In 2009, the second model of mathematics was taken in the senior high school entrance examination in Gulou District, Nanjing.

First, multiple-choice questions (24 points for this question, 4 points for each small question)

1. In the following groups, the inverse number is ().

(A)2 and (b) and 1 (C)- 1 and (D)2 and |-2|

In order to welcome the 2008 Olympic Games to be held in Beijing, China, Beijing is now implementing strict automobile exhaust emission standards, and at the same time trying to reduce the pollution caused by industrial and civilian fuels. With the annual transmission of 654.38 billion cubic meters of natural gas to Beijing, by the end of 2006, Beijing's air quality will basically reach the level of developed cities, and 654.38 billion can be expressed as () by scientific notation.

(A) 1.0× 10(B) 1.0× 10(C) 1.0× 10(D) 1.0× 10

3. The approximate location of the quadratic function image is shown in the figure, and the following judgment is wrong ()

(A) (B) (C) (D)

4. In the following surveys, the easier census method is ().

(a) to understand the annual per capita income of Guiyang residents; (2) Understand the results of physical education senior high school entrance examination for junior high school students in Guiyang.

(3) Understand the myopia rate of primary and secondary school students in Guiyang; (d) Understand the population flow leaving Guiyang on a certain day.

5. The following figure is an axisymmetric figure, but it is not a centrally symmetric figure ().

(a) isosceles trapezoid (b) rectangle (c) parallelogram (d) rhombus

6. Give the following four propositions:

(1) If the lateral expansion of a cone is a semicircle, then its axial section must be an equilateral triangle;

(2) If point A is on the straight line y=2x-3 and the distance from point A to the two coordinate axes is equal, then point A is in the first or fourth quadrant;

(3) In a circle with a radius of 5, if the chord AB=8, the distance from the circumference of four points * * * to the straight line AB is 2;

(4) If A(a, m) and b(A- 1, n)(a 0) are on the image of the inverse proportional function, then m n 。

Among them, the number of correct propositions is ()

1 (B) 2 (C) 3 (D) 4。

Two. Fill in the blanks (24 points for this question, 4 points for each small question)

7. Decomposition factor: = _ _ _ _ _ _ _ _ _ _ _ _ _ _.

8. The value range of the function independent variable is _ _ _ _ _ _ _ _ _.

9. Write a quadratic equation with two real numbers and two roots of _ _ _ _ _ _ _ _.

10. As shown in the figure, ABC is an isosceles right triangle and BC is the hypotenuse. After rotating △ABP counterclockwise around point A, it can coincide with △ACP'. If AP=3, then the length of PP' is equal to _ _ _ _ _ _ _.

1 1. In order to estimate the number of fish in the lake, we caught 100 fish from the lake, marked them and put them back in the lake. After a while, we caught 200 fish for the second time and found 25 tagged fish. Through this investigation, we can estimate that there are fish in the lake.

12. Explore with calculator: = _ _ _ _ _ _, = _ _ _ _ _, guess: = _ _ _ _ _ _.

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

13. Calculation:+Tan 60.

14. Solving inequalities

15. As shown in the figure, in the isosceles trapezoid ABCD, AD‖BC, ∠ DBC = 45. Fold the trapezoidal ABCD so that point B coincides with point D, and the crease intersects AB and BC at point F and point E respectively. If AD=2 and BC=8,

Find the length of: (1)BE; (2) Tangent of 2)CDE.

16. It is known that the ordinate of the intersection of the inverse proportional function and the linear function is -4, and the value is obtained.

17. It is known that, as shown in the figure, in a rectangular ABCD, AC and BD intersect at point O, and if point E is the midpoint of AO, point F is the midpoint of OD. Verification: Be = CF

18. As shown in the figure, it shows the images of a ship and a speedboat traveling along the same route from Port A to Port B (the images are proportional function images and linear function images respectively). Answer the following questions according to the pictures:

(1) According to the picture, the ship left _ _ _ _ hours earlier than the speedboat.

(2) Please find out the resolution function representing the running process of the ship and the speedboat respectively (the range of independent variables is not required);

(3) How long does it take for the speedboat to catch up with the ship by calculating?

Iv. Answer (20 points for this question, 19, 2 1, 5 points for each question, 4 points for 20 questions and 6 points for 22 questions)

19. It is known that ⊙O and ⊙O 1 intersect at point A and point B, O is on ⊙O 1, and BC of ⊙O intersects ⊙O 1 at point D. Verification. AD=DC。

20. An engineering team (Group A and Group B) contracted the subgrade reconstruction project of a section in the new district of our city, and agreed to complete it within a few days. It is known that the time required for group A to complete this project alone is twice as long as the specified time by 4 days, while the time required for group B to complete this project alone is twice as short as the specified time 16 days. If Group A and Group B do it in 24 days, can they be completed within the specified time?

2 1. The following picture shows two residential buildings with 10 floors in a residential area, which are 1, the second, …, and the first 10 floors from the ground. The height of the EC=h floor is 3m, and the distance between the two floors is AC=30m ... It is necessary to know the influence of Building A on the lighting of Building B in a certain period of time. suppose

(1) H is expressed by a formula containing α;

(2) When α = 30, what floor does the shadow of B on the roof of Building A fall on? From this point on, if α increases by 10 every hour, the shadow of building A will not affect the lighting of building B after several hours.

22. Junior high school students in a school launched a shuttlecock kicking competition, and each class sent five students to participate. They are ranked according to the total score of the group, and each student plays 100 or more within the specified time (including 100). The following table is the competition data of five students in Class A and Class B with the best performance (unit: one):

1 second, third, fourth and fifth total score

Grade a1009811089103500

Class b 89100 95119 97 500

Statistics show that the total scores of the two classes are equal. At this time, some students suggested that you can take other information in the materials as a reference. Please answer the following questions:

(1) Calculate the excellent rate of Class Two.

(2) Find the median of the two types of data.

(3) Calculate and compare the variance of competition data between the two classes.

(4) According to the above three pieces of information, which class do you think the champion certificate should be issued to? Briefly explain why.

Five, answer (this question ***22 points, 23 questions 6 points, 24 questions 8 points, 25 questions 8 points)

23. In a basketball game in the third grade of a school, as shown in the picture, player A is shooting. It is known that the height of the ball from the ground is m and the horizontal distance from the center of the hoop is 7m. When the horizontal distance is 4m after the ball is released, the maximum height is 4m. Let the trajectory of basketball be a parabola, and the hoop is 3 meters off the ground.

(1) Establish the plane rectangular coordinate system as shown in the figure, find the analytical formula of parabola, and judge whether the ball can be hit accurately.

(2) At this time, if the opposing player B jumps in front of A at 1m, it is known that the maximum touch height of B is 3. 1m, can it be successful?

24. Cut the rectangular paper ABCD as shown in figure 1 into two parts along the straight line CM with scissors, where m is the midpoint of AD. With these two parts of paper, we can combine some new graphics, such as Rt△BCE in Figure 2.

(1) With these two pieces of paper, not only Rt△BCE in Figure 2 can be spelled out, but also some quadrangles can be spelled out. Please try to draw quadrangles in the imaginary boxes in Figure 3 and Figure 4 respectively.

(2) If the Rt△BCE spliced by these two pieces of paper is an isosceles right triangle, let the lengths of AB side and BC side in the original rectangular paper be a cm and b cm respectively, and A and B are just two real roots of the equation about X, and try to find the area of the original rectangular paper.

25. It is known that when B takes any real number, it looks like a parabola.

(1) Now there are two statements:

① When b takes any different values, the corresponding parabolas have exactly the same shape;

② When b takes any different value, the corresponding parabola has different shapes; Which statement do you think is correct and why?

(2) If b=-1 and b=2, and the vertices of the corresponding parabola are A and B respectively, please find out the analytical formula of AB and judge whether the vertices of the corresponding parabola are on this straight line when B takes other real values. Explain why.

(3) If there is a point C on the straight line determined in (2), and the ordinate of this point C is-1, ask whether there is a point D on the X axis that makes △COD an isosceles triangle, and if so, directly write the coordinates of the point D; If it does not exist, simply explain the reason.