Systems of linear equations and inequalities

1003034506

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 \( 8 \) 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 \( 9 \) hours. How long will they mow together?
\( 2 \) hours
\( 7 \) hours
\( 6 \) hours
\( 3 \) hours

1003060501

Level: 
B
Which of the following systems of equations has no solution?
\(\begin{aligned} 2x-y+3z&=2 \\ 6x-3y+9z&=4 \\ x+y+z&=1 \end{aligned} \)
\(\begin{aligned} 2x-y+3z&=2 \\ 6x-3y+9z&=6 \\ x+y+z&=1 \\ \end{aligned} \)
\(\begin{aligned} 2x-y+3z&=2 \\ 6x-2y+z&=4 \\ x+y+z&=1 \end{aligned} \)
\(\begin{aligned} 2x-y+3z&=2 \\ 6x-3y+9z&=6 \\ 4x-4y+6z&=4 \end{aligned} \)

1103034507

Level: 
B
Consider a balance scales comprising of beam with unequal length of arms where the fulcrum is very close to one end of the beam. (Such scales are called steelyard. For instance, it is often used to weigh a catch in fisheries.) The load is hanged on the shorter arm, while the balance about the fulcrum is obtained by sliding the counterweight along the longer arm. (See the picture.) Suppose the distance of the load hanging point from the fulcrum is fixed at \( 5\,\mathrm{cm} \). If the weight of the load is \( 80\,\mathrm{N} \), the balance is achieved as the counterweight is moved to the very end of the longer arm. If the weight of the load is \( 60\,\mathrm{N} \), the balance is achieved when the counterweight is moved to the distance of \( 30\,\mathrm{cm} \) from the fulcrum. What is the length of the beam? \[ \] Hint: The steelyard is based on the law of the lever. For balanced lever is: \( F_1\cdot a=F_2\cdot b \), where \( F_1 \) is the weight of the load in the distance \( a \) from the fulcrum and \( F_2 \) is the weight of the counterweight in the distance \( b \) from the fulcrum.
\( 45\,\mathrm{cm} \)
\( 54\,\mathrm{cm} \)
\( 40\,\mathrm{cm} \)
\( 35\,\mathrm{cm} \)

2000019001

Level: 
B
Four matrices are given: \[\] $\left (\array{ 1& -1& 0\cr 2& 0& 1\cr 1& 1& -1} \right ),$ $\left (\array{ 1& -3& 0\cr 2& -5& 1\cr 1& 0& -1} \right ),$ $\left (\array{ -3& -1& 0\cr -5& 0& 1\cr 0& 1& -1} \right ),$ $\left (\array{ 1& -1& -3\cr 2& 0& -5\cr 1& 1& 0} \right )$ \[\] We want to practice Cramer's rule for solving a system of linear equations. Which of the following systems can be solved using determinants of the four matrices given above?
\[\begin{aligned} x- y = -3 & & \\2x + z = -5 & & \\x + y -z= 0 & & \end{aligned}\]
\[\begin{aligned} x- y-3z = 0 & & \\2x - 5z = 1 & & \\x + y = -1& & \end{aligned}\]
\[\begin{aligned} -3x- y = 0 & & \\-5x + z = 1 & & \\ y -z= -1& & \end{aligned}\]
\[\begin{aligned} x- y = 3 & & \\2x + z = 5 & & \\x + y -z= 0 & & \end{aligned}\]

2000019002

Level: 
B
The system of equations is given by: \[\begin{aligned} x- y = -3 & & \\2x + z = -5 & & \\x + y -z= 0 & & \end{aligned}\] When solving the system using Cramer's rule, we evaluate determinants of four matrices. Suppose we order them according to their values. What is the largest value of these determinants?
\(8\)
\(4\)
\(-4\)
\(12\)

2000019003

Level: 
B
Consider a linear system of three equations with three unknowns \(x\), \(y\), \(z\), and with the column of the right sides: \[ \left (\array{ 5\cr 17\cr 12} \right ) \] The determinants of the following two matrices were used to solve the system by Cramer's rule: \[ \left (\array{ 2& 5& 1\cr 1& 17& -3\cr 1& 12& -2} \right ),~ \left (\array{ 2& -1& 5\cr 1& 2& 17\cr 1& 1& 12} \right ) \] Which of the following systems could be solved in a specified way?
\[\begin{aligned} 2x- y +z= 5 & & \\x +2y-3 z = 17 & & \\x + y -2z= 12 & & \end{aligned}\]
\[\begin{aligned} 2x+5 y +z= -1 & & \\x +17y-3 z = 2& & \\x +12 y -2z= 1 & & \end{aligned}\]
\[\begin{aligned} 2x- y +z= -5 & & \\x +2y-3 z = -17 & & \\ x+y -2z= -12& & \end{aligned}\]
\[\begin{aligned} 2x+ y-z = 5 & & \\x-2y + 3z = 17 & & \\x - y +2z= 12 & & \end{aligned}\]

2000019004

Level: 
B
The system of equations is given by: \[\begin{aligned} 2 x-y +z=5 & & \\x +2y-3z =17& & \\x +y -2z= 12& & \end{aligned}\] When solving the system using Cramer's rule, we evaluate determinants of four matrices. What is the sum of all these determinants?
\(-14\)
\(12\)
\(0\)
\(-20\)

2000019005

Level: 
B
To solve the system of three linear equations with three unknowns, it is necessary to calculate the determinants of the matrices: \[ \left (\array{ 1& -2& 3\cr 2& 1& -7\cr -3& 1& -5} \right ),~ \left (\array{ 1& 3& -1\cr 2& -7& -3\cr -3& -5& 1} \right ). \] Which of the given ordered triples is the solution to this system?
\( [2,-2,3]\)
\( [2,2,3]\)
\( [-2,2,3]\)
\( [3,-2,2]\)

2000019006

Level: 
B
The coefficient matrix of a system of three linear equations with three unknowns is: \[ \left (\array{ 1& 2& 1\cr 3& -5& 2\cr 1& 0& -3} \right ).~ \] What is the column of the right sides if the solution is the ordered triple \([−7; 2;−1]\)?
\( \left (\array{ -4\cr -33\cr -4} \right ) \)
\( \left (\array{ -2\cr -33\cr -4} \right ) \)
\( \left (\array{ -4\cr -31\cr -4} \right ) \)
\( \left (\array{ -4\cr -33\cr -10} \right ) \)