B

2000020304

Level: 
B
Solve the given system of equations in the set of real numbers. \[\begin{aligned} x-y&=2\\ x^2-y^2&=2\\ \end{aligned}\] In the following list identify a true statement.
The system has exactly one solution.
The system has no solution.
The system has infinitely many solutions.
The quotient of numbers \(x\) and \(y\) is \(3\).

2000019207

Level: 
B
Adam went to a shop where he bought \(7\) buns and \(2\) cakes for \(64\) Kč. Mirek bought \(5\) buns, \(3\) cakes and \(4\) rolls for \(79\) Kč. Petra went to the same shop as Adam and Mirek and bought \(5\) buns and \(4\) rolls. Since it was only \(20\) minutes to closing, she got a discount for each piece of bakery goods worth \(1\) Kč and so she paid \(37\) Kč. Which of the following statements about the prices of goods before the discount is the only incorrect?
\(2\) buns and \(1\) cake altogether cost more than \(16\) rolls
The cake is more expensive than a bun and a roll altogether.
\(3\) cakes cost more than \(8\) rolls.
Buying \(10\) pieces of each (a bun, cake, and a roll) cost more than \(200\) Kč.

2000019206

Level: 
B
For what value of a real number \(a\) has the following system infinitely many solutions? \[ \begin{alignedat}{80} &x & + &2y & +& z & = 8 & & & & & & \\ &2x & & & -& z & = -1 & & & & & & \\ &7x & + & 10y & +& 4z & = a & & & & & & \\\end{alignedat}\]
\(39\)
\(73\)
\(-39\)
\(56\)

2000019205

Level: 
B
Let \([x, y, z]\) be the solution of the system of \(3\) equations with \(3\) unknowns represented by the matrix \[\left(\begin{array}{ccc|c} 1 & 2 & 1 & 6 \\ 2 & -1 & 1 & 1\\ -1 & 1 & 1 & 2 \end{array}\right). \] Which of the components \(x\), \(y\), and \(z\) is of the highest value?
\(y\)
\(x\)
\(z\)
cannot be identified

2000019204

Level: 
B
Visitors of a ZOO can buy a package with bags of goat food (blue color), sheep food (red color) and duck food (green color). The feed bags are offered in \(3\) various packages and their prices can be seen below the packages (as shown in the picture). Which of the feed is the most expensive one?
sheep food
goat food
duck food
cannot be identified

2000019203

Level: 
B
The sweet shop offers \(3\) types of confections in various packages. The price of each package can be seen below the package (as shown in the picture). How much would the sample package cost if it contained \(1\) piece of each type of confection?
\(35\) ¢
\(30\) ¢
\(34\) ¢
none of the given prices

2000019202

Level: 
B
People in Kocourkov pay by coins named groschen. Their coins are worth \(1\), \(5\) or \(7\) groschen. Martin and Petr, who live in Kocourkov, emptied their saving boxes and started counting their saved coins. They found out that Petr had \(6\) pieces of each type of a coin more than Martin, who had \(40\) coins in total. They were surprised to find out that Martin has the same number of \(1\)-groschen and \(7\)-groschen coins in total as Petr has of \(5\)-groschen ones. Petr was proud to have \(78\) groschen more than Martin, who was only \(2\) short of having \(200\) groschen. How many coins did Martin have?
\(40\)
\(58\)
\(13\)
\(50\)

2000019201

Level: 
B
People in Kocourkov pay by coins named groschen. Their coins are worth \(1\), \(5\) or \(7\) groschen. Martin and Petr, who live in Kocourkov, emptied their saving boxes and started counting their saved coins. They found out that Petr had \(6\) pieces of each type of a coin more than Martin, who had \(40\) coins in total. They were surprised to find out that Martin has the same number of \(1\)-groschen and \(7\)-groschen coins in total as Petr has of \(5\)-groschen ones. Petr was proud to have \(78\) groschen more than Martin, who was only \(2\) short of having \(200\) groschen. Which of the following systems can be used to find out how many coins of each type both boys have?
\[\begin{aligned} x +5y + 7z & = 198 & & \\ x - y+z & = 6 & & \\ x +y+z & = 40 & & \end{aligned}\]
\[\begin{aligned} x +5y+7z & = 198 & & \\x - y+z & = 6 & & \\(x+6) +5(y+6)+7(z+6) & = 276 & & \end{aligned}\]
\[\begin{aligned} x +5y+7z & = 198 & & \\x + y-z & = 6 & & \\(x+6) +5(y+6)+7(z+6) & = 276 & & \end{aligned}\]
\[\begin{aligned} x +5y+7z & = 202 & & \\x - y+z & = 6 & & \\(x+6) +(y+6)+(z+6) & = 58 & & \end{aligned}\]
\[\begin{aligned} x +5y+7z & = 198 & & \\x - y+z & = 6 & & \\x +5y+7z & = 40 & & \end{aligned}\]
\[\begin{aligned} x +5y+7z & = 198 & & \\x - y+z & = 6 & & \\(x-6) +5(y-6)+7(z-6) & = 276 & & \end{aligned}\]