1003086105
Level:
C
How many solutions does the equation \( 2\cos^2 x - \cos x - 1 = 0 \) have in the interval \( [-\pi;3\pi] \)?
\( 6 \)
\( 4 \)
\( 2 \)
\( 3 \)
C
1003086103
Level:
C
The solution set of the equation \( 2\sin x + \mathrm{tg}\,x = 0 \), \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{k\pi;\frac{2\pi}3+2k\pi;\frac{4\pi}3+2k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{2k\pi;\frac{2\pi}3+2k\pi;\frac{4\pi}3+2k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{k\pi;\frac{5\pi}6+k\pi;\frac{7\pi}6+k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{k\pi;\frac{5\pi}6+2k\pi;\frac{7\pi}6+2k\pi\right\} \)
1003158408
Level:
C
The class consists of \( 10\,\% \) long haired boys, \( 30\,\% \) short haired boys, \( 50\,\% \) long haired girls and \( 10\,\% \) short haired girls. We choose one of the students from the class randomly. Find the probability that the student has long hair if you know he is a boy.
\( 0{.}25 \)
\( 0{.}40 \)
\( 0{.}10 \)
\( 0{.}03 \)
1003158407
Level:
C
The long-term records of a car dealer show that a customer getting a new car buys within special equipment offers a parking assistant system (PAS) with the probability \( 50\,\% \) and that he buys a xenon flashlight with the probability \( 20\,\% \). Both items of special equipment (PAS and xenon lamps) are purchased by the customer with the probability \( 10\,\% \). What is the probability that the customer bought xenon lamps if you know he bought PAS?
\( 20\,\% \)
\( 60\,\% \)
\( 10\,\% \)
\( 80\,\% \)
1003158406
Level:
C
A bag contains \( 10 \) white balls and \( 5 \) black balls. Two balls are drawn at random one by one without replacement. Find the probability that two black balls are drawn.
\( \frac2{21} \)
\( \frac2{15} \)
\( \frac14 \)
\( \frac19 \)
1103158405
Level:
C
Let's roll a special die with the numbers placed according to the given scheme. Find the probability that the result is not \( 6 \) given that the result is greater than \( 2 \).
\( \frac34 \)
\( \frac56 \)
\( \frac15 \)
\( \frac14 \)
1103158404
Level:
C
A box (see the picture) contains \( 5 \) red balls and \( 7 \) green balls which are marked with numbers. A ball is drawn uniformly at random from the box. Let the event $A$ be: The drawn ball is green. And let the event $B$ be: The drawn ball is marked with the number which is greater than $6$. Find \( P(A|B) \). (Round the result to two decimal places.)
\( 0{.}50 \)
\( 0{.}43 \)
\( 0{.}25 \)
\( 0{.}83 \)
1103158403
Level:
C
A box (see the picture) contains \( 5 \) red balls and \( 7 \) green balls which are marked with numbers. A ball is drawn uniformly at random from the box. Find the probability that the ball is marked with even number given that the ball is red. (Round the result to two decimal places.)
\( 0{.}60 \)
\( 0{.}50 \)
\( 0{.}83 \)
\( 0{.}25 \)
1103158402
Level:
C
We roll red and yellow dice. Let the event $A$ be: On the red die a number greater than $2$ is rolled. And let the event $B$ be: The sum of points rolled on both dice is greater than $6$. Find \( P(A|B) \). (Note: In the table the sums of points on both dice are listed.)
\( \frac34 \)
\( \frac12 \)
\( \frac67 \)
\( \frac14 \)
1103158401
Level:
C
We roll red and yellow dice. Find the probability that the yellow die is two given that the sum of both dice is eight. (Note: In the following table, the sums of points on both dice are listed.)
\( \frac15 \)
\( \frac16 \)
\( \frac1{36} \)
\( \frac5{36} \)