1003162910
Level:
C
Find the sum of all the roots of the equation \( \left|x^2+2x\right|-6=|3x| \).
\( -3 \)
\( 9 \)
\( 0 \)
\( -2 \)
C
1003162905
Level:
C
Choose the equation that is equivalent to \( \bigl| |2-x|-1 \bigr|=1 \) for \( x\in(3;\infty) \).
\( (x-2)-1=1 \)
\( -(x-2)-1=1 \)
\( -\bigl(1-(2-x)\bigr)=1 \)
\( 1-(x-2)=1 \)
1003162904
Level:
C
Find the sum of all the roots of the equation \( \bigl| |1-x|-2 \bigr|=1 \) for \( x\in(1;\infty) \).
\( 6 \)
\( 4 \)
\( 1 \)
\( 2 \)
1003162903
Level:
C
Find the sum of all the roots of the given equation.
\[ \bigl| |x-4|-2\bigr|=0 \]
\( 8 \)
\( 12 \)
\( 4 \)
\( 6 \)
1003159201
Level:
C
The 3D printer prints a solid \( 5 \) centimetre cube in \( 2 \) hours. The printer can print the cube with the maximum edge length of \( 20\,\mathrm{cm} \). Suppose the printing time is directly proportional to the cube volume. Choose the function that describes the dependence of the number \( n \) of cubes printed in \( 1 \) day on the printed cube edge length \( a \), which is specified in centimetres. Neglect time needed for using the printer.
\( n=1500a^{-3};\ a\in(0;20] \)
\( n=60a^{-1};\ a\in(0;20] \)
\( n=300a^{-2};\ a\in(0;20] \)
\( n=2.4a;\ a\in(0;20] \)
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.
1003159101
Level:
C
Choose the function that describes the dependence of a cube volume \( V \) on the length of its space diagonal \( u \).
\( V=\frac{\sqrt3\cdot u^3}9;\ u\in(0;\infty) \)
\( V=u^3;\ u\in(0;\infty) \)
\( V=\frac{u^3}3;\ u\in(0;\infty) \)
\( V=27u^3;\ u\in(0;\infty) \)
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 \( x\in\mathbb{R} \).
\( p\in(-\infty;0) \)
\( p=0 \)
\( p\in(0;\infty) \)
\( p\in(-2;2) \)
1003162308
Level:
C
Find all the values of the real parameter \( p \) such that \( f(x)=(p-2) x^2+px+2 \) has a maximum.
\( p\in(-\infty;2) \)
\( p\in(-\infty;-2) \)
\( p\in(2;+\infty) \)
\( p\in(-\infty;0) \)
1003162307
Level:
C
Find all the values of the real parameter \( p \) such that \( f(x)=2x^2+3px+2 \) has a minimum.
\( p\in(-\infty;\infty) \)
\( p\in(-\infty;0)\cup(0;+\infty) \)
\( p=0 \)
\( p\in[0;\infty) \)