1003163002
Level:
C
Find all \( t \), \( t\in\mathbb{R} \), such that the following equation with the variable \( x \) has more than two solutions.
\[ \Bigl| |3-x|-3\Bigr|=t \]
\( t\in(0;3] \)
\( t\in\{0;3\} \)
\( t\in\{3\} \)
\( t\in[1;3] \)
C
1003108307
Level:
C
Choose the triple of points, such that the graph of any of the functions \( f(x)=ax^2+c \), where \( a\in\mathbb{R}\setminus{0} \), \( c\in\mathbb{R} \), does not pass through all three points.
\( [-2;5] \), \( [2;1] \), \( [0;3] \)
\( [-2;5] \), \( [2;5] \), \( [0;3] \)
\( [-2;5] \), \( [2;5] \), \( [0;7] \)
\( [-2;5] \), \( [0;0] \), \( [1;1] \)
1103148606
Level:
C
If an object moving with an initial velocity \( v_0 \) is slowing down at a constant deceleration \( a \), then the distance \( s \) travelled while decelerating is described by the formula \( s=v_0t-\frac12at^2 \), where \( t \) is time of decelerating. Choose the graph, which could represent the dependency of the distance \( s \) on the time \( t \).
1103148605
Level:
C
Suppose, an object, that is in rest, starts to accelerate with the constant acceleration \( a \). The distance \( s \) travelled by the object in time \( t \) is given by the formula \( s=\frac12at^2 \). You can see the graph of the distance \( s \) on the time \( t \) dependency in the picture. Find the acceleration \( a \) of the object.
\( 8\frac{\mathrm{m}}{\mathrm{s}^2} \)
\( 16\frac{\mathrm{m}}{\mathrm{s}^2} \)
\( 4\frac{\mathrm{m}}{\mathrm{s}^2} \)
\( 2\frac{\mathrm{m}}{\mathrm{s}^2} \)
1103148603
Level:
C
Consider a simple circuit in which a battery of electromotive force \( U_e \) and internal resistance \( R_i \) drives a current \( I \) through an external resistor of resistance \( R \) (see figure). The external resistor could be for example an electric light, an electric heating element, or, maybe, an electric motor. The basic purpose of the circuit is to transfer energy from the battery to the external resistor, where it actually does something useful for us (e.g. lighting a light bulb, or lifting a weight).
\[ \]
The power \( P \) transferred to the external resistor is described by the formula \( P=U_eI-R_i I^2 \). What maximum power can be transferred to the external resistor if we have the source with \( R_i=0.25\,\Omega \) and \( U_e=20\,\mathrm{V} \)?
\( 400\,\mathrm{W} \)
\( 80\,\mathrm{W} \)
\( 40\,\mathrm{W} \)
\( 790\,\mathrm{W} \)
1003148602
Level:
C
Consider an object thrown at an angle of \( 30^{\circ} \) above the horizontal with the initial velocity of \( 40\frac{\mathrm{m}}{\mathrm{s}} \). How long does it take for the object to reach its maximum height?
\[ \]
Note: The height \( y \) of an object thrown is described by the formula \( y=v_0t\sin\alpha-\frac12gt^2 \), where \( v_0 \) is the initial velocity, \( g \) is gravitational acceleration (count with the rounded value \( 10\frac{\mathrm{m}}{\mathrm{s}^2}\)), \( t \) is the time period of the object motion in seconds, and \( \alpha \) is the angle to the horizontal at which the object is thrown.
\( 2\,\mathrm{s} \)
\( 4\,\mathrm{s} \)
\( 8\,\mathrm{s} \)
\( 1\,\mathrm{s} \)
1003148601
Level:
C
Consider an object thrown upwards from the ground with the initial velocity of \( 30\frac{\mathrm{m}}{\mathrm{s}} \). The object moves upwards with decreasing vertical velocity until it stops. Then it starts moving vertically downwards. Find the greatest height above the ground the object does reach.
\[ \]
Note: The vertical distance \( y \) of a thrown object is described by the equation \( y=v_0t-\frac12gt^2 \), where \( v_0 \) is the initial velocity of the thrown object, \( g \) is gravitational acceleration (count with the rounded value \( 10\frac{\mathrm{m}}{\mathrm{s}^2}\)), and \( t \) is the time period of the object motion in seconds.
\( 45\,\mathrm{m} \)
\( 135\,\mathrm{m} \)
\( 360\,\mathrm{m} \)
\( 40\,\mathrm{m} \)
1103171201
Level:
C
Suppose a train comes to stop uniformly. The instantaneous velocity \( v \) of the train in the time \( t \) of slowing down is modelled by \( v=45-25t \). Find the graph of the function \( v \).
1003177805
Level:
C
Let \( x\in\mathbb{Z} \). Find the sum of all the solutions of the given inequality.
\[ \Bigl| \bigl| |x+1|+1 \bigr|+1\Bigr| \leq 3 \]
\( -3 \)
\( -2 \)
\( -4 \)
\( 0 \)
1003177803
Level:
C
Choose the domain of the expression.
\[ \frac1{\sqrt{|3x-9|-\sqrt2}} \]
\( \left(-\infty;3-\frac{\sqrt2}3\right)\cup\left(3+\frac{\sqrt2}3;\infty\right) \)
\( \left(-\infty;-3-\frac{\sqrt2}3\right)\cup\left(3+\frac{\sqrt2}3;\infty\right) \)
\( \left(-\infty;-3+\frac{\sqrt2}3\right)\cup\left(3+\frac{\sqrt2}3;\infty\right) \)
\( \left(-\infty;-3-\frac{\sqrt2}3\right)\cup\left(-3+\frac{\sqrt2}3;\infty\right) \)