Mathematics (Science and Engineering)

This volume is divided into two parts: the first volume (multiple-choice questions) and the second volume (non-multiple-choice questions). The score is * *150, and the examination time is 120 minutes.

volume one

Multiple choice questions: Of the four options given in each question, only one meets the requirements of the topic.

(1) is an imaginary unit, and the complex number =

(A)(B) (C) (D)

1.B

Proposition intention This exam mainly examines the concept of complex numbers and the four operations of addition, subtraction, multiplication and division of complex numbers.

Parse = = =

(2) If, then ""is "even function"

(a) Sufficient and unnecessary conditions (b) Necessary and insufficient conditions.

(c) Sufficient and necessary condition (d) is neither a sufficient condition nor a necessary condition.

2.A

Proposition intention This exam mainly examines the judgment and necessary and sufficient conditions of the parity of trigonometric functions.

Analysis ∵ is an even function, and vice versa, and ∴ ""is a necessary and sufficient condition for "being an even function".

(3) Read the program block diagram on the right and run the corresponding program. When the input value is, the output value is.

(A)(B) (C) (D)

3.C

Proposition intention This topic mainly examines the reading of algorithm block diagram and can perform operations according to the given algorithm program.

According to the algorithm program given in the figure, we can know the first time, the second time and the output.

(4) The number of zeros of the function in the interval is

(A)0 (B) 1 (C)2 (D)3

4.B

Proposition intention This exam mainly examines the ideas of functions and equations, the concept of zero point of functions, the existence theorem of zero point and the mathematical ability to draw and use graphics.

Analytical solution 1: Because, that is, the function is continuous, the number of zeros is 1.

Solution 2: Assume that the images of two functions in the same coordinate system are as shown in Figure B.

(5) In the binomial expansion of, the coefficient of is

(A) 10(B)- 10(C)40(D)-40

5.D

Proposition intention This exam mainly examines the application of the general term formula in binomial theorem, and analyzes the coefficient of this term with the help of the general term formula.

Analytic ∫ =, ∴, that is, the coefficient of ∴ is.

(6) In △ABC, the inner angle and the opposite side are respectively, and if it is known, then cosC=

(A)(B) (C) (D)

6.A

Proposition intention This exam mainly examines the sine theorem and double angle formula in trigonometric function, and examines students' ability of analysis, transformation and calculation.

Analytic:, obtained by sine theorem, and:, ∴, so it is easy to know, ∴, =.

(7) It is known that △ABC is an equilateral triangle, and points P and Q satisfy,,, if, then.

(A)(B) (C) (D)

7.A

Proposition Intention This examination takes equilateral triangle as the carrier, and mainly examines the geometric meaning of vector addition and subtraction, the basic theorem of plane vector, the theorem of * * * line vector and the comprehensive application of its scalar product.

Analysis ∵ =,

Again, again,,,, so, solution.

(8) Suppose that if a straight line is tangent to a circle, the value range of is

(A) (B)

(C) (D)

8.D

Proposition Intention This exam mainly examines the positional relationship between a straight line and a circle, the formula of the distance from a point to a straight line, the solution of important inequalities, and the ability to solve with the help of the geometric properties of tangency between a straight line and a circle.

Analysis ∵ The straight line is tangent to the circle, and the distance from the center of the circle to the straight line is, so

Then, solve

Fill-in-the-blank question: This big question has 6 small questions, each with 5 points and * * 30 points.

(9) There are 150 primary schools, 75 middle schools and 25 universities in a certain area. At present, 30 schools are selected by stratified sampling from these schools to adjust students' eyesight, including 3 primary schools and 2 middle schools.

9. 18,9

Proposition intention This exam mainly examines the concept of stratified sampling in statistics and the method and calculation of sample acquisition.

Analysis/stratified sampling, also known as proportional sampling, has a total of 250 schools.

So we should learn from primary and secondary schools.

(10)- The three views of a geometric figure are shown in the figure (unit:), then the volume of the geometric figure is.

10.

Proposition intention This exam mainly examines the drawing of simple assembly three views, the calculation of volume and the ability of spatial imagination.

Analytically speaking, it is an assembly consisting of three views and a cuboid above two tangent spheres, so its volume is: =.

(1 1) known set, set, and then.

1 1.,

Propositional intention This topic mainly examines the operation and operation properties of set intersection, and also examines the solutions of absolute inequality and unary quadratic inequality and the ideas of classified discussion.

Analysis: = and:, draw a few axes to see.

(12) It is known that the parametric equation of a parabola is (parameter), where the focus is, the directrix is, the vertical line passing through a point on the parabola is, and the vertical foot is. If the abscissa of the point is 3, then.

12.2

Proposition intention This topic mainly examines the geometric meaning of parametric equation and its parameters, the definition of parabola and its geometric properties.

According to the analysis of ∵, the standard equation of parabola is ∴ focus, and the abscissa of ∵ point is 3, then, then the point,

It is derived from the geometric properties of parabola, ∫∴,.

(13) As shown in the figure, it is known that AB and AC are two chords of a circle. The tangent passing through point B intersects with the extension line of AC at point D, the parallel passing through point C and BD intersects with the circle at point E, and intersects with AB at points F,,, and, so the length of the line segment is.

13.

Proposition Intention This exam mainly examines the positional relationship between straight lines and circles in plane geometry, the theorem of intersecting chords, the theorem of cutting lines, and the concept, judgment and nature of similar triangles.

The analysis of ⊙ is derived from the intersection theorem, so ⊙BD∨ce, ∴, =, is set, and then it is derived from the tangent theorem, that is, it is solved, so.

(14) It is known that the image of the function and the image of the function have exactly two intersections, so the value range of the real number is.

14.

Propositional Intention This topic mainly examines the image of a function and its properties, and uses the image of a function to determine the intersection of two functions, thus determining the range of parameters.

The image line of analytic function ∵ passes through a fixed point, from which we can see,, ∴.

Third, the solution: this big question is ***6 small questions, and ***80 points. The solution should be written in proof process or calculus steps.

(15) (the full mark of this question is 13) Known function,.

(i) Find the minimum positive period of the function;

(Ⅱ) Find the maximum and minimum values of the function in the interval.

The problem of propositional intention is mainly examined

Reference answer

The key to commenting on this test question is to turn the known function expression into a new mathematical model, and then solve the problem according to the image and nature of this triangular model.

(16) (full mark for this small question 13) At present, there are four people going to participate in an entertainment activity, and there are two games A and B for participants to choose from. In order to increase the interest, it is agreed that everyone will decide to participate in a game by throwing dice with uniform texture. Those with a score of 1 or 2 will participate in the A game and those with a score greater than 2 will participate in the B game.

(I) Find out the probability that exactly two of these four people will participate in Game A:

(ii) Find out the probability that the number of participants in game A among the four people is greater than the number of participants in game B:

(iii) Write and find out the distribution table and mathematical expectation of random variables on behalf of the number of four people who have participated in games A and B respectively.

The problem of propositional intention is mainly examined

Reference answer

The essay question is an important test point of the college entrance examination proposition. In recent years, using probability questions to examine, new questions often appear. For this kind of questions, we should carefully examine the questions, understand the essence of the questions from the perspective of mathematics and real life, and successfully transform the questions into classical probability models, independent events, mutually exclusive events and other probability models. Therefore, understanding is the foundation and transformation is the key to probability application problems.

(17) (the full mark of this small question is 13) As shown in the figure, among the four pyramids, the η plane,,,.

(i) Proof that:

(ii) finding the sine value of the dihedral angle;

(iii) Let E BE the point on the side, satisfy the angle formed by the non-planar straight line Be and CD, and find the length of AE.

The problem of propositional intention is mainly examined

Reference answer

From the perspective of proposition, the overall topic is similar to the one we usually practice, but the bottom is not special.

Quadrilateral is a quadrilateral with a straight line perpendicular to the bottom, so the innovation is that the position of the midpoint E in the third question is uncertain and needs to be determined by students according to known conditions, so it is hard to say that it is better to solve this problem with spatial rectangular coordinate system.

(18) (the full mark of this small question is 13) It is known that {} is arithmetic progression, and the sum of its antecedents is {} geometric progression, and =

, , .

(1) Find the general formula of the sequence {} and {};

(2) Remember and prove it.

The problem of propositional intention is mainly examined

Reference answer

Comment on the life system of this test question is relatively direct, without any implied conditions, and it is a comprehensive application of equal proportion and arithmetic progression, but there are various methods. The second question can be proved by dislocation subtraction or mathematical induction, which leaves room for students to think and conforms to the principle of selecting topics in the college entrance examination.

(19) (the full mark of this small question is 14) Let the left and right vertices of the ellipse be A and B respectively, and the point P on the ellipse is different from A and B, which is the coordinate origin.

(i) If the product of the slopes of the straight line AP and BP is, find the eccentricity of the ellipse;

(ii) If it is proved that the slope of the straight line is satisfied.

The problem of propositional intention is mainly examined

Reference answer

comment

(20) (The full score of this small question is 14) The minimum value of the known function is, where.

(i) the value of;

(ii) If there is any truth, seek the minimum value of truth number;

(iii) Evidence.

The problem of propositional intention is mainly examined

Reference answer

The review questions are divided into three questions, the questions are relatively simple, and the functions given are relatively routine, so it is not difficult for students to start. The second question, when using parameters to solve inequalities, we should pay attention to all the restrictions on parameter discussion in the stem of the question, so as not to weigh or leak; Third, the proof of inequality should be carried out by the method of derivative proof of inequality.