Math 20 13. five

Instructions for candidates 1. This paper is ***6 pages, * * * 5 big questions and 25 small questions, with full marks 120. Examination time 120 minutes.

2. Fill in the school name, name and admission ticket number accurately on the test paper and answer sheet.

3. The answers to the test questions should be filled in or written on the answer sheet, and the answers on the test paper are invalid.

4. On the answer sheet, multiple-choice questions and drawing questions are answered with 2B pencil, and other questions are answered with black pen.

At the end of the exam, return this test paper, answer sheet and draft paper together.

First, multiple-choice questions (32 points for this question, 4 points for each small question)

There are four options for each question below, and only one of them fits the meaning of the question.

The reciprocal of 1 be

A. 3 D BC

2. The largest single building "Expo Axis" in the former Shanghai World Expo Park was transformed into a comprehensive commercial center with a business area of about130,000 square meters, and130,000 was expressed by scientific notation.

a . 1.3× 105 b . 1.3× 104 c . 13× 104d . 0. 13× 106

3. As shown in the figure, AF is the bisector of ∠BAC, and EF∨AC intersects AB at point E. 。

If ∠ 1 = 25, the number of times is

A. 15

c . 25d 12.5

4. There are three red balls, two yellow balls and 1 green ball in an opaque box. There is no difference between these balls except color. Now, let's randomly pick a ball from this box. The probability of touching the yellow ball is

A. BC 1

5. If the diagonal length of the diamond is 6 and 8 respectively, the side length of the diamond is

A.5 B.6 C.8 D. 10

6. The ages of the basketball team 12 players in a middle school are as follows:

Age (year)1415161718

Number 1 4 2 3 2

Then the mode and median age of team members are

A. 16， 15 B. 15， 15.5 C. 15， 17 D. 15， 16

7. The three views of a geometric figure composed of some small cubes with the same size are shown in the figure, which constitutes this geometric figure.

The small cube * * * has

A.6

B.7

C.8

D.9

8. As shown in the figure, in the rectangular ABCD, AB=2 and BC = 4. After the rectangular ABCD rotates 90 clockwise around point C, a rectangular FGCE(A is obtained (points A, B and D correspond to points F, G and E respectively). The moving point P starts from point B and moves along BC-CE to point E and then stops. The moving point Q starts from point E and moves along EF-FG to point G..

The image function relationship between the two is roughly

II. Fill in the blanks (the score for this question is *** 16, with 4 points for each small question)

9. In the function, the range of the independent variable x is.

10. Decomposition factor: =.

1 1. As shown in the figure, in trapezoidal ABCD, ad∨BC, BD⊥DC, ∠ C = 45.

If AD=2 and BC=8, the length of AB is.

12. In the plane rectangular coordinate system xOy, there is an electronic frog at point A (1, 0).

Jump from point A to the right for the first time 1 unit, and then jump up 1 unit to A 1 point;

The second time, jump 2 units to the left from point A 1, and then jump 2 units to point A2;

Jump 3 units to the right from point A2 for the third time, and then jump 3 units up to point A3;

The fourth time, jump 4 units to the left from point A3, and then jump 4 units down to point A4;

……

According to this rule, the coordinates of point A6 are: If the coordinates of point An are (2013,2012),

Then n =

Iii. Answer the question (30 points for this question, 5 points for each small question)

13. Calculation:.

14. Solve the inequality group and find all its integer solutions.

15. As shown in the figure, point C is on line AB, and both △DAC and △DBE are equilateral triangles.

(1) verification: △ dab △ DCE;

(2) verification: da∑EC.

16. Known evaluation value.

17. As shown in the figure, in the plane rectangular coordinate system xOy, the images of proportional function and inverse proportional function are in

The second quadrant intersects with point A, and the abscissa of point A is.

(1) Find the analytical formula of inverse proportional function;

(2) The coordinate of point B is (-3,0). If the point p is on the y axis,

The area of delta δ△AOB is equal to the area of delta δ△AOP,

Write the coordinates of point P directly.

18. Solving application problems with column equations (groups):

A factory originally planned to produce 2400 air purifiers. Affected by the weather, the demand for air purifiers is on the rise, and the production task is increased by 1.200. In actual production, the factory has improved the production efficiency, producing 10 more than originally planned, and the actual number of days to complete the production task is 1.2 times that originally planned. Find out how many air purifiers were originally planned to produce every day.

Iv. Answer questions (20 points for this question, 5 points for each small question)

19. As shown in the figure, diagonal lines AC and BD of parallelogram ABCD intersect at point O,

AC⊥AB，AB=2，AC \u BD = 2 \u 3。

(1) Find the length of AC;

(2) Find the area of △AOD.

20. As shown in the figure, in △ABC, AB=AC, with AC as the diameter, let ⊙O pass through BC.

Point D, the intersection point D is the extension line of FE⊥AB of point E and AC of point F.

(1) verification: EF is tangent to ⊙O;

(2) If AE=6 and sin∠CFD=, find the length of EB.

2 1. In recent years, relying on rich natural and human resources, Beijing suburbs have vigorously developed and constructed various leisure tourism projects with agricultural sightseeing parks as the main body. Tourism in the suburbs of Beijing has risen rapidly, and farmers' income has gradually increased. The following are some statistical charts drawn according to the relevant data of Main Economic and Social Development Indicators of Beijing released by Beijing Municipal Bureau of Statistics 20 13 and 13.

Annual growth rate (accurate to 1%)

2009 12%

20 10 year

20 1 1 22%

2065438+24%

Please answer the following questions based on the above information:

(1) The annual growth rate of business income of Beijing Agricultural Sightseeing Park in 2010 is: (The result is accurate to 1%)

(2) Please complete the bar chart and indicate the corresponding data in the chart; (The result is accurate to 0. 1)

(3) If the annual business income of Beijing Agricultural Tourism Park will increase by 30% from 20 12, please

You estimate that if the annual operating income is not less than four times that of 2008, at least. (Fill in year)

22. Read the material first, then answer the questions:

When Xiao Ming studied the angles related to circles, he learned that in the same circle or in the same circle,

The circumferential angles of the same arc (or equal arc) are equal. As shown in the figure, points A, B, C and D are all

Is a point on ⊙O, then there is ∠ c = ∠ d.

Xiao Ming also found that if point E is outside ⊙O and on the same side of AB line as point D,

Have ∠ d > ∠E。

Please refer to Xiao Ming's conclusion and answer the following questions:

(1) As shown in figure 1, in the plane rectangular coordinate system xOy, the coordinates of point A are (0,7) and the coordinates of point B are (0,3).

The coordinate of point C is (3,0).

① Make the circumscribed circle of △ABC in the drawing 1 (keep the necessary drawing traces and don't write);

(2) If there is a point D on the positive semi-axis of the shaft, and ∠ACB =∠ADB, the coordinate of the point D is;

(2) As shown in Figure 2, in the plane rectangular coordinate system xOy, the coordinates of point A are (0, m) and the coordinates of point B are (0, n).

Where m>n>0. Point P is the moving point on the positive semi-axis of the shaft. When ∠APB reaches the maximum, directly write the coordinates of point P at this time.

Five, answer (this question ***22 points, 23 questions 7 points, 24 questions 7 points, 25 questions 8 points)

23. Known quadratic equation of one variable.

(1) Prove that no matter any real number, this equation always has two unequal real number roots;

(2) The abscissa of the intersection of the parabola and the axis is, where the parabola is translated by units to the right, and then translated by units upwards to obtain the parabola. Find a parabola.

Analytical formula of line;

(3)A(m, n) and B(n, m) are both on the parabola C2 in (2), and A and B are not coincident, so find the algebraic expression.

The value.

24. In Rt△ABC, ∠ ACB = 90, ∠ABC=, and point P is within △ABC.

(1) As shown in figure 1, AB=2AC, PB=3, and points M and N are on the sides of AB and BC respectively, then cos =_______ _ _,

△ The minimum value of △PMN perimeter is _ _ _ _ _ _;

(2) As shown in Figure 2, if the condition AB=2AC remains unchanged, PA=, PB=, PC= 1, find the area of △ABC;

(3) If PA=, PB=, PC=, write the degree of ∠APB directly.

25. As shown in figure 1, in the plane rectangular coordinate system xOy, the straight line L: intersects the axis and the axis at point A and point B (0,-1) respectively, the parabola passes through point B, and the other intersection point with the straight line L is c (4, n).

(1) and parabola;

(2) point d is on a parabola, and the abscissa of point d is t (0

(3) M is a point on the plane. After △AOB is rotated 90 counterclockwise around point M, △A 1O 1B 1 is obtained, and the points corresponding to points A, O and B are A 1 and O1respectively.