1003082601
Level:
C
Find the set of all solutions of the following inequality.
\[ 27^x-3^{3x}\leq 3 \]
\( (-\infty;\infty) \)
\( (-\infty;0] \)
\( (-\infty;1] \)
The inequality has no solution.
C
1003064203
Level:
C
Find the solution of the following inequality.
\[ \sqrt{2^{2x}-2^{3+x} } < 2^x-2^3 \]
No solution
\( x\in(-\infty;3) \)
\( x\in(8;\infty) \)
\( x\in[3;\infty) \)
1003064202
Level:
C
Find the solution of the following inequality.
\[ 2^x\leq 3-2^{1-x} \]
\( x\in[0; 1] \)
\( x\in[1; 2] \)
\( x\in[-2; -1] \)
No solution
1003064201
Level:
C
How many solutions does the following inequality in the set of positive integers have?
\[ 16^x-3\cdot4^x-4\leq0 \]
Exactly one solution
Infinitely many solutions
Exactly two solutions
No solutions
1003029206
Level:
C
In the hospital, \( 22 \) boys and \( 18 \) girls were born in one month. Babies were listed in the register by their date of birth. Find the probability that there are at least three boys in the first five places of the register. The results are rounded to four decimal places.
\( \frac{\binom{22}3\cdot\binom{18}2+\binom{22}4\cdot\binom{18}1+\binom{22}5\cdot\binom{18}0}{\binom{40}5} = 0{.}5982 \)
\( \frac{\binom{22}3+\binom{22}4+\binom{22}5}{\binom{40}5} = 0{.}0535 \)
\( \frac{22^3\cdot18^2+22^4\cdot18^1+22^5\cdot18^0}{40^5}=0{.}1252 \)
\( \frac{\binom{22}3\cdot\binom{18}2+\binom{22}4\cdot\binom{18}1+\binom{22}5\cdot\binom{18}0}{40^5} = 0{.}0038 \)
1003029205
Level:
C
In the hospital, \( 22 \) boys and \( 18 \) girls were born in one month. Babies were listed in the register by their date of birth. Find the probability that there are two boys and three girls in the first five places of the register. The results are rounded to four decimal places.
\( \frac{\binom{22}2\cdot\binom{18}3}{\binom{40}5}=0{.}2865 \)
\( \frac{\binom{22}2\cdot\binom{18}3}{\frac{40!}{35!}}=0{.}0024 \)
\( \frac{22^2\cdot18^3}{40^5} = 0{.}0276 \)
\( \frac{\binom{22}3\cdot\binom{18}2}{\frac{40!}{35!}}=0{.}0030 \)
1003029204
Level:
C
A class consists of \( 50 \) students including twins, Mark and Martin. For an exam students are randomly divided into two equally sized subgroups. Find the probability that Mark and Martin will be in the same subgroup. The results are rounded to two decimal places.
\( \frac{\binom{48}{23}+\binom{48}{25}}{\binom{50}{25}}=0{.}49 \)
\( \frac{\binom{48}{23}}{\binom{50}{25}}=0{.}24 \)
\( \frac{2\cdot\binom{48}{24}}{\binom{50}{25}}=0{.}51 \)
\( \frac{\binom{49}{24}}{\binom{50}{25}}=0{.}50 \)
1003029203
Level:
C
Three dice are thrown together. Find the probability that three different outcomes are rolled. Results are rounded to two decimal places.
\( \frac{\binom61\cdot\binom51\cdot\binom41}{6^3}=0{.}56 \)
\( \frac{\binom61+\binom51+\binom41}{6^3}=0{.}07 \)
\( \frac{\binom66\cdot\binom65\cdot\binom64}{6^3}=0{.}42 \)
\( \frac{\binom66+\binom65+\binom64}{6^3}=0{.}10 \)
1003067810
Level:
C
Find the solution set of the following equation.
\[ |x-4|\cdot(x+4)=4 \]
\( \left\{-2\sqrt3;2\sqrt3;2\sqrt5\right\} \)
\( \{ -4; 4 \} \)
\( \left\{ -2\sqrt3;2\sqrt3\right\} \)
\( \left\{ 2\sqrt3;2\sqrt5\right\} \)
\( \left\{-2\sqrt5;-2\sqrt3;2\sqrt3;2\sqrt5 \right\} \)
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.
\[ \left|\frac12 x^2-3\right|=\left|\frac12 x\right| \]
\( \{ -3; -2; 2; 3 \} \)
\( \{ -2; 3 \} \)
\( \{ 2; 3 \} \)
\( \left\{ -\sqrt6; -2; \sqrt6; 3 \right\} \)