20 10 national unified entrance examination for colleges and universities (Guangdong volume)

Mathematics (liberal arts)

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

Note: 1. When answering questions, candidates must fill in their name, candidate number, examination room and seat number on the answer sheet with a black pen or signature pen. Fill in the test paper type (B) in the corresponding position on the answer sheet with 2B pencil. Stick the bar code in the upper right corner of the answer sheet.

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 change 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 position in the designated area of each question 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 with 2B pencil before answering.

Candidates must keep the answer sheet clean and tidy. After the exam, 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.

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.

1. If set, set.

A.B. C. D。

Solution: unite and choose a.

2. The domain of this function is

A.B. C. D。

Solution: well, choose B.

3. If the domains of functions and are both R, then

A. sum is an even function. B. odd function is an even function.

C. sum and are odd function D. even functions, odd function.

Solution: Because it is an even function, excluding B and C.

According to the meaning of the question, the center of the circle is on the left side of the y axis, excluding a and C.

Therefore, in, select d.

7. If the major axis length, minor axis length and focal length of an ellipse are arithmetic progression, the eccentricity of the ellipse is

A.B. C. D。

10. Define two operations ○+and ○ * on the set, as shown below.

○+

○*

So ○ * ○+

A.B. C. D。

Solution: As can be seen from the above table: ○+,so ○ * ○+○ *, choose A.

2. Fill in the blanks: This topic is entitled ***5 small questions. Candidates answer 4 small questions, with 5 points for each small question and a full score of 20 points.

(1) Required questions (1 1~ 13)

1 1. The problem of water shortage in a city is quite prominent. In order to prepare water-saving pipes,

Management measures, the average monthly water consumption of urban residents in a certain year was calculated.

Sampling survey shows that the average monthly water consumption of the four residents is respectively

(unit: tons). According to the program block diagram shown in Figure 2, if it is divided into

If it is not 1, 1.5,1.5,2, the output result is.

The first step ():

The second step ():

The third step ():

The fourth step ():

Step 5 (): Output

(2) Topic selection (14, 15, candidates can only choose one)

14. (Choose the lecture and choose the problem of geometric proof) As shown in Figure 3, right angle.

In trapezoidal ABCD, DC‖AB, CB, AB=AD=, CD=,

Points e and f are the midpoint of line segment AB and AD ad respectively, then EF=

Solution: The connection DE can be known as a right triangle. Then EF is the center line on the hypotenuse, which is equal to half of the hypotenuse.

15. (Coordinate system and parameter equation are selected as questions) In the polar coordinate system, the polar coordinates of the intersection of the curve and are.

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

A TV station randomly selected 65,438+000 TV viewers in a sample survey of viewers watching art programs and news programs. Relevant data are shown in the following table:

Cultural programs, general news programs

20 to 40 years old 40 18 58

Over 40 years old 15 27 42

Total 55 45 100

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

As shown in Figure 4, arc AEC is a semicircle with radius, AC is the diameter, point E is the midpoint of arc AC, points B and C are the bisectors of line segment AD, and point F outside the plane AEC satisfies FC plane bed, FB=

(1) proof: EB FD

(2) Find the distance from point B to plane FED.

(1) Prove that point E is the midpoint of arc AC.

19. (The full mark of this question is 12)

A nutritionist asked to book lunch and dinner for a child. It is known that one unit of lunch contains carbohydrate 12 unit, protein 6 unit and vitamin C 6 unit; One unit of dinner contains 8 units of carbohydrate, 6 units of protein and 0/0 units of vitamin C/KLOC. In addition, children need at least 64 units of carbohydrate, 42 units of protein and 54 units of vitamin for two meals.

If the cost of lunch and dinner in one unit is 2.5 yuan and 4 yuan respectively, how many units should be reserved for children to meet the above nutritional requirements and spend the least?

Solution: Suppose the child has booked lunch and dinner for two units respectively, assuming that the cost is F, then F.

Draw the feasible domain:

Transformation objective function:

(2) When,

When,

When,

C. At that time,

At this point:

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

Given a curve, a point is a point on the curve.