Fill in the blanks
1, let the set A={- 1, 1, 3}, B={a+2, a2+4}, A∩B={3}, then the real number A = _ _ _ _ _ _ ▲
2. Let the complex number Z satisfy z(2-3i)=6+4i (where I is an imaginary unit), then the modulus of Z is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
5. Let the function f (x) = x (ex+AE-x) and x ∈ r be an even function, then the real number A = _ _ _ _ _ _ _ _ _ _ _ _ _
6. In the plane rectangular coordinate system xOy, if there is a point M on the hyperbola, and the abscissa of the point M is 3, then the distance from M to the right focus of the hyperbola is _ _ _ _ _ _ _ _ _.
7. The figure on the right is the flow chart of an algorithm, so the value of the output S is _ _ _ _ _ _ _ _.
8. Function y = x2(x >；; 0) The abscissa of the intersection of the tangent at point (ak, ak2) and the X axis is ak+ 1, k is a positive integer, and a 1= 16, then a1+A3+A5 = _ _ _ _ _ ▲
9. In the plane rectangular coordinate system xOy, it is known that there are only four points on the circle, and the distance from the straight line 12x-5y+c=0 is 1, so the value range of the real number c is _ _ _ _ _ _ _ _ [Source: 2 1 Century Education Network].
10, the intersection point of the image of function y=6cosx and the image of y=5tanx defined on the interval is p, the intersection point p is PP 1⊥x axis at point P 1, and the straight line PP 1 intersects with the image of y=sinx at point P2, then the line segment length is p/kloc.
1 1, given the function, the range where x satisfies the inequality is _ _ _ _ ▲ _ _
12, let the real number x and y satisfy 3≤ ≤8 and 4≤ ≤9, and the maximum value is _ _ _ _ _ _ _ [Source: 2 1 Century Education Network].
13, acute triangle ABC, where the opposite sides of A, B and C are A, B and C respectively, then _ _ ▲
14. Cut a regular triangular thin plate with a side length of 1 into two pieces along a straight line parallel to the bottom, one of which is trapezoidal. Note that S=, then the minimum value of s is _ _ _ _ _ _ _ _ _ _ _ _.
Second, answer the question.
15, (14 point) In the plane rectangular coordinate system xOy, points A (- 1, -2), B (2 2,3), C (-2,-1).
(1) Find the length of two diagonals of a parallelogram with lines AB and AC as adjacent sides.
② Let the real number t satisfy ()? =0, find the value of t.
16, (14 points) as shown in the figure, in the P-ABCD of the pyramid, PD ⊥ plane ABCD, PD=DC=BC= 1, AB=2, AB∥DC, ∠BCD=900.
(1) Verification: PC⊥BC
(2) Find the distance from point A to plane PBC.
17, (14) An interest group measures the height h (unit m) of the AE of the TV tower, as shown in the schematic diagram. The height BC of the vertically placed datum is h=4m, the elevation angle ∠ABE=α, and the elevation angle ∠ADE=β.
(1) The team measured a set of values of α and β, tan α = 1.24 and tan β = 1.20. Please calculate the value of h accordingly.
(2) After analyzing some measurement data, the team found that the measurement accuracy can be improved by properly adjusting the distance d (unit m) from the pole to the TV tower, so that the difference between α and β is larger. If the actual height of the TV tower is 125m, ask what D is, and α-β is the maximum.
18.( 16 minutes) In the plane rectangular coordinate system, as shown in the figure, the left and right vertices of the ellipse are known as A and B, and the right vertex is F. The straight lines TA and TB passing through point T () intersect with the ellipse at point M, where M >；; 0,
① Let the moving point p satisfy, and find the trajectory of the point p..
② Set and find the coordinates of point T..
③ Hypothesis and verification: The straight line MN must pass through a certain point on the X axis.
(its coordinates have nothing to do with m)
[Source: 2 1 Century Education Network]
19.( 16 points) Let the sum of the first n items of a series be all positive numbers, and it is known that the series is a arithmetic progression with tolerance.
① Find the general term formula of series (expressed by)
② If it is a real number, the inequality holds for any positive integer. Validation: The maximum value of is.
20.( 16 points) Set a function defined on the interval, and its derivative function is. If there is a real number and a function, there is >: 0, which means that the function has properties.
(1) Set a function, where is a real number.
① Verification: The function has properties.
② Find the monotone interval of the function.
(2) It is known that the function has the given, and if ||
Additional problems in science
2 1 (choose two answers from the following four questions, each with 10 score)
(1) Seminar on Geometry Proof
AB is the diameter of ⊙O, D is the point above ⊙O, and the tangent intersection point D is the extension line of ⊙O at C. If DA=DC, it is verified that AB=2BC.
(2) Moment Matrix and Transformation
In the plane rectangular coordinate system xOy, a (0 0,0), b (-3), c (-2, 1), let k≠0, k∈R, M=, N=, points A, B and C get point A65438 under the corresponding transformation of matrix MN.
(3) Parametric equation and polar coordinates
In polar coordinate system, the circle ρ=2cosθ is tangent to the straight line 3ρcosθ+4ρsinθ+a=0, and the value of number A is found.
(4) Special lecture on inequality proof
Known real numbers A and B ≥ 0, verification: [Source: 2 1 Century Education Network]
22.( 10) A factory produces two kinds of products, one of which is 80% first-class and 20% second-class; Production of B-class products, 90% of first-class products and 10% of second-class products. When producing a product, the first-class product can earn 40 thousand yuan, and the second-class product can lose 6.5438+0 million yuan; When producing a product B, the first-class product can earn 60,000 yuan, and the second-class product can lose 20,000 yuan. It is assumed that the production of various products is independent of each other.
(1) Remember that X (unit: ten thousand yuan) is the total profit that can be obtained by producing 1 products A and B, and find the distribution table of X.
(2) The probability that the profit of producing four A products is not less than 654.38+10,000 yuan.
23.( 10) It is known that all three sides of △ABC are rational numbers.
(1) proves that cosA is rational.
(2) For any positive integer n, it is proved that cosnA is also a rational number.