1003162901
Level:
B
Calculate for which value(s) of the parameter \( t \) the given equation has at least one solution.
\[ |x-5|=t \]
\( t\in[0;\infty) \)
\( t\in[5;\infty) \)
\( t\in\{0\} \)
\( t\in(-\infty;5] \)
B
1003199905
Level:
B
Let \( f(x)=\left(\frac{x^2}{\sqrt[3]x}\right)^{-0.5} \). Identify which of the following statements is true.
\( f(x)=x^{-\frac56} \)
\( f(x)=x^{\frac13} \)
\( f(x)=x^{-\frac16} \)
\( f(x)=x^{\frac76} \)
1003199904
Level:
B
Let \( f(x)=x^{\frac12} \) and \( g(x)=x^{\frac32} \). Identify which of the following statements is true.
\( \frac{f(2)}{g(2)} =0.5 \)
\( f(2)+g(2)=4 \)
\( f(4)\cdot g(4)=8 \)
\( \bigl(f(2)\bigr)^{\frac32}=4 \)
1003199903
Level:
B
Let \( f(x)=x^{-\frac32} \). Identify which of the following statements is false.
\( f(8)=0.25 \)
\( f(100)=0.001 \)
\( f(\frac1{16})=64 \)
\( f(\frac12)=2\sqrt2 \)
1003199902
Level:
B
Let \( f(x)=x^{\frac23} \). Identify which of the following statements is true.
\( f(27)=9 \)
\( f(16)=64 \)
\( f(\frac18)=4 \)
\( f(0.0125)=0.25 \)
1003199901
Level:
B
Let \( f(x)=x^{\frac12} \). Identify which of the following statements is false.
\( f(0.25)=16 \)
\( f(0.0121)=0.11 \)
\( f(\frac12)=\frac{\sqrt2}2 \)
\( f(338)=13\sqrt2 \)
1003163005
Level:
B
Find all \( t \), \( t\in\mathbb{R} \), such that the following equation with the variable \( x \) has exactly two solutions.
\[ |x-t|+1=3 \]
\( t\in\mathbb{R} \)
\( t\in(-\infty;2) \)
\( t\in(2;\infty) \)
\( t\in(-2;\infty) \)
1003163003
Level:
B
How many roots has the given equation for \( x\in(-\infty;-3] \)?
\[ |3-2x|-3|x|=-8 \]
\( 1 \)
\( 2 \)
\( 0 \)
\( 3 \)
1003163001
Level:
B
Find all \( t \), \( t\in\mathbb{R} \), such that the following equation with the variable \( x \) has exactly two solutions.
\[ |x|+t=-3 \]
\( t\in(-\infty;-3) \)
\( t\in\{-3\} \)
\( t\in(-3;\infty) \)
\( t\in\{-3;3\} \)
1003108312
Level:
B
The graph of the function \( f \) is a parabola, vertex of which is \( [6;0] \) and \( f(2)= 8 \) is given. Find the function \( f \).
\( f(x)=\frac12(x-6)^2 \)
\( f(x)=-\frac12(x-6)^2 \)
\( f(x)=\frac12(x+6)^2 \)
\( f(x)=\frac12x^2+6 \)