A

2000020302

Level: 
A
Solve the given system of equations in the set of real numbers. \[\begin{aligned} x^2+y&=2\\ 2x-y+3&=0\\ \end{aligned} \] In the following list identify a true statement.
The numbers \(x\) and \(y\) are opposite of each other.
The sum of the numbers \(x\) and \(y\) is equal to \(-2\).
The arithmetic mean of numbers \(x\) and \(y\) is equal to \(2\).
The ratio of numbers \(x\) and \(y\) is \(2:1\).

2000019305

Level: 
A
Mr. Wise buys fuel at three various gas stations – MOL, Shell and EuroOil. He always buys \(25\) liters of diesel, cafe latte and \(2\) nougat croissants. \[~\] Prices in CZK: \[ \begin{array}{|c|c|c|c|} \hline &\text{Diesel (}1~\text{liter)} &1~\text{Cafe Latte}&1~\text{Croissant}\\\hline \text{MOL}& 31.20& 57&16.90\\\hline \text{Shell}& 27.20 &52 &20 \\\hline \text{EuroOil}& 29.60 &49 &18.20 \\\hline \end{array}\] Consider the next product of the two matrices: \[ \left (\array{ 31.20& 57 & 16.90 \cr 27.20& 52 & 20 \cr 29.60& 49 & 18.20 \cr } \right ) \cdot \left (\array{ 25 \cr 1 \cr 2 \cr } \right ) \] Choose the statement which is NOT true:
The most optimal prices are at the MOL gas station.
The most optimal prices are at the Shell gas station.
The least optimal prices are at the MOL gas station.

2000019304

Level: 
A
Adam’s mom and Peter’s mom compete to see who can buy groceries cheaper. For summer weekend, they both want to buy butter, sugar, flour, and vanilla sugar and there are two stores available Aha and Praha. \[~\] The following tables show the quantity of items they both plan to buy and their prices in stores Aha and Praha. \[ \begin{array}{|c|c|c|c|c|} \hline &\begin{gathered}\text{Butter}\\\text{(kg)}\end{gathered} &\begin{gathered}\text{Sugar}\\\text{(kg)}\end{gathered} &\begin{gathered}\text{Flour}\\\text{(kg)}\end{gathered} &\begin{gathered}\text{Vanila sugar}\\\text{(number of pieces)}\end{gathered}\\\hline \text{Adam's mom}& 0.5& 2&1&8 \\\hline \text{Peter's mom}& 0.25 &2&2&5 \\\hline \end{array}\] \[ \begin{array}{|c|c|c|} \hline &\mathrm{Aha} & \mathrm{Praha}\\\hline \text{Butter (per }1\,\mathrm{kg)}& 119.6\,\mathrm{CZK}& 159.6\,\mathrm{CZK} \\\hline \text{Sugar (per }1\,\mathrm{kg)} & 12.5\,\mathrm{CZK}& 9.9\,\mathrm{CZK} \\\hline \text{Flour (per }1\,\mathrm{kg)} & 10.9\,\mathrm{CZK} & 9.5\,\mathrm{CZK} \\\hline \text{Vanilla sugar (per } \mathrm{piece)} & 5.9\,\mathrm{CZK} & 5.1\,\mathrm{CZK} \\\hline \end{array}\] What follows from the next product of the two matrices? \[ \left (\array{ 0.5& 2 & 1 & 8\cr 0.25& 2 & 2 & 5\cr } \right ) \cdot \left (\array{ 119.6& 159.6 \cr 12.5& 9.9 \cr 10.9& 9.5 \cr 5.9& 5.1 \cr } \right ) = \left (\array{ 142.9& 149.9\cr 106.2& 104.2 \cr } \right ) \]
For Adam’s mom, it is more convenient to shop at Aha, while for Peter’s mom, it is more convenient to shop at Praha.
For Adam’s mom, it is more convenient to shop at Praha, while for Peter’s mom, it is more convenient to shop at Aha.
Shopping at Aha is more convenient for both moms.
Shopping at Praha is more convenient for both moms.

2000019303

Level: 
A
Three ice cream stands of ICE company reported their July sales of four ice cream flavors in number of portions sold. All data can be seen in the table below: \[ \begin{array}{|c|c|c|c|c|} \hline &\text{vanilla} & \text{chocolate} & \text{nut} & \text{strawberry} \\\hline \text{Stand 1}& 720 & 800 & 1\,200&360 \\\hline \text{Stand 2} & 550 & 434 & 900 & 300 \\\hline \text{Stand 3} &610 &300 & 200 & 750 \\\hline \end{array}\] The profits from the sales of each specific flavor are expressed by the matrix \( P= \left (\array{ 1\cr 1\cr 3\cr 2\cr } \right ) \). Estimate what was the total profit of ICE company in July from all three stands.
more than \(12\,000\,\mathrm{CZK}\)
from \(9\,000\,\mathrm{CZK}\) to \(12\,000\,\mathrm{CZK}\)
from \(6\,000\,\mathrm{CZK}\) to \(9\,000\,\mathrm{CZK}\)
less than \(6\,000\,\mathrm{CZK}\)

2000019302

Level: 
A
Three ice cream stands of ICE company reported their July sales of four ice cream flavors in number of portions sold. All data can be seen in the table below: \[ \begin{array}{|c|c|c|c|c|} \hline &\text{vanilla} & \text{chocolate} & \text{nut} & \text{strawberry} \\\hline \text{Stand 1}& 720 & 800 & 1\,200&360 \\\hline \text{Stand 2} & 550 & 434 & 900 & 300 \\\hline \text{Stand 3} &610 &300 & 200 & 750 \\\hline \end{array}\] The profits from the sales of each specific flavor are expressed by the matrix \( P= \left (\array{ 1\cr 1\cr 3\cr 2\cr } \right ) \). Let the July sales be rewritten to the matrix \(J\). By what matrix the profits of individual ice cream stands in July are described?
\(J\cdot P\)
\(P \cdot J\)
\(J +P\)
Profits cannot be determined by any of the matrix operations.

2000019301

Level: 
A
Three ice cream stands of ICE company reported their July sales of four ice cream flavors in number of portions sold. All data can be seen in the table below. \[ \begin{array}{|c|c|c|c|c|} \hline &\text{vanilla} & \text{chocolate} & \text{nut} & \text{strawberry} \\\hline \text{Stand 1}& 720 & 800 & 1\,200&360 \\\hline \text{Stand 2} & 550 & 434 & 900 & 300 \\\hline \text{Stand 3} &610 &300 & 200 & 750 \\\hline \end{array}\] The data provided by the stands for August sales are reported in the matrix \(A\). \[ A= \left (\array{ 650& 470 & 890 & 410\cr 500& 505 & 890 & 300\cr 380& 520 & 350 & 800\cr } \right ) \] If the July sales are rewritten to the matrix \(J\), by what matrix the sales of ice cream for both summer months are described?
matrix \(J+A\)
matrix \(J-A\)
matrix \(J \cdot A\)
matrix \(2J+2A\)