Task 1. At the geometry exam, the student gets one question from the list of exam questions. The probability that this is a question on External Angles is 0.35. The probability that this is an inscribed circle question is 0.2. There are no questions that simultaneously relate to these two topics. Find the probability that a student will get a question on one of these two topics in the exam.
Solution:
Events “You will get a question on the topic Inscribed Angles” and “You will get a question on the topic Inscribed Circle” – . This means that the probability that a student will get a question on one of these two topics in the exam is equal to the sum of the probabilities of these events: 0.35 + 0.2 = 0.55.
Answer: 0.55.
Task 2. Two factories produce the same glass for car headlights. The first factory produces 70% of these glasses, the second – 30%. The first factory produces 1% of defective glass, and the second – 3%. Find the probability that glass accidentally purchased in a store will be defective.
Solution:
Situation 1:
The glass comes from the first factory (event probability 0.7) and (multiplication) it is defective (probability of event 0.01).
That is, both events must happen. In the language of probability theory, this means each of the events:
Situation 2:
The glass comes from the second factory (event probability 0.3) And it is defective (probability of event 0.03):
Because when buying glass we find ourselves in situation 1 or (amount) in situation 2, then we get:
Answer: 0.016.
Task 3. In the shopping center, two identical machines sell coffee. The probability that the machine will run out of coffee by the end of the day is 0.3. The probability that both machines will run out of coffee is 0.16. Find the probability that at the end of the day there will be coffee left in both machines.
Solution:
The probability of event A: “the coffee will run out in the first machine” P(A) is 0.3.
The probability of event B: “the coffee will run out in the second machine” P(B) is 0.3.
The probability of event AB: “the coffee will run out in both machines” P(AB) is equal to 0.16.
The probability of the sum of two joint events A+B is the sum of their probabilities without the probability of event AB:
We are interested in the probability of an event opposite to event A+B. Indeed, a total of 4 events are possible, three of them, marked in yellow, correspond to event A+B:
Answer: 0.56.
Task 4. There are two payment machines in the store. Each of them can be faulty with probability 0.12, regardless of the other machine. Find the probability that at least one machine is working.
Solution:
Both machines are faulty with probability
At least one machine is working (good+faulty, faulty+faulty, good+good) – this is an event opposite to the event “both machines are faulty”, so its probability is
Answer: 0.9856.
Task 5. A biathlete shoots at targets 5 times. The probability of hitting the target with one shot is 0.85. Find the probability that the biathlete hit the targets the first 3 times and missed the last two. Round the result to hundredths.
Solution:
Biathlete hits the target for the first time and (multiplication) second, And third:
Since the probability of hitting the target is , then the probability of the opposite event, a miss, is
The biathlete missed the fourth shot And at the fifth:
Then the probability that the biathlete hit the target the first 3 times is ( And!) the last two missed are as follows:
Answer: 0.01.
Task 6. The probability that a new vacuum cleaner will last more than a year is 0.92. The probability that it will last more than two years is 0.84. Find the probability that it will last less than two years but more than a year.
Solution:
Consider the following events:
A – “the vacuum cleaner will last more than a year, but less than 2”,
B – “the vacuum cleaner will last more than 2 years”,
C – “the vacuum cleaner will last more than a year.”
Event C is the sum of joint events A and B, that is
But, since both A and B cannot happen at the same time.
Answer: 0.08.
Task 7. The room is illuminated by a lantern with three lamps. The probability of one lamp burning out within a year is 0.07. Find the probability that at least one lamp will not burn out during the year.
Solution:
Probability of all three light bulbs burning out within a year
Then the probability of the opposite event - at least one lamp will not burn out - is
Answer: 0.999657.
Task 8. An agricultural company purchases chicken eggs from two households. 40% of eggs from the first farm are eggs of the highest category, and from the second farm - 90% of eggs of the highest category. In total, 60% of eggs receive the highest category. Find the probability that an egg purchased from this agricultural company will come from the first farm.
Solution:
Method I
Let the probability that the egg purchased from the agricultural company is from farm I be . Then the probability that the egg purchased from the agricultural company is from farm II is .
1) from farm I And Category I
2) from farm II And I category,
II method
Let be the number of eggs of the first farm, then the number of eggs of the highest category in this farm is .
Let be the number of eggs of the second farm, then the number of eggs of the highest category in this farm is .
Since, according to the condition, 60% of the eggs receive the highest category, and all eggs purchased by the agricultural company, of which the highest category, then
That is, many times more eggs are purchased from the first farm.
Then the probability that the egg purchased from this agricultural company will be from the first farm is
Answer: 0.6.
Task 9. Cowboy John has a 0.9 chance of hitting a fly on the wall if he fires a zeroed revolver. If John fires an unfired revolver, he hits the fly with probability 0.3. There are 10 revolvers on the table, only 4 of which have been shot. Cowboy John sees a fly on the wall, randomly grabs the first revolver he comes across and shoots the fly. Find the probability that John misses.
Solution:
John grabs the sighted revolver (possibly) And misses (probability). Probability of this event
John grabs an unfired revolver (possibly) And misses (probability). The probability of this event
John can grab a sighted revolver and miss or grab an unfired revolver and miss, so the required probability is:
Answer: 0.46.
Problem 10. The probability that student U will solve more than 12 problems correctly on a math test is 0.78. The probability that U will solve more than 11 problems correctly is 0.88. Find the probability that U will solve exactly 12 problems correctly.
Solution:
Let event A: “the student solves 12 problems correctly,”
event B: “the student will solve more than 12 problems”,
event C: “the student will solve more than 11 problems.”
In this case, the probability of event C is the sum of the probabilities of events A and B:
– this is the desired probability.
Answer: 0.1.
Problem 11. In the Magic Land there are two types of weather: good and excellent, and the weather, once established in the morning, remains unchanged all day. It is known that with probability 0.8 the weather tomorrow will be the same as today. On August 3, the weather in Magic Land is good. Find the probability that the weather will be great in Fairyland on August 6th.
Solution:
(We marked “X” for “good weather”, “O” for “excellent weather”)
Event D: XXXO will occur with probability
Event F: ХХОО will occur with probability
Event J: ХООО will occur with probability
Event H: XOXO will occur with probability
Answer: 0.392.
Problem 12. The picture shows a labyrinth. The spider crawls into the maze at the Entrance point. The spider cannot turn around and crawl back, so at each branch the spider chooses one of the paths along which it has not yet crawled. Assuming that the choice of the further path is purely random, determine with what probability the spider will come to exit D.
Solution:
On its way, the spider encounters four forks. And at each fork, the spider can choose the path leading to exit D with probability 0.5 (after all, at each fork, two independent equally possible events are possible: “choosing the right path” and “choosing the wrong path”). The spider will reach exit D if he chooses the “right path” at the first fork And On the second, And on third, And on the fourth, that is, the spider will come to exit D with a probability equal to
Answer: 0.0625.
Problem 13. All patients with suspected hepatitis undergo a blood test. If the test reveals hepatitis, the test result is called positive. In patients with hepatitis, the test gives a positive result with a probability of 0.9. If the patient does not have hepatitis, the test may give a false positive result with a probability of 0.01. It is known that in 6% of patients with suspected hepatitis the test gives a positive result. Find the probability that a patient admitted with suspected hepatitis actually has hepatitis. Round your answer to the nearest thousand.
Solution:
Let be the probability that a patient admitted with suspected hepatitis really sick hepatitis.
Then is the probability that a patient admitted with suspected hepatitis not sick hepatitis.
The analysis gives a positive result in cases
the patient is sick And (multiplication) the test is positive
or (addition)
the patient is not sick And the test is false positive
Since, according to the conditions of the task, in 6% of patients with suspected hepatitis, the analysis gives a positive result, then
Round to the nearest thousand: .
Answer: 0.056.
Problem 14. During artillery fire, the automatic system fires a shot at the target. If the target is not destroyed, the system fires a second shot. Shots are repeated until the target is destroyed. The probability of destroying a certain target with the first shot is 0.4, and with each subsequent shot it is 0.6. How many shots will be required to ensure that the probability of destroying the target is at least 0.98?
Solution:
Let's reformulate the problem question:
How many shots will it take for the miss probability to be less than 0.02?
With one shot, the probability of miss is 0.6.
With two shots, the probability of a miss is (the first shot is a miss and the second shot is a miss).
With three shots, the probability of a miss is –
With four shots, the probability of a miss is –
With five shots, the probability of a miss is –
We notice that .
So, five shots are enough for the probability of destroying the target to be at least 0.98.
Preparation for the unified state exam in mathematics. Useful materials and video analysis of problems in probability theory.
Useful materials
Video analysis of tasks
At a round table with 5 chairs, 3 boys and 2 girls are seated in random order. Find the probability that both girls will sit next to each other.
In the Magic Land there are two types of weather: good and excellent, and the weather, once established in the morning, remains unchanged all day. It is known that with a probability of 0.7 the weather tomorrow will be the same as today. Today is March 28, the weather in Magic Land is good. Find the probability that the weather will be great in Fairyland on April 1st.
50 athletes are competing at the diving championship, including 8 jumpers from Russia and 10 jumpers from Mexico. The order of performances is determined by drawing lots. Find the probability that a jumper from Russia will compete fifteenth.
The picture shows a labyrinth. The spider crawls into the maze at the "Entrance" point. The spider cannot turn around and crawl back, so at each fork the spider chooses one of the paths along which it has not yet crawled. Assuming that the choice of the further path is purely random, determine with what probability the spider will come to exit D.
An automatic line produces batteries. The probability that a finished battery is faulty is 0.02. Before packaging, each battery goes through a control system. The probability that the system will reject a faulty battery is 0.99. The probability that the system will mistakenly reject a working battery is 0.01. Find the probability that a randomly selected manufactured battery will be rejected by the inspection system.
The probability that the battery is defective is 0.06. A buyer in a store chooses a random package containing two of these batteries. Find the probability that both batteries are good.
Selection of problems
- Misha had four candies in his pocket - "Grilyazh", "Belochka", "Korovka" and "Swallow", as well as the keys to the apartment. While taking out the keys, Misha accidentally dropped one piece of candy from his pocket. Find the probability that the "Grillage" candy was lost.
- 4 athletes from Finland, 7 athletes from Denmark, 9 athletes from Sweden and 5 from Norway are participating in the shot put competition. The order in which the athletes compete is determined by lot. Find the probability that the athlete who competes last is from Sweden.
- Before the start of the first round of the badminton championship, participants are randomly divided into playing pairs using lots. In total, 26 badminton players are participating in the championship, including 10 participants from Russia, including Ruslan Orlov. Find the probability that in the first round Ruslan Orlov will play with any badminton player from Russia?
- There are 16 teams participating in the World Championship. Using lots, they need to be divided into four groups of four teams each. There are cards with group numbers mixed in the box: $$1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4.$$ Team captains draw one card each . What is the probability that the Russian team will be in the second group?
- The scientific conference is held over 5 days. A total of 75 reports are planned - the first three days contain 17 reports, the rest are distributed equally between the fourth and fifth days. The order of reports is determined by drawing lots. What is the probability that Professor Maksimov’s report will be scheduled for the last day of the conference?
- On average, out of 1000 garden pumps sold, 5 leak. Find the probability that one pump randomly selected for control does not leak.
- The factory produces bags. On average, for every 100 quality bags, there are eight bags with hidden defects. Find the probability that the purchased bag will be of high quality. Round the result to hundredths.
- A mechanical watch with a twelve-hour dial broke down at some point and stopped running. Find the probability that the hour hand freezes, reaching the 10 o'clock position, but not reaching the 1 o'clock position.
- In a random experiment, a symmetrical coin is tossed twice. Find the probability that the first time it lands heads, and the second time it lands tails.
- In a random experiment, a symmetrical coin is tossed twice. Find the probability that heads will appear exactly once.
- In a random experiment, a symmetrical coin is tossed three times. Find the probability that you get at least two heads.
- In a random experiment, two dice are rolled. Find the probability that the total will be 8 points. Round the result to hundredths.
- Bands perform at the rock festival - one from each of the declared countries. The order of performance is determined by lot. What is the probability that a group from Denmark will perform after a group from Sweden and after a group from Norway? Round the result to hundredths.
- There are 26 people in the class, among them two twins - Andrey and Sergey. The class is randomly divided into two groups of 13 people each. Find the probability that Andrey and Sergey will be in the same group.
- There are 21 people in the class. Among them are two friends: Anya and Nina. The class is randomly divided into 7 groups, 3 people in each. Find the probability of that. that Anya and Nina will be in the same group.
- The shooter shoots at the target once. If he misses, the shooter fires a second shot at the same target. The probability of hitting the target with one shot is 0.7. Find the probability that the target will be hit (either by the first or second shot).
- If grandmaster Antonov plays white, then he wins against grandmaster Borisov with probability 0.52. If Antonov plays black, then Antonov wins against Borisov with probability 0.3. Grandmasters Antonov and Borisov play two games, and in the second game they change the color of the pieces. Find the probability that Antonov wins both times.
- There are three sellers in the store. Each of them is busy with a client with probability 0.3. Find the probability that at a random moment in time all three sellers are busy at the same time (assume that customers come in independently of each other).
- The probability that a new DVD player will be repaired under warranty within a year is 0.045. In a certain city, out of 1,000 DVD players sold during the year, 51 units were received by the warranty workshop. How much does the frequency of the “warranty repair” event differ from its probability in this city?
- When manufacturing bearings with a diameter of 67 mm, the probability that the diameter will differ from the specified one by no more than 0.01 mm is 0.965. Find the probability that a random bearing will have a diameter less than 66.99 mm or greater than 67.01 mm.
- What is the probability that a randomly selected natural number from 10 to 19 is divisible by three?
- Before the start of a football match, the referee flips a coin to determine which team will start with the ball. The Fizik team plays three matches with different teams. Find the probability that in these games “Physicist” will win the lot exactly twice.
- Before the start of a volleyball match, team captains draw fair lots to determine which team will start the game with the ball. The "Stator" team takes turns playing with the "Rotor", "Motor" and "Starter" teams. Find the probability that Stator will start only the first and last games.
- There are two payment machines in the store. Each of them can be faulty with probability 0.05, regardless of the other machine. Find the probability that at least one machine is working.
- Based on customer reviews, Ivan Ivanovich assessed the reliability of two online stores. The probability that the desired product will be delivered from store A is 0.8. The probability that this product will be delivered from store B is 0.9. Ivan Ivanovich ordered goods from both stores at once. Assuming that online stores operate independently of each other, find the probability that no store will deliver the product.
- A biathlete shoots at targets five times. The probability of hitting the target with one shot is 0.8. Find the probability that the biathlete hits the targets the first three times and misses the last two. Round the result to hundredths
- The room is illuminated by a lantern with two lamps. The probability of one lamp burning out within a year is 0.3. Find the probability that at least one lamp will not burn out during the year.
- At the geometry exam, the student gets one question from the list of exam questions. The probability that this is an inscribed circle question is 0.2. The probability that this is a question on the topic "Parallelogram" is 0.15. There are no questions that simultaneously relate to these two topics. Find the probability that a student will get a question on one of these two topics in the exam.
- A bus runs daily from the district center to the village. The probability that there will be fewer than 20 passengers on the bus on Monday is 0.94. The probability that there will be fewer than 15 passengers is 0.56. Find the probability that the number of passengers will be between 15 and 19.
- The probability that a new electric kettle will last more than a year is 0.97. The probability that it will last more than two years is 0.89. Find the probability that it will last less than two years but more than a year.
- The probability that student O. will correctly solve more than 11 problems on a biology test is 0.67. The probability that O. will correctly solve more than 10 problems is 0.74. Find the probability that O. will solve exactly 11 problems correctly.
- To advance to the next round of competition, a football team needs to score at least 4 points in two games. If a team wins, it receives 3 points, if there is a draw, 1 point, and if it loses, 0 points. Find the probability that the team will advance to the next round of the competition. Consider that in each game the probabilities of winning and losing are the same and equal to 0.4.
- In the Magic Land there are two types of weather: good and excellent, and the weather, once established in the morning, remains unchanged all day. It is known that with probability 0.8 the weather tomorrow will be the same as today. Today is July 3rd, the weather in the Magic Land is good. Find the probability that the weather will be great in Fairyland on July 6th.
- There are 5 people in the tourist group. Using lots, they choose two people who must go to the village to buy food. Artyom would like to go to the store, but he obeys the lot. What is the probability that Artem will go to the store?
- To enter the institute for the specialty "Linguistics", an applicant must score at least 70 points on the Unified State Examination in each of three subjects - mathematics, Russian language and a foreign language. To enroll in the specialty "Commerce", you need to score at least 70 points in each of three subjects - mathematics, Russian language and social studies. The probability that Petrov will receive at least 70 points in mathematics is 0.6, in Russian - 0.8, in a foreign language - 0.7 and in social studies - 0.5. Find the probability that Petrov will be able to enroll in at least one of the two mentioned specialties
- During artillery fire, the automatic system fires a shot at the target. If the target is not destroyed, the system fires a second shot. Shots are repeated until the target is destroyed. The probability of destroying a certain target with the first shot is 0.4, and with each subsequent shot it is 0.6. How many shots will be required to ensure that the probability of destroying the target is at least 0.98?
The figure shows how the air temperature changed from April 3 to April 5. The horizontal axis indicates the time of day, the vertical axis indicates the temperature in degrees Celsius. For how many hours on April 5 was the temperature above −3 degrees Celsius?
Answer: 15.This condition is satisfied by the time from 9 to 24 (midnight), which corresponds to 15 hours.
Task 3. Training version of the Unified State Exam No. 229 Larina.
An angle is depicted on checkered paper. Find its value. Express your answer in degrees.
As you can see, the arc on which the inscribed angle rests is a quarter of the circle. Given that a circle is 360 degrees, an arc is 90 degrees. And since the magnitude of the inscribed angle is equal to half the arc on which it rests, we get 45 degrees.
Task 4. Training version of the Unified State Exam No. 229 Larina.
The picture shows a labyrinth. The beetle crawls into the maze at the “Entrance” point. The beetle cannot turn around or crawl back, so at each fork the beetle chooses one of the paths along which it has not yet crawled. Assuming that the choice is purely random, determine with what probability the beetle will come to one of the exits. Round the result to hundredths.
Taking into account the fact that the probability of going in different directions at intersections is the same, we obtain the following values (the task is simply to draw a path to each of the exits, taking into account that, for example, if there are two paths, then the probability of going in one direction is 0.5, if there are three , then 1/3, etc. There is no need to count the return path):
G: $$0.5\cdot0.5\cdot\frac(1)(3)$$
B: $$0.5\cdot0.5\cdot\frac(1)(3)\cdot0.5$$
B: $$0.5\cdot0.5\cdot\frac(1)(3)\cdot\frac(1)(3)$$
A: $$0.5\cdot0.5\cdot\frac(1)(3)\cdot\frac(1)(3)\cdot0.5$$
$$\frac(1)(3)\cdot0.25(1+0.5+\frac(1)(3)+\frac(1)(3)\cdot0.5)=$$ $$\frac (1)(12)(\frac(6)(6)+\frac(3)(6)+\frac(2)(6)+\frac(1)(6))=$$ $$\frac (2)(12)=\frac(1)(6)\approx0.17$$
Task 6. Training version of the Unified State Exam No. 229 Larina.
In triangle ABC the bisector AL is drawn. It is known that $$\angle ALC=130^(\circ)$$, and $$\angle ABC=103^(\circ)$$. Find $$\angle ACB$$. Give your answer in degrees.
$$\angle ALB=180^(\circ)-\angle ALC=50^(\circ)$$; $$\angle BAL=180^(\circ)-\angle ABL-\angle ALB=180^(\circ)-103^(\circ)-50^(\circ)=27^(\circ)$$ ; $$\angle BAC=2\cdot27=54$$; $$\angle ACB=180^(\circ)-\angle BAC-\angle ABC=23^(\circ)$$
Task 7. Training version of the Unified State Exam No. 229 Larina.
The figure shows a graph of the derivative of the function $$y=f"(x)$$, defined on the interval (−3; 9). At what point of the interval [−2; 3] does $$f(x)$$ take the greatest value?
In this task, you need to remember the following: the derivative is negative, which means the function is decreasing. In our case, the graph of an arbitrary function is located under the Ox axis on the entire segment [-2;3] (the fact that it “jumps” does not affect the decrease of the function in any way: it simply decreases somewhere faster, somewhere slower). Since the function decreases throughout the entire segment, its greatest value will be at the beginning of the segment.
Task 8. Training version of the Unified State Exam No. 229 Larina.
How many times will the volume of the octahedron decrease if all its edges are halved?
To solve these problems, you must remember that the perimeters of similar figures are related as the similarity coefficient, areas - as the square of the similarity coefficient, and volumes - as the cube of the similarity coefficient. That is, if you reduce the edge by half, the volume will change by 8 times
Task 9. Training version of the Unified State Exam No. 229 Larina.
Find the value of the expression $$\frac(\sqrt(a)\cdot\sqrt(a))(a\cdot\sqrt(a))$$ for $$a=0.1$$.
Answer: 10.$$\frac(\sqrt(a)\cdot\sqrt(a))(a\cdot\sqrt(a))=$$ $$\frac(a^(\frac(1)(4))\cdot a^(\frac(1)(12)))(a\cdot a^(\frac(1)(3)))=$$ $$a^(\frac(1)(4)+\frac( 1)(12)-1-\frac(1)(3))=$$ $$a^(-1)=\frac(1)(0,1)=10$$
Task 10. Training version of the Unified State Exam No. 229 Larina.
A diving bell located in water, containing $$v=4$$ moles of air at a pressure of $$p_(1)=1.2$$ atmospheres, is slowly lowered to the bottom of the reservoir. In this case, isothermal compression of the air occurs. The work (in joules) done by water when compressing air is determined by the expression $$A=\alpha vT\log_(2)\frac(p_(2))(p_(1))$$, where α=5.75- constant, T =300 K is the air temperature, $$p_(1)$$ (atm) is the initial pressure, and $$p_(2)$$ (atm) is the final air pressure in the bell. To what maximum pressure $$p_(2)$$ (in atm) can air be compressed in a bell if no more than 20,700 J of work is done when compressing the air?
Answer: 9.6.$$20700=5.75\cdot4\cdot300\log_(2)\frac(p_(2))(1,2)\Leftrightarrow $$$$\log_(2)\frac(p_(2))(1, 2)=\frac(20700)(23\cdot300)=3\Leftrightarrow $$$$\frac(p_(2))(1,2)=2^(3)=8\Leftrightarrow $$$$p_( 2)=1.2\cdot8=9.6$$
Task 11. Training version of the Unified State Exam No. 229 Larina.
A motor ship, whose speed in still water is 24 km/h, travels along the river and, after stopping, returns to its starting point. The current speed is 2 km/h, the stay lasts 4 hours, and the ship returns to its starting point 16 hours after departure. How many kilometers did the ship travel during the entire voyage?
Answer: 286.Let x be the one-way distance. The speed along the current is 24+2=26, against the current 24-2=22. The stay lasted 4 hours, therefore the voyage itself was 16-4=12. This time is obtained by summing the time along the current and against the current:
$$\frac(x)(26)+\frac(x)(22)=12\Leftrightarrow$$$$\frac(24x)(11\cdot13\cdot2)=12\Leftrightarrow $$$$x=\ frac(11\cdot12\cdot13\cdot2)(24)=143$$
Then the distance there/back was 143-143=286 km.
Task 12. Training version of the Unified State Exam No. 229 Larina.
Find the minimum point of the function $$y=x\sin x+\cos x-\frac(3)(4)\sin x$$, belonging to the interval $$(0;\frac(\pi)(2))$$
Answer: 0.75.$$y"=\sin x+x\cos x-\sin x-\frac(3)(4)\cos x=0 \Leftrightarrow $$$$\cos x(x-\frac(3)(4) ))=0\Leftrightarrow $$$$x=0.75 ; x=\frac(\pi)(2)+\pi*n, n \in Z$$
Let's mark the obtained points on the coordinate line and arrange the signs of the derivative (first we will consider each of the factors included in the derivative, then only the sign of the derivative itself, as a product of factors):
As we can see from the figure (F=0 is the beginning of the segment on which we are looking) the minimum point is x=0.75.
Task 13. Training version of the Unified State Exam No. 229 Larin.
A) Solve the equation $$\cos2(x+\frac(\pi)(3))+4\sin(x+\frac(\pi)(3))=\frac(5)(2)$$
B) Find the roots belonging to the segment $$[-\frac(\pi)(2);\pi]$$
Answer: $$-\frac(\pi)(6);\frac(\pi)(2)$$.Let $$x+\frac(\pi)(3)=y$$;
$$\cos2y+4\sin y=\frac(5)(2)\Leftrightarrow $$$$1-2\sin^(2)y+4\sin y-\frac(5)(2)=0\ Leftrightarrow $$$$-2\sin^(2)y+4\sin y-\frac(3)(2)=0\Leftrightarrow $$$$4\sin^(2)y-8\sin y+3 =0$$;
$$\sin y=\frac(8+4)(8)=\frac(3)(2)$$ - no solutions;
$$\sin y=\frac(8-4)(8)=\frac(1)(2)\Leftrightarrow $$$$\left\(\begin(matrix)y=\frac(\pi)(6 )+2\pi n,n\in Z\\y=\frac(5\pi)(6)+2\pi n,n\in Z\end(matrix)\right.\Leftrightarrow $$$$\ left\(\begin(matrix)x+\frac(\pi)(3)=\frac(\pi)(6)+2\pi n,n\in Z\\x+\frac(\pi)(3) =\frac(5\pi)(6)+2\pi n,n\in Z\end(matrix)\right.\Leftrightarrow $$$$\left\(\begin(matrix)x=-\frac( \pi)(6)+2\pi n,n\in Z\\x=\frac(\pi)(2)+2\pi n,n\in Z\end(matrix)\right.$$
Let's construct a unit circle, mark the roots in general and in the interval, and find special cases of roots:
Obviously, the roots falling into these segments are $$-\frac(\pi)(6);\frac(\pi)(2)$$
Task 14. Training version of the Unified State Exam No. 229 Larina.
The base of the quadrangular pyramid SABCD is a square ABCD with side AB=4. The lateral edge SC, equal to 4, is perpendicular to the base of the pyramid. The plane $$\alpha$$ passing through the vertex C parallel to the straight line BD intersects the edge SA at the point M, and SM:MA=1:2
A) Prove that $$SA\perp\alpha$$
B) Find the cross-sectional area of the pyramid SABCD by the plane $$\alpha$$
Answer: $$\frac(8\sqrt(3))(3)$$.
a) 1) $$AS=\sqrt(16+32)=4\sqrt(3)$$; $$AM=\frac(4\sqrt(3)\cdot2)(3)$$; $$MS=\frac(4\sqrt(3))(3)$$; $$MC=\frac(4\cdot4\sqrt(2))(4\sqrt(3))=\frac(4\sqrt(2))(\sqrt(3))=\frac(4\sqrt( 6))(3)$$; $$4^(2)=(\frac(4\sqrt(6))(3))^(2)+(\frac(4\sqrt(3))(3))^(2)=\frac( 16\cdot6+16\cdot3)(9)=16$$
2) $$AC\perp DB$$ $$\Rightarrow$$ $$SA\perp DB$$ $$\Rightarrow$$ $$SA\perp KN$$
b) 1) $$\frac(CE)(EM)\cdot\frac(MS)(SA)\cdot\frac(AO)(OC)=1$$; $$\frac(CE)(EM)\cdot\frac(1)(3)\cdot\frac(1)(1)=1$$; $$\frac(CE)(EM)=\frac(3)(1)$$ $$\Rightarrow$$ $$CE=\frac(3)(4)\cdot CM=\frac(3)(4 )\cdot\frac(4\sqrt(6))(3)=\sqrt(6)$$
2) $$\cos ACM=\frac(CM)(AC)=\frac(\frac(4\sqrt(6))(3))(4\sqrt(2))=\frac(\sqrt(3 ))(3)$$; $$OE=\sqrt(OC^(2)+CE^(2)-2OC\cdot CE\cdot\cos ACM)=$$ $$\sqrt((2\sqrt(2))^(2)+ (\sqrt(6))^(2)-2\cdot2\sqrt(2)\cdot\sqrt(6)\cdot\frac(\sqrt(3))(3))=$$ $$\sqrt( 8+6-\frac(4\cdot6)(3))=\sqrt(6)$$
3) $$SO=\sqrt(OC^(2)+SC^(2))=\sqrt((2\sqrt(2))^(2)+4^(2))=\sqrt(24) $$ $$\Rightarrow$$ $$SE=SO-OE=2\sqrt(6)-\sqrt(6)=\sqrt(6)$$ $$\Rightarrow$$ $$NK$$ - middle line $$\bigtriangleup SDB$$ $$\Rightarrow$$ $$NK=\frac(1)(2)DB=\frac(1)(2)\cdot4\sqrt(2)=2\sqrt(2)$ $;
4) $$S_(CKMN)=\frac(1)(2)\cdot CM\cdot NK=\frac(1)(2)\cdot\frac(4\sqrt(6))(3)\cdot2\ sqrt(2)=\frac(4\cdot\sqrt(12))(3)=\frac(8\sqrt(3))(3)$$
Task 15. Training version of the Unified State Exam No. 229 Larin.
Solve the inequality $$\log_(x-2)\frac(1)(5)\geq\log_(\frac(x-3)(x-5))\frac(1)(5)$$
Answer: $$x\in)
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