1003027303
Level:
B
Choose the function \( g \) whose derivative on \( (-\pi,0) \) is the following function:
\[ f(x)=\sin^2x\cdot\mathrm{cotg}^2\,x+\sin^2x-1\]
\( g(x)=5 \)
\( g(x)=x+3 \)
\( g(x)=\mathrm{tg}\,x \)
\( g(x)=\sin\,x \)
B
1003027302
Level:
B
Evaluate the following indefinite integral on \( \left(-\frac{\pi}2,\frac{\pi}2 \right) \).
\[ \int \left(\frac1{\cos x}-\sin x\cdot\mathrm{tg}\,x\right)\,\mathrm{d}x \]
\( \sin x+c\text{, }c\in\mathbb{R} \)
\( \cos x+c\text{, }c\in\mathbb{R} \)
\( -\sin x+c\text{, }c\in\mathbb{R} \)
\( -\cos x+c\text{, }c\in\mathbb{R} \)
1003027301
Level:
B
Evaluate the following indefinite integral on \( \left(0,\frac{\pi}2 \right) \).
\[ \int\frac{(\sin x+\cos x)^2-1}{\sin x\cos x}\,\mathrm{d}x \]
\( 2x+c\text{, }c\in\mathbb{R} \)
\( x+c\text{, }c\in\mathbb{R} \)
\( \mathrm{tg}\,x+c\text{, }c\in\mathbb{R} \)
\( c\text{, }c\in\mathbb{R} \)
1003029305
Level:
B
The production process of a particular component consists of three independent phases.
Through long-term monitoring of production quality were found success rates of individual phases \( 90\% \), \( 80\% \) and \( 85\% \). If all three phases of the process are successful, the component is of good quality. What is the probability of producing a good quality component?
\( 0{.}612 \)
\( 0{.}003 \)
\( 0{.}388 \)
\( 0{.}997 \)
1003029304
Level:
B
The probability of a man hitting a target is \( 0{.}9 \). What is the probability that he hits the target twice in a row? Round the result to two decimal places.
\( 0{.}81 \)
\( 0{.}99 \)
\( 0{.}95 \)
\( 0{.}18 \)
1003029302
Level:
B
The quality inspection found that \( 85\% \) of the items are without defect, exactly one defect has \( 10\% \) of the items, and the other items have more than one defect. We pick one item randomly. What is the probability that this item has at least one defect?
\( 0{.}15 \)
\( 0{.}10 \)
\( 0{.}95 \)
\( 0{.}01 \)
1003029301
Level:
B
Two regular dice are thrown. Find the probability that the sum is five or six. The results are rounded to two decimal places.
\( \frac14= 0{.}25 \)
\( \frac5{36}\doteq 0{.}14 \)
\( \frac2{11}\doteq 0{.}18 \)
\( \frac{10}{36}\doteq 0{.}28\)
1003049306
Level:
B
Identify which of the following functions is odd.
\( g(x)=|x|-|-x| \)
\( h(x)=|x|-x \)
\( f(x)=-|-x| \)
\( m(x)=|-1| \)
1003049305
Level:
B
Identify a real number \( a \) such that the function \( f(x)=a|-12-ax|-a \) has a maximum at \( x=3 \).
\( -4 \)
\( 3 \)
\( 4 \)
There is no such number.
1003049304
Level:
B
Identify which of the following functions is increasing in the interval \( (-\infty, +2] \).
\( m(x)=-2|5-x|+1 \)
\( h(x)=|x|+2 \)
\( g(x)=-|x|-2 \)
\( f(x)=-|x+2| \)