mathematics
First, multiple-choice questions (this big question is a small one. 3 points for each small question, * * 30 points. Of the four options given in each question.
Only one item meets the requirements of the topic)
The value of (1) sin 45 is equal to
(A)(B)(C)(D) 1
(2) In the following car signs, which one can be regarded as the central symmetrical figure is
(3) According to the statistics of the sixth national census, as of 0: 00, 20 10, 1 1, the total population of China is about
1.37 billion people, and the scientific notation of 1.37 billion people should be
(A) (B) (C) (D)
(4) The estimated value is in
(a) 1 to 2 questions (b)2 to 3 questions (c)3 to 4 questions (d)4 to 5 questions.
(5) As shown in the figure, the square paper ABCD is folded in half, so that both sides of AB and CB fall on the diagonal BD to obtain creases BE and BF, and the size of ∠EBF is
15 30 45 60
(6) The radii of ⊙ and ⊙ are known to be 3 cm and 4 cm respectively. If =7 cm, the positional relationship between ⊙ and ⊙ is
(a) Intersection (b) Separation (c) Inner Cut (d) Outer Cut
(7) The picture on the right shows a bracket (a small part), and the height and width of the two steps of the bracket are the same length. Then its three views are
(8) The following figure is a histogram of the shooting scores (loops) of A and B, so the following statement is correct.
(a) The score of A is fixed compared with that of B; (b) Compared with A, the score of B is stable.
(c) The grades of A and B are equally stable; (d) It is impossible to determine whose performance is more stable.
(9) A telecom company provides customers with two online charging methods: Method A is calculated according to the online time, and the price is 0. 1 yuan per minute; In mode B, in addition to the basic fee of 20 yuan every month, a fee of 0.05 yuan per minute will be charged according to the length of surfing the Internet. If the online time is x minutes and the charging is y yuan, as shown in the figure, in the same rectangular coordinate system, the following conclusions are drawn:
(1) Picture A describes the way of A:
② Image B describes mode B;
③ When the online time is 500 minutes, choose mode B to save money.
Among them, the number of correct conclusions is
3 (B) 2 (C) 1 (D) 0
(10) If the real numbers x, y and z are satisfied, then the following formula must hold.
(A) (B) (C) (D)
20 1 1 Tianjin junior high school graduates' academic examination papers
mathematics
Fill in the blanks (this big question is ***8 small questions. 3 points for each small question, ***24 points)
The opposite religion of (1 1) is _ _ _ _ _ _.
(12) If the value of the score is 0, the value of x is equal to _ _ _ _ _ _.
(13) It is known that the image of a linear function passes through the point (0. 1), and it is satisfied that y increases with the increase of x, then the analytical formula of the linear function can be _ _ _ _ _ _ _ _ _ _ (write only one).
(14) As shown in the figure, points D, E and F are the midpoints of the sides AB, BC and CA of △ABC respectively, connecting DE, EF and FD. Then the number of parallelograms in the graph is _ _ _ _ _ _ _.
(IS) As shown in the figure, AD and AC are the diameter chords of ⊙O respectively, and ∠ CAD = 30. OB ⊥ AD, AC intersects with point B. If OB=5, the length of BC is equal to _ _ _ _ _ _ _.
(16) Throw two dice with uniform texture at the same time. Observing the points above, the probability that two dice have the same points is _ _ _ _ _ _ _ _.
(17) As shown in the figure, the six internal angles of the hexagonal ABCDEF are all equal. If AB= 1, BC=CD=3, and DE=2, then the circumference of the hexagon is equal to _ _ _ _ _ _ _.
(18) As shown in the figure, there is a rectangular piece of paper ABCD, which is 5 in length and 3 in width. By proper cutting and splicing, a square with the same area should be obtained.
(a) the side length of a square is _ _ _ _ _ _. (Results Retain the root number)
(2) At present, only two cutting lines are needed. Please design a cutting method and draw a cutting line in the picture.
And briefly explain the process of cutting and spelling: _ _ _ _ _ _ _ _.
Third, the solution (this big question is ***8 small questions, ***68 points. The solution should be written in words, calculus steps or reasoning process)
(19) (6 points for this small question)
Solving inequality system
(20) (8 points for this small question)
Images with known linear function (b is constant) and inverse proportional function (k is constant).
Images intersect at point p (3. 1).
(i) Find the analytical expressions of these two functions;
(2) when x > 3 o'clock, try to judge the size of the sum. So I can explain why.
(2 1) (8 points for this small question)
In the reading activity of "Good books grow with me" carried out in our city, a middle school randomly investigated the reading quantity of 50 eighth-grade students in order to understand the reading situation of 300 eighth-grade students. Statistics are shown in the following table:
Album 0 1 234
No.31316171
(i) Find the average, mode and median of these 50 sample data:
(2) According to the sample data, it is estimated that 300 eighth-grade students in this school read more than two volumes in this activity.
(22) (8 points for this small question)
It is known that AB and ⊙O are tangent to point C, and OA = OB. OA, ob and ⊙O intersect at point d and point e respectively.
(i) As shown in figure 1, if the diameter ⊙O is 8AB= 10, find the length of OA (the root sign is retained as a result);
(2) As shown in Figure ②, connect CD, CE,-If the quadrilateral dODCE is a diamond, the value is obtained.
(23) (8 points for this small question)
A school interest group took a cruise to photograph the beautiful scenery on both sides of the Haihe River. As shown in the figure, the distance between the starting point A of the cruise ship and Wanghai Building B is 300 meters. In one place, Wanghai School B was measured to be 30 east of the north of A, and the cruise ship sailed north for a period of time and then arrived at C. At C, Wanghai Building B was measured to be 60 east of the north of C. Find the distance BC from the cruise ship to Wangmei Building at this time (take l.73, and the result is an integer.
(24) (8 points for this small question)
Note: In order to help students solve this problem better, we provide methods to analyze the problem. You can solve this problem according to this method. You can also choose other methods and answer according to the requirements of Class One.
At present, the price of a commodity in 35 yuan is 50 pieces per day. According to market research, if the price is adjusted, you can sell 2 pieces every day for every price reduction of 1 yuan. Please help me analyze how much each commodity can be reduced to maximize daily sales. What is the maximum sales volume?
Suppose the price of each commodity is reduced by X yuan. The daily sales amount is y yuan.
(a) analysis: according to the quantitative relationship in the question, fill in the form with the formula containing X:
(ii) (Based on the above analysis, Y is expressed by a formula containing X, and the solution of the problem is obtained)
(25) (this small problem 10)
In the plane rectangular coordinate system, the origin of the O coordinate is known. Point a (3.0) and point b (0.4). Take point A as the center of rotation, and rotate △ABO clockwise to get △ ACD. Remember that the rotation angle is α. ∠ ABO is β.
(i) As shown in Figure ①, when the rotated point D just falls on the edge of AB, find the coordinates of the point D;
(2) As shown in Figure 2, when BC∨x axis is rotated and satisfied, find the quantitative relationship between α and β;
(3) When the rotation satisfies ∠AOD=β, find the analytical formula of straight line CD (just write it out directly, that is, if).
(26) (this small problem 10)
Known parabola:. Point f (1, 1).
(i) finding the vertex coordinates of the parabola;
(Ⅱ) ① If the intersection of parabola and Y axis is A. Connect AF, extend the intersection of parabola at point B, and verify:
② Any point on the parabola p ()) (). Connect PF. And extend the intersecting parabola to point q (). Try to judge whether it is true or not. Please explain the reasons;
(iii) Properly translate the parabola to obtain the parabola: if it remains unchanged, find the maximum value of m. 。
20 1 1 Tianjin Junior High School Graduates' Academic Examination
Reference answers to math test questions
First, multiple choice questions
The title is 1 23455 6789 10.
Answer: Baba, Baba, Baba, Baba, Baba.
Second, fill in the blanks
( 1 1) 6 ( 12) 1 ( 13).
( 14)3 ( 15)5 ( 16) ( 17) 15
( 18)(Ⅰ)
(2) As shown in Figure ①, BN= (BM=4, MN= 1, ∠ MNB = 90);
② Draw two cutting lines AK and BE.
(AK=BE=。 BE⊥AK):
③ Translate △ABE and △ adk.
At this point, the quadrilateral BEF'G is what you want.
Third, answer the question (this big question is ***8 small questions, ***66 points)
(19) (6 points for this small question)
Solution:
Solve inequality ①. Get it.
Solve inequality ②. Get it.
The solution set of the original inequality group is.
(20) (8 points for this small question)
The analytical formula for solving the linear function of (i) is.
The analytical expression of the inverse proportional function is.
(ii) The reasons are as follows:
When.
At the same time, the linear function increases with the increase of x, and the inverse proportional function decreases with the increase of x,
When.
(2 1) (8 points for this small question)
Solution: (1) Observation table. It can be known that the average number of rescues in this group of samples is
The average value of this sample data is 2.
∵ In this set of sample data, .3 appeared 17 times, with the most times.
The mode of this set of data is 3.
∵ Arrange this set of sample data in order from small to large. The two numbers in the middle are both 2.
The median of this set of data is 2.
Of the 50 students, 8 have read more than two books.
According to the sample data, it can be estimated that about 108 of the 300 eighth-grade students in this school read more than two volumes in this activity.
(22) (8 points for this small question)
(ⅰ)OA =(ⅱ)
(23) (8 points for this small question)
BC≈ 173
(24) (8 points for this small question)
Solution: (1)
(2) According to the meaning of the question, the daily sales.
Keywords formula, acquisition,
When x=5, y gets the maximum value 1800.
A: When the price of each commodity is reduced to 5 yuan, the daily sales volume can be maximized, and the maximum sales volume is l 800 yuan.
(25) (this small problem 10)
Solution: (I)∫ Point A (3 3,0). B (0，4)。 Get 0A=3，OB = 4。
∴ in Rt△ABO comes from Pythagorean theorem, AB=5.
According to the meaning of the question, there is DA=OA=3
As shown in figure 1. The intersection d is the DM⊥x axis at point m,
Then MD∨ob.
∴△ADM∽△ABO。 Yes,
get
OM=OA-AM，om =。
∴ The coordinate of point D is ()
(ii) As shown in Figure 2. From what we know, we get ∠CAB=α, AC=AB,
∴∠ABC=∠ACB.
△ ∴ in ABC, which is defined by ∠ ABC+∠ ACB+∠ cab = 180.
α = 180—2 ∠ ABC，。
And ∵BC∨x axis, ∠ OBC = 90,
∠ ABC = 90-∠ ABO = 90-β。
∴α=2β.
(iii) The analytical formula of linear CD is, or.
(26) (this small problem 10)
Solution (I)*,
The vertex coordinate of a parabola is ().
(2) ① According to the meaning of the question, you can get point A (0, 1).
∫F( 1， 1)。
∴AB∥x axis. AF=BF= 1，
2 established.
The reason for this is the following：
As shown in the figure, if PM⊥AB passes through point P () at point M, then FM=, PM= ().
∴Rt△PMF has Pythagorean theorem, so.
The point p () is on a parabola,
Get, that is
∴
Namely.
QN⊥B is defined as the intersection point q (), and the extension line with AB intersects with point n,
This is also the case.
Graphics ∠ PMF = ∠ QNF = 90, ∠MFP=∠NFQ,
∴△PMF∽△QNF
have
Here,
∴
that is
(3) order,
Let the abscissa of the intersection of its image and parabola be, and
A parabola can be regarded as a parabola that translates from left to right.
Observe the image. As the parabola moves to the right, the value of increases.
When,. The constant remains the same, and the maximum value of m is.
Where appropriate, the corresponding value is the maximum value of m.
Therefore, it will be brought into,
Have a solution or (give up)
∴
At this point.
It is found that the maximum value of ∴m is 8.