(2008/06/ 15)
Choose one carefully first (this question 10 sub-question, 3 points for each sub-question, * * * 30 points).
Of the four options given in each question below, only one is correct. Please fill in the letters before the correct option in the corresponding box on the answer sheet. Please note that there are many different ways to choose the right answer.
1. The national stadium "Bird's Nest" of Beijing 2008 Olympic Games has a construction area of 258,000 square meters, which is expressed by scientific notation.
A.25.8×104m2
C.2.58×105m2
2. Known as the solution of the equation, the value of is
A. 1 B. 3 C. -3 D. - 1
3. In the rectangular coordinate system, point P(4,) is in the first quadrant, and the angle between OP and the positive and semi-axis of the shaft is 60, then the value is
A.-3d BC-1
4. As shown in the figure, if the straight line AB‖CD, ∠ C = 1 15, ∠ A = 25, then ∠E=
A.70 B. 80 C. 90 D. 100
5. As a result of simplification,
A.B. C. D。
6. Let the absolute value of the difference between an acute angle and the complementary angle of this angle be, then
A.0 & lt& lt90 B. 0 & lt≤90
C.0<& lt90 or 90?
7. In a sampling inspection, 20 bags of salt were randomly selected from a booth, and the mass of each bag was measured as (unit: g).
492 496 494 495 498 497 50 1 502 504 496
497 503 506 508 507 492 496 500 50 1 499
According to the above sampling results, the probability of buying a bag of salt with the mass of 497.5g~50 1.5g from this booth is as follows.
A.B. C. D。
8. Three views of the geometry composed of cubes of the same size are displayed on the right, so the number of cubes in the geometry is
A.6 B. 5
C.4 D. 3
9. Take the BC side of the square ABCD as the diameter, make a semicircle O, cut the semicircle at point F with a straight line passing through point D, and pass through the AB side at point E, and then get the ratio of the circumference of ADE to the right-angled trapezoidal EBCD.
A.3:4 B. 4:5 C. 5:6 D.6:7
10. As shown in the figure, remember that the intersection point between the parabolic image and the positive semi-axis is a, divide the line segment OA into n equal parts, and set the points as P 1, P2, …, Pn- 1, and the vertical lines passing through each point as the axis intersect with the parabola at points Q 1, Q2, …, Qn- respectively. Remember W=S 1+S2+…+Sn- 1. When n is getting bigger and bigger, what do you think is the closest constant of w?
A.B. C. D。
Second, fill in carefully (6 small questions in this question, 4 points each, ***24 points)
Pay attention to the conditions of the topic and the contents to be filled in carefully, and fill in the answers as completely as possible.
1 1. Write that the negative rational number greater than-1 is _ _ _ _ _; The negative irrational number greater than-1 is _ _ _
12. in rt δ ABC, ∠C is a right angle, CD⊥AB is at point d, BC=3, and AB=5. Write a pair of similar triangles as _ _ _ _ _ _ and _ _ _ _ _ _ _ _; And write their area ratio _ _ _ _ _ _ _
13. According to a bar chart reported by a media (right), Xiao Zhang wrote in his essay: "... The number of participants in the chorus competition of the middle school students' art festival in our city increased sharply this year compared with last year ..." Is Xiao Zhang right? Why? Please comment on Xiao Zhang's statement in one sentence:
______________________________________
The probability that any of the 9 natural numbers of 14. 1 to 9 is a multiple of 2 or a multiple of 3 is _ _ _ _ _ _.
15. As shown in the figure, the radius OC of the great circle O is the diameter of the small circle O 1, and there is a diameter AB. If oc is perpendicular to ⊙ O, the tangent AD of ⊙O 1 intersects with the extension line of OC at point E, and the tangent point is d, given ⊙ O/. De _ _ _ _ _ _ _
16. As shown in the figure, a 4×2 rectangle can be divided into two or five or eight small squares in three different ways, so after a 5×3 rectangle is divided in different ways, the number of small squares can be _ _ _ _ _ _ _ _ _ _ _ _ _ _.
Three, comprehensive answer (this question has 8 small questions, ***66 points)
The solution should be written in words, proof process or deduction steps. If you think some questions are a little difficult, you can write some answers that you can.
17. (Full score for this small question)
The textbook introduces China's ancient mathematical masterpiece Sun Tzu's Art of War. There is such a problem: there are chickens and rabbits in the same cage today, with 35 above and 94 below. How many chickens and rabbits are there?
If you assume that there is only one chicken and one rabbit, please list the binary linear equation about and write your method to solve this equation.
18. (Full score for this small question)
As shown in the figure, water is injected into the following four containers with the same bottom area at a constant speed (that is, the volume of injected water per unit time is the same).
(1) Please find the functional relationship image of water height h and time t corresponding to each container and connect them with straight lines;
(2) When the water in the container just reaches half the height, please mark the position of the point T corresponding to the t value on the T axis of each functional diagram.
19. (Full score for this small question)
In a convex polygon, a quadrilateral has two diagonals and a pentagon has five diagonals. After observation and induction, how many diagonals do you think a convex octagon should have? Write down your thinking process simply.
20. (The full score for this short question is 8)
As shown in the figure, ∠ α and ∠βare known, and a ∠ γ is calculated with a ruler and compasses, so that,
You just need to make correct drawings, keep drawing traces, and don't have to write down the practice. )
2 1. (Full score for this small question)
According to qianjiang evening news's report "Zhejiang people are really enthusiastic about buying cars" on May 14, 2008, by the end of 2006, the number of cars in our province is as shown in the following figure 1, in which the approximate proportion of private cars to the total number of cars can be counted by the following table (unit: 10,000 cars):
200012002, 2003, 2004, 2005 and 2006
The total number of cars is 70 90105135170.
Private car 25 30 75 135 175
Private cars account for 35.7%, 33.3% and 55.6% of the total respectively.
(1) Please complete the above table according to the information provided by the histogram of figure 1;
(2) Please show the proportion of private cars to the total number of cars with a line chart in Figure 2 below.
22. (Full score for this small question 10)
In order to prevent the flu, a school fumigated the classroom on the rest day. It is known that in the process of drug release, the drug content y (mg) per cubic meter of indoor air is directly proportional to the time t (hours); After drug release, the functional relationship between y and t is (constant). As shown in the picture, answer the following questions according to the information provided in the picture:
(1) Write the two functional relationships of y and t and the corresponding independent variable range of self-release drugs;
(2) According to the measurement, students can enter the classroom only when the drug content per cubic meter and air is reduced to below 0.25mg How many hours will it take for students to enter the classroom after the drug is released?
23. (Full score for this small question 10)
As shown in the figure, in isosceles ABC, CH is the high line at the bottom, and point P is any point on the line segment CH that does not coincide with the endpoint, connecting AP and BC at point E and connecting BP and AC at point F..
(1) Proof: ∠ CAE = ∠ CBF;
(2) Proof: AE = BF
(3) A new triangle ABG is formed with AE, BF and AB as sides (point E and point F coincide at point G), and the areas of Δ ABC and Δ ABG are sΔ ABC and sΔ ABG respectively. If there is a little p, S δ ABC = S δ ABG, the range of ∠C can be found.
24. (The full score of this short question is 12)
Set point A(0, t) and point Q(t, b) in the cartesian coordinate system xOy. The parabola f obtained by image translation of quadratic function satisfies two conditions: ① the vertex is q; (2) It intersects the X axis at two points (∣ ob ∣ < ∣OC∣), linking A and B.
(1) Is there such a parabola f? Please make a judgment and explain the reasons;
(2) If AQ‖BC, tan∠ABO=, find the analytical expression of the quadratic function corresponding to parabola F.
Mathematical reference answers and grading standards
1. Multiple choice questions (3 points for each question, 30 points for * * *)
The title is 1 23455 6789 10.
Answer C A B C A D B C D C c c
2. Fill in the blanks (4 points for each small question, 24 points for * * *)
1 1.； Wait, the answer is not unique.
12.； 9: 16 or; 9:25 or; 16:25
13. This is not right. It depends not only on the image, but also on the fact that the ordinate difference is not very big.
14. 15.16.4 or 7 or 9 or 12 or15.
Three. Problem solving (8 small questions ***66 points)
17. (6 points for this question)
The equation is as follows:, -4 points
The equation can be solved by substitution elimination and addition and subtraction elimination. -Two points.
18. (6 points for this question)
(1) The corresponding connection is as follows: -4 points.
(2) When the water in the container just reaches half the height, the position on the function diagram is as follows: -2 points.
19. (6 points for this question)
The diagonal number of convex octagon should be 20. -Two points.
Idea 1: It can be concluded through list induction and analysis:
Polygon 4 5 6 7 8
Diagonal lines 2 2+3 2+3+4 2+3+4+5 2+3+4+5+6
Idea 2: Starting from each vertex of the convex octagon, you can make 5(8-3) diagonals with 8 vertices ***40, but one diagonal corresponds to two vertices, so there are 20 diagonals. -4 points
(If you use the formula directly, you will get 20 without thinking, and only 3 points will be given for the whole question.)
20. (8 points for this question)
Draw a picture as follows, that is, for this purpose.
-4 points for correct graphics,
Trace 2 points, conclusion 2 points.
2 1. (8 points for this question)
(1) Complete the form: -4 points.
200012002, 2003, 2004, 2005 and 2006
The total number of cars is 70 90105135170 200 250.
Private car 25 30 50 75100135175
Private cars account for 35.7%, 33.3%, 47.6%, 55.6%, 58.8%, 67.5% and 70% of the total respectively.
(2) Line chart: -4 points
22. (This question 10)
(1) Substitute the point into the function relation, and the solution is, yes.
Substitution and get, so the inverse proportional function relationship is; -3 points
And then substitute it, and you get, so the relationship between the proportional functions is.
-3 points
(2) solving the inequality to obtain,
So it takes at least 6 hours for students to enter the classroom. -4 points
23. (This question 10)
(1)∫△ is isosceles△, which is the high line of the bottom edge.
∵, ∴△ ≌△,
∴, this is; -3 points
(2) ∵ , , ,
∴△ ≌△ ,∴ ; -3 points
(3) According to (2), △ is an isosceles triangle with a base, and ∴ is equivalent to,
1) When ∠ is a right angle or an obtuse angle, in △, no matter where the point is, the conclusion is not valid;
2) When ∠ is an acute angle, ∞, and to make, just make ∠ = ∞, at this time, ∠180–2 ∞,
Exactly180–2 ∞∞∞, and 60 ∠ 90 is obtained. -Four points.
You can also draw a conclusion by comparing the size of sum. )
24. (This is entitled 12 points)
(1)∫ The vertex of the parabola obtained from the translated image is,
∴ The analytical formula corresponding to parabola is: -Two points.
∫ There are two intersections between parabola and X axis, ∴.- 1 min.
Make, obtain,
∴ )( )| ,
That is, when there is a parabola. -Two points.
(2) ∫ ∴, get:
Solution. -1 min
Yes,
1) when，by，get，
When, soon, soon,
At this point, the secondary resolution function is; -2 points
When, soon, soon,
At this point, the second analytic function is++. -Two points.
2) When, by, will be replaced and available,
(Can also be obtained from generation to generation)
So the second resolution function is +-or. -Two points.