Math problems in the second day of junior high school

2.8/9 × 15/36 + 1/27

3. 12× 5/6 – 2/9 ×3

4.8× 5/4 + 1/4

5.6÷ 3/8 – 3/8 ÷6

6.4/7 × 5/9 + 3/7 × 5/9

7.5/2 -( 3/2 + 4/5 )

8.7/8 + ( 1/8 + 1/9 )

9.9 × 5/6 + 5/6

10.3/4 × 8/9 - 1/3

1 1.7 × 5/49 + 3/ 14

12.6 ×( 1/2 + 2/3 )

13.8 × 4/5 + 8 × 1 1/5

14.3 1 × 5/6 – 5/6

15.9/7 - ( 2/7 – 10/2 1 )

16.5/9 × 18 – 14 × 2/7

17.4/5 × 25/ 16 + 2/3 × 3/4

18. 14 × 8/7 – 5/6 × 12/ 15

19. 17/32 – 3/4 × 9/24

20.3 × 2/9 + 1/3

2 1.5/7 × 3/25 + 3/7

22.3/ 14 ×× 2/3 + 1/6

23. 1/5 × 2/3 + 5/6

24.9/22 + 1/ 1 1 ÷ 1/2

25.5/3 × 1 1/5 + 4/3

26.45 × 2/3 + 1/3 × 15

27.7/ 19 + 12/ 19 × 5/6

28. 1/4 + 3/4 ÷ 2/3

29.8/7 × 2 1/ 16 + 1/2

30. 10 1 × 1/5 – 1/5 × 2 1

3 1.50+ 160÷40 (58+370)÷(64-45)

32. 120- 144÷ 18+35

33.347+45×2-4 160÷52

34 (58+37)÷(64-9×5)

35.95÷(64-45)

36. 178- 145÷5×6+42 420+580-64×2 1÷28

37.8 12-700÷(9+3 1× 1 1)

38.85+ 14×( 14+208÷26)

39.(284+ 16)×(5 12-8208÷ 18)

40. 120-36×4÷ 18+35

4 1.(58+37)÷(64-9×5)

42.( 6.8-6.8×0.55)÷8.5

43.0. 12× 4.8÷0. 12×4.8

44.(3.2× 1.5+2.5)÷ 1.6

45.6- 1.6÷4= 5.38+7.85-5.37

46.7.2÷0.8- 1.2×5

47.6.5×(4.8- 1.2×4)

48. 1 0. 15- 10.75×0.4-5.7

49.5.8×(3.87-0. 13)+4.2×3.74

50.32.52-(6+9.728÷3.2)×2.5

1. In a hiking activity, students in a class were divided into two groups. The first group walked from a to b at a constant speed and returned the same way. The second group walked from a to b at a constant speed, continued to c, and returned the same way. The two groups set out at the same time, assuming that the walking time is t(h), and the distances between the two groups from B are S 1(km) and S2(km) respectively.

(1) The distance between Party A and Party B is _ _ _ _ _, and the distance between Party B and Party C is _ _ _ _ _ _ _ km;

(2) How long does it take for the second group to arrive at B for the first time from A and C respectively?

(3) Find the functional relationship between S2 and t represented by line segment AB in the figure, and write the range of independent variable T. 。

2. A large hotel in Kaili has 100 private rooms. During the business hours of dinner every day, when the charge for each private room is 100 yuan, all private rooms can be rented out. The charge for each private room is increased by 20 yuan, reduced by 10, and the charge for each private room is increased by 20 yuan, reduced by 10, thus changing the method of increasing 20 yuan every time.

(1) If the charge of each room is increased by X (yuan), the income of each room will be y 1 (yuan), but the room rent will be reduced by y2. Please write the functional relationships among y 1, y2 and x respectively.

(2) In order to get more income with less investment, after each private room is increased by X (yuan), the hotel owner assumes that the total income of eating in private rooms every day is Y (yuan). Please write down the functional relationship between Y and X, and find out how much each private room needs to increase its dinner every day to get the maximum room rate income, and explain the reasons.

3. The image of the functional relationship between the sales profit (ten thousand yuan) and the sales volume (ten thousand liters) of an oil product sold by a gas station in May is shown in the dotted line. On June 13, the sales profit of gas stations was 40,000 yuan, and on June 15, the sales profit was 55,000 yuan. (Sales profit = (price-cost price) × sales volume)

Please answer the following questions according to the information provided in the picture and all the sales records of this oil product at the gas station in May:

(1) When the sales volume is determined, the sales profit is 40,000 yuan;

(2) functional relationships corresponding to line segments AB and BC are obtained respectively;

(3) We call the profit from selling one liter of oil the profit rate. Then, in the sales information expressed in the third paragraph of OA. AB.BC, which one has the largest profit margin? (Write the answer directly)

4. In a transportation task, a car transports a batch of goods from place A to place B, and returns after unloading at place B. Suppose the car starts from place A at x(h), and the distance from the car to place A is y(km). The functional relationship between y and x is as shown in the figure.

Answer the following questions according to the picture information:

(1) Is this car at the same speed? Please explain the reasons;

(2) Find the function expression between y and x in the return journey;

(3) Find the distance to a place after the car leaves from a place for 4 hours.

Xiao Wang, the postman, set off from the county seat and rode his bike to deliver goods in Village A.. On the way, I met Li Ming, a county middle school student, and walked back to school. After completing the delivery work in Village A, Xiao Wang met Li Ming on his way back to the county seat, so he rode his bike and took Li Ming to the county seat. As a result, Xiao Wang arrived 1 minute later than expected. Their distance to the county seat (kilometers) and the time when Xiao Wang left the county seat (minutes).

(1) How many kilometers was it from the county seat when Xiao Wang and Li Ming first met? Please write the answer directly.

(2) The time it took Xiao Wang to return from the county seat.

(3) How long does it take Li Ming to get from Village A to the county seat?

6. At 8:00-8:30 on Sunday, the gas company injected natural gas into the gas storage tank of Ping 'an Gas Station. After that, a staff member added 20 cubic meters of gas to each car in turn.

Refueling many cars waiting in line at the gas station. Gas storage capacity in gas storage tank y

The functional relationship between (cubic meters) and time x (hours) is shown in Figure 2.

(1) How many liters did the gas company inject into the gas tank from 8: 00 to 8: 30?

Square meters of natural gas?

(2) When x is greater than or equal to x≥0.5, calculate the gas storage capacity y (m3) of the gas storage tank.

Resolution function with time x (hours);

(3) Please judge whether the waiting 18 car can finish filling up before 10: 30 on the same day. Please explain the reason.

7. Due to the state's key support for energy-saving and environmental protection industries, the sales market of an energy-saving product is gradually picking up. A dealer sells this product and signed a purchase contract with the manufacturer at the beginning of the year, stipulating that the purchase price will be 0. 1.0000 yuan/set within one year, and the down payment will be 50,000 yuan. He plans to reach a certain sales volume within one year, and the total purchase amount and deposit used to complete this sales volume should be controlled at not less than 340,000 yuan, but not more than 400,000 yuan. If the relationship between the selling price (ten thousand yuan/set) and the number of months in a year (integer) is as follows, it is found that there is a changing trend between the actual monthly sales volume (set) and the number of months after a year.

(1) directly write down the actual monthly sales (units) and the number of times per month.

Function relation of;

(2) Find the actual monthly sales profit of the first three months and the current month (ten thousand yuan).

Functional relationship between degrees;

⑶ Try to judge which month of the year has the highest price, and point out the highest price;

Please calculate whether he has completed the sales planned at the beginning of this year.

9. A station has a large passenger flow, so passengers often have to queue for a long time to buy tickets. According to the survey and statistics, about 300 passengers queue up to buy tickets at the beginning of ticket sales every day, and new passengers are constantly entering the ticket office to queue up to buy tickets. The functional relationship between the number of new ticket buyers y (person) and the ticketing time x (minute) is shown in Figure ①. The functional relationship between the number of votes y (person) and time x (minute) in each ticket window is shown in Figure ②. The functional relationship between the number of people waiting in line at the ticket office on a certain day y (people) and the ticketing time x (minutes) is shown in Figure ③. It is known that two ticket sales windows were opened in the first minute of ticket sales.

(1) Find the value of a;

(2) The number of passengers queuing at the ticket office at the 60th minute of ticket purchase;

(3) In studying and practicing Scientific Outlook on Development activities, the station decided to add a ticket window based on the principle of "people-oriented, convenient for passengers". If all the passengers queuing to buy tickets can buy tickets within half an hour after the ticket sales start, so that the passengers arriving at the station in the future can buy them at any time, please help us calculate how many ticket sales windows should be opened at least at the same time.

10. A refrigerator factory plans to produce 100 two types of refrigerators in response to the call of the country to "send home appliances to the countryside". According to the budget, after all the two models of refrigerators are sold, the profit can be no less than 47,500 yuan and no more than 48,000 yuan. The production cost and sales price of these two kinds of refrigerators are as follows:

Type a and type b

Cost (RMB/set) 2200 2600

Price (RMB/set) 2800-3000

(1) What kinds of production schemes does the refrigerator factory have?

(2) Which scheme produced by refrigerator factory has the lowest input cost? After "home appliances go to the countryside", farmers can enjoy a government subsidy of 13% when purchasing home appliances (refrigerators, color TVs and washing machines). How much does the government need to subsidize farmers under this plan?

(3) If the production is carried out according to the scheme in (2), the refrigerator factory plans to use all the profits to buy sports equipment, experimental equipment and office supplies to support a Hope Primary School. At most, buy four sets of sports equipment, each set of sports equipment is 6000 yuan, each set of experimental equipment is 3000 yuan, and each set of office supplies is 1.800 yuan. If all the money is used up and all three things are bought,

1 1. Class A and Class B of a certain army participated in tree planting activities. Class B planted 30 trees first, and then Class A began to plant trees with Class B. The total number of trees planted in Class A is Y A (trees), and the total number of trees planted in Class B is Y B (trees). The time for the two classes to plant trees together (from the beginning of planting trees in Class A) is (hours), y.

(1) When 0≤x≤6, find the functional relationship among Y A, Y B and x respectively. (3 points)

(2) If both Class A and Class B maintain the working efficiency in the first six hours, it is shown by calculation whether the sum of the total number of trees planted in Class A and Class B can exceed 260. (3 points)

(3) If, after six hours, Class A maintains the work efficiency of the first six hours, Class B improves the work efficiency by increasing the number of people, and continues to plant trees for two hours, the activity is over. When x=8, there are 20 trees difference between the two classes. How many trees will be planted per hour after the number of class B is increased? (4 points)

12. As shown in figure 1 1, a plane rectangular coordinate system is established on the grid paper, the two ends of the line segment AB are on the grid point, the straight line MN passes through the coordinate origin, and the coordinate of the point M is (1, 2).

(1) Write the coordinates of point A.B;

(2) Find the functional relationship corresponding to the straight line MN;

(3) Draw a line segment AB about the straight line MN with a ruler.

Symmetric graphics (keep drawing traces, don't write).

19. There is a conversion relationship between "shoe size" and shoe length (cm). The following table gives the corresponding values of several groups of "shoe size" and shoe length conversion: [Note: "shoe size" is a number representing shoe size]

Shoe length (cm) 16 19 2 1 24

Shoe Size (No.) 22 28 32 38

(1) Let the shoe length be x and the "shoe size" be y. Try to determine which function image the point (x, y) is on.

(2) Find the functional relationship between x and y;

(3) If someone wears shoes of size 44, how long are his shoes?

20. Buses and taxis of a bus company travel from Urumqi to Shihezi every day, and taxis travel more distances than buses, as shown in the figure, which shows the function image of the distance from Urumqi (unit: kilometers) and the time taken by taxis (unit: hours). It is known that the bus departs 1 hour later than the taxi and arrives in Shihezi.

(1) Please draw a function diagram of the distance (kilometers) and time (hours) from the bus to Urumqi.

(2) Find the number of times two cars meet on the way (write the answer directly)

(3) Find the distance from Urumqi when the two cars meet for the last time.

2.(20 10, Yiwu, Zhejiang) The following statement is incorrect.

A. A set of rectangles with equal adjacent sides is a square. B. A diamond with equal diagonal lines is a square.

C. A rectangle with diagonal lines perpendicular to each other is a square D. A parallelogram with right angles is a square.

1.(20 10, Wuhu, Anhui) The following propositions are correct and ().

A a quadrilateral with perpendicular and equal diagonals is a square.

Two triangles with two sides and an angle are congruent.

Two parallelograms with equal diagonals are rectangles.

D. the equilateral parallelogram is a diamond.

23. (Laiwu) In □ ABCD, AC and BD intersect at point O, the straight line passing through point O is EF and gh, and the four sides of the parallelogram intersect at four points E, G, F and H respectively, connecting EG, GF, FH and he.

(1) As shown in Figure ①, try to judge the shape of quadrilateral EGFH and explain the reasons;

(2) As shown in Figure ②, when EF⊥GH, the shape of quadrilateral EGFH is;

(3) As shown in Figure ③, under the condition of (2), if AC=BD, the shape of quadrilateral EGFH is;

(4) As shown in Figure ④, under the condition of (3), if AC⊥BD, try to judge the shape of quadrilateral EGFH and explain the reasons.

4. (Shandong Qingdao 20 10) It is known that in a square ABCD, points E and F are on BC and CD respectively, and AE = AF. ..

(1) verification: BE = DF；;

(2) Connect AC to EF at point O, extend OC to point M, make OM = OA, and connect EM and FM. What special quadrilateral is AEMF? And prove your conclusion.

2.(20 10 Qingdao, Shandong) Fold a rectangular piece of paper (rectangular ABCD) as shown in the figure, so that the vertex B coincides with the point D, and the crease is EF. If AB = 3 cm and BC = 5 cm, the area △DEF of the overlapping part is cm2. ..

(20 10 Zhongshan, Guangdong) 18. As shown in the figure, the right-angled side AC and the hypotenuse AB of Rt△ABC are made into equilateral △ACD and equilateral △ABE respectively. Known ∠BAC=30? ,

EF⊥AB, the vertical foot is f, even DF.

(1) Try to explain that AC = ef.

(2) Verification: Quadrilateral ADFE is a parallelogram.

(Changzhou, 20 10 year) 23. (The full mark of this small question is 7) As shown in the figure, in △ABC, AB=AC, D is the midpoint of BC, and the quadrilateral ABDE is a parallelogram. It is proved that the quadrilateral ADCE is a rectangle.

(Anhui 20 10) 20. As shown in the figure, AD‖FE, points B and C are on AD, ∠ 1 = ∠ 2, BF = BC.

(1) verification: quadrilateral BCEF is a diamond.

⑵ If AB = BC = CD, the verification is △ ACF △ BDE.

2.(20 10 Ningde) Full score of this question 13) As shown in the figure, the quadrilateral ABCD is a square, △ABE is an equilateral triangle, and m is any point on the diagonal BD (excluding point B). Rotate BM 60 counterclockwise around point B to get BN, connecting EN, AM and CM.

(1) verification: △ AMB △ enb;

(2) When point M is located, the value of AM+CM is the smallest;

② When the point M is located, the value of AM+BM+CM is the smallest, and the reasons are explained;

(3) When the minimum value of AM+BM+CM is 0, find the side length of the square.

(20 10 Ningxia 26. (10)

In △ABC, ∠ BAC = 45, AD⊥BC in D. Fold △ABD along the straight line where AB is located, so that point D falls on point E; Fold △ACD along the straight line where AC is located, so that point D falls on point F, and extended EB and FC intersect with point M respectively.

(1) Judge the shape of quadrilateral AEMF and give the proof.

(2) If BD= 1 and CD=2, try to find the area of quadrilateral AEMF.

(Tianjin 20 10) (6) The correct one of the following propositions is ()

(a) A quadrilateral with equal diagonals is a diamond; (b) The quadrilateral with orthogonal diagonals is a diamond.

(c) Parallelograms with equal diagonals are diamonds; (d) The parallelogram with orthogonal diagonals is a diamond.

(20 10 Ningxia 6. Points A, B and C are three points in the plane that are not on the same straight line, and point D is any point in the plane. If the four points A, B, C and D can just form a parallelogram, then the point D in the plane that meets this condition has ().

1。

2. The large square grid consists of 25 small squares with side length of 1.

Cut out the shadow in the picture and make a square with the cut shadow.

So the side length of the new square is

22.(20 10 Xianning City, Hubei Province) Problem Background

(1) As shown in figure 1, in △ABC, DE‖BC is given to AB, AC to D and E respectively.

Pass point E as EF‖AB and give it to BC at point F. Please fill in the blanks according to the data shown in the figure:

The area of DBFE quadrilateral,

△EFC area,

△ADE area.

Explore and discover

(2) In (1), if the distance from DE to BC is. Please prove it.

Extended migration

(3) As shown in Figure 2, the four vertices of □DEFG are on the three sides of △ABC, if

The areas of △ADG, △DBE and △GFC are 2, 5 and 3 respectively. Try using (2).

Find the area of △ABC according to the conclusion in.

(20 10 Jingmen city, Hubei province) 19. (The full mark of this question is 9) Fold the triangular piece of paper ABC (AB > AC) along the straight line passing through point A, so that AC falls on the edge of AB, and the crease is AD, and flatten the paper, as shown in figure (1); Fold the triangle paper again, so that point A and point D coincide, and the crease is EF. After flattening again, connect DE and DF, as shown in Figure 2, which proves that the quadrilateral AEDF is a diamond.

24.(20 10 Jinhua) (this question 12 points)

As shown in the figure, the triangle ABO with an angle of 30 is placed in a plane rectangular coordinate system, and the coordinates of points A and B are (3,0) and (0,3) respectively. The moving point P starts from point A and moves along the dotted line AO-OB-BA. The moving speed of point P on AO, OB and BA is 65,438+0 respectively. 2 (length unit/second) The upper edge L of the ruler starts from the position of the X axis and moves up in parallel at the speed of 33 (length unit/second) (that is, the l‖x axis is maintained during the movement), and the two points intersect with OB and AB at E and F respectively. Set the moving point P and the moving straight line L at the same time, and the moving time is t seconds. When the point p moves along the dotted line A0-0 b-BA,

Please answer the following questions:

(1) The analytical formula of the straight line passing through point A and point B is ▲;

(2) When t = 4, the coordinate of point P is ▲; When t = ▲, point P coincides with point E;

(3)① Make a point p', which is the symmetrical point of the point p about the straight line EF. If the quadrilateral PEP'f is rhombic in motion, what is the value of t?

② When t = 2, is there a point Q that makes △FEQ ∽△BEP? If it exists, find the coordinates of point Q; If it does not exist, please explain why.

(Lianyungang, 20 10) 27. (Full score of this question 10) If a straight line divides the area of a plane figure into two equal parts, we call this straight line the area bisector of this plane figure. For example, the straight line of a pair of lines of a parallelogram is the area bisector of the parallelogram.

(1) The lines where the median line, the height line and the angular bisector of the triangle are respectively located must be the area bisector of the triangle.

(2) As shown in figure 1, in trapezoidal ABCD, AB‖DC, if DC is extended to E, and CE = AB is connected with AE, then there is S- trapezoidal ABCD = S △ Abe. Please give the reason for this conclusion, and make the area bisector of trapezoidal ABCD through point A (don't write, keep drawing traces);

(3) As shown in the figure, if AB and CD are not parallel, and S △ ADC > S △ ABC, can the area bisector of quadrilateral ABCD be drawn when it passes through point A? If yes, please draw an area bisector and give proof; If not, explain why.

(20 10 Lanzhou) 27. (Full score of this question 10) It is known that in the parallelogram ABCD, diagonal AC and BD intersect at point O, and AC= 10.

BD=8。

(1) If AC⊥BD, try to find the area of quadrilateral ABCD;

(2) If the included angle between AC and BD is ∠AOD=, find the area of quadrilateral ABCD;

(3) Discussion: If "parallelogram ABCD" in the title is changed to "quadrilateral ABCD" and ∠AOD=

AC=, BD=, try to find the area of quadrilateral ABCD (expressed by algebraic expression containing,,,).

Question 27

(Zhenjiang 20 10) 27. Exploration and discovery (full marks for this small question)

As shown in the figure, right-angle vertices A and C in the right-angle coordinate system are always on the positive semi-axis of the X axis, B and D are in the first quadrant, point B is above the straight line OD, OC=CD, OD=2, m is the midpoint of OD, and AB and OD intersect at point E. When the position of point B changes,

Try to solve the following questions: (1) Fill in the blanks: the coordinates of point D are;

(2) If the abscissa of point B is t, please express the BD length as a function of t and simplify it;

(3) Can the equation BO=BD be established? Why?

(4) Let CM and AB intersect at F, and when △BDE is a right triangle, judge the shape of quadrilateral BDCF and prove your conclusion.

3.(20 10 Huanggang) As shown in the figure, rectangular paper ABCD, AB = 5cm, BC = 10cm, CD has a little E, ED = 2cm, AD has a little P, PD = 3cm. After passing P, do PF⊥AD, pay BC to F, and fold the paper so that point P and point E coincide.

Question 9

4.(20 10 Huanggang) (6 points) As shown in the figure, the two right-angled sides of a 45-degree triangle HBE coincide with the two adjacent sides of a square ABCD, and the bisector of the angle passing through point E is the intersection of EF⊥AE ∠DCE at point F, so as to explore the quantitative relationship between AE and EF and explain the reasons.

Map number 18

3.(20 10 Yibin, Sichuan)

As shown in the figure, point P is a point on the diagonal BD of square ABCD, PE⊥BC is at point E, PF⊥CD is at point F, and connecting EF gives the following five conclusions: ① AP = ef; ②ap⊥ef； ③△APD must be an isosceles triangle;

④∠PFE =∠BAP； ⑤ PD = 2ec。 The serial number of the correct conclusion is.

2.(20 10 Qingdao, Shandong) Fold a rectangular piece of paper (rectangular ABCD) as shown in the figure, so that the vertex B coincides with the point D, and the crease is EF. If AB = 3 cm and BC = 5 cm, the area △DEF of the overlapping part is cm2. ..

(20 10? Wenzhou, Zhejiang) 8. As shown in figure, AC; BD is the diagonal of rectangular ABCD, and the intersection point D is the extension line where DE//AC intersects BC at E, then the triangle * * * congruent with △ABC in the figure has ().

1。

(Suzhou 20 10 exam questions 14). As shown in the figure, the quadrilateral ABCD is a square, extending from AB to E,

Let AE=AC, then the number of ∠BCE is ▲.

(Lianyungang, 20 10) 18. In rectangular paper ABCD, AB = 3 and AD = 4. Fold the paper so that point B falls on B' on the edge CD, and the crease is AE. If there is a point P on the crease AE and the distance to the edge CD is equal to point B, then the equal distance is _ _ _ _ _ _ _.