C

9000007208

Level: 
C
Paul's home is \(6\, \mathrm{km}\) from the school. At the time \(t = 0\) Paul starts to walk from his home to the school along a straight street at a constant velocity \(5\, \mathrm{km}/\mathrm{h}\). Find the function which describes Paul's remaining distance to the school as a function of time.
\(s = 6 - 5t\)
\(s = 5t - 6\)
\(s = 5t\)
\(s = 5t + 6\)

9000007809

Level: 
C
The price of a goods in a shop is \(\$15\) per item. The Internet price in an e-shop is cheaper by \(\$2\) per item. The shipping cost of the e-shop is \(\$125\). What is the minimal number of items, which makes the total cost for a transaction smaller in the e-shop?
\(63\)
\(9\)
\(62\)
\(125\)
\(126\)

9000009301

Level: 
C
An automatic machine produces \(12\) components per minute and stores them in a box with capacity \(1\: 500\) components. The machine starts with an initial amount of \(240\) components in the box. In what time will be the box full?
\(1\, \mathrm{h}\) \(45\, \mathrm{min}\)
\(1\, \mathrm{h}\) \(55\, \mathrm{min}\)
\(2\, \mathrm{h}\) \(5\, \mathrm{min}\)
\(2\, \mathrm{h}\) \(15\, \mathrm{min}\)

9000009302

Level: 
C
An automatic machine produces \(12\) components per minute and stores them in a box with capacity \(1\: 500\) components. The machine starts with an initial amount of \(240\) components in the box. In what time will the box contain \(1\: 020\) components?
\(1\, \mathrm{h}\) \(5\, \mathrm{min}\)
\(55\, \mathrm{min}\)
\(1\, \mathrm{h}\)
\(1\, \mathrm{h}\) \(10\, \mathrm{min}\)

9000007210

Level: 
C
Jane has to reach the opposite shore of a lake. She has three possibilities how to cross. She can use her own boat, start immediately and sail at the velocity \(4\, \mathrm{km}/\mathrm{h}\). Another option is to wait for her friend, Peter, with a faster boat. Peter's boat is capable to sail at the velocity \(10\, \mathrm{km}/\mathrm{h}\). However, his boat will be available in \(1.5\) hours from now. The last option is to use the regular passenger transportation line which will departure in \(2.25\) hours from now and sails at the speed \(20\, \mathrm{km}/\mathrm{h}\). Find the interval of distances to the opposite shore for which the fastest option is to use Peter's boat.
between \(10\) and \(15\) kilometers
up to \(10\) kilometers
between \(15\) and \(20\) kilometers
larger that \(20\) kilometers

9000007202

Level: 
C
Consider the function \[ f(x) = [x] + 3 \] on the domain \(\mathop{\mathrm{Dom}}(f) = (1,2)\). Find the parameters \(a\) and \(b\) and a domain of the linear function \[ g\colon y = ax + b \] which ensure that \(f\) and \(g\) are identical functions. \[ \] Hint: The function \(y = [x]\) is a floor function: the largest integer less than or equal to \(x\). For positive \(x\) it is also called the integer part of \(x\).
\(a = 0\), \(b = 4\), \(\mathop{\mathrm{Dom}}(g) = (1,2)\)
\(a = 0\), \(b = 3\), \(\mathop{\mathrm{Dom}}(g) = (1,2)\)
\(a = 3\), \(b = 0\), \(\mathop{\mathrm{Dom}}(g) = (1,2)\)
\(a = -3\), \(b = 0\), \(\mathop{\mathrm{Dom}}(g) = (1,2)\)

9000007203

Level: 
C
Consider the function \[ f(x) =\mathop{ \mathrm{sgn}}\nolimits (x - 2) \] defined on \(\mathop{\mathrm{Dom}}(f) =\mathbb{R} ^{-}\). Find the parameters \(a\) a \(b\) and domain of the linear function \[ g(x) = ax + b \] which ensure that \(f\) and \(g\) are identical functions. \[ \] Hint: The function \(y =\mathop{ \mathrm{sgn}}\nolimits (x)\) is the sign function. The values of sign function is \(1\) for each positive \(x\), \(- 1\) for each negative \(x\) and \(0\) if \(x = 0\).
\(a = 0\), \(b = -1\), \(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{-}\)
\(a = 0\), \(b = 1\), \(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{+}\)
\(a = 1\), \(b = 0\), \(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{-}\)
\(a = -1\), \(b = 0\), \(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{+}\)

9000007207

Level: 
C
Identify a function which has the following three properties: it has at least one minimum or maximum, it is an increasing function and the range of this function is the set of all nonnegative numbers.
\(f(x) = 2x - 2\), \(x\in [ 1,+\infty )\)
\(f(x) = 2x + 2\), \(x\in (-1,+\infty )\)
\(f(x) = -2x + 2\), \(x\in (-\infty ,1] \)
\(f(x) = -2x - 2\), \(x\in \mathbb{R}\)

9000007209

Level: 
C
A current-voltage characteristics is shown in the graph. Find the current \(I\) as a function of the voltage \(U\).
\(I = \frac{2} {3}U -\frac{4}{3},U\in [2,\infty) \)
\(I = \frac{3} {2}U - 2,U\in [2,\infty) \)
\(I = \frac{3} {2}U + 2,U\in [2,\infty) \)
\(I = \frac{2} {3}U + 2,U\in [2,\infty) \)