First, multiple-choice questions (8 questions, 3 points for each question, ***24 points)
1. It is known that the following inequality is not necessarily correct ().
A.B. C. D。
2. The solution set of inequality group is correctly expressed as () on the number axis.
3. The solution of the fractional equation is ()
University of California, Los Angeles or
4. As shown in the figure, in □ABCD, point E is the midpoint of AD, and point F connects BE and AC, then AF: CF = ().
1:2 B. 1:3 C.2:3 D.2:5
(Figure 4) (Figure 6) (Figure 7)
A, B and C each borrowed a book from the library, and they agreed to exchange finished books with each other every Sunday. After several exchanges.
After that, they all finished reading these three books. If the third book read by B is the second book read by C, then the first book read by B is read by A ()
A. The first book B. The second book C. The third book D. Not sure.
6. If Feifei randomly throws darts into the rectangular chessboard as shown in the figure (consisting of 15 small squares, assuming that each small square is equal.
Maybe), then the probability that the dart falls in the shadow part is ()
A.B. C. D。
7. As shown in the figure, the function and the image of the function intersect at the point. If, the value range of is ().
A. or b or c or d or
8. As shown in the figure, in the regular pentagon ABCDE, diagonal lines AD, AC and EB intersect at points M and N respectively. The following propositions: ① Quadrilateral EDCN is a diamond; ② Quadrilateral MNCD is isosceles trapezoid; ③△AEN and △EDM are congruent; ④△AEM is similar to△△ CBN; ⑤ Point M is the golden section point of line segments AD, BE and ne, in which the pseudo-proposition is ().
A.0 B. 1 C.2 D.4
(No.8) (13) (17)
Fill in the blanks (10 small questions, 3 points for each small question, ***30 points)
9. The solution set of an inequality is all real numbers. This inequality can.
10. It is known that three different straight lines A, B and C are on the same plane, and there are four propositions as follows: ① If a//b, a⊥c, then B ⊥ C; 2 if b//a, c//a, then b//c; 3 if b⊥a and c⊥a, then b ⊥ c; ④ If b⊥a and c⊥a, then B//C/C.
The true proposition is _ _ _ _ _ _ _. (Fill in the serial numbers of all real questions)
1 1. Write the inverse proposition of the proposition "diagonal bisection of parallelogram":
12. The distance between two points on the map is 3cm, and the scale is 1: 1000000, so the actual distance between the two places is _ _ _ _ meters.
13. As shown in the figure, the pentagon is obtained by enlarging the pentagon ABCDE with point O as the similar center. It is known that OA= 10cm, =20cm, and the ratio of the perimeter of pentagon ABCDE to the perimeter of pentagon is _ _ _ _ _.
14. If the fractional equation about x has an increasing root, then.
15. If two numbers are randomly selected from 1, 2, 3 and 4 at one time, the probability that one number is twice that of the other number is.
16 in the plane rectangular coordinate system, if a straight line passing through the coordinate origin intersects the image of the function at p and q, the minimum length of the line segment PQ is.
17. As shown in the figure, DE is the midline of △ABC, and M and N are the midpoints of BD and CE respectively, so the ratio of the area of △ADE to the area of quadrilateral BCNM is equal to.
18. The images of two inverse proportional functions () and the first quadrant are shown in the figure. On the image with point P, the PC⊥x axis is at point C, the intersecting image is at point A, the PD⊥y axis is at point D, and the intersecting image is at point B. When point P moves on the image, the following proposition holds:
① The areas of △ ODB and △OCA are equal; ② The area of quadrilateral PAOB is always equal to; ③PA and PB are always equal; ④ When point A is the midpoint of PC, point B must be the midpoint of PD; ⑤ If the image of OA intersection is extended to point E, the value of is, in which there is a true proposition.
Iii. Answer questions (10 small questions, ***96 points)
19.(8 points) Solve the inequality group and judge whether the inequality group is solved.
2 1.(8 points) As shown in the figure, draw △ A 1 b1and △A2B2C2 in the grid composed of small squares with side length of1:
(1) shift △ABC to the right by 4 units and then shift 1 unit to get △ a1b1c1;
(2) With the point O in the figure as the potential center, transform the potential of △A 1B 1C 1 and enlarge it to twice the original value to obtain △ A2B2C2.
(Drawing No.2 1)
22.(8 points) Two boxes A and B are filled with balls with the same texture and size, and there are two white balls in box A, 1 yellow ball, 1 blue ball; There are 1 white balls, 2 yellow balls and several blue balls in box B. The probability of randomly touching a ball from box B is twice that of accidentally touching a ball from box A. 。
(1) Find the number of blue balls in box B;
(2) Randomly select a ball from box A and box B, and find the probability that both balls are blue balls by list or tree extraction.
23.( 10) In the process of road reconstruction, a city needs to lay a pipeline with a length of 1000 meters, and it is decided that two engineering teams, A and B, will complete the project. It is known that Team A can pave 20 meters more than Team B every day, and the days for Team A to pave 350 meters are the same as those for Team B to pave 250 meters.
(1) How many meters can Team A and Team B pave every day?
(2) If the required time limit for the completion of the project is no more than 65,438+00 days, how many schemes are there for the allocation of engineering quantity between the two construction teams (in 100 meters, rounded off)? Please help design it.
24.( 10) In recent years, there have been frequent coal mine safety accidents in China, among which gas is the most harmful, and its main component is Co. In a mine accident investigation, it was found that the concentration of CO in the underground air reached 4 mg/L from zero, then rose linearly, reaching the highest value of 46 mg/L in the seventh hour, and an explosion occurred; After the explosion, the concentration of carbon monoxide in the air decreased inversely. As shown in the figure, answer the following questions according to the relevant information in the questions:
(1) Find the functional relationship between CO concentration in the air before and after the explosion and time x, and write the corresponding independent variable range;
(2) When the CO concentration in the air reaches 34 mg/L, the miners 3 km underground will receive an automatic alarm signal. At this time, how many km/h must they evacuate to explode?
(3) Only when the concentration of CO in the air drops to 4 mg/L or below can the miners return to the mine to save themselves, and how many hours can the miners go down after the explosion?
(Figure 24)
25.( 10 point) As shown in the figure, the image of the proportional function and the image of the inverse proportional function intersect at this point in the first quadrant, perpendicular to the axis, with a vertical foot of 1.
(1) Find the analytical formula of inverse proportional function;
(2) If it is the point of the inverse proportional function on the first quadrant image (this point does not coincide with this point) and the abscissa of this point is 1, then find a point on the axis to minimize it.
(Figure 25)
26.( 10) In the math class, Teacher Li showed such a topic: As shown in the figure, the side length of a square is a point on the extension line of the side, the midpoint of the vertical line, and the extension line of the intersection point is. When? Compared with what?
After thinking, Xiao Ming showed a correct method to solve the problem: if a straight line is parallel to the intersection, it is:, because as shown in the figure, you can find the value of sum and then the ratio of sum.
(1) Please write the solution process according to Xiao Ming's ideas.
(2) Xiaodong further explored this issue and reached a new conclusion. Do you think this conclusion of Xiaodong is correct? If it is correct, please give proof; If not, please explain why.
27.( 12 points)
(1) Explore new knowledge: As shown in figure 1, it is known that the areas of △ABC and △ABD are equal. Try to judge the positional relationship between AB and CD, and explain the reasons.
(Figure 27)
(2) Conclusion Application: As shown in Figure 2, points M and N are on the image of inverse proportional function (k > 0), the intersection point M is ME⊥y axis, the intersection point N is NF⊥x axis, and the vertical feet are E and F respectively. Trial proof: Mn∨ef.
(3) variant exploration: as shown in fig. 3, on the image of inverse proportional function (k > 0), the intersection point m is ME⊥y axis, the intersection point n is NF⊥x axis, the intersection point m is MG⊥x axis, the intersection point n is NH⊥y axis, and the vertical scales are e and f respectively.
28.( 12 point) As shown in the figure, in the convex quadrilateral ABCD, the point E is on the edge CD, and AE and Be are connected. The following five relations are given: ① ad ∨ BC; ②DE = EC； ③∠ 1=∠2; ④∠3=∠4; ⑤AD+BC=AB。 Taking three of them as known conditions and the other two as conclusions, some propositions can be formed (the propositions in the following small questions must meet this requirement).
(1)*** meter can be a proposition;
(2) Write three true propositions:
(1) If,,, then,;
(2) if,,, then,;
③ If,,, then,.
Please choose one of the above three propositions and write down the reason why it is true:
Proof: I choose the proof proposition (fill in the serial number) for the following reasons:
(Figure 28)
(3) Please write a false proposition (no need to explain the reason):
If,,, then,.
Reference answer to eighth grade math test questions
First, multiple-choice questions (3 points for each small question, ***24 points)
Title 1 2 3 4 5 6 7 8
Answer D C C A B A D B
Fill in the blanks (3 points for each small question, 30 points for * * *)
9. The answer is not unique, such as 10. ①②④; 1 1. The quadrilateral whose diagonal lines bisect each other is a parallelogram;
12.30000 13. 1︰2; 14.7; 15.； 16.4; 17.4:7; 18.4.
Iii. Answer questions (10 small questions, ***96 points)
19. The solution set of inequality rent is (6 points), which is the solution of this inequality group (2 points).
20. Solution: Original formula =
=
= =
= =.(6 points)
When m=, the original formula =. (8 points)
2 1. As shown on the right (8 points).
22. (1) The number of blue balls in box B is 3 (4 points);
(2) Choose a ball randomly from Box A and Box B, and the probability that both balls are blue balls is. (8 points)
23.( 1) Solution: If Team A can lay rice every day, Team B can lay () meters every day.
According to the meaning of the question.
Solve.
Test: It is the solution of the original fractional equation.
Answer: Team A and Team B can cook and cook separately every day. (5 points)
(2) Solution: If the instrument is assigned to Team A, it will be assigned to Team B. 。
The solution can be obtained from the meaning of the problem.
So there are three distribution schemes.
Option 1: rice is allocated to team A and rice is allocated to team B;
Option 2: rice is allocated to team A and rice is allocated to team B;
Option 3: Distribute rice to Team A and rice to Team B.. (10)
24. Solution: (1) Because the concentration increases linearly before explosion,
Therefore, the functional relationship between y and x can be set as
Points (0, 4) and (7, 46) are known from the image.
Get a solution,
∴, at this time, the range of the independent variable is 0≤ ≤7.
(don't take =0, no points; If you don't take =7, you can put it in the second function. )
Because the concentration decreases inversely after the explosion,
Therefore, the functional relationship between y and x can be set to.
The points (7, 46) are known from the image,
∴ .∴ ,
∴, at this time, the range of independent variable x is x > 7.
(2) When =34, we get 6x+4=34 and x =5.
∴ The longest evacuation time is 7-5=2 (hours).
∴ The minimum evacuation speed is 3÷2= 1.5 (km/h).
(3) When =4, we get =80.5, 80.5-7=73.5 (hours).
At least 73.5 hours after the explosion, the miners can go down the well.
25. Solution: (1) Let the coordinates of this point be (,), then .........................................................................................................................................
∴ The analytical formula of inverse proportional function is. (4 points)
② From ∴ to (,).
Let the symmetry point of a point about the axis be, then the coordinate of the point is (,).
Let the analytical formula of a straight line be.
∫ for (,) ∴∴
The analytical formula of ∴ is.
When the ∴ point is (,). (Geometric method-similar triangles calculation is also acceptable) (10 score).
26.( 1) solution: the straight line is parallel to the intersection point, respectively at the point,
Then,,.
∵ ,∴ .
∴ , .
∴.(5 points)
(2) Correct, for the following reasons: Put forward an opinion,
Then,.
∵ ,
∴ .
∵ , ,
∴ .∴ .
. (10)
27.( 1) It is proved that CG⊥AB and DH⊥AB pass through points C and D respectively, and the vertical feet are G and H, then ∠ CGA = ∠ DHB = 90.
∴CG∨DH。
∫△ABC and △ABD have the same area, ∴ CG = DH.
∴ quadrilateral CGHD is a parallelogram. ∴ab∑CD。 (4 points)
(2)① Proof: link MF, NE.
Let the coordinates of m point be (x 1, y 1) and the coordinates of n point be (x2, y2).
∵ point m, n is on the image of inverse proportional function (k > 0),
∴ , .
∵ ME⊥y axis, NF⊥x axis, ∴ OE = y 1, of = x2. ∴ s △ EFM =，
S△EFN=。 ∴S△EFM =S△EFN。
According to the conclusion in (1), Mn∑ef. (8 points)
(3) The first method: connecting FM, EN and MN, the same as (2) can prove MN∨EF, and the same method can prove GH∨MN, so EF∨GH. ..
Method 2: directly use OE? OG=OF？ OH syndrome △OEF∽△OHG (specific process omitted) (12 score)
28. Solution: Please refer to the following table:
Is the proposition of ordinal conditional conclusion true or false?
1③∠ 1 =∠2④∠3 =∠4⑤ad+BC = Ab 1ad∨BC 2de = EC true。
2 2de = ec4 ∠ 3 = ∠ 4 χ ad+BC = ab1ad ∨ bc3 ∠1= ∠ 2 true.
32de = ec3 ∠1= ∠ 25ad+BC = ab1ad ∨ bc4 ∠ 3 = ∠ 4 true.
42DE = EC 3 ∠1= ∠ 2④ ∠ 3 = ∠ 41Dad ∨ BC5AD+BC = AB false.
5 ① AD ∨ BC 4 ∠ 3 = ∠ 4 χ AD+BC = Ab2DE = EC 3 ∠1= ∠ 2 True.
6 ① AD ∨ BC ③ ∠1= ∠ 2 ⑤ AD+BC = Ab2DE = EC4 ∠ 3 = ∠ 4 True.
7①AD∨BC③∠ 1 =∠2④∠3 =∠42de = EC5AD+BC = AB True。
8 ① AD ∨ BC2DE = EC5AD+BC = AB3 ∠1= ∠ 2④ ∠ 3 = ∠ 4 True.
9 ① ad ∨ bc2de = ec4 ∠ 3 = ∠ 4 ③ ∠1= ∠ 2 ψ 5ad+BC = ab true.
10①AD∨BC2DE = EC3∠ 1 =∠2④∠3 =∠4ψAD+BC = AB True。
According to the table, it is easy to know that the answer to this question should be:
(1) 10(3 points); (2) Choose 3 out of 9 true propositions in the table (5 points), and the reasons are omitted (8 points); (3) The pseudo-proposition is: "If DE=EC, ∠ 1=∠2, ∠ 3 = ∠ 4, then AD∨BC, AD+BC = AB." (12).