volume one

First, multiple choice questions

The reciprocal of 1 -3 Yes

A.- BC -3d. 3

2. Calculate (x2y)3, and the result is correct.

a . x5yb . x6yc . x2y3d . x6y 3

3. Among the four figures of equilateral triangle, square, diamond and isosceles trapezoid, there are figures with central symmetry.

A. 1

4. It is known that the radius of ⊙O is r, and the distance from the center of O to the straight line L is d. If the straight line L intersects with ⊙O, the following conclusion is correct.

a . d = Rb . d≤RC . d≥rd . d < r

5. When solving the fractional equation by substitution method, if it is set, the general form of transforming the original equation into a quadratic equation with one variable about y is

A.B.

C.D.

6. As shown in figure 1, in rectangular ABCD, e, f, g and h are the midpoints of AB, BC, CD and DA respectively. If ab = 2 and ad = 4, the area of the shaded part in the figure is

a . 3b . 4

c . 6d . 8

7. In a closed circuit, the power supply voltage is constant, and the current I(A) is inversely proportional to the resistance R (ω). Fig. 2 shows the functional relationship between current I and resistance R in this circuit, so the resolution function of current I represented by resistance R is

A.B.

C.D.

8. The French "Little Ninety-nine" is the same as China's "Little Ninety-nine", from "one by one" to "five by five", and the latter is changed to gesture. The following two frameworks use French "Xiao Jiu Jiu" to calculate two examples of 7×8 and 8×9. If 7×9 is calculated by the French word "Xiao Jiu Jiu", the index of the outstretched hands of the left and right hands is

A.2，3b . 3，3c . 2，4d . 3，4

9. There is such a fable in ancient times: Donkeys and mules walk together. They carry different bags of goods, each with the same weight. The donkey complained that the burden was too heavy. The mule said, "Why are you complaining? If you give me a bag, I will bear twice your weight; If I give you a bag, we will bring as much! " So the number of cargo bags that this donkey initially carried was

a . 5b . 6c . 7d . 8

10. A rope is bent into the shape shown in Figure 3- 1. As shown in Figure 3-2, when the rope is cut along the dotted line A with scissors, the rope is cut into five sections; As shown in Figure 3-3, when the rope is cut again along the dotted line b(b∑a) with scissors, the rope is cut into 9 sections. If the rope between dotted lines A and B is cut (n- 1) times with scissors (the direction of scissors is parallel to A), then the number of rope segments when this * * * is cut n times is

a . 4n+ 1b . 4n+2c . 4n+3d . 4n+5

Volume II

Second, fill in the blanks

1 1. It is known that the altitude of A is 300m and that of B is-50m, so A is m higher than B. 。

12. As shown in Figure 4, straight lines A∨b and C intersect with A and B. If ∠ 2 = 1 15, then ∠ 1 =.

13. Biologists found that the length of a virus is about 0.000 043mm, and the result of scientific counting method is 0.000 043.

14. divide a right angle n into equal parts, each part is 15, then n is equal to.

15. Decomposition factor =.

16. As shown in Figure 5, the short arm length of the railway railing is 1.2m, and the long arm length is 8m. The endpoint of the short arm drops by 0.6m, and the endpoint of the long arm rises by m (ignoring the thickness of the bar).

17. The solution set of the inequality group is.

18. Calcining limestone (CaCO3) at high temperature will produce quicklime (CaO) and carbon dioxide (CO2). If impurities and losses are not considered, it takes 25 tons of limestone to produce 14 tons of quicklime and 1 10,000 tons of limestone to produce 2.24 million tons of quicklime.

19. After the price of a drug is reduced twice, the price per box is reduced from the original 60 yuan to 48.6 yuan, so the average percentage of each price reduction is.

20. As shown in Figure 6, it is known that the generatrix length OA=8 of the cone is 8 and the radius r of the earth circle is 2. If a bug starts from point A, crawls around the side of the cone, and then returns to point A, the shortest path length of the bug crawling is (the result keeps the root formula).

Third, answer questions.

2 1. Known, the value of.

22. As shown in Figure 7, D is a point on the AB side of △ABC, and ABC and DF intersect with AC at point E, with de = ef.

Proof: AE = ce.

23. In order to check whether the size of an iron ball produced by the factory meets the requirements, the master designed the workpiece groove as shown in Figure 8- 1, in which the two bottom angles of the workpiece groove are 90, and the size is as shown in the figure (unit: cm).

Put an iron ball with regular shape into the groove, as shown in Figure 8- 1. If there are three contact points A, B and E at the same time, the size of the ball meets the requirements.

Fig. 8-2 is a schematic cross-sectional view of the ball center O and three contact points A, B and E. It is known that the diameter ⊙O is the diameter of the iron ball, AB is the chord of ⊙O, and CD is cut at the point E of bd⊥cd. ac⊥cd ⊙ O. Please refer to the data in fig. 8- 1 Calculate the diameter of this iron ball

24. In order to understand the physical training situation of athletes A and B, they were followed up and tested, and the test results for ten consecutive weeks were drawn into a broken-line statistical chart as shown in Figure 9. The coaching staff stipulates that the physical fitness test score of 70 or above (including 70 points) is qualified.

(1) Please fill in the following table according to the information provided in Figure 9:

Average median physical fitness test

Passing times

Jia 65

B 60

(2) Please judge the physical fitness test results of these two athletes from the following two different angles:

① Comparing A and B according to the average and the times of passing the test, their physical fitness test scores are better;

② Comparing A and B according to the average and median, their physical fitness test scores are better.

(3) According to the statistical chart of broken lines and the times of passing grades, analyze which athlete's physical training effect is better.

25. In the candle burning test, the relationship between the height y (cm) of the remaining parts of two candles A and B and the burning time x (hours) is shown in figure 10. Please answer the following questions according to the information provided in the picture:

(1) The height of two candles before burning is, and the time from lighting to burning is.

(2) Find the functional relationship between Y and X when candles A and B burn respectively;

(3) How long does it take to burn, and the height of two candles A and B is equal (regardless of the situation of all burning out)? In which event segment, candle A is higher than candle B? In what period, candle A was lower than candle B?

26. Example of operation

For two squares ABCD and EFGH with side length A, place them as shown in figure 1 1- 1. After cutting along the dotted lines BD and EG, it can be spliced into a quadrilateral BNED as shown in the figure.

It is easy to draw a conclusion from the sewing process:

① The quadrilateral BNED is a square;

②S-squared ABCD+S-squared EFGH = S-squared BNED.

Practice and inquiry

(1) For two squares b(a>bCD and EFGH, the side lengths are A and B (A > B) respectively, and they are placed as shown in figure 1 1-2, and connected with de. The intersection point D is DM⊥DE, the intersection point AB is M, and the intersection point M is Mn ⊥.

① Prove that quadrilateral MNED is a square, and express the area of square MNED with an algebraic expression containing A and B;

② In figure 1 1-2, the square ABCD and the square EFGH can be spliced into a square MNED after being cut along the dotted line. Please briefly explain your splicing method (for example, figure 1 1- 1, where numbers represent corresponding figures).

(2) For n(n is a natural number greater than 2) arbitrary squares, can they be spliced into a square several times? Please briefly explain your reasons.

27. A machinery leasing company has 40 machines of the same model. After running for a period of time, it was found that when the monthly rent of each machine and equipment was 270 yuan, it was just rented out. On this basis, when the monthly rent of each set of equipment increases by 65,438+00 yuan, one set of equipment will be rented out less, and a set of equipment that is not rented out will have to pay monthly fees (maintenance fees, management fees, etc.). ) 20 yuan. Assume that the monthly rent of each set of equipment is X (yuan), and the monthly income (income = rental income-expenses) of the leasing company leasing this type of equipment is Y (yuan).

(1) The number of equipment (sets) not rented out and the cost of all equipment (sets) not rented out are expressed by an algebraic expression with X.

(2) Find the quadratic function relation between y and x;

(3) The monthly rent is 300 yuan and 350 yuan respectively. What is the monthly income of the leasing company? How many sets of mechanical equipment should be rented at this time? Please briefly explain the reasons;

(4) Please formulate the quadratic function obtained in (2) into a table, and explain accordingly: What is the highest monthly income of the leasing company when X is worth? What is the highest monthly income?

28. As shown in figure 12, in the right-angled trapezoidal ABCD, AD∑BC, ∠ C = 90, BC = 16, DC = 12, and AD = 2 1. The moving point p starts from the point d and moves at a speed of 2 units per second in the direction of ray DA. The moving point Q starts from the point C and moves to the point B at the speed of 1 unit per second on the line segment CB. Point p and point q start from point d and point c respectively. When point Q moves to point B, point P stops moving. Let the time of movement be t (seconds).

(1) Let the area of △BPQ be s, and find the functional relationship between s and t;

(2) When t is what value, is a triangle with vertices b, p and q an isosceles triangle?

(3) When the line segment PQ and the line segment AB intersect at point O, and 2ao = ob, find the tangent value of ∠BQP;

(4) Is there a time t for PQ⊥BD? If it exists, find the value of t; If it does not exist, please explain why.

Mathematical answers to the college entrance examination in Hebei province in 2005

First, multiple choice questions

The title is 1 23455 6789 10.

Answer d d d d b b b b c a a

Second, fill in the blanks

1 1.350 12.65 13.4.3× 10-5 14. 12 15.(x+y)(x-y+a)

16.4 17.< x < 4 18.400 19. 10% 20。

Third, answer questions.

2 1. solution: original formula =

When x =, the original formula =

22. Proof: ∫ab∨fc, ∴∠ ADE = ∠ CFE.

* aed =∠cef，DE = Fe，∴△AED≌△CEF.

∴AE=CE

23. Solution: Connect OA and OE so that OE and AB meet at point P, as shown in the figure.

∵AC=BD,AC⊥CD,BD⊥CD

∴ Quadrilateral ABDC is a rectangle.

∵CD and⊙ O are tangent to point E, and OE is the radius of⊙ O,

∴OE⊥CD

∴OE⊥AB

∴PA=PB

∴PE=AC

∵AB=CD= 16,∴PA=8

AC = BD = 4 PE = 4

In Rt△OAP, from Pythagorean theorem,

that is

∴: The solution is OA = 10, so the diameter of this iron ball is 20 cm.

24. Solution:

Average median physical fitness test

Passing times

A 60 65 2

57.5 4

(1) See table.

(2)(2)①B； 2 a。

(3) From the line chart, both athletes' physical fitness test scores showed an upward trend, but the growth rate of B was faster than that of A, and B passed the test more times than A in the later period, so the training effect of B was better.

25. Solution: (1)30 cm, 25 cm; 2 hours, 2.5 hours.

(2) Let the functional relationship between y and x when the nail candle burns be. As can be seen from the figure, the image of the function is solved through points (2,0), (0,30) and ∴.

∴ y=- 15x+30

Let the functional relationship between y and x when candle b burns be. As can be seen from the figure, the image of the function is solved through points (2.5,0), (0,25) and ∴.

∴ y=- 10x+25

(3) From the meaning of the question-15x+30 =- 10x+25, the solution is X = 1. Therefore, when burning 1 hour, the height of two candles is equal.

Observing the image, we can know that candle A is higher than candle B when 0 ≤ x < 1; When 1 < x < 2.5, candle A is lower than candle B.

26. Solution: (1)① Prove that quadrilateral MNED is a rectangle from the drawing process.

At Rt△ADM and Rt△CDE,

AD = CD, and ADM+MDC = CDE+MDC = 90,

∴ DM = DE, ∴ quadrilateral MNED is a square.

∵ ,

What is the area of the square?

② The intersection point n is NP⊥BE and the vertical foot is P, as shown in Figure 2.

It can be proved that two triangles at 6 and 5, two triangles at 4 and 3, and two triangles at 2 and 1 are congruent.

So put 6 in the position of 5, 4 in the position of 3, and 2 in the position of 1, just splicing into a square MNED.

A: Yes.

The reason is: As can be seen from the above splicing process, any two squares can be spliced into a square, and the spliced square can be spliced into a square with the third square, and so on. Therefore, for n arbitrary squares, a square can be obtained by (n- 1) times of stitching.

27. Solution: (1) One set of equipment is not rented out, and the cost of all equipment not rented out is (2x-540) yuan;

(2)

(3) When the monthly rent is 300 yuan, the monthly income of the leasing company is 1 1040 yuan, and 37 sets of equipment are leased out at this time; When renting 350 yuan monthly, the monthly income of the leasing company is 1 1040 yuan, and 32 sets of equipment are rented out at this time. Because the income from renting 37 sets of equipment is the same as that from renting 32 sets of equipment, if we consider reducing the loss of equipment, we should choose to rent 32 sets; If you consider market share, you should choose 37 sets;

(4)

When x = 325, the maximum value of y is 1 1 102.5. But when the monthly rent is 325 yuan, the number of rented equipment is 34. 5, and 34.5 is not an integer, so the rented equipment should be 34 or 35. That is, when the monthly rent is 330 yuan (34 sets are leased) or 320 yuan (35 sets are leased), the monthly income of the leasing company is the largest, with the highest monthly income of1100 yuan.

28. The solution (1) is shown in Figure 3. If the intersection point P is PM⊥BC and the vertical foot is M, then the quadrilateral PDCM is a rectangle. ∴PM=DC= 12

∵ QB =16-t, ∴ s =×12× (16 tons) =96 tons.

(2) As can be seen from the figure, cm = PD = 2t, CQ = t .. The triangles with vertices B, P and Q are isosceles triangles, which can be divided into three situations:

(1) If pq = bq. At Rt△PMQ, from PQ2=BQ2, t =;

② If BP = BQ. In Rt△PMB, BP2=BQ2:

Namely.

Because δ =-704 < 0

∴ No solution, ∴PB≠BQ

③ If Pb = PQ. From PB2=PQ2, we get

Tidy up, please. Solve (to no avail, give up)

As can be seen from the above discussion, when t = seconds, a triangle with three vertices B, P and Q is an isosceles triangle.

(3) As shown in Figure 4, it is obtained from △OAP∽△OBQ.

∵ap=2t-2 1,bq= 16-t,∴2(2t-2 1)= 16-t。

∴t= .

Q is QE⊥AD, and the vertical foot is e,

∵PD=2t,ED=QC=t,∴PE=t。

At RT△PEQ, tan ∠QPE= =

(4) Let time t exist so that PQ⊥BD. As shown in Figure 5, point Q is QE⊥ADS, and the foot is E, which is obtained from Rt△BDC∽Rt△QPE.

, that is. The solution is t = 9.

Therefore, when t = 9 seconds, PQ⊥BD.

20 10 Hebei junior high school graduates entrance examination mathematics examination paper

I. Multiple-choice questions (This big question consists of * *12 small questions, with 2 points for each small question and 24 points for each small question). Only one of the four options given in each small question meets the requirements of the topic.

1. The result of calculating 3×( 2) is

a5b . 5c . 6d . 6

2. As shown in figure 1, in △ABC, d is a point on the extension line of BC.

∠ B = 40, ∠ ACD = 120, then ∠A is equal to

Answer 60

80-90 AD

3. In the following calculation, it is correct that

A.B. C. D。

4. As shown in Figure 2, in □ABCD, AC is divided into △ ∠DAB, AB = 3,

So the circumference of □ABCD is

A.6 B.9

c 12d 15

5. Put inequality

6. As shown in Figure 3, in a 5×5 square grid, an arc passes through points A, B and C,

So the center of this arc is

A. point Pb Q, c, r, d and m.

7. The result of simplification is that

A. BC 1

8. Abortion needs money from 48 yuan to buy books, and just used 65438+ 12 banknotes from 0 yuan and 5 yuan for payment. Assuming that the used 1 yuan banknote is X, the following equation is correct according to the meaning of the question.

A.B.

C.D.

9. A ship travels between A and B on the same route. It is known that the speed of the ship in still water is 15 km/h and the current speed is 5 km/h. The ship first sails from A to B, stays in B for a while, and then sails back to A against the current. Let's assume that the time for the ship to leave A is t(h) and the sailing distance is s.

10. As shown in Figure 4, the side lengths of two regular hexagons are both 1, and one side of one regular hexagon is just on the diagonal of the other regular hexagon, then the perimeter of the outer contour of the figure (shaded part) is

A.7 B.8

c . 9d . 10

1 1. As shown in Figure 5, the axis of symmetry of the parabola is called point A,

B is on the parabola, and AB is parallel to the X axis, where the coordinates of point A are

(0,3), the coordinates of point B are

A.(2，3)b .(3，2)

C.(3，3)d .(4，3)

12. Put the cube dice (the opposite points are 1 and 6, 2 and 5 respectively,

3 and 4) placed on a horizontal desktop, as shown in Figure 6- 1. In Figure 6-2, dice are placed.

Scroll 90 to the right, and then rotate 90 counterclockwise on the desktop, and you're done.

A change. If the initial position of the dice is as shown in Figure 6- 1, press.

After completing the 10 transformation of the above rules continuously, the number of points on the upper side of the dice is

a6 b . 5 c . 3d . 2

2. Fill in the blanks (there are 6 small questions in this big question, 3 points for each small question, *** 18 points. Write the answer on the horizontal line of the question)

/kloc-the reciprocal of 0/3. Yes.

14. As shown in Figure 7, vertices A and B of rectangular ABCD are on the number axis, CD = 6, and the number corresponding to point A is, then the number corresponding to point B is.

15. In the game of guessing the price of goods, the participant didn't know the price of goods in advance, so the host asked him to take any card from the four cards in Figure 8 and connect the remaining cards from left to right into a three-digit number, which is the price he guessed. If the price of a commodity is 360 yuan, then the probability that he can guess at a time is.

16. It is known that x = 1 is the root of a quadratic equation with one variable, then the value of is.

17. As shown in Figure 9, the space illuminated by street lamps can be considered as a cone. Its height ao = 8m, and the included angle between bus AB and bottom radius OB is,

Then the bottom area of the cone is square meters (the result π still exists).

18. Stack three square cards A, B and C with the same size on the bottom of a box with a square bottom, and the part of the bottom that is not covered by the cards is indicated by shading. If placed as shown in figure 10- 1, the area of the shaded part is s1; If the shaded area is S2, as shown in figure 10-2, then S 1 S2 (fill in ">", "

Third, the solution (this big question is ***8 small questions, ***78 points. The solution should be written in words, proof process or calculus steps)

19. (The full mark of this small problem is 8) Solve the equation:

20. As shown in figure 11,the square ABCD is a schematic diagram of an electronic screen with a 6 × 6 grid, in which the side length of each small square is1. The spot p at the midpoint of AD moves according to the program in Figure 1 1-2.

(1) Please draw the path of the light spot P in the figure11;

(2) Find the total path length of the light spot P (the result keeps π).

2 1. (The full score of this small question is 9) Both schools participated in the oral English contest for students organized by the District Education Bureau, and the number of participants in both schools was equal. After the contest, it was found that the students' scores were 7, 8, 9, 10 (full marks were 10). According to the statistical data, the following incomplete statistical map is drawn.

Scores 7, 8, 9, 10.

Number of people 1 108

(1) In the figure 12- 1, the central angle of the sector where "7 o'clock" is located.

Equal to 0

(2) Please complete the statistical chart of Figure 12-2.

(3) After calculation, the average score of school B is 8.3, and the median is 8. Please write down the average score and median of school A; And analyze which school has better grades from the perspective of average score and median.

(4) If the Education Bureau wants to organize a team of 8 people to participate in the city team competition, it is decided to choose a player from these two schools to participate in the competition for the convenience of management. Please analyze which school should I choose?

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

As shown in figure 13, in the rectangular coordinate system, the vertex o of the rectangular OABC coincides with the coordinate origin, the vertices A and C are on the coordinate axis respectively, and the coordinate of the vertex B is (4,2). Straight lines passing through points D (0 0,3) and E (6 6,0) intersect points AB and BC at points M and N, respectively.

(1) Find the analytical formula of straight line DE and the coordinates of point M;

(2) If the image of the inverse proportional function (x > 0) passes through the point m, find the analytical expression of the inverse proportional function, and judge whether the point n is on the image of the function through calculation;

(3) If the image of inverse proportional function (x > 0) has something in common with △MNB, please write the range of m directly.

23. (Full score for this small question 10)

Observe and think

Figures 14- 1 and 14-2 show some mechanical devices driven in the same plane.

It's a schematic diagram. Its working principle is that the slider q can be on the linear slider L.

Sliding left and right, in the process of Q sliding, the connecting rod PQ also moves, and

PQ drives the connecting rod OP to swing around the fixed point O. During the swing, the contact point P of the two connecting rods moves on ⊙O with OP as the radius, which is being further studied by the Mathematics Interest Group.

Investigate the mathematical knowledge contained in it, do OH ⊥l from H point to O point, and measure it.

OH = 4 decimeters, PQ = 3 decimeters, OP = 2 decimeters.

solve problems

(1) The minimum distance from point Q to point O is decimeter;

The maximum distance between point Q and point O is decimeter;

Point q slides between the leftmost position and the rightmost position on L.

The distance is decimeter.

(2) As shown in figure 14-3, Xiao Ming said, "When point Q slides to the position of point H"

PQ and ⊙O are tangent. "Do you think his judgment is correct?

Why?

(3)① Xiaoli found: "When point P moves to OH, point P goes to L..

In fact, there is another point where the distance from P to L is the largest.

At this time, the distance from point P to L is decimeter;

② When the OP swings around the O point, the swept area is fan-shaped.

Find the degree of the central angle when the sector area is the largest.

24. (Full score for this small question 10)

In the figure 15- 1 to 15-3, the straight line MN intersects the line segment AB.

At o point ∠ 1 = ∠ 2 = 45.

(1) as shown in figure 15- 1, if AO = OB, please write down AO and BD.

The relationship between quantity and position;

(2) Rotate MN in the diagram 15- 1 clockwise around the O point to get the result.

Figure 15-2, where AO = OB. ..

Verification: AC = BD, AC ⊥ BD; ;

(3) lengthen the OB in figure 15-2 to k times that of AO.

The value of figure 15-3.

25. (The full score of this short question is 12)

As shown in figure 16, in the right-angled trapezoidal ABCD, AD∑BC, AD = 6, BC = 8, and point m is the midpoint of BC. Point P starts from point M, moves from MB to point B at a speed of 1 unit per second, and immediately returns to point B at the original speed. The point q starts from the point m and moves at a constant speed per second 1 unit length on the ray MC. During the movement of point P and point Q, an equilateral triangle EPQ is made with PQ as the edge, so that it and trapezoidal ABCD start on the same side of ray BC. When point P returns to point M, it stops moving and point Q stops moving.

The time for P and Q to move is t seconds (t > 0).

(1) Let the length of PQ be y. When point P moves from point M to point B, write the functional relationship between y and t (it is not necessary to write the range of t).

(2) When BP = 1, find the overlapping area of △EPQ and trapezoidal ABCD.

(3) With the change of time t, a part of the line segment AD will be covered by △EPQ, and the length of the covered line segment will reach the maximum at a certain moment. Please answer: Can this maximum value last for some time? If yes, write the range of t directly; If not, please explain why.

26. (The full score of this short question is 12)

A company sells a new energy-saving product, and now it is ready to choose one of the two sales schemes at home and abroad.

If it is only sold in China, the functional relationship between the sales price y (yuan/piece) and the monthly sales volume x (piece) is y = x+ 150.

Cost 20 yuan/unit. No matter how much you sell, you still need to spend 62500 yuan on advertising every month, and the monthly profit is within W (yuan) (profit = sales-cost-advertising fee).

If it is only sold abroad, the selling price is 150 yuan/piece. Due to various uncertainties, the cost is one yuan/piece (A is

Constant, 10≤a≤40). When the monthly sales volume is X (pieces), the monthly surcharge is x2 yuan, and the monthly profit is W (yuan) (profit = sales-cost-surcharge).

(1) when x = 1000, y = yuan/piece, W = yuan;

(2) Find the functional relationships among w, w and x respectively (it is not necessary to write the range of x);

(3) When is the value of X, and the monthly profit of domestic sales is the largest? If the monthly maximum profit of foreign sales is the same as that of domestic sales, find the value of A;

(4) If you want to sell all 5,000 products in one month, please help the company make a decision through analysis and choose whether to sell them at home or abroad, so as to make a bigger monthly profit.

Reference formula: the vertex coordinates of parabola are.

20 10 senior high school entrance examination for junior high school graduates in Hebei province.

Reference answers to math test questions

First, multiple choice questions

The title is123455678911112.

Answer d c c a b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

Second, fill in the blanks

13. 14.5 15. 16. 1 17.36 π 18.=

Third, answer questions.

19. Solution:

It is the solution of the original equation.

20. solution: (1) as shown in figure1;

Note: If students draw without compasses, and the drawn route is smooth and basically accurate, 4 points will be given.

(2)∵ ,

∴ The total length of the path passed by point P is 6π.

2 1. Solution: (1)144;

(2) As shown in Figure 2;

(3) The average score of a school is 8.3, and the median is 7;

Because the average scores of the two schools are equal, the average score of school B is greater than that of school A..

The median of the school, so from the average score and median,

The results of school B are better.

(4) A school won because eight students were selected to participate in the city oral English team competition.

Eight people took the 10 test, while only five people in school B took the 10 test, so school A should be chosen.

22. Solution: (1) Let the analytical formula of straight line DE be,

* The coordinates of points D and E are (0,3), (6,0), ∴.

Get a solution.

Point m is on the side of AB, B (4 4,2), and the quadrilateral OABC is a rectangle.

The ordinate of point m is 2.

Point m is on a straight line,

∴ 2 = .∴ x = 2。 ∴ M(2，2)。

(2) ∵ (x > 0) passes through points m (2 2,2), ∴. ∴.

Point n is on the side of BC, and the abscissa of point n is 4.

Point n is on a straight line, ∴.∴ n (4, 1).

When y = = 1, point n is on the image of the function.

(3)4≤ m ≤8。

23. Solution: (1) 456;

(2)No. 。

∵OP = 2, PQ = 3, OQ = 4, and 42≠32+22, that is, OQ2≠PQ2+OP2.

∴OP and PQ are not perpendicular. PQ and O are not tangent.

(3)① 3;

(2) from (1), there is a little p on ⊙O, and the distance to L is 3. At this point, the OP will not be able to rotate downward, as shown in Figure 3. The largest sector that OP sweeps in the process of swinging around point O is OP. 。

Connect p and cross OH at point D.

∫PQ, all perpendicular to L, and PQ =,

∴ Quadrilateral PQ is a rectangle. ∴ Oh ⊥ P, PD = D.

From OP = 2, OD = OH HD = 1, we get ∠ DOP = 60.

∴∠PO = 120。

The degree of the maximum central angle is 120.

24. solution: (1)AO = BD, ao ⊥ BD;

(2) Proof: As shown in Figure 4, point B is ∑ca and point E is ∴∠ACO = ∠BEO. ..

AO = OB，∠AOC = ∠BOE，

∴△aoc?△ BOE. ∴AC = BE。

∫≈ 1 = 45，∴∠ACO = ∠BEO = 135 ..

∴∠DEB = 45。

∠∠2 = 45，∴BE = BD，∠ EBD = 90。 ∴ AC = BD。 Extend the extension line of AC to DB at F, as shown in Figure 4. ∫be∑AC，∴∠ AFD。

(3) As shown in Figure 5, the intersection point B is be∨ca, DO is E, ∴∠BEO = ∠ACO. ..

And ? ≈BOE =∠AOC,

∴△BOE ∽ △AOC。

∴ .

OB = kAO，

Be = BD can be easily obtained by the method of (2).

25. Solution: (1) y = 2t; (2) When BP = 1, there are two situations:

① As shown in Figure 6, if point P moves from point M to point B, MB = = 4, MP = MQ = 3,

∴ PQ = 6。 Connect EM,

∵△EPQ is an equilateral triangle, ∴ EM ⊥ PQ.

Ab =, ∴ point e is on AD.

∴△EPQ and trapezoidal ABCD overlap with △EPQ, whose surface

The product is.

(2) If point P moves from point B to point M, it is derived from the question.

PQ = BM+MQ BP = 8, PC = 7 ... Let PE and AD intersect at point F, QE and AD or AD.

The extension line passes through point G, and point P is PH⊥AD at point H, then

HP =，ah = 1。 At Rt△HPF, ∠ HPF = 30,

∴HF = 3，PF = 6。 ∴FG = iron = 2 ... and ∵FD = 2,

As shown in Figure 7, Point G and Point D coincide. At this time △EPQ and trapezoidal ABCD.

The overlapping part of is trapezoidal FPCG with an area of.

(3) can .4 ≤ t ≤ 5。

26. Solution: (1)140 57500;

(2) within w = x (y-20)-62500 = x2+ 130x,

W = x2+( 150) x .

(3) When x = = 6500, it is the largest in W; minute

Judging from the meaning of the question,

The solution method is a 1 = 30 and a2 = 270 (irrelevant, discarded). So a = 30 ..

(4) When x = 5000, W = within 337500, and W = outside.

If w is less than w, then a is less than a < 32.5；;

If W = w, then a = 32.5；;

If within w > outside w, a > 32.5.

Therefore, when 10 ≤ A < 32.5, choose to sell abroad;

When a = 32.5, sales abroad and at home are the same;

When 32.5< a ≤40, choose to sell in China.