Examination questions and answers of college entrance examination papers

20 1 1 National Unified Entrance Examination for Colleges and Universities (Hubei Volume)

Mathematics examination questions (literature and history)

This test paper is ***4 pages, with three major questions and 2 1 minor questions. The full score of the whole paper is 150, and the test time is 120 minutes.

★ Good luck with the exam ★

Precautions:

1. Before answering questions, candidates must fill in their names and admission ticket numbers on the test paper and answer sheet. And paste the bar code of the admission ticket number on the designated position of the answer sheet. Black the box behind the A-type test paper on the answer sheet with 2B pencil.

2. Multiple-choice answers: After choosing the answer for each question, use 2B pencil to blacken the answer label of the corresponding question option on the answer sheet. If you need to change it, clean it with an eraser, and then choose to apply other answer labels. The answers on the test paper and draft paper are invalid.

3. Answer the blanks and answer the questions: use 0.5mm black ink pen to directly enter in the corresponding answer area on the answer sheet. The answers on the test paper and draft paper are invalid.

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

First, multiple-choice questions: This big question is a small question of *** 10, with 5 points for each small question and 50 points for * * *. Of the four options given in each question, only one meets the requirements of the topic.

1. Known rules

A.B.

C.D.

2. If it is a vector, the included angle between 2a+b and is equal to

A.B. C. D。

3. If the even function defined on R and odd function satisfy, then =

A.B. C. D。

4. If two vertices are on a parabola and the other vertex is the focus of this parabola, the number of regular triangles is recorded as

A.B.

C.D.

5. There is a sample with a capacity of 200, and its frequency distribution histogram is as shown in the figure. According to the frequency distribution histogram of samples, it is estimated that the frequency of sample data falling in the interval is

A. 18

C.54 D.72

6. Given a function, if, the value range of x is

A.B.

C.D.

7. Let the volume of a ball be and the volume of its inscribed cube be. The most appropriate of the following statements is

A. it's half as much as point B. Two and a half times more than point B.

C. it is about twice as much as d, and about one and a half times as much as d.

8. The common points of plane areas represented by straight lines and inequality groups are as follows

A.0b.1c.2d. Countless.

9. The problem of "Nine-section Bamboo" in Nine Chapters Arithmetic: There is a bamboo with nine sections, the volume of each section from top to bottom is arithmetic progression, the volume of the upper four sections is ***3 liters, and the volume of the lower three sections is ***4 liters, so the volume of the fifth section is

A. 1 liter

10. If real numbers A and B are satisfied, then A and B are complementary, then A and B are complementary.

A. Necessary but not sufficient conditions B. Sufficient and unnecessary conditions

C. Sufficient and necessary conditions D. Conditions that are neither sufficient nor necessary

Fill-in-the-blank question: This big question is ***5 small questions, each with 5 points and ***25 points. Please fill in the answer in the position corresponding to the question number on the answer sheet. There are two blank questions, fill in the answers in order, answer the wrong position, and do not score if the handwriting is unclear or vague.

1 1. There are 200 large supermarkets, 400 medium-sized supermarkets and 1400 small supermarkets in a city. In order to know the operation of each supermarket, a sample with a capacity of 100 was sampled by stratified sampling, and _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

The extension of 12. With a coefficient of _ _ _ _ _ _. (Results are expressed in numerical values)

13.30 bottles of drinks, 3 bottles have passed the shelf life. If two bottles are randomly selected from these 30 bottles, the probability of getting at least 1 bottle of expired beverage is _ _ _ _ _ _ _. (The result is expressed in the simplest score)

14. If the chord length of the straight line L passing through the point (-1, -2) is, the slope of the straight line L is _ _ _ _ _ _ _.

15. The formula for calculating the Richter scale m is:, where a is the maximum amplitude of the seismic curve recorded by the seismograph and the amplitude of the corresponding standard earthquake. Suppose that in an earthquake, the maximum amplitude recorded by the seismograph is 1000, and the amplitude of the standard earthquake is 0.00 1, then the magnitude of this earthquake is magnitude; The maximum amplitude of an earthquake of magnitude 9 is twice that of an earthquake of magnitude 5.

Third, the solution: this big topic is ***6 small questions, and the score is ***75 points. The solution should be written in words, proof process or calculus steps.

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

Let inner angles a, b and c face sides a, b and c respectively. as everyone knows

(1) Find the circumference of ...;

(II) the value of.

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

The sum of three positive numbers in arithmetic progression is equal to 15. Add 2, 5 and 13 to these three numbers respectively, and you will get,, in geometric progression.

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

(2) The sum of the first n items in the series is, which proves that the series is geometric progression.

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

As shown in the figure, it is known that the base length of regular triangular prism-is 2, the side length is, the point E is on the side, and the point F is on the side, and.

(i) Verification:

(2) Find the size of dihedral angle.

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

Improving the capacity of the river-crossing bridge can improve the traffic conditions of the whole city. Generally, the traffic speed V (unit: km/h) on the bridge is a function of the traffic density X (unit: vehicle /km). When the traffic density on the bridge reaches 200 vehicles /km, it will cause congestion, and the traffic speed is 0. When the traffic density does not exceed 20 vehicles /km, the traffic speed is 60 km/h, and the research shows that the traffic speed V is a linear function of the traffic density X.

(i) When, find the expression of function v(x);

(2) When the traffic density x is large, the traffic flow (the number of vehicles passing through an observation point on the bridge in unit time, unit: vehicles/hour) can reach the maximum, and the maximum value can be obtained. (accurate to 1 vehicle/hour).

20. (The full score of this short question is 13)

Let a function, where a and b are constants, and it is known that the curve has the same tangent L at point (2,0).

(i) Find the values of A and B and write the equation of tangent L;

(II) If the equation has three mutually different real roots 0, 0, 0, where for any constant, the value range of the real number m is true.

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

The curve c formed by the product of the slopes of two fixed points in the connecting plane and () is equal to the locus of points with non-zero constant m, and A2 can be a circle, an ellipse or a hyperbola.

(1) Find the equation of curve C and discuss the relationship between the shape of C and the value of M;

(ii) When the corresponding curve is; For a given, the corresponding curve is, for example, two focal points. Question: Is there a point in the world that makes the area of △? The value of, if it exists; If it does not exist, please explain why.

Reference answer

First, multiple-choice questions: This question mainly examines basic knowledge and basic operations. 5 points for each small question, out of 50 points.

Volume A:1-5 ACDCB6-10 ADBBC

Volume B:1-5 dcbc6-10 adbbc

Fill-in-the-blank question: This question mainly examines the basic knowledge and basic operation, with 5 points for each small question, out of 25 points.

11.2012.1714.1or 15.6, 10000.

Third, answer: This big question is ***6 small questions, ***75 points. The solution should be written in words, proving the process or calculation steps.

16. This small question mainly examines the basic formula of trigonometric function and the basic knowledge of solving oblique triangles, and at the same time examines the basic operation ability. (in 12)

Solution: (1)

The circumference of is

(Ⅱ)

, so a is an acute angle,

17. This topic mainly examines the basic knowledge of arithmetic progression, geometric progression and their summation formulas, and also examines the basic calculation ability. (in 12)

Solution: (1) Let the three positive numbers of arithmetic progression be

According to the meaning of the question, get

So the order is

According to the meaning of the question, there is (giving up)

So the third term is 5 and the common ratio is 2.

pass by

Therefore, it is a geometric series with the ratio of the first term to 2, and its general term formula is

(Ⅱ) the sum of the first few terms of the series, namely

therefore

Therefore, it is the first term with a common ratio of 2 in geometric series.

18. This small question mainly examines the positional relationship between spatial straight lines and planes, the solution of dihedral angles, and the ability of spatial imagination and reasoning. (in 12)

Solution 1: (1) From the known.

So there is

therefore

and

pass by

(II) In the middle, it can be obtained from (i)

So there is EF2+CF2=CE2, so

CF C 1E is also known from (I), therefore, cf plane C 1EF,

And the plane C 1EF, so cf c1f.

So it is the plane angle of dihedral angle E-CF-C 1

From (i), we know that it is an isosceles right triangle, so the dihedral angle E-CF-C 1 is.

Scheme 2: Establish the spatial rectangular coordinate system as shown in the figure, which can be obtained from the known.

(Ⅰ)

(Ⅱ) Let a normal vector of plane CEF be

pass by

that is

Let the normal vector of BC 1 edge be

Let dihedral angle E-CF-C 1 be θ, so θ is an acute angle.

, so

That is, the dihedral angle e-cf-c 1 is.

19. This small question mainly examines the basic knowledge such as function and maximum value, and at the same time examines the ability to solve practical problems by using mathematical knowledge. (in 12)

Solution: (1) by the meaning of the question: when; while

And then by the known

So the expression of this function is

(ii) According to the meaning of the question and obtained from (i)

When it is an increasing function, its maximum value is 60× 20 =1200;

When,

The equal sign holds if and only if, that is, if.

Therefore, when the maximum value is obtained in the interval [20, 200]

To sum up, when the maximum value is obtained in the interval [0,200].

That is, when the traffic density is 100 vehicles /km, the traffic flow can reach the maximum, and the maximum value is about 3333 vehicles/hour.

20. This question mainly examines basic knowledge such as functions, derivatives, inequalities, etc., and at the same time examines the ability to comprehensively use mathematical knowledge for reasoning and argumentation, as well as function equations and special and general ideas, (full score 13).

Solution: (1)

Because the curves have the same tangent at point (2,0),

So there is

From this

So, the tangent equation is

(II) is derived from (I), so

According to the meaning of the question, the equation has three different real numbers.

So they are two different real roots of the equation.

therefore

And create arbitrary,

Especially timely, certain and obtained.

From Vieta's theorem.

Yes, at will.

rule

So the maximum value of the function is 0.

So for any constant,

To sum up, the value range of is

20. This small topic mainly examines the basic knowledge such as curves and equations and conic curves. At the same time, the ability of reasoning and operation, as well as the idea of classification and integration and the combination of numbers and shapes are investigated. (in 14)

Solution: (i) Let the moving point be m and the coordinates be,

If conditions permit.

That is to say,

Coordinate your satisfaction.

So according to the meaning of the question, the equation of curve C is

When the equation of curve C is an ellipse whose focus is on the Y axis;

When the equation of curve c is, c is a circle with the center at the origin;

When, the equation of curve C is, C is an ellipse whose focus is on the X axis;

When, the equation of curve C is that C is a hyperbola whose focus is on the X axis.

(2) According to (1), when m=- 1, the equation of C 1 is

When,

The two focuses of C2 are