Mathematics topic network
A. Subject network of required questions
Theme network
1. Fill in the blanks: This big question is a *** 14 small question, with 5 points for each small question and 70 points for * * *. Theme network
1. If the set U={ 1, 2, 3, 4, 5, 6, 7} is a complete set, then the set = ▲. Theme network
2. If the function is known, the minimum positive period is ▲.
3. The equation of the straight line passing through point (-2,3) and parallel to the straight line is ▲. Theme network.
4. If the plural number is satisfied, ▲. Theme network
5. The process is as follows: Topic network
T 1 subject network
I/2 theme network
And i≤4 topic network.
T←t×i Subject Network
I I+ 1 subject network
End of topic network
Print test topic network
The output of the above program is ▲. Theme network.
6. If the variance of is 3, the variance topic network of is.
For ▲. Theme network
7. If the side length of the cube ABCD-a1b1c1d1is, the volume of the circumscribed sphere of the tetrahedron is ▲.
8. If the circle with the left focus of the ellipse as the center and the circle with the radius of c intersect with the left directrix of the ellipse at two different points, the eccentricity range of the ellipse is ▲. Theme network.
9. let a > 0, let A={(x, y)|} and B={(x, y) |}. If point P(x, y)∈A is a necessary and sufficient condition for point P(x, y)∈B, then the range of a is ▲.
10. If any two real numbers are taken in the closed interval [- 1, 1], the probability that their sum is not greater than 1 is ▲.
1 1. In the series,, and (,), the general theme network of this series.
▲. Theme network
Theme network
12. According to the following set of equations: topic network
Theme network
………………………………………………………………………………………………………………….
Available ▲. Theme network
13. In △ABC,, d is any point on the side of BC (D does not coincide with B and C) and is equal to ▲. Theme network.
14. Set a function, and remember that if the function has at least one zero, the range of the real number m is ▲. Theme network
Theme network
Second, the solution: this big question is ***6 small questions, ***90 points. The solution should be written in words, proof process or calculus steps. Theme network
15. (This small question 14 points) Subject Network
As shown in the figure, in the regular triangular prism ABC-A1B1C1,point D is on the side of BC, and AD ⊥ C1D.
(1) verification: AD⊥ plane BC c1b1; Theme network
(2) Let e be a point on B 1C 1. When the value is, the main network
A 1E‖ planar ADC 1? Please prove it. Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
16. (This small question is 14 points) Subject Network
As shown in the figure, in quadrilateral ABCD, AD=8, CD=6, AB= 13, ∠ ADC = 90, and topic network.
(1) Find the value of sin∠BAD; Theme network
(2) Let the area of △ABD be S△ABD and the area of △ BCD be S△BCD, and evaluate. Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
17. (This small question is 15 points) Subject Network
An agricultural research institute analyzed the relationship between the temperature difference between day and night in winter and the germination of a new anti-season soybean variety. They recorded the daily temperature difference between day and night from 65438+February 1 to 65438+February 5, respectively, and the number of germination per 100 seeds in the laboratory every day, and obtained the following information: Subject Network
Date: 65438+February165438+February 2nd 65438+February 3rd 65438+February 4th 65438+February 5th.
Temperature difference (℃)101113128
Number of germinated seeds (23 25 30 26 16)
The research scheme determined by the Institute of Agricultural Sciences is as follows: firstly, select two groups from these five groups of data, use the remaining three groups of data to find the linear regression equation, and then test the selected two groups of data. Theme network
(1) Find the probability that the selected two groups of data are just two days' data that are not adjacent; Theme network
(2) If two sets of data are selected, namely, 65438+February 1 and 65438+February 5th, please work out the linear regression equation of Y about X according to the data from 65438+February 2nd to 65438+February 4th; Theme network
(3) If the error between the estimated data obtained from the linear regression equation and the selected test data is less than 2, the obtained linear regression equation is considered to be reliable. (2) Is the linear regression equation obtained in (2) reliable? Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
18. (This small question is 15 points) Subject Network
The focus of parabola is f, and f has a real number λ on parabola, which makes 0,.
(1) Find the equation of straight line AB; Theme network
(2) Find the equation of △AOB circumscribed circle. Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
19. (This small question is 16 points) Subject Network
The known function is increasing function on [1, +∞), and θ∈(0, π), m∈ R.
(1) Find the value of θ; Theme network
(2) If it is a monotone function on [1, +∞], find the value range of m; Theme network
(3) Assume that if there is at least one range of values required to make it true on [1, e]. Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
Theme network
20. (This topic 16) Subject Network
It is known that the first term of arithmetic progression is a, the tolerance is b, the first term of geometric progression is b, and the common ratio is a, where a and b are positive integers greater than 1, and.
(1) Find the value of a;
(2) If it always exists for any one, so that it holds, find the value of b;
(3) Order, ask whether there are three consecutive geometric series terms in the series. If it exists, find out all three consecutive terms that constitute the geometric series; If it does not exist, please explain why.
B. Other issues
2 1. (topic selection) Choose two from A, B, C and D, with 10 for each question and ***20 for each question. Write a written explanation, proof process or calculus steps when answering.
A. Elective course 4- 1 (Selected lecture on geometric proof)
As shown in the figure, AB is the diameter of a semicircle, and C is a point on the extension line of AB and CD.
Cut a semicircle at point D, CD=2, DE⊥AB, the vertical foot is E, and E is
Find the midpoint of OB and the length of BC.
B. Elective Course 4-2 (Matrix and Transformation)
Rotate the curve 45 counterclockwise around the coordinate origin, and find the equation of the obtained curve.
C. Elective Course 4-4 (Coordinate System and Parameter Equation)
Find the chord length of a straight line (t is a parameter) cut by a circle (α is a parameter).
D. Elective course 4-5 (lecture on inequality)
It is known that both x and y are positive numbers, and x > y, so the proof is:
22. (required) known equation, in which
AI (I = 0, 1, 2, …, 10) is a real constant. Find:
The value of (1);
The value of (2).
23. Read first: As shown in the figure, let the lengths of the upper and lower bottoms of the trapezoidal ABCD be A and B (A < B) and the height be H, and find the area of the trapezoid.
In 2009, the first survey was conducted in the third grade of senior high school in Nantong.
Mathematical reference answers and grading opinions
A. Required questions
1. Fill-in-the-blank question: This big question is entitled *** 14, with 5 points for each question and 70 points for * * *.
1. If the complete set U={ 1, 2, 3, 4, 5, 6, 7} is known, then the set = ▲.
2. If the function is known, the minimum positive period of is ▲.
3. The equation of the straight line passing through point (-2,3) and parallel to the straight line is ▲.
4. If the plural number is satisfied, ▲.
5. The process is as follows:
t← 1
I ←2
When i≤4
t←t×i
i←i+ 1
end time
Print t
The output of the above program is ▲.
6. If the variance of is 3, the variance of is.
For ▲.
7. If the side length of the cube ABCD-a1b1c1d1is, the volume of the circumscribed sphere of the tetrahedron is ▲.
8. If the circle with the left focus of the ellipse as the center and the circle with the radius of c intersect with the left directrix of the ellipse at two different points, the eccentricity range of the ellipse is ▲.
9. let a > 0, let A={(x, y)|} and B={(x, y) |}. If point P(x, y)∈A is a necessary and sufficient condition for point P(x, y)∈B, then the range of a is ▲.
10. If you take two real numbers in the closed interval [- 1, 1], the probability that their sum is not greater than 1 is ▲.
1 1. In the series,, and (,) are the general formulas of this series.
▲ .
12. According to the following set of equations:
…………
Available ▲
13. In △ABC,, d is any point on the side of BC (D does not coincide with B and C) and is equal to ▲.
14. Set a function and remember that if the function has at least one zero, the range of the real number m is ▲.
Answer: 1. {6,7} 2.3.4.5.24 6.27 7.8.
9.0 y, so the proof is:
Solution: Because x-y>0, y>0, X-Y > 0,
Three points
= ... 6 points.
Nine points
Therefore, .......................................... 10.
22. (required) known equation, in which
AI (I = 0, 1, 2, …, 10) is a real constant. Find:
The value of (1);
The value of (2).
Solution: (1) in,
Order and get 2 points for ...................................................
Order and get 4 points.
So ... five points.
(2) Take the derivative of x on both sides of the equation and get 7 points.
Yes,
Make x=0, finish machining, and get ........................... 10.
23. Read first: As shown in the figure, let the lengths of the upper and lower bottoms of the trapezoidal ABCD be A and B (A < B) and the height be H, and find the area of the trapezoid.
Method 1: Extend the intersection of DA and CB at point O, and cross point O as the vertical line of CD to intersect AB and CD at points E and F respectively.
Suppose.
.
Method 2: Parallel line MN with AB intersects with AD and BC in M and N respectively, and parallel line AQ with BC intersection point A intersects with MN and DC in P and Q respectively.
Let the height of trapezoid AMNB be,
.
Then solve the following problems:
It is known that the areas of the upper and lower bottom surfaces of the four prisms ABCD-A ′ B ′ C ′ D ′ are respectively, and the height of the prism is h. By analogy with the above two methods, the volumes of the prisms (the volume of the pyramid = the height of the bottom area) are calculated respectively.
Solution 1: The quadrangular prism ABCD-A ′ B ′ C ′ D ′ is complemented by a pyramid V-ABCD, and the distance from the point V to the plane A ′ B ′ C ′ D ′ is H ′.
therefore
Therefore, the volume ABCD-a' b' c' d' of a quadrangular prism is ...................................................................................................................................................
Solution 2: make a plane parallel to the upper and lower bottom surfaces, cut the area of the quadrilateral into s, and the distance from it to the upper bottom surface is x,
.
.................................. 10.