1. restriction

The value of is ().

A: 0

B. 1

Chinese version

D.∞

Correct answer: C.

Reference analysis:

2. It is known that the included angle between vectors A and B is π/3, and |a|= 1, |b|=2. If m=λa+b and n=2a-b are perpendicular to each other, λ is ().

Artificial intelligence 2

B. 1 1

C. 1

D.2

Correct answer: D.

Reference analysis: Because M and N are vertical, mn=0, that is, (λ a+BN) (2a-b) = 0,2λ | a | 2+(2-λ) | a ||| b | cos π/3 | b | 2 = 0, and λ=2 is obtained.

3. Let f(x) and g(x) be defined as increasing function in the same interval, and the following conclusion must be correct ().

A.f(x)+g(x) is increasing function.

B.f(x)-g(x) is a decreasing function.

C.f(x)g(x) is increasing function.

D.f(g(x)) is a decreasing function.

Correct answer: a

Reference analysis: According to the increase or decrease of the function, increase+increase = increase, so we can know that f(x)+g(x) is increasing function. Therefore, choose a in this question.

4. It must be correct to assume that A and B are square matrices of order n ().

A.A+B=B+A

B.AB=BA

C.

D.

Correct answer: a

Reference analysis: Since both A and B are known to be square matrices of order N, we can know that A+B=B+A, so we choose A in this question.

Two students, A and B, went to different companies for interviews. The probability of a student being selected is 1/7, and that of b student is 1/5, so the probability of at least one of the two students being selected is ().

A. 1/7

B.2/7

C. 1 1/35

D. 12/35

Correct answer: C.

Reference analysis: At least 1 The opposite of students being selected is that neither of them was selected. Obviously, the probability of opposing events is easier to calculate, and the probability that neither student is selected is:

6. If the vectors A = (1, 0, 1), A2 = (0, 1, 1) and A3 = (2, λ, 2) are linearly related, then the value of λ is ().

A. 1 1

B.0

C. 1

D.2

Correct answer: b

Reference analysis: the necessary and sufficient condition for the linear correlation of vector groups is that the determinant value they form is equal to 0, so

=0, the solution is λ=0.

7. The following statement is propositional ().

①2x & lt； 1

②x-3 is an integer.

③ There is an x∈z, which makes 2x- 1=5.

④ For any irrational number x, x+2 is also irrational.

A.①②

B.①③

C.②③

D.③④

Correct answer: D.

Reference analysis: From the concept of proposition, a declarative sentence that can judge its truth or falsehood is called a proposition. For ①, it is not a declarative sentence, so it is not a proposition; For ②, because we don't know the specific scope of X, we can't judge whether it is true or not, so it is not a proposition; For ③ and ④, it is a declarative sentence that can judge whether it is true or not, and it is a proposition. Therefore, choose D in this question.

8. The following mathematical achievements are China's famous achievements ().

① Pythagorean theorem ② Logarithm ③ Circumcision ④ Multiphase subtraction.

A.①②③

B.①②④

C.①③④

D.②③④

Correct answer: C.

Reference analysis: ①, ③ and ④ all belong to China's ancient mathematical achievements, and the logarithm mentioned in ② was invented by British scientist John Napier. Therefore, this question chooses C.

9.

known function

Find the monotone interval and extreme value of function f(x).

Reference analysis: the monotonic increasing interval is [0, 1] [2, 1 ∞], and the monotonic decreasing interval is (1 ∞, 0) and (1, 2); The maximum value is 2 and the minimum value is 1.

10. Find a straight line

And parallel to the straight line.

Plane equation of.

Reference analysis: 2x-3y-z+7=0

analyse

1 1. It is known that 80% of girls and 90% of boys in a class choose skating, and 60% of the students in this class are girls.

(1) Randomly select a student from this class to find out the probability of this student taking skating class; (3 points)

(2) Choose a student randomly from the students who choose skating in this class to find out the probability that this student is a girl. (4 points)

Reference analysis: (1) 0.84; (2)4/7。

analyse

12. Briefly describe two methods to study the geometric properties of ellipses.

Reference analysis: two methods to study the geometric properties of ellipses;

① Using curve equation to study geometric properties, such as studying the range of X and Y, the range of path and focal radius through elliptic equation, can explain the geometric meaning of elliptic standard equations A, B and C. This method is a model of mathematical thinking method combining numbers with shapes.

② Using algebraic method to study geometric properties. In the research process, the general method of studying geometric properties by algebraic method is extracted by intuitively abstracting the process of geometric properties from graphics, and the eccentric model is established.

13. Briefly describe the design intention of setting the following exercises in the teaching design content of the textbook (only answer two). Known 0

And explain its design significance.

Reference analysis: design intent:

The distance on the left side of the inequality (1) is (x, y) to (0,0), (0, 1, (1, 0), (1, 1), which can improve students' understanding of the distance formula between two points.

(2) The sum of the distances from (x, y) to these four points can be analyzed by combining the positions of these four points on the plane. The range of xy corresponds to the square range with the side length of the first quadrant of 1. In the process of solving this problem, students' ability of combining numbers and shapes has been improved.

14. Known parabola

(1) Find the tangent equation of parabola at point (2, 1) (5 points)

(2) As shown in the figure, the tangent PT(XO, yo)(xo ≠0) of the parabola at point P intersects with the Y axis at point M, the light source is at the parabola focus F (0, 1), and the incident light FP is PQ after being reflected by the parabola, that is, ∠FPM=∠QPT. Verification. (5 points)

Reference analysis: (1) y = x-1; (2) Thinking: By constructing a diamond, it is concluded that it is parallel to the Y axis.

15. Discuss the role of the history of mathematics in all stages of mathematics teaching (introduction, formation and application).

Reference analysis: The introduction can introduce mathematicians in history. For example, Euclid defined the tangent of a circle as "a straight line that intersects the circle but does not intersect after extension" in the Elements of Geometry.

Formation part: Let students recall the tangent definition of the circle, guide them to improve the tangent definition, and guide them to get a new tangent definition with the help of related propositions in the original geometry.

Application part: from form to number, guide students to get the definition of derivative.

Answer the questions according to the materials given.

16. The following are the teaching clips of Teacher A and Teacher B.

[Teacher A]

Teacher A: What is the symmetry point of the point (x, y) in the plane rectangular coordinate system about the Y axis?

Student 1: (an x, y).

Teacher A: In order to study the symmetry of functions, please fill in the following table and observe the relationship between the corresponding function values when the independent variable X of a given function is the opposite number.

Student 2: Through calculation, it is found that when the independent variables are in opposite directions, the corresponding function values are equal and can be expressed analytically.

Teacher A: Usually, we call the functions with the above characteristics even functions. Please try to give the definition of even function.

[Teacher B]

Teacher B: We have studied the monotonicity of functions and described them accurately in symbolic language. Today, we will learn other properties of functions. Please draw the images of functions f(x)=x2 and g(x)=|x| and observe their * * * characteristics.

(Students found through observation that the images of this function are all symmetrical about the Y axis. )

Teacher B: Can the monotonicity of analog function accurately describe the concept of "the number image is symmetrical about Y axis" in symbolic language?

(Students discover through observation that F (an x)=f(x))

Teacher B: Usually, we call functions with the above characteristics even functions. Please try to give the definition of even function.

Question:

(1) Write the definition of even function and briefly describe the function of parity; ( 1)

(2) Evaluate the teaching of teachers A and B .. (10)

Reference analysis: (1) Definition of even function: Let the domain of function f(x) be D. If Vx∈D has an x∈D and f (an x)=f(x), then function f(x) is called even function. The function of parity is studied: the parity of a function is closely related to the symmetry of its image, odd function is symmetric about the origin, and even functions are symmetric about the Y axis; For the function with parity, we only need to know the properties of one side of the Y axis, so we can deduce the properties of the other side of the Y axis, which can simplify the operation and analysis of the function properties.

(2) In the new teaching process of binary function, Teacher A focuses on guiding students to get the definition of even function through the analysis of calculation results, lacking the process of students' active exploration, and directly giving the research topic of this lesson is symmetry, which is too straightforward; In the teaching process, teacher B guides students to observe images and explore conclusions, which is more in line with the concept that students are the main body of learning in the new curriculum reform, combined with the monotonous introduction of what they have learned before. In the definition, they guide students to try to combine the knowledge they have learned before, so that students can consolidate the old knowledge while learning new knowledge.

Answer the questions according to the materials given.

17. The following is part of the high school textbook "The positional relationship between straight lines and planes in space".

According to the above, complete the following tasks:

(1) Draw a schematic diagram of the positional relationship between a straight line and a plane, and illustrate these three positional relationships in life with examples; (12)

(2) Write this part of the teaching design, including teaching objectives, teaching priorities and teaching process (including activities to guide students to explore and design intentions). (18)

Reference analysis:

(1) Three positional relationships between a straight line and a plane are shown in the following figure:

Examples that can reflect these three positional relationships in life: ① The straight line is in the plane: the straight line where one long side of the blackboard is located is included in the plane where the blackboard is located; ② Intersection of lines and planes: the straight line where the door axis is located intersects the plane where the ground is located; ③ Line-plane parallelism: the straight line of one long side of the blackboard is parallel to the plane of the ground.

(2) The positional relationship between a straight line and a plane in space.

Teaching design. The positional relationship between a straight line and a plane in space.

First, the teaching objectives

1. Knowledge and skill goal: Understand the positional relationship between straight lines and planes in space.

2. Process and Method Objective: Students can draw the relationship between straight line and plane correctly by operating models or observing examples, and cultivate basic drawing ability and spatial concept.

3. Emotion, attitude and values Goal: Feel the connection between mathematics and real life, and strengthen the team consciousness of cooperation and communication.

Second, the difficulties in teaching

1. Teaching emphasis: Understand the positional relationship between straight lines and planes in space.

2. Teaching difficulties: learn to express three positional relationships with graphic language and symbolic language.

Third, the teaching process

1. Review lead-in: review the positional relationship of straight lines in space and guide students to review old knowledge to get (1) intersection points; (2) parallel; (3) Different surfaces. Thus, the positional relationship between the straight line and the plane in the object space is derived.

Teach new knowledge

(1) Explain the situation and give life examples (1) What is the positional relationship between the straight line where a pen is located and the plane where an exercise book is located? (2) What is the positional relationship between the straight line of the front diagonal in the cuboid and the six planes of the cuboid? Organize students to have a group discussion.

(2) cooperative inquiry

After group cooperation and communication, the teacher asked questions and summed up that there are only three kinds of positional relationships between straight lines and planes in space: (1) A straight line is in the plane (there are countless things in common); (2) The straight line intersects the plane (there is a common point); (3) When a straight line is parallel to the plane (there is no common point), when the straight line is parallel to or intersects with the plane, it is collectively called "straight line out of the plane". What the teacher emphasizes here is that the straight line is out of the plane, and there may be a common point or zero common point between the straight line and the plane, but the situation just presented specifically describes the positional relationship between the straight line and the plane.

(3) Emphasis on representativeness

Teachers encourage students to try to give three kinds of graphic and symbolic languages of positional relationship, and encourage students to perform on stage. Finally, the teacher makes a perfect supplement (pictured), and emphasizes its reading and writing methods and their corresponding relationship with written language. When drawing, the teacher reminds the students to draw a straight line in the parallelogram representing the plane when the straight line is in the plane.

Consolidation exercise

(1) PPT shows pictures, and students can quickly judge the position relationship between straight lines and planes in each picture.

(2) Show the textbook example 1 (correct in the following proposition) and explain.

Summarize your homework.

Class (1) summarizes that the positional relationship between a straight line and a plane can be divided by position or by the number of intersections.

(2) The positional relationship between the straight line and the operation plane can be divided according to the position or the number of intersection points.

First, you must do questions 5 and 6 in the textbook;

Second, thinking: If a straight line is parallel to a plane, what is the positional relationship between the plane where the straight line is located and the plane? If a straight line intersects a plane, what is the positional relationship between the plane where the straight line is located and the plane?

Fourth, blackboard design.

The positional relationship between straight line and plane in space