1003162309
Level:
C
Find all the values of the real parameter \( p \) such that \( f(x)=px^2-4px+4p-3 \) is a negative quadratic function for all \( x\in\mathbb{R} \).
\( p\in(-\infty;0) \)
\( p=0 \)
\( p\in(0;\infty) \)
\( p\in(-2;2) \)
Quadratic functions
1003206002
Level:
C
We are given three quadratic functions:
\[ \begin{aligned}
f_1(x)&=ax^2+2ax+a-3, \\
f_2(x)&=a(x-1)^2+2, \\
f_3(x)&=ax^2,
\end{aligned} \]
where \( a\in(-\infty;0) \). If possible, determine which of the given functions has the highest output value for \( x = 0.5 \).
\( f_2 \)
\( f_3 \)
\( f_1 \)
Given information is insufficient to decide.
1103067801
Level:
C
Given the graph of the function \( f(x)=x^2 + 2x - 4 \), find the solution set of the following equation.
\[ \left|x^2 + 2x - 4\right|= 4 \]
\( \{ -4; -2; 0 ; 2 \} \)
\( \{ -4; 2 \} \)
\( \{ -4; 0; 2 \} \)
\( \left\{ -1-\sqrt5;-1+ \sqrt5 \right\} \)
1103067809
Level:
C
Given the graphs of the functions \( f(x)=\frac12x^2-3 \) and \( g(x)=\frac12x \), find the solution set of the following equation.
\[ \left|\frac12 x^2-3\right|=\left|\frac12 x\right| \]
\( \{ -3; -2; 2; 3 \} \)
\( \{ -2; 3 \} \)
\( \{ 2; 3 \} \)
\( \left\{ -\sqrt6; -2; \sqrt6; 3 \right\} \)
1103082702
Level:
C
Function \( f \) is given completely by the graph. Identify which of the following statements is true.
\( f(x)=\left|x^2-1\right|;\ x\in[-2;2] \)
\( f(x)=\left|x^2\right|-1;\ x\in[-2;2] \)
\( f(x)=-\left|x^2+1\right|;\ x\in[-2;2] \)
\( f(x)=\left|-x^2\right|+1;\ x\in[-2;2] \)
1103120009
Level:
C
In the picture there are two parabolas. One parabola can be mapped onto the other by shifting. These parabolas are graphs of the quadratic functions
\[ f(x)=-(x-a)^2+b\ \text{ and }\ g(x)=-(x-c)^2+d, \]
where \( a \), \( b \), \( c \), \( d\in\mathbb{R} \).
Following statements describe the relations between the pairs of the coefficients \( a \), \( b \), \( c \) and \( d \). Choose the true statement.
\( a=c-1\wedge b=d+4 \)
\( a=c+1\wedge b=d-4 \)
\( a=c-4\wedge b=d+1 \)
\( a=c+4\wedge b=d-1 \)
1103120010
Level:
C
Let \( f(x)=(x+a)^2+b \) and \( g(x)=(x+a-2)^2+b+3 \), where \( a \), \( b\in\mathbb{R} \). In which of the next four given pictures are the graphs of both functions \( f \) and \( g \)?
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 \( U_e=20\,\mathrm{V} \)?
\( 400\,\mathrm{W} \)
\( 80\,\mathrm{W} \)
\( 40\,\mathrm{W} \)
\( 790\,\mathrm{W} \)
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.
\( 8\frac{\mathrm{m}}{\mathrm{s}^2} \)
\( 16\frac{\mathrm{m}}{\mathrm{s}^2} \)
\( 4\frac{\mathrm{m}}{\mathrm{s}^2} \)
\( 2\frac{\mathrm{m}}{\mathrm{s}^2} \)
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 \( t \).
