Knowledge points of mathematics college entrance examination
Trajectory contains two problems: all points on the trajectory meet the given conditions, which is called the purity of trajectory (also called inevitability); None of the points that are not on the trajectory meet the given conditions, that is, the points that meet the given conditions must be on the trajectory, which is called the completeness (also called sufficiency) of the trajectory.
First, the basic steps of finding the moving point trajectory equation.
1. Establish an appropriate coordinate system and set the coordinates of the dispatching point m;
2. Write a set of points m;
3. List the equation = 0;
4. Simplify the equation to the simplest form;
5. check.
Second, the common methods to solve the trajectory equation of the moving point: There are many methods to solve the trajectory equation, such as literal translation, definition, correlation point method, parameter method, intersection method and so on.
1. Literal translation method: the conditions are directly translated into equations, and the trajectory equation of the moving point is obtained after simplification. This method of solving trajectory equation is usually called literal translation.
2. Definition method: If it can be determined that the trajectory of the moving point meets the definition of the known curve, the equation can be written by using the definition of the curve. This method of solving trajectory equation is called definition method.
3. Correlation point method: use the coordinates x and y of moving point Q to represent the coordinates x0 and y0 of related point P, then substitute them into the curve equation satisfied by the coordinates (x0, y0) of point P, and simply sort them out to get the trajectory equation of moving point Q.. This method of solving trajectory equation is called correlation point method.
4. Parametric method: When it is difficult to find the direct relationship between the coordinates x and y of the moving point, the relationship between x and y and a variable t is often found first, and then the equation is obtained by eliminating the parameter variable t, that is, the trajectory equation of the moving point. This method of solving trajectory equation is called parameter method.
5. Trajectory method: eliminate the parameters in the two dynamic curve equations, and get the equation without parameters, which is the trajectory equation of the intersection of the two dynamic curves. This method of solving trajectory equation is called trajectory method.
The general steps of finding the moving point trajectory equation;
(1) Establish a system-establish a suitable coordinate system;
② set point-set any point on the trajectory P(x, y);
(3) Formula —— List the relationship that the moving point P satisfies;
④ Substitution-according to the characteristics of conditions, the distance formula and slope formula are selected, converted into equations about X and Y, and simplified;
⑤ Proof —— Prove that the equation is a moving point trajectory equation that meets the requirements.
Summary of mathematics knowledge points in college entrance examination
Forgetting an empty assembly leads to an error.
Since an empty set is the proper subset of any non-empty set, B=? Also met B. Answer: When solving the set problem with parameters, we should pay special attention to the situation that a given set may be empty when the parameters take values within a certain range.
Ignoring three attributes of a collection element will cause an error.
The elements in the set are deterministic, disordered and different from each other, and the difference of the three elements of the set has the greatest influence on solving the problem, especially the set with letter parameters, which actually implies some requirements for letter parameters.
Negative proposition of confusing proposition
The negation of proposition and the negation of proposition are two different concepts. The negation of proposition P is the judgment of denying a proposition, and the negation of proposition P is the negation of both conditions and conclusions of a proposition in the form of "if P is, then Q".
The inversion of sufficient and necessary conditions will lead to errors.
For two conditions a and b, if a? B is established, then A is the sufficient condition of B, and B is the necessary condition of A; If b? If A holds, A is a necessary condition for B, and B is a sufficient condition for A; If a? B, then A and B are necessary and sufficient conditions for each other. The most common mistake in solving problems is to reverse sufficiency and necessity, so when solving problems, we must make accurate judgments according to the two concepts of sufficiency and necessity.
The understanding of "or" and "not" is not allowed to go wrong.
Is the proposition p∨q true? Is p true or q true, and the proposition p∨q false? P false, q false (summed up as one true is true); Is the proposition p∧q true? P is true, Q is true, and the proposition p∧q is false? P false or q false (summarized as one false is false); Really? P fake, won P fake? P true (summed up as one true and one false). The problem of finding the range of parameters can also be understood by the correspondence of sets or, and, not and union, intersection and complement, and can be solved by the operation of sets.
Monotonous interval understanding of functions is not allowed to lead to errors.
When studying function problems, we should always think of "function images" and learn to analyze problems from function images and find solutions. For several different monotone increasing (decreasing) intervals of a function, it is forbidden to use union, as long as it is indicated that these intervals are monotone increasing (decreasing) intervals of the function.
Judging the parity of a function ignores the domain, which leads to errors.
To judge the parity of a function, we must first consider the domain of the function. The necessary condition for a function to have parity is that the domain of the function is symmetric about the origin. If this condition is not met, the function must be a parity function.
Improper use of function zero theorem leads to errors.
If the image of the function y=f(x) in the interval [a, b] is a continuous curve with f (a) f (b); 0, there is no denying that the function y=f(x) has zero in (a, b). The zero point of a function includes "sign-changing zero point" and "sign-changing zero point", but the zero point theorem of the "sign-changing zero point" function is "powerless", so we should pay attention to this problem when solving the zero point problem of the function.
Error caused by monotonicity judgment of trigonometric function
For the monotonicity of the function y=Asin(ωx+φ), when ω >; 0, because the internal function u=ωx+φ is monotonically increasing, the monotonicity of this function is the same as that of y=sin x, so it can be solved completely according to the monotonic interval of function y=sin x; But when ω
Ignoring zero vectors leads to errors.
The zero vector is the most special vector in the vector. It is stipulated that the length of zero vector is 0 and the direction is arbitrary, and both zero vector and arbitrary vector are * * * lines. Its position in the vector is just like the position of 0 in the real number, but it is easy to cause some confusion. If you don't consider it, you will make mistakes, so candidates should pay enough attention to it.
Error caused by unclear range of vector included angle.
When solving problems, we should consider them comprehensively. Mathematics test questions often contain some factors that are easily overlooked by candidates. Whether these factors can be taken into account when solving problems is the key to success, such as when A B.
Unclear relationship between an and Sn will lead to errors.
In the problem of sequence, the general term an and its first n terms of sequence have the following relations with s n: an=S 1, n= 1, Sn-Sn- 1, n≥2. This relationship is valid for any sequence, but it should be noted that this relationship is segmented. When n= 1, n≥2, this relationship has completely different forms, which is also the place where mistakes are often made in solving problems. When using this relationship, we should keep in mind its "segmentation" characteristics.
Misunderstanding of the definition and properties of sequences.
The sum of the first n terms of arithmetic progression is a quadratic function of the zero constant term about n when the tolerance is not zero; Generally, it is concluded that "if the sum of the first n terms of the sequence {an} is Sn=an2+bn+c(a, b, c∈R), then the necessary and sufficient condition for the sequence {an} to be arithmetic progression is c = 0"; In arithmetic progression, Sm, S2m-Sm, S3m-S2m(m∈N_) are arithmetic progression.
Error in the maximum value in the sequence.
In the sequence problem, the general formula, the first n terms and the formula are all functions about positive integer n, so we should be good at understanding and understanding the sequence problem from the perspective of function. The relationship between the general item an and the first n items and Sn is the focus of the college entrance examination. When solving problems, we should pay attention to discussing n= 1 and n≥2 separately, and then see if they can be unified. In a quadratic function about a positive integer n, the point at which the maximum value is taken depends on the distance from the positive integer to the symmetry axis of the quadratic function.
Dislocation subtraction and improper item handling will lead to errors.
The applicable conditions of the dislocation subtraction summation method: the sequence consists of the product of the counterpart of a arithmetic progression and a geometric progression, and the sum of the first n items is found. The basic method is to set this sum as Sn, and at the same time multiply this sum by the common ratio of geometric progression at both ends to get another sum. Subtracting one bit from the two sums will turn the problem into the sum of the top n geometric progression or the top n- 1. The most likely problem here is the treatment of the remaining term after dislocation subtraction.
Improper application of inequality properties will lead to errors.
When using the basic properties of inequality for reasoning, we must be accurate, especially when both ends of inequality are multiplied or divided by a number at the same time, when two inequalities are multiplied, and when both ends of an inequality are N-power at the same time, we must pay attention to the conditions that enable it to do so. If we ignore the preconditions for the establishment of inequality properties, errors will occur.
Ignoring the application conditions of basic inequalities will lead to errors.
When using the basic inequality a+b≥2ab and the variant ab≤a+b22 to find the maximum value of a function, we must pay attention to the fact that both A and B are positive numbers (or both A and B are non-negative), and one of ab or a+b should be a constant value, especially the condition that the equal sign holds. For shapes such as y=ax+bx(a, b >); 0) function, when using basic inequality to find the maximum value of the function, we must pay attention to the symbols of ax and bx, and discuss them in categories if necessary. In addition, we should pay attention to the range of the independent variable x, and whether the equal sign can be obtained within this range.
Senior three mathematics knowledge points
There are generally four three-dimensional geometry questions in the college entrance examination (multiple-choice questions and fill-in-the-blank questions, solving problems 1 question). * * * The total score is about 27 points, and the knowledge points examined are within 20. Choose fill-in-the-blank questions to investigate the calculation problems in the classroom, and the analytical questions focus on the logical reasoning problems in the classroom. Of course, both of them should be based on correct spatial imagination. With the further implementation of the new curriculum reform, the test questions of solid geometry are developing in the direction of "thinking more and calculating less". Judging from the changes of examination questions over the years, the demonstration of the positional relationship between line and surface and the exploration of angle and distance based on simple geometry are always hot topics.
Knowledge integration
1. Problems about parallelism and verticality (straight line, straight line and plane) are repeatedly encountered in the process of solving solid geometry problems, and are indispensable contents in various problems (including argumentation, angle calculation, distance, etc.). Therefore, in the general review of subject geometry, we must first solve the problems about parallelism and verticality. By analyzing and summarizing the problems, we can master the law of solving problems in solid geometry-make full use of the idea of mutual transformation between lines (vertical) and lines (vertical), and improve our logical thinking ability and spatial imagination ability.
2. The method of judging the parallelism of two planes:
(1) According to the definition, it is proved that two planes have no common point;
(2) Judgment Theorem-Prove that two intersecting straight lines in one plane are parallel to another plane;
(3) Prove that two planes are perpendicular to a straight line.
3. The main properties of two parallel planes:
(1) According to the definition, "two parallel planes have nothing in common";
(2) Derived from the definition: "Two planes are parallel, and the straight line in one plane must be parallel to the other plane";
(3) The property theorem of two parallel planes: "If two parallel planes intersect the third plane at the same time, their intersection lines are parallel";
(4) The straight line is perpendicular to one of the two parallel planes and also to the other plane;
(5) The parallel lines sandwiched between two parallel planes are equal;
(6) Only one plane passing through a point outside the plane is parallel to the known plane.
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