2008-20 10 Tianjin Mathematics Entrance Examination.

20 10 Tianjin junior high school graduates' academic examination papers

mathematics

This paper is divided into two parts: the first volume (multiple choice questions) and the second volume (non-multiple choice questions). Page 1 in volume 1 to page 3, and page 4 to page 8 in volume 2. The full mark of the test paper is 120. Examination time 100 minutes.

Before answering the questions, candidates must fill in their names, candidates' numbers, test sites, examination room numbers and seat numbers on the "answer sheet", and paste the bar code of the test in the specified position. When answering questions, be sure to scribble the answers on the "answer sheet". The answer on the test paper is invalid. After the exam, return this paper together with the "answer sheet".

I wish all candidates good luck in the exam!

The first volume (multiple choice questions ***30 points)

Precautions:

After selecting the answer for each question, use 2B pencil to blacken the information points on the "Answer Sheet" corresponding to the answer label of the question. If you need to change it, clean it with an eraser, and then select the information point with other answer labels.

1. Multiple-choice question: This topic is entitled *** 10, with 3 points for each question and 30 points for each question. Only one of the four options given in each small question meets the requirements of the topic.

The value of (1) equals to

(1)

(2)

(3)

(2) The following figures can be regarded as both axisymmetric figures and centrally symmetric figures.

(A) (B) (C) (D)

(3) The Shanghai World Expo is the first comprehensive Expo held in China. According to statistics, 1, 2065438 was opened in May.

From the end of the curtain to May 3 1, the cumulative number of visitors was about 8.03 million, which was expressed by scientific notation.

it should be

(1)

(2)

(3)

(4)

(4) In a shooting competition, the average scores of two athletes A and B 10 are all 7 rings, in which the variance of A's score is 1.2 1, and the variance of B's score is 3.98.

(a) A is more stable than B.

(B)B's performance is more stable than A's.

(d) It is impossible to determine whose performance is more stable.

(5) The picture on the right is a three-dimensional figure composed of four identical cubes, and its three views are as follows.

(A) (B)

(6) The following proposition is correct

A quadrilateral with equal diagonal lines is a diamond.

(b) A quadrilateral with diagonal lines perpendicular to each other is a diamond.

A parallelogram with equal diagonal lines is a diamond.

Parallelograms with diagonal lines perpendicular to each other are diamonds.

(7) As shown in the figure, in ⊙O, the chord and intersect at the point, and if,, it is equal to.

(1)

(2)

(3)

(4)

(8) Comparing the sizes of 2, 0 and 0, it is correct that

(1)

(2)

(3)

(4)

(9) As shown in the figure, it is a schematic diagram of an ancient timer-"Leaky Pot". The pot contains a certain amount of water, and the water leaks from a small hole under the pot. The scale is painted on the wall of the pot, and people calculate the time according to the position of the water surface in the pot. If it is expressed by time and the height from the bottom of the pot to the water surface, the following figure is suitable for expressing the functional relationship between and in a short time (regardless of the influence of water quantity change on pressure).

(A) (B)

(10) Given the image of the quadratic function (), the following conclusions are drawn:

① ;

② ;

③ ;

④ .

Among them, the number of correct conclusions is

1 (B)2

20 10 Tianjin junior high school graduates' academic examination papers

mathematics

Volume 2 (non-multiple choice questions ***90 points)

Precautions:

Write the answers on the "answer sheet" in black ink with a pen or signature pen.

Fill in the blanks: this big question is ***8 small questions, each with 3 points and ***24 points.

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

(12) It is known that the image of the sum of linear functions intersects a point.

Then the coordinates of this point are.

(13) As shown in the figure, it is known that points A, D, B and F are in one.

On a straight line, you need to add a condition to make △△.

This condition will do.

(14) As shown in the figure, it is known that the side length of a square is 3, which is a point on the side.

Turn △ clockwise around the point.

Delta, connected, the length of is equal to.

(15) There are three ping-pong balls in box A, numbered 1, 2, 3 respectively. Box B contains 2 ping-pong balls, marked as

1, 2. Now we randomly take out 1 balls from each box, so the probability that the sum of the labels of the two balls taken out is 4 is.

(16) Some corresponding values of independent variables and function values in the known quadratic function () are as follows:

…

Then the analytic expression of quadratic function is.

(17) As shown in the figure, in an equilateral triangle, there are,, and on each side.

The points of,, and intersect at the point,

The value of is.

(18) There is a rectangular piece of paper ABCD, which is folded according to the following steps:

Step 1: As shown in Figure 1, fold the rectangular piece of paper, so that point B and point D coincide, and point C falls on this point to obtain the crease EF;

Step 2: As shown in Figure 2, fold the Pentagon so that AE and AE overlap to get the crease DG, and then open it;

Step 3: Fold further, as shown in Figure ③, so that AE and QP both fall on DG, point A and point F fall on point, and point E and point F fall on point, to obtain creases MN and QP.

In this way, a Pentagon can be folded.

(1) Please write a set of equal line segments in Figure ① (only one set);

(ii) If the Pentagon DMNPQ thus folded (as shown in Figure ③) happens to be a regular Pentagon, the following conclusions can be drawn:

① ; ② ;

③ ; ④ .

Among them, the serial number of the correct conclusion is (fill in the serial number of the correct conclusion you think).

Third, the solution: this big question is ***8 small questions, with ***66 points. The solution should be written in words, calculation steps or proof process.

(19) (6 points for this small question)

Solving inequality system

(20) (8 points for this small question)

The inverse proportional function (constant) is known.

(i) If the point is on the image of the function, the value;

(ii) If on each branch of this function image, it decreases with the increase of, then the value range is;

(iii) If yes, try to determine whether the point is on the image of the function and explain the reason.

(2 1) (8 points for this small question)

China is one of the countries with serious water shortage in the world. In order to advocate "saving water starts with me", Xiao Gang randomly surveyed the average monthly water consumption (unit: t) of 0/0 students in his class, and plotted the survey results into the following histogram.

(i) Find the average, mode and median of this 10 sample data;

(2) According to the sample data, it is estimated that there are about 50 households whose monthly water consumption does not exceed 7 t in Xiaogang class.

(22) (8 points for this small question)

It is known that the diameter of ⊙ is the tangent of ⊙, which is the tangent point and intersects ⊙.

(i) As shown in figure 1, if, and the length of the solution (the result retains the root sign);

(ii) As shown in Figure ②, if it is the midpoint, verify that the straight line is tangent to ⊙.

(23) (8 points for this small question)

Yongle Bridge Ferris wheel is one of the landmark landscapes in Tianjin. The math interest group of a school wants to measure the height of the Ferris wheel. As shown in the figure, they measure the elevation of the highest point A of the Ferris wheel at point C, then walk 50 m to D in the direction of the Ferris wheel, and measure the elevation of the highest point A as.

Find the height AB (,

The result is an integer).

(24) (8 points for this small question)

Note: In order to let students answer this question better, we provide a solution idea. You can fill in the blanks according to this idea and the following requirements to complete the solution of this problem. You can also choose other problem-solving schemes. At this time, there is no need to fill in the blanks, just answer according to the general requirements of solving problems.

The average yield of rice planted in Qingshan Village in 2007 and 2009 was 8000kg/hm2 and 9680kg/hm2, respectively, in order to understand the average annual growth rate of rice yield in this village.

Solution:

Suppose that the average annual growth rate of rice yield per hectare in this village is.

(i) Represented by an algebraic expression containing:

① The average rice yield per hectare in 2008 was:

② The average yield per hectare of rice planted in 2009 is:

(2) According to the meaning of the problem, list the corresponding equations;

(iii) Solve this equation and get:

(iv) Inspection:

(5) A: The average annual growth rate of rice yield per hectare in this village is%.

(25) (this small problem 10)

In the plane rectangular coordinate system, the vertex o of the rectangle is at the coordinate origin, and the vertices a and b are on the axis respectively.

On the positive semi-axis of the shaft, d is the midpoint of the OB side.

(i) If it is a moving point on the side, when the perimeter of △ is the smallest, find the coordinates of the point;

(ii) If the sum is two moving points on the side, when the perimeter of the quadrilateral is the smallest, find the coordinates of the sum of points.

(26) (this small problem 10)

In the plane rectangular coordinate system, it is known that the parabola intersects the axis at a point (the point is on the left side of the point), intersects the positive semi-axis of the axis at a point, and the vertex is.

(i) If the coordinates of the vertex of the parabola at this time are found;

(ii) Translate the parabola in (i) downwards, and if it is translated, it is satisfied in the quadrilateral ABEC.

S△BCE = S△ABC, and find the analytical formula of the straight line at this time;

(iii) Properly translate the parabola in (i), and if it is translated, it is satisfied in the quadrilateral ABEC.

S△BCE = 2S△AOC, and the vertex just falls on a straight line, find the analytical formula of parabola at this time.

20 10 Tianjin junior high school graduates' academic examination

Reference answers and grading standards of mathematics test questions

Rating description:

1. Each question is graded according to the reference answer and the grading standard.

2. If the candidate's answer to the multiple-choice question is not exactly the same as the reference answer, but it is reasonable, it can be graded as appropriate, but it shall not exceed the score assigned to the question.

First, multiple-choice questions: this big question * * 10 small questions, 3 points for each small question, * * 30 points.

( 1)A (2)B (3)C (4)A (5)B

(6)D (7)C (8)C (9)B ( 10)D

Fill in the blanks: this big question is ***8 small questions, each with 3 points and ***24 points.

( 1 1)

( 12)(3,0)

(13) (The answer is not unique, it can also be or)

( 14)

( 15)

( 16)

( 17)

(18) (Ⅰ) (The answer is not unique, but it can be equal); (Ⅱ)①②③

Third, answer: This big question is ***8 small questions, ***66 points.

(19) (6 points for this small question)

Solution:

Solve inequality ①, score .........................................................................................................................................................................

If you solve inequality ②, you will get ......................................................................................... 4 points.

∴ The solution set of the original inequality group is ................................................................. 6 points.

(20) (8 points for this small question)

Solution: (i) Point ∵ On the image of this function,

∴. Shed. .............................., two points.

(ii) on each branch of the function image, it follows,

∴. Shed. .............................. scored four points.

(iii) Yes.

∴ The analytical formula of inverse proportional function is.

By substituting the coordinates of points, we can know that the coordinates of points satisfy the functional relationship.

Focus on the image of the function.

Substituting the coordinates of points, we can see that the coordinates of points do not meet the functional relationship.

The focus is not on the image of the function. .............................. scored eight points.

(2 1) (8 points for this small question)

Solution: (1) Observing the histogram, we can know that the average value of this group of sample data is

The average value of this set of sample data is.

∵ In this set of sample data, it appeared four times, the most frequent.

The pattern of this set of data is.

∵ Arrange this set of sample data in order from small to large, in which the middle two numbers are both,

Yes,

The median of this set of data is. .............................. scored six points.

(2) Among the 0/0 households in Jiong/Kloc, there are 7 households whose monthly water consumption does not exceed 7 t..

Yes

According to the sample data, it can be estimated that among the 50 students in Xiaogang's class, there are about 35 households whose average monthly water consumption does not exceed 7 t. .............................. scored 8.

(22) (8 points for this small question)

Solution: (I) ∵ is the diameter of ∵, which is the tangent.

∴ .

In rt delta,,,

∴ .

From Pythagorean Theorem, .................. scored 5 points.

(ii) As shown in the figure, connect,

∵ is the diameter of ⊙,

Yes.

In rt delta, yes, the midpoint,

∴ .

Say it again,

∴ .

∵ ,

∴ .

Namely.

∴: This straight line is tangent to ⊙. .............................. scored eight points.

(23) (8 points for this small question)

Answer: according to the meaning of the question, it can be seen that,

At rt delta, from, from.

In Rt△, by,

.............................. scored six points.

Say it again,

That is ∴.

∴ .

A: The height of the Ferris wheel measured by the interest group is about118 m.. ..................... is 8.

(24) (8 points for this small question)

Solution: (Ⅰ) ①; ② ;

(Ⅱ) ; ........................, 4 points.

(Ⅲ) , ;

(Ⅳ), this is the root of the original equation, but it does not meet the meaning of the question, so it is only taken;

㈤ 10。 ........................ scored eight points.

(25) (this small problem 10)

Solution: (1) As shown in the figure, make point D a symmetrical point about the axis, connect the axis at point E, and connect them.

If you take any point on the edge (not coincident with point E), connect,,.

By,

It is known that the circumference of △ is the smallest.

In a rectangle,,, is the midpoint,

∴ , , .

In 700 BC,

∴ Rt△ ∽Rt△, yes.

∴ .

∴ The coordinate of this point is (1, 0). ................................ scored six points.

(2) As shown in the figure, make a symmetrical point about the shaft, cut it at the side, connect it with the shaft at this point, and cut it at the top.

∫GC‖EF，，

A quadrilateral is a parallelogram.

The length of sum is a constant value,

∴ The points obtained at this time minimize the perimeter of the quadrilateral.

In 700 BC,

∴ Rt△ ∽Rt△, yes.

∴ .

The coordinate of this point is (0), and the coordinate of this point is (0). ............... 10.

(26) (this small problem 10)

Solution: (1) When, when, the analytical formula of parabola is, namely.

The coordinate of the parabola vertex is (1, 4). ................. scored two points.

(ii) translating the parabola in (i) downwards, with the vertex on the axis of symmetry,