2010011503
Level:
B
Assuming \(x\in \mathbb{R}\),
find the solution set of the following system.
\[
3 x -1 \leq 2x + 7 \leq 7x-8
\]
\(x\in [ 3;8] \)
\(x\in (-\infty ;3] \)
\(x\in [ 8;\infty )\)
\( x \in \mathbb{R}\setminus (3;8)\)
B
2010011502
Level:
B
Solve the following inequality:
\[
26 -2\left (9x +4 \right ) > 6\left (3-3x\right )
\]
\( x \in \emptyset\)
\(x\in (-\infty ;0) \)
\(x\in \left (-\infty;\infty \right )\)
\(x\in (0;\infty )\)
2010011501
Level:
B
Solve the following inequality:
\[
6 -3\left (2x +4 \right ) \leq 2\left (3-3x\right )
\]
\(x\in \left (-\infty;\infty \right )\)
\(x\in (-\infty ;-1]\)
\(x\in [ -1;\infty )\)
\( x \in \emptyset\)
2000011302
Level:
B
Find the solution of the equation \(\log_{16}x+\log_4x+\log_2x=7\).
\(x=16\)
\(x=4\)
\(x={\frac12}\)
\(x=2\)
2000011301
Level:
B
Let \( x \in (0;1) \cup (1;+\infty)\). Find the value of \(m \in \mathbb{R}\) if \(2\log_m x=\frac32 \log_2 x\).
\(m=2^{\frac43}\)
\(m=2^{\frac34}\)
\(m={\frac34}\)
\(m={\frac43}\)
2010011204
Level:
B
Kamil is able to mow a meadow in \( 12 \) hours. Zdeněk has a better lawn mower and he is able to mow the same meadow in \( 9 \) hours. They have agreed that Kamil starts to mow alone sooner and Zdeněk will join him later so that the total time of mowing is \( 8 \) hours. How long will they mow together?
\( 3 \) hours
\( 5 \) hours
\( 2 \) hours
\( 1 \) hour
2010011203
Level:
B
The March price of a T-shirt and shorts was \( 900\,\mathrm{CZK} \) together. In April there was on store price adjustment. The price of the shorts decreased by \( 20\% \) and the price of the T-shirt increased by \( 20\% \). So the April price of both together the shorts and the T-shirt was by \( 40\,\mathrm{CZK} \) lower. What was the April price of the T-shirt?
\( 420\,\mathrm{CZK} \)
\( 350\,\mathrm{CZK} \)
\( 440\,\mathrm{CZK} \)
\( 550\,\mathrm{CZK} \)
2000010606
Level:
B
For which values of parameter \(p\) is the function \(f(x)=(p^2-4p+3)^x\) an increasing exponential function?
\(p \in \left(-\infty;2-\sqrt{2}\right) \cup \left(2+\sqrt{2};\infty\right)\)
\(p \in \left(2-\sqrt{2};2+\sqrt{2}\right)\)
\(p \in \left(2-\sqrt{2};1\right) \cup \left(3;2+\sqrt{2}\right)\)
2000010603
Level:
B
Find the coordinates of the point of intersection of the graphs of functions \( f(x)=\left(\frac35\right)^x\) and \(g(x)=\left(\frac{\sqrt{15}}{5}\right)^{x-1}\).
\( \left[-1;\frac53\right]\)
\( \left[-3;\frac{25}9\right]\)
The graphs of functions \(f\) and \(g\) have no points in common.
2000010602
Level:
B
For which values of the parameter \(k\) has the equation \(|2^x-3|=k\) two solutions the product of which is a negative number?
\( k \in (2;3)\)
\( k \in [ 2;3 ] \)
\( k \in (-\infty;2) \cup (3;+\infty)\)