mathematics

Note: 1. The whole volume is ***8 pages, with 28 questions, and the full score is 120, and the examination time is 120 minutes.

2. Fill in the items in the sealed line clearly before answering the questions, and fill in the seat number in the position specified in the test paper.

Fill in the answers directly on the test paper with a blue or black pen or ballpoint pen.

4. Candidates can use CZ 1206 and HY82 function calculators in the process of answering questions. If the calculation results of the test questions are not required to be approximate, the calculation results are accurate (the sum of the root numbers is reserved).

1. Fill in the blanks (each blank in this big question is 1, *** 18. Fill in the answers on the lines in the questions)

The reciprocal of 1 Yes, the absolute value of is, and the number equal to the cube is.

2. The coordinates of the point about the axisymmetric point are; The coordinates of a point symmetrical about the origin are.

3. If, the complementary angle is 0,

4. In the campus singer contest, seven judges rated a singer as follows: 9.8, 9.5, 9.7, 9.6, 9.5, 9.5, 9.6, so the average value of this set of data is, and the interval is.

5. Given a sector with a radius of 2cm, an area of 0, an arc length of cm and a central angle of 0.

6. If the mirror passing point of the linear function is known, then.

7. As shown in the figure, it is known that,,,

Then,,.

8. Some corresponding values of quadratic function are as follows:

…

…

…

…

The symmetry axis of quadratic function image is the corresponding function value.

Second, multiple-choice questions (the following questions give four answers, code A, B, C and D, and one and only one is correct. Fill in the code () of the correct answer at the end of the question, 2 points for each small question, *** 18 points)

9. Among the following real numbers, the irrational number is ()

A.B. C. D。

10. In the function, the range of independent variables is ().

A.B. C. D。

1 1. Among the following symmetrical figures, the figure with the least number of symmetrical axes is ().

A. circle B. regular hexagon C. square D. equilateral triangle

12. There are three red balls and two white balls in the schoolbag. If/kloc-0 balls are randomly drawn from the bag, the probability of drawing white balls is ().

A.B. C. D。

13. As shown in the figure, the image (dotted line) describes the functional relationship between the speed and time of the car during driving, and the following statement is incorrect ().

A the speed in the third minute is 40 km/h.

B the speed is 0km/h 12.

C From the 3rd minute to the 6th minute, the car went 120km.

D From the 9th minute to12nd minute, the vehicle speed decreased from 60km/h to 0 km/h..

14. The following figure consists of six squares with the same size, in which () can be folded into a cube along the sides of the square.

15. Xiaoming and Xiaoli were born in 1998 12. Their birthdays are not on the same day, but they are both Friday. Xiaoming was born earlier than Xiaoli. The sum of their birth dates is 22, so Xiaoli's birth date is ().

15， 16， 17， 18

16. If the image of the quadratic function (constant) is as follows, the value of is ().

A.B. C. D。

17. As shown in the figure, in,, and, the motion circle passing through the point and the motion circle tangent to the edge intersect with the point respectively, so the minimum length of the line segment is ().

A.B. C. D。

Third, solve the problem (this big problem is two small problems, * *** 18 points. Solve the steps to write calculus)

18. (Full score for this small question 10) Simplification:

( 1) ; (2) .

19. (Full score for this small problem) Solve the equation:

( 1) ; (2) .

Fourth, answer the question (this big question is ***2 small questions, *** 12 points. The answer should be written in the proof process)

20. (The full score for this short question is 5)

As shown in the figure, in, the bisector of intersects with the point.

Verification:

2 1. (The full score for this short question is 7)

As we all know, as shown in the figure, the extended sides are connected in turn to form an equilateral triangle.

Verification: (1);

(2) It is an equilateral triangle.

Verb (abbreviation of verb) solving problems (this big question is ***2 small questions, *** 15 points. The solution should be written in words or calculus steps)

22. (The full score for this short question is 7)

Figure 1 is a broken-line statistical chart of the daily minimum temperature in a city from February 5 to June 5, 2007.

(1) Figure 2 is the frequency distribution histogram of daily maximum temperature in Japan from February 5, 2007 to February 4, 2007. According to the information provided in figure 1, the frequency distribution histogram in figure 2 is completed.

(2) In this 10 day, the mode of the lowest temperature is, the median is, and the variance is.

23. (The full score for this short question is 8)

There are two small balls in the pocket, marked with numbers and respectively; There are three balls in the pocket, marked with numbers and. Every ball is the same except the number. A and B play games and randomly take out 1 balls from each pocket. If the sum of the numbers on the two balls is even, A wins. If the sum is odd, B wins. Is this game fair to both sides? Please explain the reason.

Vi. Exploration and reference (this big topic is ***2 small topics, *** 13 points)

24. (The full score for this short question is 6)

As shown in the figure, the shapes of diamonds, rectangles and squares are all different. We call the closeness of diamonds, rectangles and squares "proximity". When studying "proximity", we should ensure that the similarity of similar graphics is equal.

(1) Let the degrees of two adjacent inner angles of a diamond be and respectively, and define the "proximity" of the diamond as, therefore, the smaller the diamond is, the closer it is to a square.

① If the inner angle of the diamond is 0, the "proximity" of the diamond is equal to;

② When the "proximity" of the diamond is equal to, the diamond is a square.

(2) Let the lengths of two adjacent sides of the rectangle be and () respectively, and define the "proximity" of the rectangle as, so the smaller it is, the closer it is to the square.

Do you think this statement is reasonable? If it is unreasonable, give a reasonable definition of rectangle "approximation".

25. (The full score for this short question is 7)

It is known that the image passing through four points and a linear function is a straight line, which intersects the axis at that point.

(1) is drawn in the plane rectangular coordinate system on the right, and the coordinates of the intersection of the straight line and are;

(2) If there is an integral point (a point whose abscissa and ordinate are integers is called an integral point), it is an isosceles triangle, and the coordinates of all points that meet the conditions are:

(3) When the unit is translated to the right along the axis, it is tangent to.

Seven, solve the problem (this big problem ***3 small problems, ***26 points. The answer should be a written explanation, proof process or calculus step)

26. (The full score for this short question is 7)

The school held the "Welcome to the Olympics" knowledge contest, and won the first, second and third prizes *** 12. The prize distribution scheme is as follows:

First prize, second prize, third prize

1 Boxfuwa 1 badge 1 Boxfuwa 1 badge

The total cost of purchasing prizes is not less than 1000 yuan, but not more than 1 100 yuan. Before buying Fuwa and stamps, Xiao Ming learned the following information:

(1) How much is a box of Fuwa plus badges?

(2) If there are two first prizes in this activity, how many second prizes and third prizes should there be?

27. (The full score for this short question is 9)

As we all know, as shown in the figure, the side length of a square is 6, and the three vertices of a diamond are connected to the side of the square.

(1) when, the area;

(2) Let be the area represented by the contained algebraic expression;

(3) Determine whether the areas are equal and explain the reasons.

28. (Full score for this small question 10)

The known sum is an inverse proportional function of two points on the image.

( 1);

(2) If there is a point, is there a point on the inverse proportional function image, which makes the quadrilateral with four vertices a trapezoid? If it exists, find out the coordinates of the point; If it does not exist, please explain why.

Changzhou junior high school graduation entrance examination in 2007.

Reference answers and grading standards of mathematics test questions

1. Fill in the blanks (every space 1 minute, *** 18)

1., , ; 2., ; 3., ; 4.9.6,0.3;

5., ; 6., ; 7., , ; 8., .

Second, multiple-choice questions (this big topic ***9 small questions, 2 points for each small question, *** 18 points)

Title: 9101121314151617.

Answer b c d b c d b c d b c d bCD BCD B

Third, the solution (this big question is ***2 questions, 18 questions 10 points, 19 questions 8 points, *** 18 points. Solve the steps to write calculus)

18. Solution: (1) The original formula scored 3 points.

.5 points

(2) Deduct 2 points for the original formula

3 points

4 points

.5 points

19. Solution: (1) is divided by the denominator. 1.

Solution, .2 points

Prove to be the root of the original equation.

The root of the original equation is 0.4 point.

(2), 2 points

.3 points

, .4 points

Fourth, solve the problem (this big question is ***2 small questions, the 20th question is 5 points, 2 1 question is 7 points, *** 12 points. The proof process should be written for the solution).

2 1. Prove that the quadrilateral is a parallelogram.

. 1 point

Divide equally, 0.2 points

.3 points

.4 points

0.5 points again

2 1. Proof: (1),. 1.

This is an equilateral triangle, 0.2 points.

0.4 points again

② From, get,

, is an equilateral triangle,

,

Similarly, you can get 0.5 points.

Medium, 0.6 points

This is an equilateral triangle. 7 points

Verb (abbreviation of verb) solving problems (7 points in question 22, 8 points in question 23, *** 15 points)

22.( 1) Draw correctly. 2 points

7℃, 7.5℃, 2.49(℃)2 (mode 1, median 2, variance 2) .7.

23. Solution: Draw a tree diagram or list;

3 4 5

1 ( 1,3)

The total is 4 (1, 4)

The total is 5 (1, 5)

The total is 6.

2 (2,3)

The total is 5 (2,4)

The total is 6 (2,5)

The total is 7.

4 points

There are six possible situations for the sum of numbers * * *, of which three are even and three are odd.

Six points.

This game is fair to both sides. 8 points

Inquiry and extraction of intransitive verbs (6 points in question 24, 7 points in question 25, *** 13)

24. Solution: (1) ① 40. 2 points

② 0.4 points

(2) Unreasonable. For example, for two similar but unequal rectangles, they are the same as the square, but they are not equal. There is no unique definition, for example, the smaller the definition, the closer the rectangle is to the square; The bigger it is, the greater the shape difference between rectangle and square; When, the rectangle becomes a square. Six points.

25. Solution: (1) drawing, 0.3 points.

(2), .5 points

(3) .7 points

VII. Answer questions (26.7 points, 27.9 points, 28 10 points, ***26 points)

26. Solution: (1) Set a box of "Fuwa" yuan and a badge yuan, depending on the meaning of the question.

2 points

3 points for the solution

A: A box of Fuwa 150 yuan and a badge 15 yuan.

(2) If there are second and third prizes,

5 points

Get 0.6 points.

It's an integer,, .7 points

Answer: There are 4 second prizes and 6 third prizes.

27. Solution: In (1) square, …

Therefore, the side length of a diamond is also the same.

In and,,

, ,

. .

, ,

In other words, a diamond is a square.

It can also be proved.

So, that is, the point is on the side and available at the same time,

Therefore. 2 points

(2) used for hanging feet and connecting,

, ,

, .

.

In and,,,

.

That is, no matter how the diamond changes, the distance from the point to the straight line is always a constant value of 2.

So 0.6 points

(3) If, by, get, at this time, in,.

Accordingly, there is, that is, the point is no longer on the side.

So it is impossible to get a score of .9.

Another method: because the point is on the side, the side length of the diamond is at least,

When the side length of the diamond is 4, the point is on the side, satisfying. At this point, when the point moves to the right gradually, the length of the diamond (that is, the side length of the diamond) will gradually increase, with the maximum value of.

So at this time.

And the function value decreases with the increase of,

Therefore, when, the minimum value is.

Because, therefore, the area of cannot be equal to 1.9 points.

28. solution: (1) By, get, so .2 points.

(2) As shown in figure 1, if the axis is vertical, then,,, therefore.

Because the abscissa of the point is the same as the abscissa of the point, that is, the axis, therefore.

When it is the bottom, because there is only one common point between the straight line passing through and parallel to this point and the hyperbola,

So it doesn't meet the problem. 3 points

When it is the bottom, the parallel line passing through this point intersects the hyperbola at this point.

The intersection point is the parallel lines of the axis and the axis, which intersect at the point.

Because, if, then,

Start from the main points and get the main points.

Therefore,

Get the solution (give up), so get the point.

At this point, the lengths of sum are not equal, so the quadrilateral is a trapezoid. Five points.

As shown in Figure 2, when it is the bottom, the intersection of the parallel line passing through this point and the hyperbola in the first quadrant is.

Because, therefore, as an axis, it is a vertical foot.

So, if, then,

From one point to another,

Therefore.

Get the solution (give up), so get the point.

At this point, the lengths of sum are not equal, so the quadrilateral is a trapezoid. Seven points

As shown in fig. 3, when the intersection of the parallel line passing through this point and the hyperbola in the third quadrant is,

Similarly, points and quadrilaterals are trapezoid. 9 points

To sum up, there are some points on the function image, which makes the quadrilateral with four vertices become a trapezoid, and the coordinates of the points are: or. 10 points.