First, multiple-choice questions (concise results express your keen thinking and need to be careful! 3 points for each small question, ***30 points)

1. If the value of the score is 0, the value of is ().

A. 1 or B. or1c.d. or

2. The image of inverse proportional function and direct proportional function in the same coordinate system can't be ().

A B C D

3. physical education class, Class 8 (65,438+0). There are 65,438+00 people in two groups taking part in the standing long jump. To judge which group is more orderly, you usually need to know (). A. frequency distribution B. mean C. variance D. these two groups of standing long jump performance patterns.

4. The time (hours) for students of a school 10 to participate in the environmental protection practice activities in the west in April are: 3, 3, 6, 4, 3, 7, 5, 7, 4 and 9 respectively. The mode and median of this set of data are ().

A.3 and 4.5 B.9 and 7 C.3 and 3 D.3 and 5

5. A township needs to rebuild a 4000-meter-long power line to transform the rural power grid. In order to reduce the influence of construction on farmers' electricity consumption, the daily work efficiency during construction is higher than the original plan. As a result, the task was completed two days ahead of schedule. How long is the wire installed every day in actual construction? If the original plan is to set up x meters of conductor every day, then the equation listed is ().

A.―2 b .―2 c .―2d .―2

6. As shown in figure 1, in the isosceles trapezoid, ABCD, AD‖BC, AE‖DC, ∠B=60o, BC=3,

When the circumference of △ABE is 6, the circumference of isosceles trapezoid is ().

10

Figure 1

7. With the following groups as side lengths, a right triangle can be formed ().

A.，b，2，c，32，42，52 D， 1，2，3

8. The quadrilateral whose diagonal is vertically bisected and equal must be ().

A. square B. diamond C. rectangle D. isosceles trapezoid

9. It is known that, as shown in Figure 2, in the rhombic ABCD, diagonal AC and BD intersect at point O, OE‖DC intersects at point E BC, and AD = 6 cm, then the length of OE is ().

6 cm long, 4 cm wide, 3 cm high and 2 cm wide.

Figure 2

10. There are 500 ninth-grade students in a school. To know how many of them got A, B, C and D in the academic proficiency test, what needs to be done is ().

A. find the average score B. carry out frequency distribution C. find the interval D. calculate the variance

Fill in the blanks (4 points for each small question, 40 points for * * *)

1 1. The solution of the equation is.

12. Simplify:

13. If the image of the inverse proportional function passes through this point, then.

14. Within 2km around Mount Qomolangma, there are also the famous Luozi Peak (8516m above sea level), Zhuoqiong Peak (7589m above sea level), Makaru Peak (8463m above sea level), Zhang Zifeng Peak (7543m above sea level) and Nuz Peak (7855m above sea level).

15. As shown in Figure 3, point P is a point on the inverse proportional function image, PD is perpendicular to the X axis, and at point D, the area of △POD is.

Figure 3

16. In quadrilateral ABCD, diagonal AC and BD intersect at point O, from (1) ab = cd; (2)AB‖CD； (3)OA = OC； (4)OB = OD； (5)ac⊥bd； (6) In the six conditions of ∠BAD equal division of 6)AC, select three to deduce that the quadrilateral ABCD is rhombic. For example, (1)(2)(5) ABCD is a diamond, and then write two that meet the requirements _ _ _ _ _ _ _ _ ABCD is a diamond; _ _ _ _ _ _ _ ABCD is a diamond.

17. Fold the rectangular piece of paper ABCD in Figure 4 in half, with points B and C just overlapping on point P on the side of AD (as shown in Figure 5). It is known that ∠ MPN = 90, PM = 3 and PN=4, so the area of rectangular paper ABCD is _ _ _ _ _ _ _.

Figure 4

Figure 5

18. The following propositions: ① the vertex angles are equal; ② The two base angles of isosceles triangle are equal; (3) Two straight lines are parallel and have the same angle. Among them, the inverse proposition is true: (please fill in all serial numbers that match the meaning of the question).

19. As shown in Figure 6, if a rectangular wooden frame made of four pieces of wood is changed into a parallelogram, its area is half that of the rectangle, then the minimum internal angle of the parallelogram is equal to.

Figure 6

20. 10 Students bought shoes in the following sizes:

20, 20, 2 1, 22, 22, 22, 23, 24 (unit: cm) Among the three indicators of average, median and mode, shoe store owners like _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

Iii. Answering questions (***50 points)

2 1.(6 points) Simplify the score first, and then please choose an appropriate value for x to find the value of the original formula.

22.(6 points) It is known that both the images of the proportional function and the inverse proportional function pass through the point (2, 1). Find out the relationship between these two functions.

23.(6 points) In a 4×4 square grid, the side length of each small square is 1. Line segments AB and EA are diagonal lines of two rectangles 1×3 in Figure 7. Please prove that AB⊥EA.

Figure 7

24. As shown in Figure 8, when △ ABC and ∠ ACB = 90, points D and E are the midpoint of AC and AB respectively, and point F is on the extension line of BC's weir, and ∠CDF=∠A, which proves that the quadrilateral DECF is a parallelogram.

Figure 8

25. As shown in Figure 9, in ∠ABC, AB = BC, D, E and F are the midpoint of BC, AC and AB respectively;

(1) Verification: the quadrilateral BDEF is a diamond;

(2) If AB =, find the perimeter of rhombic BDEF.

Figure 9

26. Xiao Ming and Xiao Bing participated in a physical training, and their latest eight test scores (scores) are as follows:

Test 1 second, third, fourth, fifth, sixth, seventh and eighth.

Xiaoming101010161417

Xiao Bing1131214131513.

(1) Fill in the following table according to the data provided in the above table:

Mean (Minute) Mode (Minute) Mean (Minute) Variance

Xiaoming 10 8.25

Xiao Bing 13 13

(2) If you choose one person to participate in the city middle school sports meeting, who do you think is suitable? Please explain the reason.

27. As shown in figure 10, it is a cubic paper box with no cover on it. Now it is cut and expanded into a plan, as shown in figure 1 1. It is known that the side length of each square in the expanded view is 1.

(1) Find the length of the longest line segment that can be drawn in the expansion diagram? How much can such a line segment be drawn?

(2) Try to compare the size relationship between ∠BAC in the three-dimensional diagram and ∠ B ′ a ′ c ′ in the plane expansion diagram.

Figure 10 Figure 1 1

28. As shown in figure 12, let the quadrilateral ABCD be a square with a side length of 1, make the second square ACEF with the diagonal AC of the square ABCD as the side, and then make the third square AEGH with the diagonal AE of the second square as the side, and so on.

(1) Remember that the side length of the square ABCD is A 1 = 1, and the side length of the square made by the above method is a2, a3, a4, ..., and then find an and the values of a2, a3 and a4.

(2) Write the expression of the side length an of the nth square according to the above rules.

Figure 12

Reference answer:

I. 1. C2 d 3 . C4 . a5 . B6 . a7 . A8 . a9 . c 10 . b

Second,11.x = 5; 12.； 13.-6; 14. 137 1; 15. 1 ; 16.( 1)(2)(6); (3)(4)(5) or (3)(4)(6) meets the requirements; 17.； 18.②③; 19.30 ; 20. Average, mode.

Third,

2 1. solution: original formula =, when x=0, original formula = 1.

22. Substitute x=2 and y= 1 into two relations, and get k 1= and k2=2.

Therefore, the direct proportional function relationship is y= x, and the inverse proportional function relationship is y=.

23. proof: connect BE, according to the characteristics of the grid, EF=AG=3, get ∠ f = ∠ g = ∠ BCE = 90,

Then in Rt△EFA, AE2 = EF2+AF2 =10 is obtained by Pythagorean theorem; In Rt△ABG, AB2 = Ag2+GB2 =10 is obtained from Pythagorean theorem; In Rt△EBC, BE2=BC2+EC2=20,

So ae2+ab2 =10+10 = 20 = be2, and from the inverse theorem of Pythagorean theorem, we get ∠ BAE = 90, so AB⊥EA.

24. It is proved that since point D and point E are the midpoint of AC and AB respectively, DE//BC,

Because < ACB = 90 degrees,

So CE= AB=AE, so ∠A=∠ECA,

Because < CDF = < A,

So ∠CDF=∠ECA, so DF//CE, so the quadrilateral DECF is a parallelogram.

25.( 1) Because D, E and F are the midpoint of BC, AC and AB respectively,

So de AB, EF BC,

So the quadrilateral BDEF is a parallelogram.

Because DE = AB, EF = BC, AB = BC.

So DE = EF

So the quadrilateral BDEF is a diamond;

(2) Because AB =, F is the midpoint of AB, so BF =, so the circumference of diamond BDEF is

26. Solution: (1)

Mean (Minute) Mode (Minute) Mean (Minute) Variance

Xiaoming131012.58.25

Xiao Bing1313131.25

(2) Both of them have the same per capita value, and the mode and median of Xiao Bing's performance are higher than that of Xiao Bing, with a small variance, indicating that Xiao Bing's performance is stable, but Xiao Bing's performance fluctuates greatly, and he has been better than Xiao Bing in recent times. Therefore, from the development trend, Xiao Ming should be chosen for the competition.

27. Analysis: (1) As shown in A'C' in Figure ①,

In rt △ a ′ c ′ d ′, c ′ d ′ =1,a ′ d ′ = 3,

From Pythagorean Theorem:

In other words, the longest line segment can be drawn in the plane expansion diagram. Four such line segments can be drawn (the other three are marked with dotted lines).

① ②

(2) Because ∠B'a'c' in the stereoscopic view is the acute angle of a plane isosceles right triangle,

∠b′a′c′= 45，

Connect the line segment B'C' in the flat expansion diagram, as shown in Figure ②.

From the Pythagorean theorem, we can get: a ′ b'c'=, b ′ c ′ =.

Because A'B'2+B'C'2=A'C'2,

From the inverse theorem of Pythagorean theorem, it can be concluded that △ a ′ b ′ c ′ is a right triangle.

And because A'B'=B'C', △A'B'C' is an isosceles right triangle.

So ∠ BAC = 45, so ∠ b' a' c' = ∠ BAC.

28. solution: (1) in Rt△ABC, because ∠ B = 90, so ac2 = ab2+bc2 =1+1= 2, so AC=, AE=2, EH=2, so a2 =

(2) because A 1 = 1 = () 0, A2 = () 1, A3 = 2 = () 2, A4 = (2) = () 3, an=( )n- 1.

(n≥ 1, where n is an integer).