Multiple choice question:
1.A.
2.B.
3.D.
4.B。
5.C
6.C
7.A
8.D
9.D
10.C
1 1.A
12.B
Volume 2 (non-multiple choice questions, ***90 points)
2. Fill in the blanks: This big question has four small questions, each with 5 points and * * 20 points. Fill in the answers on the answer sheet.
13.6
14.9 .
15.8
16.
Third, answer: This big question is ***6 small questions, ***70 points. The solution should be written in words, proving the process or calculation steps.
17 (the full mark of this small question is 10)
Let the relative lengths of internal angles, and be,, and respectively.
Analysis: from, it is easy to think of substitution. Then use the cosine formula of the sum and difference of two angles to expand. There is also the use of sine theorem to make the corners mutually, and then get. Most candidates ignore the test here. In fact, when, by, and then, contradictions should be given up.
You can also use if to give up. But this method is not easy for students to think of.
Comments: This small question is easy for candidates to score, but it is difficult to get full marks.
18 (the full mark of this small question is 12)
As shown in the figure, in a straight triangular prism, sum is the midpoint of the plane of sum and sum, respectively.
(i) Proof that:
(2) Let the dihedral angle be 60, and find the included angle with the plane.
(I) analysis 1: the connecting BE is a triangular prism,
Yes, the midpoint. And flat,
(two diagonal lines with equal projective are equal) and plane,
Equal diagonals have equal projections.
Analysis 2: Take the midpoint, prove that the quadrilateral is a parallelogram, and then prove that ∨,, can be obtained.
Analysis 3: Using the method of space vector. The specific solution is abbreviated.
(2) Analysis 1: To find the included angle between a straight line and a plane, only the distance from the point to the plane is needed.
Because even numbers, then, are the plane angles of dihedral angles. Then it can be settled. Easily available.
The distance from the point to the surface is, and the angle with the plane is. Use, available, available.
That is to say, the angle with the plane is
Analysis 2: Make an angle with the plane, and then solve. The picture can prove it, and so can the face. It is easy to know from the analysis that the quadrilateral is square and connected, and the intersection point is, then it is a projection on the plane. . The following is omitted.
Analysis 3: Use the method of space vector to find the normal vector of the surface, and the included angle with the plane is the complementary angle with the normal vector. The specific solution can be found in the reference answers to the college entrance examination questions.
In a word, at present, the two main processing methods in solid geometry: traditional method and vector method, still account for half each. The proponent will consider the interests of both parties here.
19 (the full mark of this small question is 12)
The sum of the first items of a sequence is known.
(i) Assume that the series is a geometric series.
(2) Find the general term formula of the sequence.
Solution: (1) From the sum, there is.
By, ... ① Then when, there is ... ②.
②-① Germany
There is also the first term, the geometric series of common ratio 2.
(II) obtainable from (i),
The sequence is a geometric series, the first term is and the tolerance is.
,
Comments: Question (1) The idea is clear, so you can use the known conditions to find it.
Question (2) can be easily obtained from (1), and this recursive formula is obviously a model for constructing a new series: the main treatment means is division by two sides.
Generally speaking, in 2009, the two sets of questions in National College Entrance Examination I and II put the number series question in the first place, mainly to examine the structure of the new number series (National I also investigated the method of finding the sum of the first n items by dislocation subtraction), and changed the proposition mode of combining the number series with inequality scale method in previous years into the central axis question. It has the guiding role of making candidates and front-line teachers pay attention to teaching materials, basic knowledge, basic methods and skills, and attach importance to the two cardinal guides. It can also be seen that the proposer is consciously reducing the difficulty and seeking change.
20 (the full score of this small question is 12)
Group A workers in a workshop 10, including 4 women workers. There are five workers in Group B, including three women workers. At present, the stratified sampling method (simple random sampling, not put back into the layer) is used to extract 3 workers from group A and group B for technical assessment.
(i) Find out the number of people drawn from Group A and Group B respectively;
(II) Find out the probability that there are exactly 1 female workers among the workers selected from Group A;
(3) Pay attention to the number of male workers among the selected three workers, the distribution list and the mathematical expectation.
Analysis: (1) This question is relatively simple. The key is to grasp the meaning of the question and understand the principle of stratified sampling. Also note that this stratified sampling has nothing to do with gender.
(2) According to the first question, this problem is not difficult to deal with.
There is a probability that there are exactly 1 female workers among the workers selected from group A.
The possible values of (III) are 0, 1, 2, 3.
, ,
,
Distribution table and its expectation.
Comments: This question is more routine and easier than the probability statistics question in 2008. In the calculation, you can use classification or direct method, but it is more complicated, so candidates should be more flexible.
(2 1) (the full score of this small question is 12)
It is known that the eccentricity of an ellipse is that a straight line passing through the right focus F intersects with two points. When the slope of is 1, the distance from the coordinate origin is.
(1) Seeking the value of;
(II) Is there a point p on it, so that it holds when turning around F to a certain position?
If it exists, find out the equation of the coordinate sum of all p; If it does not exist, explain why.
Solution: (I) Let it be a straight line, and the distance from the origin of coordinates is
So, the solution is. Here we go again.
(II) From (i), we know that the equation of ellipse is. Jean,
Judging from the meaning of the question, the slope must not be 0, so we might as well set it.
Substitute it into the equation of ellipse, and it is obvious.
According to Vieta's theorem: ...
Suppose there is a point p if and only if it is:
Point p is on the ellipse, that is.
Tidy up.
Which is on the ellipse.
Therefore, ................................ ②.
Substitute the sum ① into ② to get the solution.
=, that is.
When;
When?
Comments: When students deal with analytic geometry problems, they are mainly not careful in "calculation". The so-called "calculation" mainly talks about arithmetic and algorithms. Algorithm is a calculation method to solve problems, and arithmetic is the basis and reason for adopting this algorithm. One is the exterior, the other is the interior, the other is the phenomenon and the other is the essence. Sometimes arithmetic and algorithm are not completely different. For example, is the area of a triangle calculated by multiplying the base by the half height or the half sine of the included angle between the two sides, or is it divided into several parts? When dealing with specific problems, we should make adjustments according to the specific problems and the meaning of the questions, and find suitable breakthroughs and breakthrough points.
22. (The full score of this short question is 12)
Let a function have two extreme points.
(i) Find the range of values and discuss monotonicity;
(ii) Proof that:
Solution: (1)
Order, its symmetry axis is. From the meaning of the question, we can see that the two unequal real roots of the equation are greater than each other, if and only if, we get.
(1) when, including adding functions;
(2) If is a decreasing function;
(3) When, including adding functions;
(II) by deleting item (i),
Settings,
rule
(1) When monotonically increasing;
(2) When,, monotonously decreasing.
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