Problem description:

Find math problems, geometry and algebra that are difficult in the middle school entrance examination. . . big problem

Analysis:

Third, answer questions (***9 small questions, 72 points. The solution should be written)

17. (The full score of this question is 5)

Solve the fractional equation:

Solution: ................................. (1)

...................................... (4 points)

Proved to be the solution of the original equation.

The solution of the original equation is .......................................... (5 points).

18. (The full mark of this question is 6)

Look at the chart in the table below and answer the following questions:

(1) Move the image on the left in the grid to the right in the horizontal direction, so that the point A moves to this point, and make the translated image:

(2) The figure made in (1) and the original figure on the right form a new figure. Is this new graph central symmetric or axisymmetric?

Solution: (1) as shown in the figure. (3 points for correct drawing)

(2) The new graph is axisymmetric. ............................... (6 points)

19. (The full mark of this question is 7)

The fiscal revenue of Shaanxi Province from 2003 to 2005 is shown in the figure. According to the information in the picture, answer the following questions:

(1) What is the fiscal revenue of Shaanxi Province in recent three years?

(2) What is the annual growth rate of fiscal revenue in Shaanxi Province from 2004 to 2005? (accurate to 1%)

(3) If the annual growth rate of fiscal revenue in Shaanxi Province in 2005-2006 is basically the same as that obtained in (2). Please estimate that the fiscal revenue of Shaanxi Province in 2006 is about several hundred million yuan. (accurate to 1 100 million yuan)

Solution: (1)∵ (100 million yuan)

The fiscal revenue of Shaanxi Province in the past three years is * * *

654.38+026.9 billion yuan (2 points)

(2)∵

The average annual growth rate of fiscal revenue in Shaanxi Province in 2004-2005 was 27% (4 points).

(3)√ (100 million yuan)

∴ The fiscal revenue in 2006 was about 67 1 100 million yuan (7 points).

20. (The full mark of this question is 8)

As shown in the figure. O is the midpoint of diagonal AC, and the straight line passing through point O intersects AB and CD at points M, N, E and F respectively, while

Solution: (1) There are four pairs of congruent triangles .............. (1).

They are △ AMO △ CNO and △ OCF △ OAE.

△ AME△ CNF, △ ABC△ CDA .................. (5 points)

(2) Proof: ∫,

∴△AME≌△CNF，

∴ 。 ............... (7 points)

Inches, AB‖CD

∴ ,

................ (8 points)

2 1. (The full mark of this question is 8)

Two cars, A and B, start from Place A and travel along the same expressway to Place B, which is 400 kilometers away from Place A, respectively showing the relationship between the distance (kilometers) and time (hours) of the two cars (as shown in the figure). According to the information provided in the picture, answer the following questions:

Function expression of (1) (no need to write the value range)

(2) Which of the two cars, A and B, gets to B first? How long does this bus arrive at B before the other one?

The function expression of the solution: (1) is, then

.................. (2 points)

If you solve it, you will get ... (4 points).

(2) Car B first arrives at .............., location B (5 points).

................... (6 points)

Let the function expression be,

∵ Image intersection (300),

That's it.

When,,, Ⅷ

∴ (hours) ∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴87

22. (The full mark of this question is 8)

There are two unified turntables A and B that can rotate freely, which are divided into three equal parts, and each part is marked with numbers, as shown in the figure. The rules are as follows:

① Turn the turntables A and B respectively.

(2) After the two turntables stop, multiply the numbers of the parts pointed by the two pointers (if the pointer stops at the bisector, turn it again until the pointer points to a certain part).

(1) Use the list method (or tree diagram) to find the probability that the products of numbers are multiples of 3 and multiples of 5, respectively;

(2) Liang Xiao and Xiao Yun want to play games with these two turntables. They stipulate that when the product of numbers is a multiple of 3, Liang Xiao gets 2 points; When the product of numbers is a multiple of 5, Xiaoyun gets 3 points. Is this game fair to both sides? Please explain the reasons; If it is unfair, try to modify the scoring rules to make both sides fair.

Solution: (1) The list of all possible results in each game is as follows:

Dial the number of B.

Dial a 4 5 6 number.

1 ( 1,4) ( 1,5) ( 1,6)

2 (2,4) (2,5) (2,6)

3 (3,4) (3,5) (3,6)

So, table * * * has nine possible outcomes.

The product of five numbers is a multiple of 3, and the probability is; ........................ (2 points)

The product of three numbers is a multiple of 5, and the probability is. ........................ (4 points)

(2) The game is unfair to both sides. ......................................................................... (5 points)

Xiao Liang scored an average score (points) and Xiaoyun scored an average score (points) every time.

This game is unfair to both sides. .............................. (6 points)

The revised score is as follows: if the product of numbers is a multiple of 3, Liang Xiao gets 3 points; If the product of numbers is a multiple of 5, Xiaoyun will get a ............................................................................................................ of 5 points (8 points).

23. (The full mark of this question is 8)

As shown in the figure, the diameter ⊙O, the midpoint of line BC is at D,

(1) Try to judge the positional relationship between point D and point ⊙O, and explain the reasons;

(2) Make it pass through point D, and the vertical foot is point E, and verify that the straight line DE is tangent to ⊙ O. ..

Solution: (1) Point D is on ⊙O, .................. (1).

Connect the outside diameter, and connect point F through point O. ..... (2 points)

In Rt△BOF,

∴ 。

∵ ,∴ 。

At Rt△ODF, ∫,

∴ Point D is on ⊙ O. ........................ (5 points)

(2)∫D is the midpoint of BC, O is the midpoint of AB,

∴OD‖AC

∵, ∴,

And ∵OD is the radius of ⊙O, and ∴DE is the tangent ................... of ⊙ o (8 points).

24. (The full mark of this question is 10)

A unit needs to send a letter to each of the five schools by "registered mail" or "express mail". The weights of these five letters are 72,90,215,340,400 respectively. According to the addresses of these five schools and the weight range of letters, the relevant postage standards found in the post office are as follows:

Business Type Billing Unit Charge Standard (Yuan) Registration Fee (Yuan/Seal) Special Envelope (Yuan/Piece)

The first weight of registered letter is 100, and each weight is 20.

0.8 3 0.5

Continued weight 10 1 ~ 2000, per weight 100.

2.00

The first weight of express mail is within 1000.

5.00 3 1.0

(1) If you send a letter with a weight of 90 by registered mail, what is the postage? What if I send it by express mail?

(2) What is the most cost-effective way to send these five letters? Please provide a justification for the answer.

(3) What did you get by answering the above questions? Please explain it in one or two sentences.

Solution: (1) If you send a letter with a weight of 90 by registered mail, the postage will be (yuan);

By express mail, the postage is

(Yuan) ............. (2 points)

(2) The weight of these five letters is less than 1000.

What's the postage if it's express mail?

(yuan)

According to (1), the cost of sending a registered letter with a weight of 90 is less in 7.5 yuan than in 9 yuan.

∵ ,

∴ A letter weighing 72 is sent by a "registered letter" less than 9 yuan; ................... (4 points)

If the letter with a weight of 2 15 is sent by registered mail, the mailing fee is

(Yuan) .......................................... (6 points)

,

Letters with weights of 400 and 340 are sent by "registered mail", and the cost is higher than that of 9 yuan.

So the first two of these five letters are sent by registered mail, and the last three by express mail is the most cost-effective. ..................................... (8 points)

(3) As long as the students are reasonable ............................... (10)

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

Master Wang has two scraps, one is a square plate with a side length of 60; The other is a right-angled trapezoidal plate with an upper bottom 30, a lower bottom 120 and a height of 60 (as shown in Figure ①). Master Wang is going to cut these two boards into two rectangular boards of equal size. He stacked the two boards together, so that the two right-angled vertices of the trapezoid coincided with the two vertices of the square, and the overlapping part of the two boards was the area surrounded by the pentagonal ABCDE (Figure ②). Due to the limitation of materials, it is required that the cut rectangle take point B as the vertex.

(1) Find the length of FC;

(2) Using Figure ②, what is the distance from the vertex opposite to the vertex B of the rectangle to the BC side, and what is the largest area of the rectangle? What is the maximum area?

(3) If you want to make the cut rectangle a square, try to find out the side length of the square with the largest area.

Solution: (1) Get △DEF∽△CGF from the meaning of the question.

∴ ,∴

.................................... (3 points)

(2) As shown in the figure, if the opposite vertex of the rectangular vertex B is P, then

① When the vertex p is on AE,

The maximum value of is ................................... (4 points).

(2) When the vertex p is on EF, the passing point p is made at point n and point m respectively.

According to the meaning of the question, you get △GFC∽△GPN.

∴ ,∴ ,∴

∴

When appropriate, the maximum value is 2400 () ........................ (7 points).

③ When the vertex p is on FC, the maximum value of is. ..... (8 points)

Based on ① ② ③, the rectangular area is the largest, with a maximum area of 2400.

............................................ (9 points)

(3) According to the meaning of the question, the functional expression of square area and side length is:

When the square has the largest area, ⅷ.

Solve it, get (give up), ().

The side length of the largest square is 48. .......................... (12)