\)
\(A \cap B=(-\infty;-1)\cup(2;6)\)

2010011504

Level: 
B
We are given the inequality \( 2x+\frac{3-4x}2 < \frac72 \). Decide which of the following inequalities is equivalent to the given inequality, i.e. which of the following inequalities was obtained from the given inequality by equivalent transformations.
\( 0\cdot x < 4 \)
\( 0\cdot x < -4\)
\( 0\cdot x > 4 \)
\( 2\cdot x > -4 \)