Systems of linear equations and inequalities

2000019007

Level: 
B
The system of equations is given by: \[\begin{aligned} x+2z= 3 & & \\2x -y+ z = 2& & \\3x -2 y -z= 1 & & \end{aligned}\] When solving the system using Cramer's rule, we evaluate determinants of four matrices. What is the arithmetic mean of all these determinants?
\(2 \)
\(3.5 \)
\(\frac73 \)
\(\frac83 \)

2010006702

Level: 
B
The augmented matrix of a system of three equations with three unknowns is row equivalent with the following matrix \(A'\). Find the solution of the system. \[ A' = \left(\begin{array}{ccc|c} 2 & 3 & 1 & 7\\ 0 & 3 & 4 & 0\\ 0 & 0 & 5 & 45 \end{array}\right) \]
\([17;-12;9]\)
\([12;10;-9]\)
\([-19;12;9]\)
\([7;0;45]\)

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?
\( 420\,\mathrm{CZK} \)
\( 350\,\mathrm{CZK} \)
\( 440\,\mathrm{CZK} \)
\( 550\,\mathrm{CZK} \)

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

9000019904

Level: 
B
The coefficient matrix of a \(3\times 3\) linear system is \(A\) and the augmented matrix \(A'\). Find \(\mathop{\mathrm{rank}}(A)\) and \(\mathop{\mathrm{rank}}(A')\). \[ A = \begin{pmatrix} -1 & 3 & 2 \\ 0 & 4 & -5 \\ 0 & 0 & 2 \end{pmatrix} \qquad A' = \left(\begin{array}{ccc|c} -1 & 3 & 2 & 5 \\ 0 & 4 & -5 & 10\\ 0 & 0 & 2 & 0 \end{array}\right) \]
\(\mathop{\mathrm{rank}}(A) = 3,\ \mathop{\mathrm{rank}}(A') = 3\)
\(\mathop{\mathrm{rank}}(A) = 2,\ \mathop{\mathrm{rank}}(A') = 3\)
\(\mathop{\mathrm{rank}}(A) = 3,\ \mathop{\mathrm{rank}}(A') = 2\)
\(\mathop{\mathrm{rank}}(A) = 2,\ \mathop{\mathrm{rank}}(A') = 2\)

9000019905

Level: 
B
Let \(A\) and \(A'\) be the coefficient matrix and the augmented matrix of the following linear system, respectively. Find the ranks of these matrices. \[ \begin{array}{cl} \phantom{ -} 3x + 5y +\phantom{ 2}z =\phantom{ -}10& \\ - 2x - 3y + 2z = -10& \\ \phantom{ - 2}x +\phantom{ 2}y - 5z =\phantom{ -}10& \end{array} \]
\(\mathop{\mathrm{rank}}(A) = 2,\ \mathop{\mathrm{rank}}(A') = 2\)
\(\mathop{\mathrm{rank}}(A) = 3,\ \mathop{\mathrm{rank}}(A') = 3\)
\(\mathop{\mathrm{rank}}(A) = 3,\ \mathop{\mathrm{rank}}(A') = 2\)
\(\mathop{\mathrm{rank}}(A) = 2,\ \mathop{\mathrm{rank}}(A') = 3\)

9000019906

Level: 
B
Consider a linear system of four equations with four unknowns. The rank of the coefficient matrix \(A\) is \(\mathop{\mathrm{rank}}(A) = 3\). The rank of the augmented matrix \(A'\) is \(\mathop{\mathrm{rank}}(A') = 4\). Identify a true statement on this system.
The system does not have any solution.
The system has infinitely many solutions.
The system has a unique solution.
It is not possible to draw any conclusion from this information.