mathematics

Matters needing attention in mathematics:

1. This paper is ***6 pages, with full mark 120, and the examination time 120 minutes. The candidates' answers are all on the answer sheet, and the answers on this test paper are invalid.

2. Please carefully check whether all the names and test certificate numbers pasted by the invigilator on the answer sheet are consistent with me, and then fill in my name and admission ticket number on the answer sheet and this paper with 0.5mm black ink pen.

3. For multiple-choice questions, the corresponding answer label on the answer sheet must be blacked out with 2B pencil. If you need to change, please clean it with an eraser first, and then paint other answers. Non-multiple choice questions must be written in the designated position on the answer sheet with 0.5mm black ink pen, and the answers in other positions will be invalid.

4. The drawing must be answered with 2B pencil, and please describe it clearly in black and bold.

1. Multiple choice questions (This big question is ***6 small questions, each with 2 points, *** 12 points. Only one of the four options given in each small question meets the requirements of the topic. Please fill in the letter code of the correct option in the corresponding position on the answer sheet).

The value of 1. be equal to

a . 3 B- 3 c . 3d

2. The following operation is correct

A.a2+a3=a5 B.a2？ a3=a6 C.a3÷a2=a D.(a2)3=a8

According to the sixth national census, the permanent population of Nanjing is about 8 million, of which 9.2% are people aged 65 or above. Then the population aged 65 and over in this city is expressed by scientific notation as about

A.0.736× 106 people B.7.36× 104 people C.7.36× 105 people D.7.36× 106 people.

4. In order to know the eyesight of a junior high school student, some students need to be selected for investigation. The following methods are the most suitable.

A. Randomly select students from a class in this school.

B. Randomly select students of one grade in our school.

C. Randomly select some boys in our school.

D. Randomly select 10% students from each class of Grade One, Grade Two and Grade Three in our school.

5. The picture shows a triangular prism. In the picture below, what can be folded into a triangular prism is

6. as shown in the figure, in the plane rectangular coordinate system, if the center ⊙P is (2, a) (a > 2), the radius is 2, and the image of function y=x is the length of the chord AB of ⊙P, then the value of a is

A.B. C. D。

2. Fill in the blanks (this topic is entitled *** 10, with 2 points for each topic and 20 points for * * *, and there is no need to write the answer, please fill in the answer directly in the corresponding position on the answer sheet).

The reciprocal of 7. -2 is _ _ _ _ _.

8. As shown in the figure, if the vertex A of the regular pentagon ABCDE is a straight line l∨CD, then ∠1= _ _ _ _ _.

9. Calculation = _ _ _ _ _ _ _ _ _.

10. If the waist length of an isosceles trapezoid is 5㎝ and the circumference is 22㎝, the length of its midline is _ _ _ _ _ _ \

1 1. As shown in the figure, draw an arc with O as the center and any length as the radius, intersect with ray OM at point A, then draw an arc with A as the center and AO as the radius, and draw ray OB when two arcs intersect at point B, then the value of cos∠AOB is equal to _ _ _ _ _ _ _ _.

12. As shown in the figure, the company commander of rhombic ABCD is 2㎝, e is the midpoint of AB, and DE⊥AB, so the area of rhombic ABCD is _ _ _ _ _ _ _ _ \ 2.

13. As shown in the figure, there are two lighthouses A and B at the seaside, and the reefs are distributed in the arc area passing through point A and point B (arc arc is a part of ⊙O), and ∠ AOB = 80. In order to avoid hitting the rocks, the maximum opening angle ∠APB of P, A and B is _ _.

14. As shown in the figure, e and f are the points on the side BC and CD of the square ABCD respectively, and BE=CF connects AE and BF. Turn △ABE to △BCF counterclockwise around the center of the square, and the rotation angle is a (0 < A < 180), then ∠ a.

15. Let the meshing coordinates of the image of the function sum be (a, b), then the value of is _ _ _ _ _ _ _.

16.A, B, C and D form a circle and count off in turn. It is stipulated that:

① The numbers quoted by Party A, Party B, Party C and Party D for the first time are 1, 2, 3 and 4, followed by Party A 5 and Party B 6 ... According to this rule, the number reported by the last classmate is greater than that reported by the previous classmate 1, and the number of reports ends at 50;

(2) If the quoted number is a multiple of 3, the student who quoted the number needs to clap his hands once. In this process, the number of times a student needs to clap his hands is _ _ _ _ _ _ _ _.

Three. Solution (this big question is * *12, and the score is ***88. Please answer in the designated area of the answer sheet, and write a written explanation, proof process or calculus steps when answering.

17.(6 points) Solve the inequality group and write the integer solution of the inequality group.

18.(6 points)

19.(6 points) Solve the equation X2-4x+ 1 = 0.

20.(7 points) Some boys in a school are divided into three groups for pull-up training, and the results before and after the training are statistically analyzed. The statistical chart of the corresponding data is as follows.

(1) found that the average score of the first group increased after training compared with that before training;

(2) Xiao Ming claimed to have found a mistake after analyzing the chart: "The number of boys in the second group has not changed after training, accounting for 50% of the whole group, so it is impossible to increase the average number of boys in the second group by three." Do you agree with Xiao Ming? Please explain the reasons;

(3) Which group do you think has the best training effect? Please give an explanation to support your point of view.

2 1.(7 points) As shown in the figure, extend the side DC of □ABCD to point E, so that CE=DC, then AE, and then BC to point F. 。

(1) Verification: △ ABF △ ECF

5] If ∠AFC=2∠D, connect AC and BE. Prove that the quadrilateral ABEC is a rectangle.

22.(7 points) Xiaoying and Liang Xiao went up the mountain to play. Xiaoying took the cable car and Liang Xiao walked. They met at the end of the cable car at the top of the mountain. It is known that the distance from Liang Xiao to the end of the cable car is twice as long as that from the cable car to the top of the mountain. Xiaoying takes the cable car 50 minutes after leaving Liang Xiao, and the average speed of the cable car is 180 m/min. Let Liang Xiao walk on1000m. ..

(1) Xiao Liang walked _ _ _ _ _ _ _ kilometers in total, and he had a _ _ _ _ _ _ _ minute rest on the way.

(2) ① When 50≤x≤80, find the functional relationship between Y and X;

(2) When Xiaoying reaches the end of the cable car, what is the distance between Liang Xiao and the end of the cable car?

23.(7 points) 20 14 Nanjing Youth Olympic Games volunteers were randomly selected from 3 boys and 2 girls. Find the probability of the following events:

(1) Choose 1 student, who happens to be a girl;

(2) Two students were selected, which happened to be boys 1 and girls 1.

24.(7 points) Known function y = mx2-6x+ 1 (m is constant).

(1) Verification: No matter what the value of m is, the image of this function passes through a fixed point on the Y axis;

⑵ If the image of the function has only one intersection with the X axis, find the value of m. 。

25.(7 points) As shown in the figure, a math extracurricular activity group measures the height of the TV tower AB, and they measure it with the help of a 30-meter-high building CD. At point C, the elevation of tower B is 45, and at point E, the elevation of tower B is 37 (B, D and E are in a straight line). Find the height h of TV tower.

(Reference data: sin37 ≈0.60, cos37 ≈0.80, tan37 ≈0.75)

26.(8 points) As shown in the figure, in Rt△ABC, ∠ ACB = 90, AC=6㎝, BC=8㎝, and P is the midpoint of BC. The moving point q starts from the point p and moves at a speed of 2㎝/s along the direction of the ray PC with the point p as the center.

(1) When t= 1.2, judge the positional relationship between straight line AB and ⊙P, and explain the reasons;

⑵ O is known as the circumscribed circle of △ABC. If υp is tangent to υo, find the value of t 。

27.(9 points) As shown in figure 1, P is a point in △ABC, connecting PA, PB and PC. In △PAB, △PBC and △PAC, if there is a triangle similar to △ABC, then P is called the self-similarity point of △ABC.

(1) As shown in Figure ②, it is known that in Rt△ABC, ∠ ACB = 90, ∠ ACB > ∠ A, CD is the center line on AB, intersection B is ⊥ CD, and vertical foot is E. Try to explain that E is the self-similarity point of △ABC.

(2) In △ABC, ∠ A < ∠ B < ∠ C.

① As shown in Figure ③, use a ruler to make the self-similarity point P of △ABC (write out the practice and keep the drawing traces);

② If the internal P of △ABC is the self-similar point of the triangle, find the degrees of the three internal angles of the triangle.

28.( 1 1)

Problem situation

It is known that the area of a rectangle is a(a is a constant and a > 0). When the length of a rectangle is what, its circumference is the smallest? What is the minimum value?

mathematical model

Let the length of a rectangle be x and the circumference be y, then the functional relationship between y and x is.

exploratory research

⑴ We can learn from the previous experience in studying functions, and discuss the image properties of functions first.

① Fill in the table below and draw the image of the function:

②

x

……

1

2

three

four

……

y

……

……

② Observe the image and write two different types of properties of the function;

③ when finding the maximum (minimum) value of the quadratic function y = ax2+bx+c (a ≠ 0), we can find the minimum value of the function (x > 0) by observing the image.

solve problems

⑵ Use the above methods to solve the problems in the "problem situation" and write the answers directly.

Answer:

First, multiple-choice questions: ACCDBB

Two. Fill in the blanks:

7.2 8.36 9.10.61.12.13.4014.9015.16.

17. Solution:

Solve inequality ①:

Solve inequality ②:

Therefore, the solution set of the inequality group is.

The integer solution of the inequality group is 0, 1.

18.

19. Solution 1: shift item, get.

Keywords formula, acquisition,

From this, you can get

Solution 2:

,.

20. Solution: (1) After training, the average score of the first group was 67% higher than that before training.

(2) I don't agree with Xiao Ming, because the average score of the second group increased by 8×10%+6× 20%+5× 20%+0× 50% = 3.

(3) The answer to this question is not unique. I think the training effect of the first group is the best, because the average score of the first group after training is greater than that before training.

2 1. Prove: (1) (4) Quadrilateral ABCD is a parallelogram, ABCD, ab = CD. ?? ABF = ?。

* EC = ∴ab=ec. SAR

In △ABF and △ECF, ∫∠ABF =∠ECF, ∠AFB=∠EFC, AB=EC,

∴⊿ABF≌⊿ECF.

(2) the solution 1:∫ab = EC, ABEC, ∴ quadrilateral abec is a parallelogram. ∴ af = ef，BF = cf

∵ Quadrilateral ABCD is a parallelogram, ∴∠ABC=∠D, and ∵∠AFC=2∠D, ∴∠ AFC = 2 ∠ ABC.

∵∠AFC=∠ABF+∠BAF,∴∠ABF=∠BAF.∴FA=FB.

∴ FA = Fe = FB = FC，∴ AE = BC。 Abeck is rectangular.

Solution 2:∫ab = EC, ABEC, ∴ quadrilateral abec is a parallelogram.

∵ quadrilateral ABCD is a parallelogram, ∴AD∥BC, ∴ D = ∠ BCE.

And ∵∠AFC=2∠D, ∴∠AFC=2∠BCE,

∵∠AFC=∠FCE+∠FEC,∴∠FCE=∠FEC.∴∠D=∠FEC.∴AE=AD.

* ce = DC，∴ AC ⊥ DE。 That is, ∠ ACE = 90. Abeck is rectangular.

22. Solution (1) 3600, 20.

At this time, let the functional relationship between y and x be.

According to the meaning of the question, at that time,; When?

Therefore, the functional relationship with is.

② The length from the cable car to the top of the mountain is 3600÷2= 1800 (),

The time required for the cable car to reach its destination is1800 ÷180 =10 ().

When Xiaoying reached the end of the cable car, Liang Xiao left 10+50 = 60 ().

If substituted, y = 55× 60-800 = 2500.

Therefore, when Xiaoying arrives at the cable car terminal, the distance from Liang Xiao to the cable car terminal is 3600-2500 = 1 100 ().

23. The probability that the students who know (1) 1 happen to be girls is.

⑵ The five students were represented by male 1, male 2, male 3, female 1 and female 2, and two students were randomly selected. All possible outcomes are: (male 1, male 2), (male 1, female 3), (male 1. (male 3, female 1), (male 3, female 2), (female 1, female 2), *** 10, their possibilities are the same. In all the results, it is enough to choose 2, which is exactly 1 boys and 1 girls (.

24. Solution: (1) When x=0,

So no matter what value is taken, the image of the function passes through a fixed point (0, 1) on the axis.

(2) At that time, there was only one intersection point between the image and the axis of the function;

② At this time, if the image of the function has only one intersection with the axis, the equation has two equal real roots, so.

To sum up, if the image of the function has only one intersection with the axis, the value of is 0 or 9.

25. yes, =

∴EC=≈().

In ∴, ∠ BCA = 45

Yes, = ∴.∴ ().

A: The height of the TV tower is about 120.

26. Solution (1) The straight line is tangent to ⊙ p. 。

As shown in the figure, the intersection point P is PD⊥AB, and the vertical foot is D.

In Rt△ABC, ∠ ACB = 90, ∫AC = 150 px, BC=200px,

∴.∫p is the midpoint of BC, ∴ Pb = 100px.

∠∠PDB =∠ACB = 90，∠PBD=∠ABC。 ∴△PBD∽△ABC.

∴, that is ∴ PD = 2.4 (cm).

At that time, (cm)

∴, that is, the distance from the center of the circle to the straight line is equal to the radius ⊙ P.

A straight line is tangent to p.

5] ⑸ACB = 90°, ∴AB is the diameter of the circumscribed circle of △ABC.

The connection point op .∫p is the midpoint of ∴. BC province.

∵ Point P is within ⊙O, ∴⊙P and ⊙O can only be inscribed.

∴ or ∴= 1 or 4.

When ∴⊙P is tangent to O, the value of t is 1 or 4.

27. Solution (1) When Rt △ABC, ∠ ACB = 90, CD is the midline on AB, ∴, ∴ CD = BD.

∴∠BCE=∠ABC.∵BE⊥CD,∴∠BEC=90 ,∴∠BEC=∠ACB.∴△BCE∽△ABC.

∴E is the self-similarity of △ABC.

(2) sketch.

The practice is as follows: (i) in ∠ABC, make ∠ CBD = ∠ A;

(ii) in ACB, BCE = ∠ ABC; BD intersects CE at point p.

Then p is the self-similarity of △ABC.

② Connect PB and PC. ∫p is the heart of ∴, △ ABC.

∵P is the self-similarity point of △ABC, ∴△BCP∽△ABC. ..

∴∠PBC=∠A,∠BCP=∠ABC=2∠PBC =2∠A，

∠ACB = 2∠BCP = 4∠A .∠A+∠ABC+∠ACB = 180。

∴∠A+2∠A+4∠A= 180。

The degrees of the three inner angles of a triangle are,, respectively.

28. Solution (1) ①,,,, 2,,.

The image of this function is shown in the figure.

The answer to this question is not unique. The following solutions are for reference.

At that time, it was increasing and decreasing; At that time, it rose with the increase; The minimum value of the function at that time was 2.

③

=

=

=

When =0, the minimum value of the function is 2.

When the length of a rectangle is, its perimeter is the smallest, and the minimum value is.

20 12 mathematics examination questions in Nanjing, Jiangsu province

First, multiple-choice questions (this topic is entitled ***6 small questions, with 2 points for each small question, *** 12 points)

1, the following four numbers, the negative number is

A.B. C. D。

2.PM 2.5 refers to particulate matter with a diameter less than or equal to 0.0000025 m in the atmosphere, and 0.0000025 is expressed by scientific symbols as follows

A.B. C. D。

3. The result of calculation is

A.B. C. D。

4. The negative square root of12 is between

A.-5 and -4, B- 4 and -3, c-3 and -2, d-2 and-1.

5. If there is no intersection between the images of inverse proportional function and linear function, the value of can be

A.-2 B. - 1 C. 1 D. 2

6. As shown in the figure, in the diamond-shaped paper ABCD, when the paper is folded, point A and point D fall at a' and d' respectively, A'D' passes through b, and EF is the crease. When D'FCD, the value is

A.B. C. D。

II. Fill in the blanks (this big question * * 10 small question, 2 points for each small question, 20 points for * * *)

7. The meaningful range of values is

8, the result of the calculation is

9, the solution of the equation is

10, as shown in figure,,, are the four outer corners of pentagonal ABCDE, if, then.

1 1. If the image of the linear function is known to pass through point (2,3), the value of is

12. The following functions are known: ① ② ③, in which the image of the function can be obtained by translation (fill in the serial numbers of all correct options).

13. The annual salary details of all employees in a company are as follows:

Annual salary/ten thousand yuan

30

14

nine

six

four

3.5

three

Number of employees/person

1

1

1

2

seven

six

2

Then the average annual salary of all employees is 10 thousand yuan more than the median.

14 as shown in the figure, put it on a scale, the vertex o coincides with the endpoint of the lower edge of the scale, OA coincides with the lower edge of the scale, and the intersection point b of OB and the upper edge of the scale reads 2cm on the scale. If you put it on the scale in the same way, the intersection point C between OC and the upper edge of the scale will read about cm on the scale.

(The result is accurate to 0. 1 cm, reference data:,,)

15, as shown in the figure, in the parallelogram ABCD, AD= 10cm, CD=6cm, E is a point above AD, BE=BC, CE=CD, then DE= cm.

16, (6 points) In the plane rectangular coordinate system, it is stipulated that a triangle should be folded along the X axis first, and then translated by two units to the right as the primary transformation. As shown in the figure, the coordinates of vertices B and C of equilateral triangle ABC are (-1,-1) and (-3, -65438) respectively.

Third, solve the problem (this big problem * *11,***88 points)

17, (6 points) to solve the equation

18, (9 points) Simplify the algebraic expression and judge the symbol of the algebraic expression when x satisfies the inequality group.

19, (8 points) as shown in the figure, in the right triangle ABC, the point D is on the extension line of BC, BD=AB, and B is BEAC, which intersects with the vertical line DE of BD at the point E.

(1) Verification:

(2) Rotate the triangle ABC to get the triangle BDE, and use a ruler to make the rotation center O (leaving traces without writing).

20.(8 points) There are 450 seventh-grade students in a middle school, including 250 boys and 200 girls. The school conducted a physical education test for all students in Grade 7, and immediately analyzed the test results of 50 boys and 40 girls as samples, and obtained the following statistics:

achievement

frequency

per cent

fail

nine

10%

pass

18

20%

okay

36

40%

excellent

27

Thirty percent

total

90

100%

(1) Please explain the rationality of "randomly selecting 50 boys and 40 girls";

(2) Select a column from the data of "frequency" and "percentage" in the above table, and represent it with an appropriate statistical chart;

(3) Estimate the number of students who failed the physical education examination in grade seven.

2 1, (7 points) Four students A, B, C and D play a badminton singles match. Two students should be selected to play the first game and find out the probability of the following events.

(1) The first game of A has been determined, and then 1 student is randomly selected from the other three students, just to select the second student;

(2) Randomly select 2 students, including student B. 。

22.(8 points) As shown in the figure, in trapezoidal ABCD, AD//BC, AB=CD, diagonal AC and BD intersect at point O, and ACBD, E, F, G and H are the midpoint of AB, BC, CD and DA respectively.

(1) Verification: quadrilateral EFGH is a square;

(2) If AD=2 and BC=4, find the area of the quadrilateral EFGH.

23.(7 points) Look at the pictures and tell stories.

Please make up a story, so that a pair of variables X and Y in the story situation satisfy the functional relationship shown in the diagram. The requirements are: ① point out the meaning of X and Y; ② Use the data in the diagram to illustrate the practical significance of this to the process of variable change, in which the design quantity "speed" is needed.

24.(8 points) A toy consists of a circular area and a fan-shaped area, as shown in the figure. In the fan-shaped area, it is tangent to two intersections of straight lines of A, B, E and F, EF=24cm, and the radius is x cm.

① Represent the radius of the sector with an algebraic expression containing x;

(2) If the manufacturing cost of two sectors is 0.45 yuan and 0.06 yuan respectively, what is the minimum cost of toys when the radius is 0?

25.(8 points) A car sales company sold a manufacturer's car in June. Within a certain range, the purchase price of each car has the following relationship with the sales volume. If only 1 car is sold in that month, the purchase price of this car is 270,000 yuan. For every additional car sold, the purchase price of all cars sold will be reduced by 0. 1 10,000 yuan. At the end of the month, the manufacturer will give the sales company a one-time rebate according to the sales volume. The sales volume is less than 10, of which 10, each with a rebate of 5,000 yuan, and each with a rebate of 10 yuan.

(1) If the company sells three cars in that month, the purchase price of each car is 10000 yuan;

(2) If the car sales price is 280,000 yuan/car, and the company plans to make a profit of 1.2 million yuan in that month, how many cars will it sell? (Profit = sales profit+rebate)

26.(9 points) " "thinking

The box below is Xiao Ming's answer to a question and the teacher's correction.

My results are also correct.

Xiao Ming found that his solution was correct, but the teacher drew a horizontal line on his solution and opened a "?"

Why is the result correct?

(1) Please point out the problems in Xiao Ming's solution and supplement the missing process:

What will happen if you change? ...

(2) As shown in the figure, the rectangle is inside the rectangle, and the distances between and, and, and and are respectively. What are the conditions for making a rectangle? Please provide a justification for the answer.

27.( 10) As shown in the figure, A and B are two fixed points on the gyro, and P is the moving point on the gyro (P does not coincide with A and B), which is called the sliding angle of A and B on the gyro.

(1) is called the sliding angle of point A and point B.

① If AB is the diameter, then

② If the radius is 1 and AB=, find the degree.

(2) Known as the outer point, make the center of the circle intersect with points A ANd B, which is the sliding angle of points A and B, and straight lines PA and PB intersect with points M and N respectively (point M and point A and point N do not coincide), connecting an, and trying to explore the quantitative relationship between and.