Mathematics examination for science students
Precautions:
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 the first volume, after choosing the answer for 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, which 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. given sets A={ 1, 2, 3, 4, 5} and B={(x, y)|x∈A, y∈A, x-y∈A}, the number of elements contained in b is
A.3 B.6 C.8 D. 10
2. Divide two teachers and four students into two groups and arrange them to participate in social practice activities in Party A and Party B respectively. Each group consists of 65,438+0 teachers and two students. The different arrangements are as follows.
A. 12 species B. 10 species C.9 species D.8 species.
(3) The following are four propositions about the complex number z=
P 1:=2 p2: =2i
The complex number of the * * * yoke of P3: z is 1+IP4: z, and the imaginary part is-1.
The real proposition is
P3 P2 P2 P2 P4 P3 P4
(4) let F 1, F2 be the left and right focus of ellipse e: += 1 (a > b > 0), and p be a point on the straight line x=.
△F2PF 1 is an isosceles triangle with a base angle of 30, then the eccentricity of e is
A B C D
(5) If it is known that {an} is a geometric series, A4+A 1 = 2A5A6 =-8, then a 1+a 10 =
A.7 B.5 C-5 D.-7
(6) If the program chart on the right is executed and positive integer N(N≥2) and real number a 1.a2, …an are input, then
(A)A+B is the sum of a 1a2, …, an.
(b) It is a 1a2. The average value of the formula.
(C)A and B are the largest and smallest numbers in a 1a2, respectively, …an.
(D)A and B are the smallest and largest numbers in a 1a2, …an, respectively.
(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 (B)9 (C) 12 (D) 18
(8) The center of the equilateral hyperbola C is at the origin and the focus is on the X axis. C and parabola y? The directrix = 16x intersects at point A and point B, and the real axis length of C is
(A)(B)(C)4(D)8
(9) assuming that w > 0 and the function is monotonically decreasing, the range of w is
(A)(B)(C)(D)(0，2)
(10) Given the function, the image of y=f(x) is approximately
(1 1) It is known that all vertices of the triangular pyramid S-ABC are on the spherical surface of the ball O, △ABC is a regular triangle with a side length of 1, SC is the diameter of O, and SC=2, then the volume of this pyramid is.
(A)(B)(C)(D)
(12) If point P is on the curve, point Q is on the curve, and y=ln(2x), then the minimum value of |PQ| is
(A) 1-ln2(B)
Volume II
This volume consists of two parts: mandatory questions and multiple-choice questions. The question 13 ~ 2 1 is called a required question, and every candidate must answer it. 22-24 is entitled "Select Examination Questions" and answer as required.
Two. Fill-in-the-blank question: This big question has four small questions, each with 5 points.
(13) Given that the included angle between directional quantities A and B is 45, and |a|= 1, |2a-b|=, then | B | = _ _ _ _ _ _ _ _ _
(14) If x and y satisfy the constraints, the range of z=x-2y is _ _ _ _ _ _ _.
(15), one component consists of three electronic components, which are connected as shown in the figure below. If component 1 or component 2 works normally and component 3 works normally, then the component works normally. Let the service life (unit: hours) of three electronic components obey the normal distribution n (1000,502), and whether each component can work normally is independent of each other, then the probability that the service life of the component exceeds1000 hours is _ _ _ _ _ _ _ _.
(16) series {an} satisfies an+1+(-1) nan = 2n-1,then the sum of the first 60 items of {an} is _ _ _ _ _.
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 the opposite sides of the three internal angles A, B and C of △ABC respectively.
(i) Find one;
(ii) 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 16 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:
The frequency of each demand recorded in 100 days is taken as the probability of each demand.
(1) If a flower shop buys 16 roses a day, X represents the profit of that day (unit: yuan), and find the distribution table, mathematical expectation and variance of X;
(2) If the flower shop plans to buy 16 or 17 roses a day, do you think it should buy 16 or 17 roses? Please provide a justification for the answer.
(19) (the full score of this small question is 12)
As shown in the figure, in the straight triangular prism ABC-a1b1c1,AC=BC=AA 1, D is the midpoint of the side AA 1, and DC 1⊥BD.
Proof: BC DC1⊥;
Find the size of dihedral angle A 1-BD-C 1.
(20) (The full score of this small question is 12)
Let the focus of parabola C: x2 = 2py (p > 0) be F, the directrix be L, and A be a point on C. It is known that the circle F with F as the center and FA as the radius intersects L at B and D.
If ∠ BFD = 90, and the area of △ABD is, find the value of P and the equation of circle F;
If the three points A, B and F are on the same straight line M, and the straight line N is parallel to M, and N and C have a common point, 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)
It is known that the function f(x) satisfies f (x) = f ′ (1) ex-1-f (0) x+x2.
Find the analytic formula and monotone interval of f(x);
If f(x)≥x2+ax+b, find the maximum value of (a+1) b.
Please answer any of questions 22, 23 and 24. If you do more, you will be graded according to the first question you do. Please write down the question number clearly when you answer.
(22) (full score for this small question 10) Take 4-1; Special lecture on geometric proof
As shown in the figure, D and E are the midpoint of AB and AC on the side 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；
(2) △BCD △GBD.
(23) (full score for this small question 10) Elected 4-4; Coordinate system and parametric equation
Given the parametric equation of curve C 1 (as a parameter), the coordinate system is established with the origin of coordinates as the pole and the positive half axis of X axis as the polar axis, and the polar coordinate equation of curve C2 is 2. The vertices of square ABCD are all on C2, A, B, C and D are arranged in counterclockwise order, and the polar coordinates of point A are.
(i) Find the rectangular coordinates of points A, B, C and D;
(Ⅱ) Let p be any point on C 1 and find the value range.
(24) (Full score for this small question 10) Elective courses 4-5; Seminar on inequality
known function
(i) When a=-3, find the solution set of inequality (x) 3;
(2) If the solution set of f(x)≤ contains, find the value range of A. ..