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. A. 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 are in 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 we must make an accurate judgment according to the concepts of sufficiency and necessity when solving problems.

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 is true (summed up as one true and one false). To find the range of parameters, we can also know the OR, AND, NOT, Union, Intersection and Complement of a set, and solve them through the operation of the set.

Monotonous interval understanding of functions is not allowed to lead to errors.

When studying function problems, we should always think of "the image of function" and learn to analyze problems and find solutions from the image of function. For several different monotone increase (decrease) intervals of a function, as long as we indicate that these intervals are monotone increase (decrease) intervals of the function, we should never use union.

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 an odd or even 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, and f (a) f (b) < 0, then the function y = f (x) has zero in the interval (a, b), but when f (a) f (b) > 0, the function cannot be denied.

The geometric meaning of derivative is unknown, which leads to error.

The derivative value of a function at a point is the slope of the tangent of the function image at that point. But in many problems, it is often necessary to solve the problem that a point outside the function image leads to the tangent of the function image. The basic idea to solve this kind of problem is to set the tangent coordinates, write the tangent equation according to the geometric meaning of the derivative, and then solve the equation (group) according to other conditions given in the title. Therefore, it is necessary to distinguish between "tangency at a certain point" and "crossing a certain point" when solving problems.

The relationship between derivative and extreme value is unclear, which leads to errors.

F ′ (x0) = 0 is only a necessary condition for the derivative function f(x) to obtain the extreme value at x0, that is, this condition is necessary, but it is not enough. It is also necessary to consider whether the signs of f'(x) are different on both sides of x0. In addition, it is necessary to check when the extreme point is known to obtain parameters.

Error caused by monotonicity judgment of trigonometric function

For the monotonicity of the function y = asin (ω x+φ), when ω > 0, because the inner function u = ω x+φ is monotonically increasing, the monotonicity of the function is the same as that of y = sin x, so it can be completely solved according to the monotonic interval of the function y = sin x; But when ω< 0, the internal function U =ωx+φ is monotonically decreasing. At this time, the monotonicity of the function is opposite to the monotonicity of the function Y = sin x, so it can no longer be solved according to the monotonicity of the function y = sin x. Generally, the coefficient of the inner function is turned into a positive number according to the parity of the trigonometric function, and then the trigonometric function with absolute value is intuitively judged according to the image.

The direction of image transformation is not allowed to cause errors.

The image of the function y = asin (ω x+φ) (where a > 0, ω > 0, x∈R) can be regarded as obtained by (1) moving all points on the sine curve to the left (when φ > 0) or to the right (when φ < 0) in parallel by | φ | unit lengths. (2) shortening (when Ω >1) or lengthening (when Ω <1) the abscissa of each point is1Ω times (the ordinate is unchanged); (3) then lengthen (when a > 1) or shorten (when 0 < a < 1) the ordinate of each point to the original a times (the abscissa remains unchanged), that is, do phase transformation first, then do periodic transformation, and finally do amplitude transformation. If you do periodic transformation first and then phase transformation, it should be left (right) translation.

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, and it will make mistakes if you don't consider it. Candidates should pay enough attention to it.

Error caused by unclear range of vector included angle.

When solving problems, we should consider them comprehensively. There are often some factors that are easily overlooked by candidates in mathematics test questions. Whether these factors can be taken into account when solving problems is the key to solving problems successfully. For example, when A B < 0, the included angle between A and B is not necessarily obtuse, so it should be noted that θ = π.

Unclear relationship between an and Sn will lead to errors.

In the sequence problem, the first n terms of the general term an of the sequence have the following relations with Sn: an = S 1, n = 1, Sn-Sn- 1, and n≥2. This relationship is valid for any sequence, but it should be noted that this relationship is divided into N = 1.

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.

The general formula, the first n terms and the formula in the sequence problem 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 term an and the first n terms of the sequence and Sn is the focus of the college entrance examination. When solving problems, we should pay attention to discussing n = 1 and n≥2 respectively, 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 a arithmetic progression and a geometric progression counterpart, and the first n items are summed. The basic method is to set this sum as Sn, and multiply the common ratio of geometric progression at both ends of this sum to get another sum. If one bit is subtracted from these two sums, the problem is transformed into the sum of the first n terms or the first n- 1 terms of the geometric series. The problem is most likely to appear here.

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, when two inequalities are multiplied, and when both ends of an inequality are n-power, we must pay attention to the conditions that enable it to do so. If we ignore the preconditions for the establishment of inequality properties, we will make mistakes.

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 the function whose shape is Y = ax+bx (A, B > 0), the basic inequality is applied to find the maximum value of the function.

Improper classification of inequality with parameter solution

When solving the inequality in the form of Ax2+BX+C > 0, we should first discuss the coefficient of x2. When a = 0, this inequality is linear, and B and C should be further classified and discussed when solving. When a≠0 and δ > 0, the inequality can be transformed into a (X-x 1) (X-X2) > 0, where x 1 and X2 (X 1 < X2) are two roots of the equation AX2+BX+C = 0. If A > 0,

Errors caused by the establishment of inequality constants

The conventional method to solve the problem of inequality constancy is the monotonicity of corresponding function, among which the main methods are the combination of numbers and shapes, the separation of variables and the principal component method. The conclusion comes from the maximum value. Pay attention to the difference between constancy and existence, such as for any x∈[a, b], f(x)≤g(x), that is, f (x)-g.

Ignore the real and dotted lines in the three views to make mistakes.

According to the principle of orthographic projection, the three views are drawn in strict accordance with the rules of "long alignment, high horizontal alignment and equal width". If the surfaces of two adjacent objects intersect, the intersection line of the surfaces is their original dividing line. The dividing line and the visible contour line are drawn with solid lines, and the invisible contour line is drawn with dotted lines, which is easy to be ignored.

The conversion of area and volume calculation is not flexible, which will lead to errors.

The calculation of area and volume not only requires students to have solid basic knowledge, but also needs to use some important thinking methods. It is an important question in the college entrance examination. So we should master the following common ways of thinking. (1) Back to the pyramid idea: This is a common thinking method when dealing with pyramid bodies. (2) Digging and filling method: it is commonly used in calculating irregular figure area or geometric volume. (3) Equal product transformation method: make full use of the characteristics of any side of the triangular pyramid as the bottom surface, and flexibly solve the volume of the triangular pyramid. (4)

Error caused by conclusion in arbitrary generalization of plane geometry

Some concepts and properties in plane geometry may not be valid when extended to space. For example, properties such as "only one straight line can be perpendicular to a known straight line when intersecting with points outside the straight line" and "two straight lines perpendicular to the same straight line are parallel" are not valid in space.

Unclear understanding of folding and unfolding leads to errors.

Folding and unfolding are common thinking methods in solid geometry. In this kind of problems, we should pay attention to the variables and invariants in plane and space graphics when folding or unfolding, not only to what has changed and what has not changed, but also to the changes in positional relations.

The positional relationship between points, lines and surfaces is not clear, which leads to errors.

The combination judgment question about the spatial position relationship between points, lines and planes is an ideal question to comprehensively examine the candidates' judgment on the spatial position relationship and their grasp of nature in the college entrance examination, and it has always been favored by proposers. There are two basic ideas to solve such problems: one is to find counterexamples one by one to make negative judgments or to make positive judgments one by one through logical proof; The second is to make a judgment based on the cuboid model or the actual spatial position (such as desks and classrooms), but we should pay attention to the accurate application of the theorem and the comprehensive and meticulous consideration of the problem.

Ignoring the slope will not lead to errors.

When solving the related problems of two parallel lines, if we use l 1∑L2? When solving K 1 = k2, we should pay attention to the premise that two straight lines do not coincide and the slope exists. If we ignore the non-existence of K 1 and K2, it will lead to the wrong solution. This kind of problem can also be solved by the following conclusions. That is, the straight line l1:a1x+b1y+c1= 0 is parallel to L2: A2X+B2Y+C2 = 0, provided that a1B2-a2b1= After finding the specific value, we will enter the test and look at two lines. When k 1 k2 =- 1, it should be noted that k 1 and k2 must exist at the same time. Use straight lines l1:a1x+b1y+c1= 0 and L2: A2X+B2Y+.

Ignoring the zero intercept will lead to an error.

When solving the intercept problem of a straight line, we should pay attention to two points: first, we can't ignore the special situation that the intercept is zero; Secondly, it needs to be clear that a straight line with zero intercept cannot be written in intercept form. Therefore, when solving this kind of problems, we should discuss them in categories and don't miss the situation that the intercept is zero.

Ignoring conditions in the definition of conic section will lead to errors.

When using the definitions of ellipse and hyperbola to solve problems, we should pay attention to the definition forms of the two curves and their restrictive conditions. For example, in the definition of hyperbola, two points are indispensable: one is absolute value; Secondly, 2a < | f 1f2 |. If the first condition is not satisfied, the difference between the distances of a moving point and two fixed points is constant, but the absolute value of the difference is constant, then its trajectory can only be a hyperbola.

Misjudge that positional relationship between straight line and conic curve

The two counting principles are unclear, resulting in errors.

No mistakes can be made in arrangement or combination.

Confusing binomial coefficient and binomial coefficient will lead to errors.

Error is not allowed in loop end judgment.

Conditional structure is not allowed to make mistakes in conditional judgment.

Unclear concept of complex number leads to errors.