Set the correspondence between inequalities and the solution. Tests and tasks for preparation for the exam in mathematics
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The apartment consists of room, kitchen, corridor and bathroom (see drawing). The room has dimensions of 5 m × 3.5 m, the corridor - 1.5 m × 6.5 m, the length of the kitchen is 3.5 m. Find the area of \u200b\u200bthe bathroom (in square meters).
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In the circle with the center O segment AC and BD - diameters. The inscribed ACB angle is 53 °. Find the AOD angle. Answer in degrees.
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In the ABC triangle, it is known that AB \u003d BC \u003d 80, AC \u003d 96. Find the length of the median BM.
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In the circle with the center O segment AC and BD - diameters. The inscribed ACB angle is 71 °. Find the AOD angle. Answer in degrees.
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Find the inscribed angle based on the arc, the length of which is equal to 16 circumference lengths. Answer in degrees.
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In the ABC triangle, it is known that AB \u003d BC \u003d 65, AC \u003d 50. Find the length of the median BM.
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The country area has a rectangle shape, the sides of which are equal to 30 m and 20 m. The house located on the plot has a square shape with a side of 6 m. Find the area of \u200b\u200bthe remainder of the site. Give the answer in square meters.
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The pyramid of the Snofer has the form of the correct four-grade pyramid, the side of which is 220 m, and the height is 104 m. The base side of the exact museum copy of this pyramid is 110 cm. Find the height of the museum copy. Answer give
in centimeters.
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The area of \u200b\u200bthe terrain is divided into cells. Each cell denotes a square 1 m × 1 m. Find the area of \u200b\u200bthe site highlighted on the plan. Give the answer in square meters.
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In the ABC triangle, it is known that AB \u003d BC \u003d 37, AC \u003d 24. Find the length of the median BM.
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The country area has a rectangle shape with 24 meters and 36 meters side. The owner plans to escalate him and divide the same fence into two parts, one of which has the shape of the square. Find the total length of the fence in meters.
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Two cylinders are given. The radius of the base and the height of the first is equal, respectively, 9 and 8, and the second - 12 and 3.
How many times the side surface area of \u200b\u200bthe first cylinder more square side surface of the second?
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The area of \u200b\u200bthe terrain is divided into cells. Each cell denotes a square 1 m × 1 m. Find the area of \u200b\u200bthe site highlighted on the plan. Answer give |
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The figure shows how the wheel with 7 spokes looks like. How many spokes will be in the wheel if the angle between adjacent knitting needles in it will be equal to 36 °?
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In the ABC triangle, it is known that AB \u003d BC \u003d 80, AC \u003d 128. Find the length of the median BM.
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Apartment consists of room, kitchen, corridor
and bathroom (see drawing). The kitchen has dimensions of 3 m × 4 m, the bathroom - 1.5 m × 2 m, length
corridor 6 m. Find the room
(in square meters).
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In the ABC triangle, it is known that AB \u003d BC \u003d 65, AC \u003d 104. Find the length of the median BM.
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The plan indicates that the rectangular room has an area of \u200b\u200b15.2 square meters. The accurate measurements showed that the width of the room is 3 m, and the length is 5.1 m.
How many square meters the room is different from the value indicated on the plan?
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In the ABCD trapezion, it is known that ad \u003d 6, Bc \u003d 5, and its area is equal to 22. Find the ABC triangle area.
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In the ABC triangle, it is known that AB \u003d BC \u003d 5, AC \u003d 8. Find the length of the median BM.
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In the ABC triangle, it is known that AB \u003d BC \u003d 82, AC \u003d 36. Find the length of the median BM.
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There are two balls with radius 6 and 1. How many times the surface area of \u200b\u200bthe larger ball is more than the surface area of \u200b\u200banother? |
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What the smallest angle (in degrees) form a minute and hour arrows hours at 16:00?
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The country area has a rectangle shape with parties of 25 meters and 40 meters. The owner plans to escalate him and divide the same fence into two parts, one of which has the shape of the square. Find the total length of the fence in meters.
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Dana two balls with radius 9 and 3. How many times the surface area of \u200b\u200bthe larger ball is more than the surface area of \u200b\u200bthe other? |
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PR№5, Tasks on the topic "Cone", option-1.
1. The height of the cone is 57, and the diameter of the base is 152. Find a forming cone.
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7. The height of the cone is 4, and the diameter of the base is 6. Find the forming cone.
8. The base area of \u200b\u200bthe cone is equal to 16, the height is 6. Find the area of \u200b\u200bthe axial cross section of the cone.
9. The circumference of the base of the cone is equal to 3, forming equal to 2. Locate the side surface area of \u200b\u200bthe cone.
12. The height of the cone is equal to 6, forming equal to 10. Find the area of \u200b\u200bits complete surface divided by.
PR№5, Tasks on the topic "Cone", option-2
2. The base area of \u200b\u200bthe cone is 18. The plane, parallel plane of the base of the cone, divides its height on the segments of length 3 and 6, counting from the vertex. Find the area of \u200b\u200bthe cone cross section by this plane.
10. Which time the side surface area of \u200b\u200bthe cone increases, if it is formed to increase 36 times, and the radius of the base will remain the same?
11. Which time the side surface of the cone decreases if the radius of its base is 1.5 times?
13. The area of \u200b\u200bthe full surface of the cone is equal to 108. In parallel, the base of the cone was carried out by a height in half. Find the area of \u200b\u200bthe full surface of the cut-off cone.
14. The radius of the base of the cone is 3, the height is 4. Find the area of \u200b\u200bthe full surface of the cone divided by.
15. The area of \u200b\u200bthe side surface of the cone is four times larger than the base area. Find what is equal to the cosine of the angle between the cone forming and the base plane.
16. The area of \u200b\u200bthe full surface of the cone is 12. In parallel, the base of the cone was carried out by a height in half. Find the area of \u200b\u200bthe full surface of the cut-off cone.
17. The area of \u200b\u200bthe side surface of the cone is twicecred the base area. Find the angle between the cone forming and the base plane. Answer in degrees.
Task analysis
P2. The base area of \u200b\u200bthe cone is 18. The plane, parallel plane of the base of the cone, divides its height on the segments of length 3 and 6, counting from the vertex. Find the area of \u200b\u200bthe cross section of the cone with this plane.
The cross section is a circle.
It is necessary to find the area of \u200b\u200bthis circle.
We will construct an axial cross section:
Consider the triangles AKL and AOC - they are similar. It is known that in such figures the relationship of the corresponding elements is equal. We will look at the relationship of heights and cathets (radii):
OC is a radius of the base, it can be found:
So
Now we can calculate the cross section area:
* This is an algebraic method of calculation without using the properties of such bodies relating to their area. It was possible to judge the following:
Two cones (source and cut-off) are similar, which means the area of \u200b\u200btheir foundations are similar figures. For the areas of such figures there is a dependence:
The likeness ratio in this case is 1/3 (the height of the original cone is 9, cut off 3), 3/9 \u003d 1/3.
Thus, the area of \u200b\u200bthe foundation of the resulting cone is equal to:
Answer: 2.
P3.The height of the cone is 8, and the length of forming - 10. Find the area of \u200b\u200bthe axial cross section of this cone.
Find the diameter of the base and using the formula of the triangle area is calculated area. According to Pythagore's theorem:
Calculate the cross section:
Answer: 48.
P4. The diameter of the base of the cone is 40, and the length of forming - 25. Find the area of \u200b\u200bthe axial cross section of this cone.
Let forming it L, height is H, the radius of the base is R.
The radius of the base is equal to half the diameter, that is, 20.
Calculate the cross section:
Answer: 300.
P1. The height of the cone is 57, and the base diameter is 152. Find the forming cone.
Answer: 95.
P5.The height of the cone is 21, and the length of forming - 75. Find the diameter of the base of the cone.
The diameter of the base of the cone is equal to two radius. Radius we can find on the Pythagore Theorem from rectangular triangle:
Consequently, the diameter of the base of the cone is 144.
Answer: 144.
P6.The diameter of the base of the cone is 56, and the length of forming - 100. Find the height of the cone.
Consider the axial cross section of the cone. According to Pythagore's theorem:
Answer: 96.
P7. The height of the cone is 4, and the base diameter is 6. Find the forming cone.
P8.The base area of \u200b\u200bthe cone is equal to 16, the height is 6. Find the area of \u200b\u200bthe axial cross section of the cone.
The axial cross section of the cone is a triangle with the base to be equal to the diameter of the base of the cone and the height of equal to the height of the cone. Denote the diameter as D, height as n, we write the formula of the triangle area:
The height is known, calculate the diameter. We use the formula of the area of \u200b\u200bthe circle:
So, the diameter will be equal to 8. Calculate the cross section area:
Answer: 24.
P9. The length of the cone base circumference is 3, forming equal to 2. Locate the side surface area of \u200b\u200bthe cone.
We substitute the data:
Answer: 3.
P10.How many times the side surface of the cone increases, if it is formed to increase 36 times, and the radius of the base will remain the same?
Side view of the cone:
The forming increases by 36 times. The radius remained the same, which means the bottom circumference length has not changed.
So, the area of \u200b\u200bthe side surface of the changed cone will be viewed:
Thus, it will increase 36 times.
* Dependence straightforward, so this task is easily able to solve orally.
Answer: 36.
P11.How many times does the side surface of the cone decrease, if the radius of its base is 1.5 times?
The area of \u200b\u200bthe side surface of the cone is:
Radius decreases 1.5 times, that is:
Received that the side surface area decreased by 1.5 times.
Answer: 1.5
P12.The height of the cone is 6, forming equal to 10. Find the area of \u200b\u200bits complete surface divided by.
Full cone surface:
It is necessary to find a radius.
The height and forming, according to the Pythagora theorem, we calculate the radius:
In this way:
The result is separated by and write the answer.
Answer: 144.
P13.The area of \u200b\u200bthe full surface of the cone is equal to 108. In parallel, the base of the cone was carried out by a height in half. Find the area of \u200b\u200bthe full surface of the cut-off cone.
Formula of the full surface of the cone:
The section passes through the middle of the height parallel to the base. So, the radius of the base and the forming cut-off cone will be 2 times less than the radius and forming the source cone. We write what is equal to the surface area of \u200b\u200bthe cut-off cone:
In the seventeenth task, we need to compare the data with the position on the coordinate direct or decide and compare the solutions of inequalities with the area on the line. In this task, you can use the rule of exception, so it is sufficiently correct to determine three solutions from four, choosing primarily simple. So, we will proceed to the analysis of 17 tasks of the basic eME mathematics.
Analysis of typical options of tasks №17 EGE in mathematics of the baseline
Option 17MB1
On the coordinate direct point A, B, C and D.
| Points | Numbers |
Performance algorithm:
- Analyze next to what of the integers is this point.
- Analyze at what interval is the number from the right column.
- Compare the intervals and put in line with the intervals.
Decision:
- Consider the point A. It is greater than 1 and less than 2.
- Consider the point B. Its value is greater than 2 and less than 3.
- Consider the point C. Its value is greater than 3 and less than 4.
- Consider the point D. Its value is greater than 5 and less than 6.
- Recall what Logarithm is.
The logarithm on the base A from the X argument is the degree in which the number A is to be taken to get the number x.
Designation: Log. A. x. = b.where a. - reason, x. - argument, b. - Actually, what is equal to logarithm.
In our case, a \u003d 2, x \u003d 10.
That is, we are interested in the number 2 b \u003d 10. 2 3 \u003d 8 and 2 4 \u003d 16, therefore, B lies between 3 to 4.
Consequently, 7/3 more 2 and less than 3.
Consider √26. √25 \u003d 5, √36 \u003d 6. So √26 more than 5 and less than 6.
That is (3/5) -1 greater than 1 and less than 2.
We put together the intervals obtained.
A - (3/5) -1 - 4
In - 7/3 - 2
C - LOG 2 10 - 1
D - √26 - 3
Answer: 4213.
Option 17MB2.
| Inequalities | SOLUTIONS |
Performance algorithm:
- Represent the right and left parts of inequalities in the form of the same number.
- Compare degrees because the foundations are equal.
- Put in accordance with the proposed intervals.
Decision:
The inequality will take the form:
that is, the option at number 2.
The inequality will take the form:
The bases of degrees are the same, therefore, degrees correlate in the same way.
that is, the option at number 1.
Similarly with option B.
The number 0,5 can be represented as, it means (0.5) x \u003d (2 -1) x \u003d 2 -x
The inequality will take the form:
The bases of degrees are the same, therefore, degrees correlate in the same way.
If you multiply and the right and left part of the inequality on -1, the sign will change to the opposite.
that is, the option at number 4.
Imagine 4 as a degree with the base 2. 2 2 \u003d 4.
The inequality will take the form:
The bases of degrees are the same, therefore, degrees correlate in the same way.
and - option at number 3.
Answer: 2143.
Option 17MB3
The direct numbers M and N are noted.
Each of the four numbers in the left column corresponds to a segment to which it belongs. Install the correspondence between the numbers and segments from the right column.
| Numbers | Segments |
Performance algorithm:
- Find the gaps in which the numbers M and N are located.
- Assess the intervals in which expressions are in the left column.
- Put them in accordance with the intervals from the right column.
Decision:
It can be seen from the figure that the number n is slightly less than 0, and the number M is much more from 1. Consequently, their sum M + n will give a number within - an answer version 3.
The number M\u003e 1, therefore, when dividing 1, we get positive Less 1. When adding a small negative value of N, remain in the range. Answer version 2.
The work of Mn positive and negative numbers give a negative number. Only one option is suitable [-1; 0] at number 1.
D) the square of the number M is much larger than the number of N numbers, so their difference will be positive and belong to the range - an option at number 4.
Answer: 3214.
Option 17MB4
Each of the four inequalities in the left column corresponds to one of the solutions in the right column. Set correspondence between inequalities and solutions.
Consider the first inequality:
imagine 4 as 2 2, then:
The remaining inequalities are solved in a similar way, it suffices to recall that 0.5 \u003d ½ \u003d 2 -1:
Answer: A-4, B-3, B-2, A-1.
Option 17MB5
Algorithm execution
- We solve each of the inequalities (AA). If necessary (for clarity), displays the solution obtained on the coordinate direct.
- We write down the results in the form, which is proposed in the "Solution" column. We find the corresponding pairs of the "letter number".
Decision:
A. 2 --x + 1< 0,5 → 2 –x+1 < 2 –1 → –x+1 < –1 → –x < –2 → x > 2. Answer: X ε (2; + ∞). We get: A-3..
B. 
The transformation inequality does not require, so immediately use the interval method, displaying the roots of inequality on the coordinate direct.
The roots in this case are x \u003d 4 and x \u003d 5. We mean that the inequality is strictly, i.e. The values \u200b\u200bof the roots in the interval for the response do not turn on. At point x \u003d 5, the sign transition does not occur, because By condition (X-5) is given in the square. Since we need a gap where x<0, то ответ в данном случае: х ϵ (–∞; 4).
Accordingly, we have: B-4..
B. Log 4 x\u003e 1 → Log 4 x\u003e log 4 4 → x\u003e 4. Those: x ε (4; + ∞). Answer: IN 1.
G. (X-4) (X-2)< 0. Здесь так же, как и в неравенстве Б, нужно сразу отобразить решение на координатной прямой.
The inequality is given square, its roots - x \u003d 2 and x \u003d 4. To obtain gaps with positive and negative values, schematically depict a parabola crossing the coordinate direct at the root points. The interval "inside" parabola is negative, the gaps "outside" is positive. Because In the inequality is given "<0», то для ответа следует взять промежуток отрицательных значений. Учитываем, что неравенство строгое. Получаем: х ϵ (2; 4).
Answer: M-2..
Option 17MB6.
The number M is equal to √2.
Each point corresponds to one of the numbers in the right column. Set the match between the specified points and numbers.
Algorithm execution
For each of the expressions of the right column, we do the following:
- We substitute instead of my numeric value (√2). Calculate the approximate value.
- Focusing on the integer part of the resulting number, we find the appropriate value on the coordinate direct.
- Fix a pair of "letter-number".
Decision:
This value on the straight is between the values \u200b\u200bof -3 and -2 and corresponds to the point A. Received: A-1..
The number is between the values \u200b\u200b2 and 3 and corresponds to the point D. We have: D-2.
The number is on the line between 0 and 1. It is point S. We have: C-3..
The number is placed on a straight line between the values \u200b\u200bof -1 and 0, which displays that T.V. We get: AT 4.
Option 17MB7
Each of the four inequalities in the left column corresponds to one of the solutions in the right column. Establish compliance between inequalities and solutions.
Algorithm execution
- Consistently solve each inequality (AA), receiving the values \u200b\u200bin response. We find the corresponding graphical display in the right column (solutions).
- When solving inequalities, we take into account that: 1) when removing the logarithm signs with the base, smaller 1, the sign of inequality changes to the opposite; 2) The expression under the logarithm is always greater than 0.
Decision:
The resulting gap-response is displayed on the 4th coordinate direct. Therefore, we have: A-4..
The resulting gap is represented on the 1st straight line. From here we have: B-1..
B. This inequality is similar to the previous one (b) with the difference exclusively in the sign. Therefore, the answer will be similar to the only difference that in the final inequality will be the opposite sign. Those. We get: h. ≤ 3, h. \u003e 0 → X ε (0; 3]. Accordingly, we get a pair: AT 2.
This inequality is similar to the 1st (a), but with the opposite sign. Therefore, the answer here will be: h. ≥ 1/3, h. \u003e 0 → X ε. Answer: B-4..
Number B. This number is: 1.8 + 1 \u003d 2.8, which corresponds to the segment. Answer: AT 2.
The number of G. Here we receive: 6 / 1.8≈3.33. This value corresponds to the segment. Answer: Mr..
Option 17MB13
The number m is equal to √0.15.
Each of the four numbers in the left column corresponds to a segment to which it belongs. Install the correspondence between the numbers and segments from the right column.
Algorithm execution
- We transform the number M so to make a value from the root.
- We substitute the sequentially obtained value for M into each of the expressions in the left column. The results obtained by correlate with a suitable segment from the right.
Decision:
The number √0.15 is very few different from √0.16, and from 0.16 you can accurately extract the root. Making this approach is only 0.01 - we do not go beyond the acceptable absolute error. Therefore, we have the right to accept that √0.15≈√0,16 \u003d 0.4.
We find the values \u200b\u200bof the expressions of AA and determine their correspondence to segments:
A. -1 / 0.4 \u003d -2.5. The result corresponds to the segment [-3; -2]. Answer: A-1..
B. 0.4 2 \u003d 0.16. The number is in the interval. Answer: B-3..
B. 4 · 0.4 \u003d 1.6. This number is in the interval. Answer: AT 4.
0.4-1 \u003d -0.6. The result falls on the segment [-1; 0]. Answer: M-2..
Option of the seventeenth task of 2019 (10)
On the coordinate direct number M and points A, B, C and D.
Each point corresponds to one of the numbers in the right column. Set the match between the specified points and numbers.
Algorithm execution
- We define the approximate value for m..
- Calculate the values \u200b\u200bof expressions 1-4, we find the correspondence between the results obtained and the points A-D on the coordinate direct.
Decision:
Point M is located almost in the middle between 1 and 2, but is slightly closer to 1 than to 2. The maximum approximate to real in this case should be considered M \u003d 1.4.
Determine the correspondence of the numbers and points on the straight line.
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