20 12 National Unified Examination for Enrollment of Ordinary Colleges and Universities (New Curriculum Standard Volume)
Liberal arts mathematics
New Curriculum Standard (Ning, Ji, Hei, Jin, Yu and Xin) Examination Paper
Interest note:
1. This paper is divided into two parts: Volume I (multiple choice questions) and Volume II (non-multiple choice questions). Before answering questions, candidates must fill in their names and admission ticket numbers in the corresponding positions of this paper and the answer sheet.
2. When answering questions about Volume I, After selecting the answer to each small question, black the answer label of the corresponding question on the answer sheet with a pencil. If you need to change it, clean it with an eraser, and then choose to apply other answer labels. It is invalid to write on this test paper.
3. When answering Volume 2. Write the answer on the answer sheet. Writing on this piece of paper is invalid.
4. After the exam, return this paper together with the answer sheet.
volume one
1. Multiple choice question: This big question is a small question of *** 12, with 5 points for each small question. Only one of the four options given in each small question meets the requirements of the topic.
1, known set a = {x | x2-x-2 < 0}, b = {x |-1< X< 1}, then
(A)AB (B)BA (C)A=B (D)A∩B=？
(2) the complex number z = of the complex number yoke is
(A)2+I(B)2-I(C)- 1+I(D)- 1-I
3. In the scatter plot of a set of sample data (x 1, y 1), (x2, y2), …, (xn, yn)(n≥2, x 1, x2, …, xn is not completely equal), if all the sample points (xi
(A) 1(B)0(C)0(D) 1
(4) let F 1 and F2 be ellipses e:+=1(a >; B>0), p is the upper point of a straight line x=, and △F 1PF2 is an isosceles triangle with a base angle of 30, then the eccentricity of e is ().
(A) (B) (C) (D)
5. The vertices A( 1, 1) and B( 1, 3) of the regular triangle ABC are known, and the vertex C is in the first quadrant. If the point (x, y) is within △ABC, the range of z =-x+y is
(A)( 1-，2) (B)(0，2) (C)(- 1，2) (D)(0， 1+)
(6) If the program block diagram on the right is executed, positive integer N(N≥2) aNd real number a 1, a2, …, an, and outputs A and B, then
(A)A+B is the sum of a 1, a2, …, aN.
(b) is the arithmetic average of a 1, a2, …, aN.
(C)A aNd B are the largest and smallest numbers in a 1, a2, …, an respectively.
(D)A aNd B are the minimum number and the maximum number of a 1, a2, …, an, respectively.
begin
A=x
B=x
x>A
no
Outputs a, b
be
Enter n, a 1, a2, …, aN.
end
x & ltB
k≥N
k= 1，A=a 1，B=a 1
k=k+ 1
x =ak
be
no
no
be
(7) As shown in the figure, the side length of the small square on the grid paper is 1, and the thick line draws three views of a certain geometry, so the volume of this geometry is
(A)6
9.
(C) 12
18
(8) If the radius of the circle obtained by cutting the spherical surface of the ball O with plane α is 1 and the distance from the center of the ball O to plane α is, then the volume of the ball is
(A)π (B)4π (C)4π (D)6π
(9) known ω > 0，0 & lt； φ& lt； π, straight lines x= and x= are two adjacent symmetry axes of an image with the function f(x)=sin(ωx+φ), then φ =
(A) (B) (C) (D)
(10) The center of the equilateral hyperbola C is at the origin, the focus is on the X axis, the directrix of C and parabola y2= 16x intersects at points A and B, and |AB|=4, then the real axis length of C is
(A) (B)2 (C)4 (D)8
(1 1) When 0
(A)(0)，(B)( 1)(C)( 1)，(D)(2)
(12) series {an} satisfies an+1+(-1) n an = 2n-1,then the sum of the first 60 items of {an} is
(A)3690(B)3660(C) 1845(D) 1830
Volume II
This volume consists of two parts: mandatory questions and multiple-choice questions. Question 13- question 2 1 The topic is required, and candidates must answer every question. 22-24 is entitled multiple-choice questions, and candidates answer as required.
2. Fill in the blanks: This big question has four small questions, each with 5 points.
The tangent equation of (13) curve y=x(3lnx+ 1) at (1, 1) point is _ _ _ _ _ _ _ _.
The sum of the first n terms of (14) geometric series {an} is Sn. If S3+3S2=0, the common ratio Q = _ _ _ _ _ _
(15) Given that the included angle between directional quantities A and B is 45, and |a|= 1, | 2a-b | =, then |b|=
(16) Let the maximum value of the function f(x)= be m and the minimum value be m, then m+m = _ _ _
Third, problem solving: the idea of solving problems should be clearly written, explaining the process or calculus steps.
(17) (the full score of this small question is 12)
It is known that A, B and C are opposite sides of three internal angles A, B and C of △ABC, and c = asinC-ccosA.
(1) search
(2) if a=2 and the area of △ABC is, find b and c.
18. (The full score of this small question is 12)
A flower shop buys several roses from the farm at the price of 5 yuan every day, and then sells them at the price of 10 yuan. If it is not sold out that day, the remaining roses will be disposed of as garbage.
(1) If a flower shop buys 17 roses a day, find the analytic function of the profit y (unit: yuan) and the demand n (unit: branches, n∈N) of that day.
(2) The flower shop recorded the daily demand of roses (unit: branches) on 100 day, and compiled the following table:
Daily demand
14
15
16
17
18
19
20
frequency
10
20
16
16
15
13
10
(1) Suppose the flower shop buys 17 roses every day on this 100 day, and find the daily average profit (unit: yuan) on this 100 day;
(2) If the flower shop buys 17 roses a day, take the frequency of each demand recorded by 100 as the probability of each demand, and get the probability that the profit of that day is not lower than that of 75 yuan.
(19) (the full score of this small question is 12)
As shown in the figure, in the triangular prism ABC-a1b1,the side is perpendicular to the bottom, ∠ ACB = 90, AC=BC=AA 1, and d is the midpoint of the side AA 1.
Evidence: BDC 1⊥ aircraft BDC aircraft.
(Ⅱ) Plane BDC 1 Divide this prism into two parts and find the volume ratio of these two parts.
(20) (The full score of this small question is 12)
Let parabola c: x2 = 2py (p > The focus of 0) is F, the directrix is L, and A is a point on C. It is known that a circle with F as the center and FA as the radius intersects L at B and D..
(i) If ∠ BFD = 90, and the area of △ Abd is 4, find the value of p and the equation of circle F;
(2) If the three points A, B and F are on the same straight line M, the straight lines N and M are parallel, and there is only one common point between N and C, find the ratio of the coordinate origin to the distance between M and N. ..
(2 1) (the full score of this small question is 12)
Let the function f (x) = ex-ax-2.
(i) Find the monotone interval of f(x)
(ii) if a= 1, k is an integer, and when x >; 0，(x-k) f？ (x)+x+ 1 & gt； 0, find the maximum value of k.
Please answer any of questions 22, 23 and 24. If you do too much, score according to the first question. Please write down the question number clearly when you answer.
(22) (Full score for this small question 10) Elective course 4- 1: Selected lectures on geometric proof.
As shown in the figure, D and E are the midpoint of AB and AC on the edge of △ABC, and the circumscribed circle of the straight line d E intersecting △ABC is at F and G points. If CF//AB, it is proved that:
(ⅰ)CD = BC；
(ⅱ)△BCD∽△GBD
(23) (full score for this small question 10) Elected 4-4; Coordinate system and parametric equation
It is known that the parameter equation of curve C 1 is (φ is the parameter), and the polar coordinate system is established with the coordinate origin as the pole and the positive and semi-axis of X axis as the polar axis, and the polar coordinate equation of curve C2 is ρ=2. The vertices of square ABCD are all on C2, A, B, C and D are arranged in counterclockwise order, and the polar coordinate of point A is (2,).
(i) Find the rectangular coordinates of points A, B, C and D;
(2) Let p be any point on C 1 and find the range of |PA| 2+ |PB|2+|PC| 2+ |PD|2.
(24) (The full mark of this small question is 10) Elective course 4-5: Selected lectures on inequality.
The function f (x) = | x+a |+| x-2 | is known.
(i) When a =-3, find the solution set of inequality f(x)≥3;
(ii) If it contains the solution set of f (x) ≤ | x-4 |, find the value range of A. ..
Reference answer