Which four inequalities in the left column. Tests and tasks for preparing for the exam in mathematics
PR No. 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 the generatrix of the cone.
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7. The height of the cone is 4, and the diameter of the base is 6. Find the generatrix of the cone.
8. The area of the base of the cone is 16, the height is 6. Find the area of the axial section of the cone.
9. The circumference of the base of the cone is 3, the generatrix is 2. Find the area of the lateral surface of the cone.
12. The height of the cone is 6, the generatrix is 10. Find the area of its total surface divided by.
PR No. 5, tasks on the topic "CONUS", option-2
2. The area of the base of the cone is 18. A plane parallel to the plane of the base of the cone divides its height into segments of length 3 and 6, counting from the top. Find the cross-sectional area of the cone by this plane.
10. How many times will the area of the lateral surface of the cone increase if its generatrix is increased by 36 times, and the radius of the base remains the same?
11. How many times will the area of the lateral surface of the cone decrease if the radius of its base is reduced by 1.5 times?
13. The total surface area of the cone is 108. A section is drawn parallel to the base of the cone, dividing the height in half. Find the total surface area of the truncated cone.
14. The radius of the base of the cone is 3, the height is 4. Find the total surface area of the cone divided by.
15. The area of the lateral surface of the cone is four times the area of the base. Find the cosine of the angle between the generatrix of the cone and the plane of the base.
16. The total surface area of the cone is 12. A section is drawn parallel to the base of the cone, dividing the height in half. Find the total surface area of the truncated cone.
17. The area of the lateral surface of the cone is twice the area of the base. Find the angle between the generatrix of the cone and the plane of the base. Give your answer in degrees.
ANALYSIS OF TASKS
R2. The area of the base of the cone is 18. A plane parallel to the plane of the base of the cone divides its height into segments of length 3 and 6, counting from the top. Find the sectional area of the cone by this plane.
The section is a circle.
You need to find the area of this circle.
Let's build an axial section:
Consider triangles AKL and AOC - they are similar. It is known that in similar figures the ratios of the corresponding elements are equal. We will consider the relationship of heights and legs (radii):
OC is the radius of the base, it can be found:
Means
Now we can calculate the cross-sectional area:
*This is an algebraic way of calculating without using the area property of similar bodies. It could be argued like this:
Two cones (original and cut off) are similar, which means that the areas of their bases are similar figures. For the areas of similar figures, there is a dependence:
The similarity coefficient in this case is equal to 1/3 (the height of the original cone is 9, cut off 3), 3/9=1/3.
Thus, the area of \u200b\u200bthe base of the resulting cone is:
Answer: 2
P3.The height of the cone is 8, and the length of the generatrix is 10. Find the area of the axial section of this cone.
Find the diameter of the base and use the formula for the area of a triangle to calculate the area. According to the Pythagorean theorem:
We calculate the cross-sectional area:
Answer: 48
P4. The diameter of the base of the cone is 40, and the length of the generatrix is 25. Find the area of the axial section of this cone.
Let the generatrix be L, the height be H, and the radius of the base be R.
The radius of the base is half the diameter, that is, 20.
We calculate the cross-sectional area:
Answer: 300
P1. The height of the cone is 57, and the diameter of the base is 152. Find the generatrix of the cone.
Answer: 95
P5.The height of the cone is 21, and the length of the generatrix is 75. Find the diameter of the base of the cone.
The diameter of the base of a cone is equal to two radii. We can find the radius using the Pythagorean theorem from right triangle:
Therefore, 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 the generatrix is 100. Find the height of the cone.
Consider the axial section of the cone. According to the Pythagorean theorem:
Answer: 96
P7. The height of the cone is 4, and the diameter of the base is 6. Find the generatrix of the cone.
R8.The area of the base of the cone is 16, the height is 6. Find the area of the axial section of the cone.
The axial section of the cone is a triangle with a base equal to the diameter of the base of the cone and a height equal height cone. Let's denote the diameter as D, the height as H, write down the formula for the area of a triangle:
The height is known, we calculate the diameter. We use the formula for the area of a circle:
This means that the diameter will be equal to 8. Calculate the cross-sectional area:
Answer: 24
P9. The circumference of the base of the cone is 3, the generatrix is 2. Find the area of the lateral surface of the cone.
Plugging in the data:
Answer: 3
P10.How many times will the area of the lateral surface of the cone increase if its generatrix is increased by 36 times, and the radius of the base remains the same?
The area of the lateral surface of the cone:
The generatrix is increased by 36 times. The radius remains the same, which means the circumference of the base has not changed.
So, the area of the lateral surface of the modified cone will look like:
Thus, it will increase by 36 times.
*The dependence is straightforward, so this problem can be easily solved orally.
Answer: 36
R11.How many times will the area of the lateral surface of the cone decrease if the radius of its base is reduced by 1.5 times?
The area of the lateral surface of the cone is:
The radius is reduced by 1.5 times, that is:
It was found that the lateral surface area decreased by 1.5 times.
Answer: 1.5
R12.The height of the cone is 6, the generatrix is 10. Find the area of its total surface divided by.
Full surface of the cone:
You need to find the radius.
The height and generatrix are known, by the Pythagorean theorem we calculate the radius:
In this way:
Divide the result by and write down the answer.
Answer: 144
R13.The total surface area of the cone is 108. A section is drawn parallel to the base of the cone, dividing the height in half. Find the total surface area of the truncated cone.
The formula for the total surface of a cone is:
The section passes through the mid-height parallel to the base. This means that the radius of the base and the generatrix of the truncated cone will be 2 times less than the radius and generatrix of the original cone. Let's write down what the surface area of the cut-off cone is equal to:
In the seventeenth task, we need to compare these numbers with the position on the coordinate line or solve and compare the solutions of inequalities with the area on the line. In this task, you can use the rule of exclusion, so it is enough to correctly determine three solutions out of four, choosing first of all the simple ones. So, let's proceed to the analysis of the 17 tasks of the basic version of the exam mathematics.
Analysis of typical options for tasks No. 17 USE in mathematics of a basic level
Option 17MB1
Points A, B, C and D are marked on the coordinate line.
| POINTS | NUMBERS |
Execution algorithm:
- Analyze next to which of the integers is this point.
- Analyze on what interval the numbers from the right column lie.
- Compare the resulting intervals and match.
Solution:
- Consider point A. Its value is greater than 1 and less than 2.
- Consider point B. Its value is greater than 2 and less than 3.
- Consider point C. Its value is greater than 3 and less than 4.
- Consider point D. Its value is greater than 5 and less than 6.
- Let's remember what a logarithm is.
The logarithm to the base a of the argument x is the power to which the number a must be raised to get the number x.
Designation: log a x = b, where a- base, x- argument, b What exactly is the logarithm.
In our case, a = 2, x = 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 in the interval from 3 to 4.
Therefore, 7/3 is greater than 2 and less than 3.
Consider √26. √25 = 5, √36 = 6. Hence, √26 is greater than 5 and less than 6.
That is, (3/5) -1 is greater than 1 and less than 2.
Let's match the obtained intervals.
A - (3/5) -1 - 4
B - 7/3 - 2
C - log 2 10 - 1
D - √26 - 3
Answer: 4213.
Option 17MB2
| INEQUALITIES | SOLUTIONS |
Execution algorithm:
- Express the right and left parts of the inequalities as a power of the same number.
- Compare powers because the bases are equal.
- Match the suggested intervals.
Solution:
The inequality will take the form:
that is, option number 2.
The inequality will take the form:
The bases of the powers are the same, therefore, the degrees are related in the same way.
that is, option number 1.
Same with option B.
The number 0.5 can be represented as , so (0.5) x = (2 -1) x = 2 -x
The inequality will take the form:
The bases of the powers are the same, therefore, the degrees are related in the same way.
If you multiply both the right and left sides of the inequality by -1, then the sign will change to the opposite.
that is, option number 4.
Let's represent 4 as a power with base 2. 2 2 = 4.
The inequality will take the form:
The bases of the powers are the same, therefore, the degrees are related in the same way.
and - option number 3.
Answer: 2143.
Option 17MB3
Numbers m and n are marked on the straight line.
Each of four numbers in the left column corresponds to the segment to which it belongs. Set the correspondence between numbers and segments from the right column.
| NUMBERS | LINES |
Execution algorithm:
- Find the gaps in which the numbers m and n lie.
- Evaluate the intervals in which the expressions in the left column lie.
- Match them with intervals from the right column.
Solution:
It can be seen from the figure that the number n is slightly less than 0, and the number m is much more distant from 1. Therefore, their sum m + n will give a number within - answer option number 3.
The number m>1, therefore, when dividing by 1, we get a positive number less than 1. By adding a small negative value of n, we will remain in the range. Answer option 2.
The product of mn positive and negative numbers gives 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 square of the number n, so their difference will be positive and belong to the range - option 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. Establish a correspondence between inequalities and their solutions.
Consider the first inequality:
represent 4 as 2 2 , then:
The remaining inequalities are solved in a similar way, it is enough to recall that 0.5 \u003d ½ \u003d 2 -1:
Answer: A-4, B-3, C-2, A-1.
Option 17MB5
Execution algorithm
- We solve each of the inequalities (A–D) in turn. If necessary (for clarity), we display the resulting solution on the coordinate line.
- We write down the results in the form that is proposed in the "Solutions" column. We find the corresponding pairs "letter-number".
Solution:
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 inequality of transformations does not require, therefore, we immediately apply the method of intervals, displaying the roots of the inequality on the coordinate line.
The roots in this case are x=4 and x=5. We mean that the inequality is strict, i.e. we do not include the values of the roots in the interval for the answer. At the point x=5, there is no sign transition, 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 > 1 → log 4 x > log 4 4 → x > 4. That is: 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 are x=2 and x=4. To obtain gaps with positive and negative values, we schematically depict a parabola intersecting the coordinate line at the points of the roots. The interval "inside" the parabola is negative, the intervals "outside" it are positive. Because in the inequality given "<0», то для ответа следует взять промежуток отрицательных значений. Учитываем, что неравенство строгое. Получаем: х ϵ (2; 4).
Answer: G–2.
Option 17MB6
The number m is equal to √2.
Each point corresponds to one of the numbers in the right column. Set the correspondence between the specified points and numbers.
Execution algorithm
For each of the expressions in the right column, do the following:
- We substitute instead of m its numerical value (√2). We calculate the approximate value.
- Focusing on the integer part of the resulting number, we find the corresponding value on the coordinate line.
- We fix the pair "letter-number".
Solution:
This value on the straight line is between -3 and -2 and corresponds to point A. We got: A–1.
The number is between 2 and 3 and corresponds to point D. We have: D–2.
The number is on a straight line between 0 and 1. This is point C. We have: C-3.
The number is placed on a straight line between the values -1 and 0, which represents e.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 a correspondence between inequalities and their solutions.
Execution algorithm
- We successively solve each inequality (A–D), receiving in the answer a range of values. We find the graphic display corresponding to it in the right column (Solutions).
- When solving inequalities, we take into account that: 1) when removing the signs of the logarithm with a base less than 1, the sign of the inequality changes to the opposite; 2) the expression under the sign of the logarithm is always greater than 0.
Solution:
The resulting gap-response is displayed on the 4th coordinate line. Therefore we have: A-4.
The resulting interval is presented on the 1st straight line. Hence we have: B–1.
C. This inequality is similar to the previous one (B) with a difference only in the sign. Therefore, the answer will be similar with the only difference that the final inequality will have the opposite sign. Those. we get: X ≤ 3, X> 0 → x ϵ (0; 3]. Accordingly, we obtain a pair: IN 2.
D. This inequality is similar to the 1st (A), but with the opposite sign. So the answer here would be: X ≥ 1/3, X> 0 → x ϵ . Answer: B–4.
Number B. This number is: 1.8 + 1 \u003d 2.8, which corresponds to the segment. Answer: IN 2.
Number G. Here we get: 6 / 1.8≈3.33. This value corresponds to the segment . Answer: G–3.
Option 17MB13
The number m is √0.15.
Each of the four numbers in the left column corresponds to the segment to which it belongs. Set the correspondence between numbers and segments from the right column.
Execution algorithm
- Let's transform the number m so as to take the value out from under the root.
- We substitute the sequentially obtained value for m into each of the expressions in the left column. The results obtained are correlated with a suitable segment from the right.
Solution:
The number √0.15 differs very little from √0.16, and 0.16 can be exactly rooted. Making such an approximation - by only 0.01 - we do not go beyond the acceptable absolute error. Therefore, we have the right to accept that √0.15≈√0.16=0.4.
We find the values of the expressions A–D and determine their correspondence to the segments:
A. -1 / 0.4 \u003d -2.5. The result corresponds to the segment [–3; –2]. Answer: A–1.
B. 0.4 2 = 0.16. The number is in the range. Answer: B–3.
B. 4 0.4 = 1.6. This number is in the interval . Answer: AT 4.
D. 0.4–1=–0.6. The result falls on the segment [–1; 0]. Answer: G–2.
Variant of the seventeenth task of 2019 (10)
The number m and the points A, B, C and D are marked on the coordinate line.
Each point corresponds to one of the numbers in the right column. Set the correspondence between the specified points and numbers.
Execution algorithm
- We determine the approximate value for m.
- We calculate the values of expressions 1–4, find a correspondence between the results obtained and points A–D on the coordinate line.
Solution:
The point m is located almost in the middle between 1 and 2, but a little closer to 1 than to 2. In this case, the value m=1.4 should be considered as close as possible to the real one.
We determine the correspondence of numbers and points on a straight line.
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The apartment consists of a room, a kitchen, a corridor and a bathroom (see drawing). The room has dimensions of 5 m × 3.5 m, the corridor is 1.5 m × 6.5 m, the length of the kitchen is 3.5 m. Find the area of the bathroom (in square meters).
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In a circle with center O, segments AC and BD are diameters. The inscribed angle ACB is 53°. Find the AOD angle. Give your answer in degrees.
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In triangle ABC it is known that AB=BC=80, AC=96. Find the length of the median BM.
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In a circle with center O, segments AC and BD are diameters. The inscribed angle ACB is 71°. Find the AOD angle. Give your answer in degrees.
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Find the inscribed angle subtended by an arc whose length is 16 times the circumference of the circle. Give your answer in degrees.
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In triangle ABC it is known that AB=BC=65, AC=50. Find the length of the median BM.
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The summer cottage has the shape of a rectangle, the sides of which are 30 m and 20 m. The house located on the site has the shape of a square with a side of 6 m. Find the area of the remaining part of the plot. Give your answer in square meters.
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Pyramid of Sneferu has a regular shape quadrangular pyramid, the side of the base of which is 220 m, and the height is 104 m. The side of the base of the exact museum copy of this pyramid is 110 cm. Find the height of the museum copy. Give an answer
in centimeters.
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The terrain plan is divided into cells. Each cell represents a square 1 m × 1 m Find the area of the plot marked on the plan. Give your answer in square meters.
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In triangle ABC it is known that AB=BC=37, AC=24. Find the length of the median BM.
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The suburban area has the shape of a rectangle with sides of 24 meters and 36 meters. The owner plans to enclose it with a fence and divide it into two parts with the same fence, one of which has the shape of a square. Find the total length of the fence in meters.
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Given two cylinders. The radius of the base and the height of the first are 9 and 8, respectively, and the second - 12 and 3.
How many times greater is the lateral surface area of the first cylinder than the lateral surface area of the second?
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The terrain plan is divided into cells. Each cell represents a square 1 m × 1 m Find the area of the plot marked on the plan. Give an answer |
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The picture shows what a wheel with 7 spokes looks like. How many spokes will there be in a wheel if the angle between adjacent spokes in it is 36°?
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In triangle ABC it is known that AB=BC=80, AC=128. Find the length of the median BM.
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The apartment consists of a room, a kitchen, a corridor
and a 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 area of the room
(in square meters).
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In triangle ABC it is known that AB=BC=65, AC=104. Find the length of the median BM.
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The plan indicates that the rectangular room has an area of 15.2 square meters. m. Accurate measurements showed that the width of the room is 3 m, and the length is 5.1 m.
By how many square meters does the area of the room differ from the value indicated on the plan?
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In trapezoid ABCD we know that AD=6, BC=5, and its area is 22. Find the area of triangle ABC.
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In triangle ABC it is known that AB=BC=5, AC=8. Find the length of the median BM.
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In triangle ABC it is known that AB=BC=82, AC=36. Find the length of the median BM.
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Given two balls with radii 6 and 1. How many times is the surface area of the larger ball greater than the surface area of the other? |
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What is the smallest angle (in degrees) between the minute and hour hands of a clock at 4:00 pm?
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The suburban area has the shape of a rectangle with sides of 25 meters and 40 meters. The owner plans to enclose it with a fence and divide it into two parts with the same fence, one of which has the shape of a square. Find the total length of the fence in meters.
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Given two balls with radii 9 and 3. How many times is the surface area of the larger ball greater than the surface area of the other? |
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