1003162802
Level:
C
Find the solution set of the following inequality.
\[ \log_{\frac12}(x) \leq 1 \]
\( \left[\frac12;\infty\right) \)
\( \left(-\infty;\frac12\right] \)
\( \left(0;\frac12\right] \)
\( [1;\infty) \)
C
1003162801
Level:
C
Find the solution set of the following inequality.
\[ \log_{0.3}(x)\geq \log_{0.3}3 \]
\( (0;3] \)
\( (-\infty;3] \)
\( [3;\infty) \)
\( [0;3] \)
1003034502
Level:
C
Petr would like to buy a new smartphone. If he starts a temporary job at the electro shop he gets payed \( 120\,\mathrm{CZK} \) per hour plus he gets \( 20\% \) discount on the smartphone bough in the shop. He calculated that for \( 24 \) hours of work he would not earn even half of the phone price. Another employer pays \( 150\,\mathrm{CZK} \) per hour. If Petr gets a temporary job with the other employer he is not eligible for the discount in the electro shop anymore, however he can buy the smartphone from e-shop for the price by \( 600\,\mathrm{CZK} \) lower than in the electro shop and for \( 20 \) hours of work he earns more than one third of the electro shop smartphone price. Determine as closely as possible the smartphone price in the electronics shop.
greater than \( 7\,200\,\mathrm{CZK} \) and less than \( 9\,600\,\mathrm{CZK} \)
greater than \( 7\,200\,\mathrm{CZK} \) and less than \( 10\,800\,\mathrm{CZK} \)
greater than \( 4\,800\,\mathrm{CZK} \) and less than \( 9\,600\,\mathrm{CZK} \)
greater than \( 4\,800\,\mathrm{CZK} \) and less than \( 10\,800\,\mathrm{CZK} \)
1003034501
Level:
C
Two different aquarium fish shops have special price on Congo Tetra. The price is \( 42\,\mathrm{CZK} \) for one fish. Further, the discount of \( 50\,\mathrm{CZK} \) from a purchase over \( 300\,\mathrm{CZK} \) is offered in shop A. In shop B a customer is given \( 5\% \) discount of the price of any purchase. How many of Congo Tetra must one buy to make the total price in shop A lower than the total price in shop B?
greater than \( 7 \) and less than \( 24 \)
less than \( 24 \)
greater than \( 23 \)
less than \( 7 \)
1003187106
Level:
C
Consider expressions \( |3x-12| \), \( 3|x|+12 \), \( |3x|-|-12| \), \( 3|x-4| \). Which of the expressions has the greatest value if we substitute any \( x \) from the interval \( (0;+\infty) \)?
\( 3|x|+12 \)
\( |3x-12| \)
\( |3x|-|-12| \)
\( 3|x-4| \)
1003187103
Level:
C
Find the relation that does not hold for any \( x \), \( y\in\mathbb{R} \).
\( \left| |x|-|y| \right| > |x+y| \)
\( |xy|=|x| |y| \)
\( \left|\frac xy \right|=\frac{|x|}{|y|}\text{, } y\neq0\text{ .} \)
\( \left| (xy)^2 \right|=|xy|^2=(xy)^2 \)
1003187102
Level:
C
For \( x \), \( y\in\mathbb{R} \) consider \( |x+y|=|x|+|y| \).
The equality holds if and only if the sign of \( x \) and \( y \) is the same.
The equality does not hold for any \( x \) and \( y \).
The equality holds if and only if \( x \) and \( y \) are all positive.
The equality holds if and only if \( x \) and \( y \) are both nonpositive.
1003187101
Level:
C
Which of the following relations is correct for all \( x \), \( y\in\mathbb{R} \)?
\( |x+y| \leq |x|+|y| \)
\( |x+y|=|x|+|y| \)
\( |x-y| < |x|-|y| \)
\( |x-y|=|x|-|y| \)
1003032404
Level:
C
Simplifying the rational expression \( \frac{x^6-y^6}{x^2-y^2} \) we get:
\( \left(x^2-xy+y^2\right)\left(x^2+xy+y^2\right) \)
\( x^4-y^4 \)
\( x^3 - y^3 \)
\( \left(x^2-2xy+y^2\right)\left(x^2+2xy+y^2\right) \)
1003032403
Level:
C
Reducing the rational expression \( \frac{4m^2-4mn+n^2}{8m^3-n^3} \) we get:
\( \frac{2m-n}{4m^2+2mn+n^2} \)
\( \frac{m-4mn+1}{2m-n} \)
\( \frac{2m-n}{4m^2-4mn+n^2} \)
\( \frac{2m-n}{4m^2+4mn+n^2} \)