Shaanxi anzhenping
Fill-in-the-blank problem is one of the three basic questions in mathematics college entrance examination, and its solving methods are divided into: direct operational reasoning, assignment calculation, law discovery, number-shape mutual assistance and so on. When solving a problem, it is necessary to have reasonable analysis and judgment, require that every step of reasoning and operation are correct, and also require that the answer is accurate and complete. Reasonable reasoning, optimized thinking and less thinking will be the basic requirements for solving fill-in-the-blank questions quickly and accurately. The following are the basic requirements.
Functions and inequalities.
Example 1 known function, then
Explain why, why, you should fill in 4.
Please think about why you don't have to ask.
The proper subset number of the set in Example 2 is
Explain that there are 90 elements in the set M, and the number of proper subset is, to be filled in.
To answer this question quickly, you need to remember the small conclusion; For a finite set with n elements, the number of proper subset is
Example 3 If the image of a function is symmetrical about a straight line, then
Explain that the symmetry axis of parabola is known as, and, and, so fill in 6.
Example 4 If the function, then
The explanation is easy to find, which is a useful rule we have found, so
Original type =, should be filled in.
This question is the national college entrance examination in 2002. Interestingly, there was a similar question in the spring exam in Shanghai in 2003:
Suppose, with the method of deducing the sum formula of arithmetic progression's first n terms in the textbook, we can get
2. Triangles and Complex Numbers
Example 5 If point P is in the third quadrant, the terminal edge of the angle is in the fourth quadrant.
Explain what is known.
So the terminal edge of the angle is in the second quadrant, so you have to fill in two.
Example 6 The solution set of inequality () is.
The explanation points out that the original inequality can be transformed into
Therefore, it should be filled in.
Example 7 If the image of a function is symmetrical about a straight line, then
Explain, among them.
Is the symmetry axis of a known function,
,
That is to say,
Therefore, it should be filled in.
In the process of solving the problem, we use the following small conclusions:
The image of the sum of functions is symmetrical about a line passing through the maximum point and perpendicular to the X axis.
Example 8 If a complex number corresponds to a vector on a complex plane, rotate this vector clockwise to get this vector. If the corresponding complex number is, then
Explain the geometric meaning of applying complex multiplication, and get
,
therefore
Therefore, it should be filled in
Example 9 If a non-zero complex number is satisfied, the value of the algebraic expression is _ _ _ _ _ _ _ _.
Explain how to transform the known equation into,
To solve this unary quadratic equation, you must
Obviously there is, and, then
Original formula =
=
=
In the above scheme, the method of dividing by two sides achieves the purpose of concentrating variables, which is the best policy to reduce variables and deserves attention.
3. Sequence, permutation and combination and binomial theorem
The known example 10 is an arithmetic series with non-zero tolerance. If it is the sum of the first n terms, then
Explain the special belt, there is, so there is.
Therefore, you should fill in 2.
Example 1 1 series, then
Explain the classification and summation, and get
, so it should be filled in.
Example 12 has the following four propositions:
①
②
③ The sum of internal angles of convex N-polygon is
(4) The number of convex N-sided diagonals is
Among them, the number of propositions that satisfy "If the proposition holds, the proposition also holds when n=k+ 1" but do not satisfy "If the proposition holds (which is the initial value of n given in the problem)" is.
Explanation ① When n=3, the inequality holds;
(2) when n= 1, but assuming that the equation holds when n=k, then
;
(3), but suppose, then
(4) Suppose, then
Therefore, ② ③ should be filled in.
Example 13 A shopping mall launched a promotion campaign and designed a lottery ticket with numbers ranging from 000000 to 999999. If the odd digits of a number are different odd digits, and even digits are even digits, it is a winning number, then the winning face (that is, the percentage of winning numbers in all numbers) is.
Explain the arrangement method of winning numbers: there are different odd numbers on odd numbers and even numbers on even numbers, so there are several winning numbers, so the winning surface is
Therefore, it should be filled in
Example 14 expansion coefficient is
By knowing that the coefficient should be the sum of the coefficient of the X term and the coefficient of the X term, it is explained that there are
So fill in 1008.
4. Solid geometry
Example 15 The lengths of three sides passing through a vertex of a cuboid are 3, 4 and 5 respectively, and its eight vertices are all on the same sphere. The surface area of this sphere is _ _ _ _ _ _.
Explain that the diagonal of the cuboid is the diameter of the circumscribed sphere, that is, there is
Therefore, it should be filled in.
Example 16 If the length of each side of a tetrahedron is 1 or 2, and this tetrahedron is not a regular tetrahedron, then its volume is (write only one possible value).
Explaining this question is a good open-ended question. The starting point of solving the problem is: how are the three sides of each face constructed? According to "the sum of two sides of a triangle is greater than the third side", {1, 1, 2} can be negated, and thus {1, 1 can be obtained.
Example 17 As shown in the right figure, e and f are the centers of the faces ADD 1A 1 and BCC 1B 1+0 of a cube, then the projection of the quadrilateral BFD 1E on the cube face may be. (Requirement: Fill in all possible figures)
Because the cube is a symmetrical geometry, the projection of the quadrilateral BFD 1E on the cube surface can be divided into three directions, namely, the projection on the surface ABCD, the surface ABB 1A 1 and the surface AD 1A 1.
The quadrilateral BFD 1E has the same projection on surface ABCD and surface ABB 1A 1, as shown in figure 02.
The quadrilateral BFD 1E is within ABC 1D 1 of the diagonal face of the cube, and its projection on the face ADD 1A 1 is obviously a line segment, as shown in Figure 03, so it needs to be filled.
4. Analytic geometry
Example 18 The midpoint coordinate of the segment cut by parabola is _ _ _ _ _ _ _ _.
Explain by eliminating y and simplifying it.
Let the two roots of this equation be and the midpoint coordinates of the cutting line segment be, then
So it should be filled in.
Example: The product of the distance from a point P on the 19 ellipse to two focal points is m, so when m is the maximum, the coordinate of the point P is _ _ _ _ _ _ _ _ _ _ _ _ _.
The two focuses of explaining the ellipse are: there are
Zezhi
Obviously, when the point P is located at the vertex of the minor axis of the ellipse, the maximum value of m is 25.
Therefore, you should fill in or.
The axial section of a glass is a part of a parabola, and its analytical function is that if a glass ball is placed in the glass, the radius r of the glass ball should be _ _ _ _ _ _ _ _ _.
According to the symmetry of parabola, the center of the great circle is on the Y axis, and both the circle and the parabola are tangent to the vertex of the parabola, so the equation of the great circle can be set as
pass by
Eliminate x and get (*)
Solve or
Let formula (*) have one and only one real root, if and only if.
Combined radius should be filled in.
The types of fill-in-the-blank questions can generally be divided into cloze, multiple-choice fill-in-the-blank and open-ended fill-in-the-blank questions. This shows that the fill-in-the-blank question is the experimental field for the reform of the mathematics college entrance examination proposition, and innovative fill-in-the-blank questions will appear constantly. So when preparing for the exam, we should not only pay attention to this new trend, but also prepare for the exam.