1, function

Function is the focus, set, function definition range, value range, image, parity, monotonicity and week of college entrance examination propositions over the years.

The common problems in college entrance examination include duration, maximum value, inverse function, images and properties of specific functions. So be sure to pay attention to the following points.

(1) set is one of the most basic concepts in modern mathematics, and the view of set permeates all aspects of middle school mathematics content. Therefore, it is necessary to understand the concept of set, master the properties of set elements, and skillfully perform intersection, union and complement operations of sets. At the same time, we should accurately understand the mathematical language and symbols in the form of set.

(2) Function is one of the most important contents in middle school, which is mainly studied from three aspects: definition, image and nature. Every knowledge point in the review should be fully mastered and thoroughly understood. In order to improve the quality of the review, we ask the following questions:

① Master the common methods of image transformation (refer to the section on image transformation in the first semester textbook of Nanjing Normal University). Special note: all transformations are carried out on independent variables.

② Finding the maximum value of a function is an important problem. To master the method of finding the maximum value of a function, we should pay special attention to the maximum value of a quadratic function in a certain interval. Some problems may hide the range, so the range problem is the key to the maximum value of a quadratic function. In addition, we should also pay attention to the maximum value of a quadratic fractional function, and its basic solution is the ""method. Of course, some of them can be transformed into the form of functions, and then linked with basic inequalities, or solved by using the monotonicity of functions.

(3) Learn to solve simple function equations, and take seriously the solution of problems with parameters in exponents or logarithms, paying special attention to the fact that the real number of logarithms must be ">"; 0 ",pay attention to equivalence when solving equations.

Step 2: Triangle

Trigonometry consists of two parts: trigonometric function and trigonometric function of sum and difference of two angles. Trigonometric function mainly investigates the properties of trigonometric function, image transformation, resolution function, minimum positive period and so on. There are many formulas in the trigonometric function of the sum and difference of two angles. We should understand and be familiar with these formulas on the basis of mastering their internal relations and derivation process. Please pay special attention to the following questions:

The formulas of (1) sum, difference, multiple and half angle are all trigonometric functions that express complex angles (sum, difference, multiple and half angle) with single-angle trigonometric functions, which determines the universality of these formulas, that is, these formulas can unify trigonometric functions into single-angle trigonometric functions.

(2) Understand the range of angles in the formula. Any angle that makes a trigonometric function or formula meaningless in the formula is not suitable for the formula. For example:

() There are some similar ones, please pay attention to yourself.

(3) The choice of the symbol before the irrational formula in the half-angle formula is determined by the range of the angle in the trigonometric function at the left end of the formula. In the rational expression of the half-angle tangent formula, it is not necessary to choose coincidence, but it is consistent with.

(4) Mastering the positive, negative, deformation and application of formulas under certain conditions can improve the starting point of thinking and shorten the thinking line, thus making the operation smooth and natural. For example:

= ; ;

; .

(5) Simplification and evaluation of trigonometric function is one of the important contents in middle school mathematics, which is combined with solving triangles, and some of them are related to trigonometric operation of complex numbers. Therefore, we should pay attention to the commonly used methods and skills: string cutting, power raising and lowering, mutualization and product, mutualization "1", auxiliary element method and so on.

3. Inequality

The college entrance examination questions about inequality are widely distributed. In the objective questions, we mainly investigate the nature of inequality, the solution of simple inequality and the preliminary application of mean inequality. They often appear in the form of comparison size, solution set of inequality, definition range of function, value range and maximum value. In the intermediate problem, solving inequality is related to classification discussion. Especially in recent years, the ability of logical reasoning has been emphasized, and an algebraic reasoning problem has been added, which is also related to the proof of inequality. In the finale, whether it is a function problem or an analytic geometry problem, we often need to use the related knowledge of inequality. In the review, we should pay attention to the following issues:

(1) Common methods to master comparison size: difference, quotient, square difference and mirror method.

(2) To master how to find the maximum value with mean inequality skillfully, we must pay attention to three conditions: one is positive; Second, setting; Three phases are equal. All three are indispensable.

(3) master the matters needing attention in solving inequalities with parameters.

When solving inequalities with parameters, we should first pay attention to whether it is necessary to discuss them in categories. If you encounter the following situations, you generally need to discuss them:

① When multiplying and dividing a formula with parameters at both ends of an inequality, it is necessary to discuss the positive, negative and zero properties of this formula.

(2) In the process of solving, when monotonicity of exponential function and logarithmic function is needed, their cardinality should be rounded.

Offline discussion.

③ When the boundary value of the solution set contains parameters, the order of zero values should be discussed.

Step 4: Order

This chapter is one of the main contents of the college entrance examination proposition, which should be reviewed comprehensively and deeply, and on this basis, the following problems should be highlighted:

The proof of (1) arithmetic and geometric series must be proved by definition. It is worth noting that if the sum of the first few terms of a series is given, its general terms can be written if they are satisfied.

(2) The calculation of series is the central content of this chapter. It is the key content of the college entrance examination proposition to skillfully use the general formula of arithmetic progression and geometric progression, the sum formula of the preceding paragraph and their properties to calculate.

(3) When solving the problems about series, we often use various mathematical ideas. Being good at using various mathematical ideas to solve a series of problems is our goal to review.

① Function Thought: The summation formula of the general term formula of arithmetic geometric progression can be regarded as a function, so some problems of arithmetic geometric progression can be solved as function problems.

(2) Classification discussion ideas:

The summation formula of equal proportion sequence should be divided and summed;

When the time is known, it should also be classified;

When calculating, it is necessary to divide time, time and time;

When calculating the sum of general series, we should also consider the parity of letters or items.

④ Holistic thinking: When solving the sequence problem, we should pay attention to getting rid of the rigid thinking mode solved by formulas and use integers.

Body and mind solutions.

(4) When solving the related application problems of series, we should carefully analyze and abstract the actual problems into mathematical problems, and then use the knowledge and methods of series to solve them. Solving this kind of application problem is a comprehensive application of mathematical ability, and it is by no means a simple imitation and application. Pay special attention to the items of geometric series related to years.

5. Complex number

The complex questions in the college entrance examination questions are scattered, some are about the concept of complex numbers, some are about the operation of complex numbers, and some are about the geometric meaning of complex numbers. Moreover, each question has a certain comprehensiveness, and even simple objective questions contain 3-4 knowledge points. Since 1994, complex problems are mainly distributed in objective problems and intermediate solutions. Therefore, we should comprehensively review the basic knowledge and basic problem-solving methods in a down-to-earth manner.

(1) Understand the related concepts of complex numbers accurately, and don't be specious, or you will make mistakes in solving problems. For example, the algorithm of power that is applicable in the range of real numbers is not applicable in the set of complex numbers, the concept of pure imaginary number and so on.

(2) Master the methods of finding the principal value of the complex number and the maximum value of the angle. The common methods of finding the maximum value of the complex number are: changing the complex number into a triangular form and finding the maximum value of the trigonometric function (trigonometric method); Using the algebraic form of complex numbers, the maximum problem of algebraic functions is solved (algebraic method); Using the geometric meaning of complex number, it is transformed into a geometric problem on the complex plane (mirror image method); The main methods to use or find the maximum or principal radial angle of complex numbers are geometric method and trigonometry method.

(3) Mastering the solution methods of quadratic equation and binomial equation in complex set: All quadratic equations can be solved by root formula, and so can Vieta's theorem. Only the quadratic equation with real coefficient can be used to judge the root of the equation, and the quadratic equation with complex coefficient can only be solved by the condition that complex numbers are equal.

(4) Complex knowledge is closely related to many contents in middle school mathematics, which provides a basis for the two-way transformation of complex numbers and real numbers, complex numbers and trigonometric functions, and complex numbers and geometry. Therefore, reviewing complex numbers is an excellent opportunity to train us to change our minds.

6, solid geometry

(1) The chapter "Lines and Planes" is the basis of solid geometry. When reviewing, we should comb the knowledge system repeatedly, master the essential attributes of each concept, and understand the preconditions and conclusions of each judgment theorem and property theorem.

(2) When studying the positional relationship among lines, lines and planes, the parallel and vertical relationships are mainly studied, and the research method is transformation.

(3) The three vertical theorems and their inverse theorems are widely used in solid geometry. As long as there is a straight line perpendicular to the plane, the three perpendicular theorems and their inverse theorems are often used. This theorem is tested every year in the college entrance examination. Three vertical theorems and their inverse theorems are mainly used to prove the vertical relationship and measurement of spatial graphics. For example, to prove that straight lines in different planes are vertical, determine the plane angle of dihedral angle and determine the perpendicular line from point to line.

(4) When solving problems related to solid geometry, we should pay attention to the idea of transformation:

① By constructing rectangle, right triangle and right trapezoid, the problems about prism, pyramid and frustum are transformed into plane figures to solve.

(2) The related problems of the rotating body are transformed into plane figures by using the shaft section to solve them.

③ Expanding space graphics is a common method to transform solid geometry problems into plane graphics problems.

(4) Because the platform is a geometric body cut by a plane parallel to the bottom of the cone, some problems of the platform are often solved by transforming it into a cone cut by the platform.

⑤ Irregular graphics are transformed into regular graphics and complex graphics are transformed into simple graphics by cutting and filling method.

⑥ Using the self-equivalence of the volume of the triangular pyramid, the problem of finding the distance from a point to a plane is transformed into the problem of finding the height of the triangular pyramid.

(5) Solving solid geometry problems generally includes three steps: doing, proving and seeking, which are indispensable. When using the theorem in proof, the conditions of the theorem must be written completely, especially the obvious "straight line in the plane" and "intersection of two straight lines" must be explained clearly.

6. Plane analytic geometry

The college entrance examination questions about linear equations can be divided into two parts. Some of them are independent questions, mostly in objective questions, and there is only one question every year. The difficulty belongs to the basic problem. In addition to symmetry, there are also necessary and sufficient conditions for solving the equation of straight lines and the parallelism or perpendicularity of two straight lines. The other part is the comprehensive problem of analytic geometry. For example, the relationship between conic curve and straight line is often involved. In this case, the inclination angle of straight line is generally used.

(1) Pay attention to prevent solution loss caused by "zero intercept" and "no slope"

(2) Learn to use the formula of the distance between two points, and when the slope of a straight line is known, deform the formula into or; When the inclination of a straight line is known, or can also be obtained.

(3) Flexible use of the fractional formula can simplify the operation.

(4) The linear equation will be solved under any conditions.

(5) Pay attention to the combination of numbers and shapes to study the properties of plane graphics.

The distribution characteristics of analytic geometry in college entrance examination questions are that there are four objective questions, that is, there is a finale question in solving problems. That is to say, there is no intermediate problem in analytic geometry, and the finale of analytic geometry examines the trajectory problem, the positional relationship between straight lines and conic curves, and the maximum value of conic curves. The most important thing is the positional relationship between a straight line and a conic curve. In the process of review, we should pay attention to the following issues:

(1) When solving problems about conic curves, we should first consider the position of the focus of conic curves and pay attention to the opening direction of parabola, which is a key to reduce or avoid mistakes.

(2) When investigating the positional relationship between a straight line and a quadratic curve or the positional relationship between two quadratic curves, the quadratic equation can be obtained by eliminating the equations, and the discriminant can be used to judge. But when the straight line is parallel to the parabolic symmetry axis and the hyperbolic asymptote is parallel, the discriminant can't be used. In order to avoid complicated operations and accurately judge special situations, we can use the idea of combining numbers with shapes to draw curves expressed by equations and solve them by graphics.

(3) The undetermined coefficient method is usually used to solve the conic equation. If we can find the definition of conic curve according to the conditions, it is very simple to solve the conic curve equation with the definition. The definition of conic can also be used to simplify the operation or proof process when dealing with the problems related to the focus and directrix of conic.

(4) When solving the proposition related to the focus triangle (the triangle formed by any point of an ellipse or hyperbola and two focuses is called the focus triangle), it is generally necessary to use the sine and cosine theorem, the sum fraction theorem and the definition of conic curve.

(5) Be familiar with the discriminant of the roots of quadratic equations in one variable and the application of Vieta's theorem in finding chord length, midpoint chord, fixed-point chord and chord at right angles to a fixed point.

(6) Finding the trajectory equation of the moving point is one of the key contents of analytic geometry, which is a comprehensive application of various knowledge and has great flexibility. The essence of finding the trajectory equation of the moving point is to change "curve" into "equation" and "shape" into "number", and understand the properties of the curve through the study of the equation. The common methods for solving the moving point trajectory equation are: direct method, definition method and geometric method.

(7) For the contents of parametric equations and polar coordinates, please master the formulas skillfully, and then transform them into ordinary equations with the idea of simplification.