C

1103021602

Level: 
C
The side of an equilateral triangle is \( 6\,\mathrm{cm} \) long. Find the area of the annulus between the incircle and circumcircle of the given triangle. (See the picture.)
\( 9\pi\,\mathrm{cm}^2 \)
\( 6\pi\,\mathrm{cm}^2 \)
\( 12\pi\,\mathrm{cm}^2 \)
\( 8\pi\,\mathrm{cm}^2 \)

1003164304

Level: 
C
Which of the following situations could arise for suitable functions \( f \) and \( g \)?
\( \lim\limits_{x\to2} f(x)=\infty\ \wedge\ \lim\limits_{x\to2}g(x)=-\infty\ \wedge\ \lim\limits_{x\to2}[f(x)+g(x)]=-\infty \)
\( \lim\limits_{x\to2} f(x)=13\ \wedge\ \lim\limits_{x\to2}g(x)=0\ \wedge\ \lim\limits_{x\to2}\frac{f(x)}{g(x)}=13 \)
\( \lim\limits_{x\to2} f(x)=-\infty\ \wedge\ \lim\limits_{x\to2}g(x)=\infty\ \wedge\ \lim\limits_{x\to2}[f(x)-g(x)]=0 \)
\( \lim\limits_{x\to2} f(x)=\infty\ \wedge\ \lim\limits_{x\to2}g(x)=-\infty\ \wedge\ \lim\limits_{x\to2}[f(x)\cdot g(x)]=\infty \)

1003164303

Level: 
C
Which of the following situations could arise for suitable functions \( f \) and \( g \)?
\( \lim\limits_{x\to5}f(x)=0\ \wedge\ \lim\limits_{x\to5}g(x)=\infty\ \wedge\ \lim\limits_{x\to5}[f(x)\cdot g(x)]=13 \)
\( \lim\limits_{x\to5}f(x)=1\ \wedge\ \lim\limits_{x\to5}g(x)=\infty\ \wedge\ \lim\limits_{x\to5}[f(x)\cdot g(x)]=13 \)
\( \lim\limits_{x\to5}f(x)=\infty\ \wedge\ \lim\limits_{x\to5}g(x)=\infty\ \wedge\ \lim\limits_{x\to5}[f(x)\cdot g(x)]=13 \)
\( \lim\limits_{x\to5}f(x)=-\infty\ \wedge\ \lim\limits_{x\to5}g(x)=\infty\ \wedge\ \lim\limits_{x\to5}[f(x)\cdot g(x)]=13 \)

1003164302

Level: 
C
Which of the following situations could arise for suitable functions \( f \) and \( g \)?
\( \lim\limits_{x\to2}⁡f(x)=\infty\ \wedge\ \lim\limits_{x\to2}g(x)=\infty\ \wedge\ \lim\limits_{x\to2}[f(x)-g(x)]=\infty \)
\( \lim\limits_{x\to2}⁡f(x)=1\ \wedge\ \lim\limits_{x\to2}g(x)=\infty\ \wedge\ \lim\limits_{x\to2}\frac{f(x)}{g(x)}=1 \)
\( \lim\limits_{x\to2}⁡f(x)=-\infty\ \wedge\ \lim\limits_{x\to2}g(x)=1\ \wedge\ \lim\limits_{x\to2}[f(x)+g(x)]=1 \)
\( \lim\limits_{x\to2}⁡f(x)=-\infty\ \wedge\ \lim\limits_{x\to2}g(x)=-\infty\ \wedge\ \lim\limits_{x\to2}[f(x)\cdot g(x)]=-\infty \)

1003164301

Level: 
C
Which of the following situations could arise for suitable functions \( f \) and \( g \)?
\( \lim\limits_{x\to3} f(x)=\infty\ \wedge\ \lim\limits_{x\to3} g(x)=\infty\ \wedge\ \lim\limits_{x\to3}\frac{f(x)}{g(x)}=5 \)
\( \lim\limits_{x\to3} f(x)=1\ \wedge\ \lim\limits_{x\to3} g(x)=\infty\ \wedge\ \lim\limits_{x\to3}\frac{f(x)}{g(x)}=5 \)
\( \lim\limits_{x\to3} f(x)=\infty\ \wedge\ \lim\limits_{x\to3} g(x)=1\ \wedge\ \lim\limits_{x\to3}\frac{f(x)}{g(x)}=5 \)
\( \lim\limits_{x\to3} f(x)=0\ \wedge\ \lim\limits_{x\to3} g(x)=\infty\ \wedge\ \lim\limits_{x\to3}\frac{f(x)}{g(x)}=5 \)

1003197408

Level: 
C
The painter \( A \) takes \( 15 \) hours to paint a flat. The painter \( B \) takes \( 12 \) hours to paint the same flat and the painter \( C \) takes \( 10 \) hours to complete the same task. The painter \( A \) starts to paint alone and after \( 2 \) hours joins him the painter \( B \). After the next hour the painter \( C \) starts to paint with them too. How long will all three painters paint together before completing the work?
\( 2\,\mathrm{h}\ 52\,\mathrm{min} \)
\( 3\,\mathrm{h}\ 8\,\mathrm{min} \)
\( 3\,\mathrm{h}\ 24\,\mathrm{min} \)
\( 4\,\mathrm{h} \)

1003197407

Level: 
C
A hose can fill the garden pool in \( 20 \) hours. The owners bought a pump that guarantees filling the pool in \( 8 \) hours. However, a crack appeared in the pool's shell which can drain all the water from the full pool within \( 5 \) days. How long will it take to fill the pool with the hose and the pump at the same time as the water runs off through the crack?
exactly \( 6 \) hours
approximately \( 5.7 \) hours
approximately \( 5.5 \) hours
approximately \( 6.8 \) hours

1003197406

Level: 
C
Each of two companies should deliver the same amount of raw material. While inspecting it was found out that the company \( A \) delivered \( 150\,\mathrm{kg} \) and the company \( B \) delivered \( 194\,\mathrm{kg} \). At the moment of inspection, company \( A \) has to deliver yet three times more than what is left to be delivered by company \( B \). Choose the equation which does NOT correspond to the described situation.
\( 3(x-150)=x-194 \), where \( x \) represents the total planned delivery of both companies.
\( x-150=3(x-194) \), where \( x \) represents the total planned delivery of both companies.
\( 150+3x=194+x \), where \( x \) represents the amount of raw material which is left to be delivered by company \( B \).
\( 150+x=194+\frac x3 \), where \( x \) represents the amount of raw material which is left to be delivered by company \( A \).

1003197405

Level: 
C
Nine people travel by bus. The same number of people gets off the bus at each of three bus stops and then so many people get in to double the number of remaining bus passengers. After the third bus stop there are \( 30 \) people in the bus. How many passengers get off at each bus stop?
\( 3 \)
\( 2 \)
\( 1 \)
\( 6 \)

1003197404

Level: 
C
The stock of the monitored company lost \( 12\,\% \) of its value during a week. Its fall continued over the next week, and the value declined by another \( 4\,\% \). Let’s denote the original value of the stock by an \( x \). From the following possibilities, choose the expression that gives the value of the stock at the end of the monitored term.
\( 0.96\cdot0.88x \)
\( (0.96+0.88)x \)
\( 0.04\cdot0.12x \)
\( [1-(0.04+0.12)]x \)