1. Let a set, z is an integer set, then the number of elements in it is [].
2. Let I be the imaginary unit, then the item containing x4 in the expansion is [].
3. In order to get the image of the function, just put all the points [] on the image of the function.
4. Use the numbers 1, 2, 3, 4 and 5 to form five digits, which are not complex, and the odd number is [].
In order to stimulate innovation, a company plans to increase R&D investment year by year. If the company invested RMB 0.3 million in R&D in 20 15 years, and on this basis, the annual investment in R&D increased by 12% compared with the previous year, then the year when the company invested more than RMB 2 million in R&D in 20 15 years was [].
(Reference data: lg 1. 12≈0.05, LG 1.3 ≈ 0. 1, lg2≈0.30).
6. Qin was a mathematician in China during the Southern Song Dynasty (now Anyue County, Sichuan Province). Qin Jiushao algorithm for polynomial evaluation proposed by him in Shu Shu Jiu Zhang is still an advanced algorithm. The program block diagram shown in the figure gives an example of finding polynomial value by Qin Jiushao algorithm. If the values of n and x are 3 and 2, respectively, it is judged that the value of v is [].
7. Let p: real number x and y satisfy (x- 1)2-(y- 1)2≤2, and q: real number x and y satisfy, then p is q [].
8. Let O be the coordinate origin, P be any point on the parabola with F as the focus, M be the point on the line segment PF, and
When =2, the maximum value of the slope of the straight line OM is []
9. Let straight lines l 1 and l2 be functions f(x)= point P 1 on the image is tangent to P2, l 1 and l2 vertically intersect at point P, and l 1 and l2 intersect with Y axis at point A and point B respectively, then the range of △PAB is [].
10. In the plane, if the fixed points A, B, C and D satisfy = =, and the fixed points P and M satisfy = 1, =, the maximum value is [].
1 1 . cos 2–sin 2 =。
12. When two coins with uniform texture are thrown at the same time, when at least one coin faces up, the experiment is said to be successful, and the average number of successful times of the two experiments is [].
13. It is known that all four faces of a triangular prism are isosceles triangles with a waist length of 2. The front view of triangular prism is shown in the figure, so the volume of triangular prism is [].
14. It is known that the function f(x) is a odd function with a period of 2 and is defined on R. When 0 < x < 1 and f(x)=, then f()+f( 1)= 1
15. In the plane rectangular coordinate system, when P(x, y) is not the origin, the "adjoint point" of p is defined as;
When P is the origin, the adjoint point of P is defined as itself, and the curve formed by the adjoint points of all points on the plane curve C is defined as the adjoint curve of curve C. There are the following propositions:
① If the "adjoint point" of point A is a point, then the "adjoint point" of point A is a point.
② The "adjoint curve" of the unit circle itself;
(3) If curve C is axisymmetrical about X, its "adjoint curve" is axisymmetrical about Y;
The adjoint curve of a straight line is a straight line.
The true proposition is _ _ _ _ _ _ (the order in which all true propositions are written).
16. (The full score of this small question is 12)
China is a country with serious water shortage in the world. In order to encourage residents to save water, a municipal government plans to adjust the residential water charging scheme and determine a reasonable monthly water consumption standard (tons). Residents' monthly water consumption does not exceed the part charged at parity, and the excess part is charged at bargaining price. In order to understand the water consumption of residents, the monthly water consumption (unit: tons) of residents in a certain year 100 was obtained through sampling survey, and the data was calculated according to [
(i) finding the value of a in the histogram;
(two) assuming that there are 300 thousand residents in the city, it is estimated that the average monthly water consumption of urban residents is not less than 3 tons, and the reasons are explained;
(three) if the municipal government wants to make the monthly water consumption of 85% residents not exceed the standard (tons), estimate the value and explain the reasons.
17. (The full score of this small question is 12)
In △A, B and C, the opposite sides of angles A, B and C are A, B, C and.
(i) Proof that:
(II) If yes, please ask questions.
18. (The full score of this small question is 12)
As shown in the figure, in the quadrangular cone P-ABCD, ADC, ADC = PAB = 90, BC=CD=AD. E is the midpoint of the side AD, and the angle between the straight line PA and CD is 90.
(i) Find a point m in the plane PAB to make the straight line CM∨ a plane PBE, and explain the reasons;
(II) If the dihedral angle P-CD-A is 45, find the sine value of the angle formed by the straight line PA and the plane PCE.
19. (The full score of this small question is 12)
It is known that the first term of the sequence {} is 1, which is the sum of the first n terms of the sequence {}, where q >；; 0, .
(i) If it becomes arithmetic progression, find the general formula of an;
(ii) Let the eccentricity of hyperbola be, and prove that:
20. (The full score of this short question is 13)
It is known that the two focal points of ellipse E and one endpoint of the short axis are the three vertices of a right triangle, and the straight line l:y=-x+3 and ellipse E have only one common point T. 。
(i) Find the equation of ellipse E and the coordinates of point T;
(2) Let O be the coordinate origin, the straight line L' is parallel to OT, intersects the ellipse E at two different points A and B, and intersects the straight line L at point P, and prove that there is a constant λ, so that ∣ Pt ∣ 2 = λ ∣ Pa ∣∣ Pb ∣, and find λ.
2 1. (The full score of this small question is 14)
Let the function f(x)=ax2-a-lnx, where
(i) Discuss the monotonicity of f(x);
(II) Determine all possible values of a so that it holds in the interval (1, +∞) (e=2.7 18… is the base of natural logarithm).