Mathematics discipline
I. Guiding ideology of the proposition
According to the requirements of ordinary colleges and universities for freshmen's cultural quality, the proposition of 2001KLOC-0/Mathematics Subject of National Unified Entrance Examination for Ordinary Colleges and Universities (Jiangsu Volume) will be based on the Mathematics Curriculum Standard for Ordinary High Schools (Experiment) issued by the Chinese People's Republic of China and the Ministry of Education, with reference to the Outline of National Unified Entrance Examination for Ordinary Colleges and Universities (Course Experimental Edition), and combined with the teaching requirements of ordinary high schools in Jiangsu, two middle schools will
Highlight the basic knowledge, skills and methods of mathematics.
The examination of basic knowledge and skills of mathematics is close to the teaching practice, paying attention to comprehensiveness, highlighting key points, paying attention to the examination of internal relations of knowledge and the examination of mathematical thinking methods contained in middle school mathematics.
2. Pay attention to the examination of basic mathematics ability and comprehensive ability.
The basic abilities of mathematics mainly include imagination, abstract generalization, reasoning, operation and data processing.
(1) The examination requirements of spatial imagination ability are: being able to imagine and make correct plane intuitive graphics according to the conditions set by the topic, and being able to imagine spatial graphics according to the plane intuitive graphics; It can correctly analyze the basic elements and their relationships in graphics, and can decompose and combine spatial graphics.
(2) The requirements for the ability of abstract generalization are: being able to discover the essence of the research object through the inquiry of examples; Can summarize some conclusions from the given information materials and use them to solve problems or make new judgments.
(3) The ability of reasoning and argumentation requires that the truth value of a mathematical proposition can be reasoned and demonstrated by induction, analogy and deduction according to the known facts and the correct mathematical proposition that has been obtained.
(4) The requirements for checking the operation and solving ability are: being able to operate and deform according to laws and formulas; Be able to find and design a reasonable and simple operation mode according to the conditions of the problem; Ability to estimate or approximate data as required.
(5) The requirement of the data processing ability test is to be able to use basic statistical methods to sort out and analyze the data in order to solve the given practical problems.
The examination of mathematics comprehensive ability is mainly reflected in the examination of problem analysis and problem solving ability, which requires the comprehensive use of relevant knowledge and methods to solve difficult or comprehensive problems.
3. Pay attention to the examination of mathematics application consciousness and innovation consciousness.
The examination of mathematical application consciousness requires that we can use the mathematical knowledge, ideas and methods we have learned to build a mathematical model and turn some simple practical problems into mathematical problems and solve them.
The test requirement of innovation consciousness is to be able to solve problems creatively by comprehensively and flexibly using the learned mathematical knowledge and thinking methods.
Second, the examination content and requirements
Mathematics test questions consist of compulsory questions and additional questions. Candidates who choose to take the history test only need to answer the required questions in the test questions; Candidates who choose to take the physics test should answer the required questions and additional questions in the test questions. The required questions are the compulsory content of senior high school and the content of elective series L; The contents of the additional questions are the contents of elective series 2 (excluding elective series 1) and the contents of four topics of elective series 4- 1 "Selected Lectures on Geometry Proof", 4-2 "Matrix and Transformation", 4-4 "Coordinate System and Parameter Equation" and 4-5 "Selected Lectures on Inequality" (candidates only need to take two of them).
The examination requirements of knowledge are divided into three levels: understanding, understanding and mastering (the following table is represented by A, B and C respectively).
Understanding: It requires a basic understanding of the meaning of the listed knowledge and the ability to solve related simple problems.
Understanding: It requires a deep understanding of the listed knowledge and can solve some comprehensive problems.
Proficiency: It requires the system to master the internal relations of knowledge, and can solve comprehensive or difficult problems.
The specific examination requirements are as follows:
1 required parts
Demand for internal content
B.C.
1. set set and its representation √.
Subset √
Pay, merge and make up √.
2. The concept of function and the concept of basic elementary function I function √.
Basic properties of functions √.
Exponential sum logarithm √.
Images and properties of exponential function √.
Images and properties of logarithmic functions √.
Power function √
Functions and equations √.
Function model and its application √.
3 Basic elementary function Ⅱ
Trigonometric identity transformation
Some concepts of trigonometric functions √.
The basic relationship of trigonometric functions with the same angle √ 0
Inductive formula of sine and cosine √.
Images and properties of sine function, cosine function and tangent function √.
Images and properties of function y=Asin(ωx+φ) √.
Sine, cosine and tangent of the sum (difference) of two angles √.
Sine, cosine and tangent of double angle √.
Product sum and difference, product of sum and difference, half-angle formula √.
4. Solve triangle sine theorem and cosine theorem and their applications √.
5. The concept of plane vector √.
Addition, subtraction, multiplication and division of plane vectors.
Coordinate representation of plane vector √.
The product of plane vectors √.
Parallelism and verticality of plane vectors √.
Application of plane vector √.
6. The concept of sequence sequence √.
Arithmetic series √.
Geometric series √.
7. Inequality Basic Inequality √.
Unary quadratic inequality √.
Linear programming √
8. The concept of complex number √.
Four operations of complex numbers √.
Geometric meaning of complex numbers.
9. The concept of derivative and its application.
Geometric meaning of derivative √.
The operation of derivative √.
Study on monotonicity and extremum of derivative function
The application of derivative in practical problems.
sequential
Internal capacity requirement
B.C.
10. The significance of the preliminary algorithm √.
Flowchart √
Basic algorithm statement √.
1 1. Four propositional forms in common logical terms √.
Sufficient condition, necessary condition, necessary and sufficient condition.
Simple logical conjunction √.
Full name quantifier and existential quantifier √.
12. Reasoning and
certificate
Rational reasoning and deductive reasoning √.
Analytical method and synthetic method √.
Reduction to absurdity √
13. Probability and statistical sampling methods √.
Population distribution estimation √.
Estimation of population characteristic number √.
Correlation of variables √.
Random events and probability √.
Classical probability √
Geometric probability √
Mutually exclusive events and its occurrence probability √.
14. Space geometry cylinder, cone, platform, ball and their simple combinations √.
Surface area and volume of column, cone, platform and ball √.
15. The plane of position relation between points, lines and planes and its basic properties √.
Determination of parallelism and perpendicularity between a straight line and a plane and its properties.
Determination and properties of parallelism and verticality of two planes.
16. Plane analysis
Slope and inclination of geometric preliminary straight line.
Linear equation √
Parallel relationship and vertical relationship of straight lines √.
The intersection of two straight lines √.
Distance between two points, distance from point to straight line √.
Standard equation and general equation of circle √.
The positional relationship between straight line and circle, circle and circle √.
Spatial rectangular coordinate system √
17. Standard equation and geometric properties of conic curves and ellipses with the center of the equation at the coordinate origin √.
Standard equation and geometric properties of hyperbola whose center is at coordinate origin √.
Standard equation and geometric properties of parabola with vertex at coordinate origin √.
2. Other issues
Content requirements
B.C.
Elective series 2: excluding elective series.
1
1. Conic curves and equations
Curves and equations √.
Standard equation and geometric properties of parabola with vertex at coordinate origin √.
2. Space vector
And solid geometry
The concept of space vector √.
Necessary and Sufficient Conditions of Space Vector * * * Straight Line and * * * Plane
Condition √.
Addition, subtraction, multiplication and division of space vectors.
Coordinate representation of space vector √.
Product of space vector √.
* * * Linear and vertical √ of space vector
Direction vector of straight line and normal vector of plane √.
Application of space vector √.
3. Derivative and its application √ Derivative of simple compound function
Definite integral √
4. Reasoning proof of mathematical induction principle √.
Simple application of mathematical induction √.
5. Counting principle addition principle and multiplication principle √.
Arrangement and combination √
Binomial theorem √
6. Probability and statistics discrete random variables and their distribution list √.
Hypergeometric distribution √
Conditional Probability and Independent Events √
Model and binomial distribution of n independent repeated tests.
Mean and variance of discrete random variables.
Elective series
four
Hanzhong
four
special subject
7. similar triangles's Judgment and Property Theorem √.
Projective theorem √
Determination and property theorem of tangent of circle √.
Circumferential angle theorem, chord tangent angle theorem √
Secant theorem, secant theorem, secant theorem √.
Determination and property theorem of quadrilateral inscribed in a circle.
8. Concepts of matrix and transformation matrix √.
Second order matrix and plane vector √.
Coplanar transformation √.
Matrix composition and matrix multiplication.
Second order inverse matrix √.
Eigenvalues and eigenvectors of second-order matrices √.
Simple application of second-order matrix √.
9. Related concepts of coordinate system and parametric equation coordinate system √.
Polar coordinate equation of simple graph √.
Conversion between polar coordinate equation and rectangular coordinate equation.
Parameter equation √.
Parameter equation of straight line, circle and ellipse √.
Mutual transformation between parametric equation and ordinary equation.
Simple application of parametric equation √.
10. Selected lectures on inequalities √ Basic properties of inequalities
Solution of inequality with absolute value √.
Proof of inequality (comparison method, synthesis method, analysis method) √.
Arithmetic-geometric mean inequality, Cauchy inequality.
Find the maximum (minimum) value with inequality √.
Prove the inequality by mathematical induction.
Third, the examination form and examination paper structure
(1) examination form
Closed book and written test. The test questions are divided into two parts: required questions and additional questions. The full score of required questions is 160, and the examination time is 120 minutes. The full score of additional questions is 40 points, and the examination time is 30 minutes.
Examination questions
1. Required questions are composed of fill-in-the-blank questions and solution questions. Among them, the fill-in-the-blank question is 14, accounting for about 70 points; Answer 6 questions, accounting for about 90 points.
2. Additional questions Additional questions are composed of solution questions and **6 questions. Among them, question 2 is required to be a small topic, and the contents in elective series 2 (excluding elective series 1) are examined; Choose to do ***4 questions and examine the contents of 4- 1, 4-2, 4-4 and 4-5 in elective series 4 in turn. Candidates choose 2 questions to answer.
Fill-in-the-blank questions only require writing the results directly, not writing the calculation or reasoning process; The solution should be written in words, proof process or calculation steps.
(3) the difficulty ratio of the test questions.
Required questions are composed of easy questions, intermediate questions and difficult questions. The proportion of easy questions, medium questions and difficult questions in the test questions is roughly 4: 4: 2.
Additional questions consist of simple questions, medium questions and difficult questions. The proportion of easy questions, medium questions and difficult questions in the test questions is about 5: 4: 1.
Iv. Examples of typical problems
A. Required questions
1. function y=Asin(ωx+φ)(A, ω, φ are constants, A >；; 0，ω& gt； 0)
If the image on the closed interval is as shown in the figure, then ω =.
The analysis of this topic mainly investigates the image and period of trigonometric function, which is an easy problem.
Answer 3.
2. If the dice with uniform texture are thrown twice (cube toys with points of 1, 2, 3, 4, 5, 6 on each side), the probability that the sum of upward points is 4 is.
The analysis of this topic mainly investigates classical probability, which is an easy problem.
The answer.
3. If it is an imaginary unit), the value of the product is
The analysis of this problem mainly focuses on the basic concept of complex number, which is an easy problem.
Answer -3
4. Set a set, then there are elements in set A. 。
It is easy to analyze this problem by solving basic knowledge such as unary quadratic inequality and set operation.
Answer 6
5. The figure on the right is the flow chart of an algorithm, and the final output is w =.
This topic analysis mainly examines the basic knowledge of algorithm flow chart, which is an easy topic.
Answer 22
6. Assuming that the straight line is tangent to the curve,
Then the real number b=
The analysis of this topic mainly investigates the geometric meaning of derivative and the solution of tangent.
The answer.
7. In the rectangular coordinate system, the vertex of parabola C is the coordinate origin, the focus is on the X axis, and the straight line y=x intersects with parabola C at points A and B. If p (2,2) is the midpoint of line segment AB, the equation of parabola C is.
The analysis of this topic mainly examines the basic knowledge such as midpoint coordinate formula and parabolic equation, and this topic belongs to the middle level.
answer
8. The equation of a circle tangent to a straight line with the point (2,-1) as the center is.
The analysis of this topic mainly examines the basic knowledge such as the equation of the circle and the positional relationship between the straight line and the circle.
answer
9. If the sum of the preceding paragraphs of the known sequence {0} is satisfied, then.
The analysis of this topic mainly examines the relationship between the first n terms of a sequence and its general terms, as well as basic knowledge such as simple inequalities. This problem is a medium one.
Reference answer
10. If the known vector is perpendicular to, the value of the real number is _ _ _ _ _.
The analysis of this topic mainly examines the basic knowledge of addition, subtraction, multiplication, division and product of plane vectors expressed by coordinates, and this topic is medium.
answer
1 1. Let it go.
The analysis of this topic mainly examines the basic knowledge such as algebraic deformation and basic inequality, and the topic is medium.
Answer 3
12. The maximum area of a triangle that meets the conditions is _ _ _ _ _ _ _ _.
Analysis of this topic mainly examines the ability to solve problems flexibly by using relevant basic knowledge. This topic is difficult.
answer
Second, answer the question.
13. in ABC, C-A =, SINB =.
(1) Find the value of Sina;
(2) Let AC= and find the area of ABC.
The analysis of this topic mainly examines the basic knowledge such as trigonometric identity transformation and sine theorem, and examines the ability of operation and problem solving.
The reference answer (1) is given by and,
∴,∴,
∴, once again, ∴
(2) As shown in the figure, it is obtained by sine theorem.
Here we go again.
∴
14. As shown in the figure, in the straight triangular prism ABC? In A 1B 1C 1, e and f are the midpoint of A 1B and A 1C, respectively, and point d is at B 1C 1 and a1.
Verification: (1)EF‖ plane ABC；;
(2) plane A 1FD plane bb1c1c.
The analysis of this topic mainly examines the basic knowledge of parallel lines, planes and vertical planes, and examines the ability of spatial imagination and reasoning.
Reference answer
(1) because e and f are the midpoint of A 1B and A 1C, EF‖BC, and EF plane ABC, BC plane ABC,
∴EF‖ aircraft abc；;
(2) ABC in a straight triangular prism? In A 1B 1C 1,
A1d plane A 1B 1C 1
Also, BB 1B 1C=B 1, ∴.
Again, so plane A 1FD plane bb1c1c.
15. It is known that the center of the ellipse is the origin of the rectangular coordinate system, the focus is on the axis, and one of its vertices points to two.
The focal distances are 7 and 1 respectively.
(1) Equation for Finding Elliptic Circle
(2) If the moving point is an ellipse, it is a point on a straight line passing through and perpendicular to the axis.
(e is the eccentricity of ellipse c), find the trajectory equation of this point and explain what curve the trajectory is.
This topic mainly examines some basic contents and methods in analytic geometry, and examines the ability of operation and problem solving.
The reference answer (1) lets the semi-axes of the ellipse be a and c respectively, which are given by the known w.w.w.k.s.5.u.c.o.m
{the solution is a=4, c=3,
So the equation of ellipse c is
(2) Let M(x, y) and P(x,), where it is known that
Therefore (1)
Get w.w.w.k.s.5.u.c.o.m from point p on ellipse C.
Substitute into formula ① and simplify it.
So the trajectory equation of point M is that the trajectory is two line segments parallel to the X axis. 5.u.c.o.m
16. Set the function, and the tangent equation of the curve at this point is.
Analytical formula of (1);
(2) Prove that the area of the triangle surrounded by tangent, straight line and straight line at any point on the curve is a constant, and find this constant.
The analysis of this topic mainly examines the geometric meaning of derivatives, the operation of derivatives and the basic knowledge of linear equations, and examines the ability of operation and solution, reasoning and demonstration. This topic is medium.
The equation of reference answer (I) can be simplified as.
At that time,
Here we go again.
So the solution is
Therefore.
(II) Let it be any point on the curve, and the tangent equation of the curve at that point is
,
Namely.
So the coordinates of the intersection of tangent and straight line are.
So the coordinates of the intersection of tangent and straight line are.
So the area of the triangle surrounded by tangent and straight line at this point is
.
Therefore, the tangent of any point on the curve and the triangle area surrounded by straight lines are fixed.
Value, fixed value is 6.
17.( 1) Let n () be a arithmetic progression with non-zero tolerance. If one item is deleted from this series, this series (in the original order) is geometric progression:
(1) value at that time; (2) All possible values;
(2) It is proved that for a given positive integer, there exists a arithmetic progression whose term and tolerance are not zero, and any three of them (in the original order) cannot form a geometric series.
Based on geometric progression and arithmetic progression, this paper mainly examines students' exploration and reasoning ability.
The reference answer first proves a "basic fact":
In a arithmetic progression, if three consecutive terms become geometric series, the tolerance d0=0 of this series is 0.
In fact, let three consecutive terms a- d0, A, a+ d0 in this series become geometric series, then
This makes d0=0.
(1) (i) When n=4, it is only possible to delete or,
If you Delete it, it will be geometric series, de and Cause, so you can get it from the above formula, that is. At this time, the sequence is -4d, -3d, -2d, -d, which satisfies the problem.
If you delete it, it will become a geometric series.
Because, from the above formula, that is, at this time, the sequence is d, 2d, 3d, 4d, satisfying the problem.
To sum up, what you get is still.
(2) When n≥6, the series obtained by deleting one item from the series satisfying the topic must have three consecutive items in the original series, so that these three items become arithmetic progression and geometric progression. Therefore, from the "basic facts", we can know that the tolerance of the sequence must be 0, which contradicts the topic. Therefore, the number of items in the series that satisfy the problem is satisfied. Because of the topic, n=4 or 5.
When n=4, we can see from the discussion in (i) that there is a sequence that satisfies the problem.
When n=5, if there is a series that satisfies the problem, we can know from the "basic fact" that the deleted item can only be a geometric series, so
And ...
Simplify the above two equations respectively, and get, so d=0, which is contradictory. Therefore, there is no arithmetic progression with 5 items satisfying the problem.
To sum up, n can only be 4.
(2) Suppose that for a positive integer n, there is an n-term arithmetic progression with a tolerance of d, three of which are geometric series. Here, there is.
Simplify (*)
By knowing, and or both are 0, or neither is 0.
If, moreover, there is,
In other words, get, which contradicts the topic.
Therefore, and at the same time is not 0, so use (*).
Because they are all non-negative integers, the right side of the above formula is rational, so it is rational.
Therefore, for any positive integer, as long as it is irrational, the corresponding sequence is the sequence that meets the requirements of the question.
For example, take, then, the series of n items 1,,,, meets the requirements.
B. Other issues
1. 200 pieces of a product from a factory were randomly selected. After quality inspection, there are 26 first-class products/kloc-0, 50 second-class products, 20 third-class products and 4 defective products. It is known that the profit of producing 1 piece of first, second and third-class products is 60,000 yuan, 20,000 yuan and 1 10,000 yuan respectively.
(1);
(2) Find the average profit of 1 product (i.e. mathematical expectation);
(3) After technological innovation, there are still four grades of products, but the rate of defective products has decreased to and the rate of first-class products has increased to. If the average profit of 1 product is not less than 47300 yuan, what is the highest rate for third-class products?
analyse
Reference answer
All possible values of (1) are 6,2, 1,-2; ,
Therefore, the distribution list is:
6 2 1 -2
0.63 0.25 0. 1 0.02
(2)
(3) Let the rate of third-class products after technological innovation be, then the average profit of 1 product at this time is
According to the meaning of the problem, that is, the solution
So the rate of third-class products is the highest.
2. As shown in the figure, the known points are in the cube.
On the diagonal, when, remember, it is an obtuse angle, the range of values is found.
2. Solution (1/3, 1)
3. Elective course 4- 1 geometric proof.
As shown in the figure, let the tangent AE of the circumscribed circle of △ABC intersect with the extension line of BC at point E, and the bisector of △ ∠BAC and BC intersect at point D.
analyse
Reference answers prove that, as shown in the figure, because it is the tangent of a circle,
So,,,
Because of this bisector,
therefore
therefore
Because,
So, therefore.
Because it is the tangent of a circle, we can know from the tangent theorem.
,
Therefore,
4. Elective 4-2 Matrix and Transformation
In the plane rectangular coordinate system, the known vertex coordinates are the area of the graph obtained by the matrix, where the matrix.
analyse
Reference answer. 1
5. Elective 4-4 coordinate system and parameter equation.
In the plane rectangular coordinate system, the point is the moving point on the ellipse, and the maximum value is obtained.
analyse
This topic mainly investigates the basic knowledge of parametric equation of curve and the ability to solve mathematical problems by using parametric equation.
The reference answer is that the parametric equation of ellipse is
Therefore, the coordinates of the moving point can be set as, where.
therefore
So, at this time, take the maximum value of 2.
6. Elective Course 4-5: Special Lecture on Inequality
Setup verification:
analyse
Reference answer