20 10 Guangzhou senior high school graduation class comprehensive test (1) What are the answers to liberal arts mathematics?

Test paper type: a

20 10 comprehensive test for graduating classes of senior middle schools in Guangzhou (1)

Mathematics (liberal arts)

20 10.3

This test paper is ***4 pages, 2 1 question, full mark 150. Examination time 120 minutes.

Precautions:

1. Before answering the paper, candidates must fill in "Candidate Number" with 2B pencil, and fill in their city, county/district, school, name, candidate number, examination room number and seat number on the answer sheet with black pen or signature pen. Fill in the test paper type (A) in the corresponding position on the answer sheet with 2B pencil.

2. After choosing the answers for each multiple-choice question, use 2B pencil to blacken the answer information points of the corresponding question options on the answer sheet. If you need to modify it, clean it with an eraser and choose another answer. The answer can't be answered on the paper.

3. Non-multiple choice questions must be answered with a black pen or signature pen, and the answers must be written in the corresponding positions in the designated areas of each topic on the answer sheet; If you need to change, cross out the original answer first, and then write a new answer; Pencils and correction fluid are not allowed. Answers that do not answer according to the above requirements are invalid.

4. When answering the selected question, please fill in the information points corresponding to the selected question (or question group number) with 2B pencil before answering. If omitted, wrongly painted or painted too much, the answer is invalid.

Candidates must keep the answer sheet clean. After the exam, they should return the test paper and answer sheet together.

Reference formula: the volume formula of cone, where is the bottom area of cone and the height of cone.

The volume formula of the ball, where is the radius of the ball.

Cubic difference formula of two numbers.

1. Multiple choice question: This big question is a small question of *** 10, with 5 points for each small question, out of 50 points. Only one of the four options given in each small question meets the requirements of the topic.

The complex number of 1.* * * yoke is

A.B. C. D。

2. The solution set of inequality is

A.B.

C.D.

3. Let the center of a globe be the origin of the space rectangular coordinate system, and there are two points on the sphere, whose coordinates are, and.

18。

4. If known, the value of is

A.B. C. D。

5. It is known that a straight line is perpendicular to countless straight lines on the plane, and a straight line is perpendicular to the plane.

A. Sufficient and unnecessary conditions B. Necessary and insufficient conditions

C. Necessary and sufficient conditions D. It is neither a sufficient condition nor a necessary condition

6. In a cube with a side length of 2, this point is the center of the bottom surface and is in the cube.

If a point is randomly selected, the probability that the point-to-point distance is greater than 1 is

A.B. C. D。

7. According to the Road Traffic Safety Law of the People's Republic of China, drivers whose blood alcohol concentration is between 20-80 mg/ 100 ml (excluding 80) are drunk drivers, and their driver's license is temporarily suspended for more than 1 month and less than 3 months, and they are also fined from 200 yuan to 500 yuan; If the blood alcohol concentration is above 80mg/ 100ml (including 80), it belongs to drunk driving, and the driver's license shall be detained for less than 15 days and suspended for more than three months and less than six months, and a fine of more than 500 yuan and less than 2,000 yuan shall be imposed.

According to the Legal Evening News, from August 6th to August 8th, 2009,

On August 28th, the whole country investigated and dealt with drunk driving and drunk driving.

28,800 people, as shown in Figure 2, are drinking the blood of these 28,800 people.

Square of the frequency distribution of the result obtained by detecting the refined content.

According to statistics, the number of drunk drivers is about

2 160

C.4320 D.8640

8. In the middle, the point is at the top and the point is the midpoint. If,, then

A.B. C. D。

9. If the known function is monotonically increasing, the range of real numbers is

A.B. C. D。

10. The triangle array shown in Figure 3 is called "Leibniz Harmonic Triangle".

They consist of the reciprocal of an integer, and the number in the first row has two ends.

The number of is, and each number is two adjacent numbers in the next row.

The sum of, such as,,,,

Then the fourth number in line 7 (from left to right) is

A.B.

C.D.

Fill-in-the-blank question: There are 5 small questions in this big question. Candidates answer 4 small questions, with 5 points for each small question and 20 points for 10.

(1) Required questions (1 1 ~ 13)

1 1. In geometric series, the common ratio of, if, then the value of.

Because.

12. The program block of the algorithm is shown in Figure 4. If the output result is 0, the input real number.

The value of is _ _ _ _ _.

(Note: The assignment symbol "=" in the block diagram can also be written as "↓"; :=”)

13.△, the angles of the three sides,, and are,, and respectively.

If so, the size of the angle is.

(2) choose to do the problem (14 ~ 15, candidates can only choose to do one of them)

14. (geometric proof, choose to talk and choose to do the problem) As shown in Figure 5, it is the diameter of a semicircle, and the key points are

On the semicircle, the vertical foot is,

The value of is.

15. In the polar coordinate system, if the polar coordinate sum of two points is known as, then the area of △ (where it is the pole) is.

Third, the solution: this big question ***6 small questions, out of 80 points. The solution must be written in words, proof process and calculation steps.

16. (The full score of this small question is 12)

Known function (where,).

(1) Find the minimum positive period of the function;

(2) If the point is on the image of the function, find the value of.

17. (The full score of this small question is 14)

As shown in fig. 6, the plane of the square and the plane of the triangle intersect on the plane, and.

(1) verification: plane;

(2) Find the volume of convex polyhedron.

18. (The full score of this small question is 12)

Known straight line:, straight line:, where,.

(1) Probability of finding a straight line;

(2) Find the probability of the intersection of a straight line and the first quadrant.

19. (The full score of this small question is 14)

The ratio of the distance from a moving point to a fixed point to the distance from a point to a fixed straight line is known.

(1) The equation for finding the locus of the moving point;

(2) Let sum be two points on a straight line, which are symmetrical about the origin. If so, find the minimum value.

20. (The full score of this short question is 14)

It is known that this function is a decreasing function on the ground and an increasing function on the ground. The function has three zeros on the ground, and 1 is one of them.

( 1);

(2) Value range;

(3) Try to explore the number of image intersections between lines and functions, and explain the reasons.

2 1. (The full score of this small question is 14)

The known sequence satisfies the requirements of arbitrary, and.

(1);

(2) Find the general term formula of the sequence;

(3) Let the sum of the antecedents of the sequence be, and the inequality holds for any positive integer, which is in line with reality.

Range of values.

20 10 comprehensive test for graduating classes of senior middle schools in Guangzhou (1)

Reference answers and grading standards of mathematics (liberal arts) test questions

Note: 1. Reference answers and grading standards point out the main knowledge and ability to be tested in each question, and give one or several solutions for reference. If the candidate's answer is different from the reference answer, the corresponding score can be given according to the knowledge points and abilities mainly examined in the question type.

2. For the calculation questions in the analytical questions, if the candidate's answer in a certain step is wrong, if the content and difficulty of the subsequent part of the question are not changed, the score of the subsequent part can be determined according to the degree of influence, but the score given shall not exceed half of the score because of the correct answer of this part; If there are serious mistakes in the subsequent answers, no more points will be given.

3. Answer the score on the right hand side, which means that the candidate should get the accumulated score if he does this step correctly.

4. Only integer scores are given, and multiple-choice questions and fill-in-the-blank questions are not given intermediate scores.

First, multiple-choice questions: this big question examines basic knowledge and basic operations. * * * 10 small questions, 5 points for each small question, out of 50 points.

The title is 1 23455 6789 10.

Answer d D D C C B B C A C A

Fill-in-the-blank question: This big question examines basic knowledge and basic operation, showing selectivity. * * * 5 small questions, 5 points for each small question, out of 20 points. Among them, 14~ 15 is optional, and candidates can only choose one question.

11.712.13. (or) 14.6438+05.3.

Third, the solution: this big question ***6 small questions, out of 80 points. The solution must be written in words, proof process and calculation steps.

16. (The full score of this small question is 12)

(This small question mainly examines the knowledge of the nature and basic relationship of trigonometric functions, the mathematical thinking method of reduction and reduction, and the ability of operation and solution. )

(1) solution: ∵,

The minimum positive period of a function is.

(2) solution: ∵ function,

Click the image of this function again,

∴ .

Namely.

∵ ,∴ .

17. (The full score of this small question is 14)

(This small topic mainly examines the relationship between straight lines and planes in space, the volume of geometry and other knowledge, as well as the mathematical thinking method of the combination of numbers and shapes and the transformation of transformation, as well as the ability of spatial imagination, reasoning and argumentation, and operational solution. )

(1) proof: ∫ plane, plane,

∴ .

In a square,

Airplane.

∵ ,

Airplane.

(2) solution 1: in △,,,

∴ .

Do something excessive,

Aircraft, aircraft,

∴ .

∵ ,

Airplane.

∵ ,

∴ .

And a square area,

∴

Therefore, the volume of convex polyhedron is.

Solution 2: In △,,,

∴ .

Connect and divide the convex polyhedron into triangular pyramids.

And a triangular pyramid.

It is known by (1).

∴ .

Again, planes, planes,

Airplane.

The distance from a point to a plane is.

∴ .

∫ plane,

∴ .

Therefore, the volume of convex polyhedron is.

18. (The full score of this small question is 12)

(This small question mainly examines the knowledge of probability, solving equations and inequalities, the mathematical thinking method of combining numbers and shapes, and the ability to solve problems by operation. )

(1) solution: slope of straight line, slope of straight line.

Suppose the event is a straight line.

The total number of events is,,,,,,,,,, * * 36.

If, then, that is, that is.

There are three pairs of real numbers that meet the conditions: 0, * * *.

So ...

Answer: The probability of a straight line is.

(2) Solution: Let the event be "the intersection of a straight line and the first quadrant". Since the straight line and the first quadrant intersect, then.

Solutions of simultaneous equations

Because the intersection of the straight line and the first quadrant, then

We'll get it soon.

The total number of events is,,,,,,,,,, * * 36.

There are six pairs of real numbers that meet the conditions:,,, and * *.

So ...

Answer: The probability that the intersection of a straight line sum is in the first quadrant is.

19. (The full score of this small question is 14)

This small topic mainly examines the knowledge of ellipses and basic inequalities, the combination of numbers and shapes, the transformation of reduction, the mathematical thinking methods of functions and equations, and the ability of reasoning and operation. )

(1) solution: set point,

According to the meaning of the question, there are

Tidy it up and bring it here.

So the equation of moving point trajectory is.

(2) Solution: ∵ Points and points are symmetrical about the origin,

∴ The coordinates of the point are.

∫ is two points on a straight line,

∴: Yes, (me too).

∵ ,

∴ .

That is. That is.

Because, then,

∴ .

The equal sign is true if and only if.

Therefore, the minimum value of is.

20. (The full score of this short question is 14)

(This small topic mainly examines functions, derivatives, equations and other knowledge. , and examine the combination of numbers and shapes, transformation, classification and discussion of mathematical thinking methods, as well as problem-solving ability. )

(1) solution: ∞, ∴.

∫ is the decreasing function in the field and also the increasing function in the field.

When, get the minimum value, that is.

∴ .

(2) Solution: According to (1),

∫ 1 is a zero of the function, that is, ∴.

The two roots of ∵ are.

∵ is the increasing function on, and there are three zeros on this function.

That is ∴.

∴ .

Therefore, the range of values is.

(3) Solution: It is known from (2) and.

In order to discuss the intersection number between a straight line and a function image,

That is, find the number of solutions of the equation.

By,

Yes

Namely.

Or.

Through the equation, (*)

Yes

∵ ,

If, that is, it is understood, then equation (*) has no real number solution.

If, that is, it is understood, then equation (*) has a real number solution.

If, that is, a solution is obtained, then equation (*) has two real number solutions, namely,.

When ...

To sum up, when the images of lines and functions intersect.

When or, the straight line of the function and the image have two intersections.

When sum, the straight line of the function and the image have three intersections.

2 1. (The full score of this small question is 14)

This small topic mainly examines the knowledge of general terms, summation and inequality of sequence, the mathematical thinking method of transformation, and the ability of abstract generalization, operational solution and innovative consciousness. )

(1) Solution: If, if,

Because, so.

When, when,

Will be substituted into the above formula, because, so.

(2) Solution: Because, ①

And then there is. ②

②-①，get，

Because, therefore. ③

Also, (4)

③-④ Yes.

So ...

Because, that is, the series has always been a arithmetic progression with the first term of 1 and the tolerance of 1.