2. The trunk of a plant will grow many branches, and each branch will grow the same number of small branches. The total number of trunks, branches and twigs is 9 1. How many twigs are there on each branch?
There is a match between every two teams participating in the football league. * * * There are 45 matches. * * * How many teams are taking part in the competition?
Every two teams participating in a football league have two games, * * * 90 games. * * * How many teams are taking part in the competition?
Students in the biology interest group donated a specimen they collected to other members of the group, and the whole group donated 182 pieces to each other. How many students are there in this group?
6. How many people send greeting cards to each other in groups during the Chinese New Year? If the whole group is dealt 72 cards, how many people are there in this group?
7. Quantity before change × (1 x) n = quantity after change.
8. The average yield of rice planted in Qingshan Village was 7,200 kg per hectare in 2006 and 5,438+0 and 8,450 kg per hectare in 2003, so as to seek the average annual growth rate of rice yield per hectare.
9. After two consecutive price reductions, the price of a commodity dropped from 90 yuan to 40 yuan. What is the average price reduction every time?
10 Original price of a commodity in 50 yuan. Due to the financial crisis, the price was reduced by 10% in June, and the price increased in February. The price in March is 64.8 yuan, and the average price growth rate in February and March is sought.
10 The retail price of a drug was reduced to half of the original price after two price reductions. Assuming that the percentage of price reduction is the same, what is the percentage of price reduction each time?
1 1 In order to green the campus, a middle school planted 400 trees in 2007, and plans to make the total number of trees planted in these three years reach 1324 by the end of 2009, so as to seek the average annual growth rate of trees planted in this school.
Selling price-purchase price = profit
Profit of a commodity × sales volume = total profit × unit price = sales volume.
12 A store bought a product, and the purchase price was 30 yuan. In the process of trial sale, it is found that the daily sales volume P of this commodity satisfies the relationship with the sales price X (yuan) of each commodity: P= 100-2X sales volume P. If the store sells this commodity every day to make a profit in 200 yuan, what should be the price of each commodity? How many pieces of this commodity are sold every day?
A toy factory plans to produce a toy panda, with a maximum output of 40 pandas a day, and all the products produced every day are sold. It is known that the cost of producing eight pandas is R (yuan), and the price of each panda is P (yuan). The relationship between R P and X is R=500+30X, and P = 170-2x respectively.
When the daily output is (1), what is the daily profit of 1750 yuan?
(2) If the maximum profit available is 1950 yuan, what should be the daily output?
13. A fruit wholesale mall sells a high-grade fruit. If the profit per kilogram is 10 yuan, it can sell 500 kilograms every day. According to market research, if the price per kilogram increases by 1 yuan, the daily sales will be reduced by 20 kilograms. At present, commodities should ensure a daily profit of 6000 yuan, and at the same time make customers get benefits, so how much is the price increase per kilogram?
14. In the sales of clothing cabinets, it was found that a certain brand of children's clothing could sell 20 pieces every day on average, and each piece earned 40 yuan. In order to welcome Children's Day on June 1st, the shopping mall decided to take appropriate measures to reduce prices, expand sales, increase profits and reduce inventory. According to market research, if the price of each piece of children's clothing is reduced by 4 yuan, on average, eight more pieces can be sold every day. If you want to make an average profit 1200 yuan by selling this kind of children's clothing every day, how much will the price of each child's clothing be reduced?
15. Watermelon business owners buy a batch of small watermelons at the price of 2 yuan/Jin and sell them at the price of 3 yuan/Jin, which can sell 200 Jin every day. In order to promote sales, the business owner decided to reduce the price. According to the survey, every time the price of this small watermelon is reduced by 0. 1 yuan/kg, it can sell 40 kg more every day. In addition, the daily rent and other fixed costs ***24 yuan. If business owners want to make a profit in 200 yuan every day, how much will the price of small watermelons be reduced per catty?
16 ... The desktop of a table is 6 meters long and 4 meters wide, and the tablecloth area is twice that of the desktop. If the tablecloth is laid on the table and each side is the same length, find the length and width of the tablecloth.
17. There is a rectangle with an area of 1350px2. One set of its opposite sides is shortened by 125px, and the other set is shortened by 50px, just becoming a square. What is the side length of this square?
18 Zhang Grandpa bought a rectangular piece of iron sheet from the market. After he cut a square with a side length of 1 m from each of the four corners of the rectangular iron sheet, the remaining part can just enclose a cuboid box with an uncovered volume of 15 cubic meters, and the bottom of the cuboid box is 2 meters longer than the width. It costs 20 yuan money per square meter to buy this tin now. Let uncle Zhang buy this tin.
19 The sales of Hengli Commercial Building in September was 2 million yuan, which decreased by 20% in 10. Since 1 1 month, the management of commercial buildings has been strengthened, the operation has been improved, and the sales volume has steadily increased. 1February sales reached19.36 million yuan, and the average growth rate of these two months was found.
Some kind of computer virus spreads very fast. If a computer is infected, 8 1 computer will be infected after two rounds of infection. Please analyze it with what you have learned. How many computers will be infected by one computer in each round? If the virus is not effectively controlled, will there be more than 700 infected computers after three rounds of infection?
2 1 Wang Hongmei implicitly deposited the lucky money of 1000 yuan into the children's bank for the first time. After the expiration, he took out the principal and interest, donated 500 yuan to Project Hope, and saved all the rest for one year. At this time, the annual interest rate of the deposit has been lowered to 90% of the annual interest rate of the first deposit. After the maturity, he can get the principal and interest * * 530 yuan.
On the 22nd, Mr. Zhou Jiazhong implicitly deposited the lucky money of 1 0,000 yuan into the children's bank for the first time in a one-year way. After the expiration, he took out the principal and interest, donated 500 yuan to Project Hope, and saved all the rest for one year. At this time, the annual interest rate of the deposit has been lowered to 60% of the annual interest rate of the first deposit, so that he can get the principal and interest * * 530 yuan when it expires. When looking for the first deposit,
In order to solve the problem of people's difficulty in seeing a doctor, the municipal government decided to reduce the price of drugs. After two consecutive price reductions of a drug, the 200 yuan per box is reduced to 128 yuan, so the average price reduction percentage of the drug is
Commodity pricing: 1. Yiqun Boutique buys a batch of goods at the price of 2 1 yuan each, and can set its own price. If the price of each commodity is one yuan, you can sell (350- 10a) pieces, but the price bureau limits the profit of each commodity to no more than 20%. How many pieces do you need to purchase if the store plans to make a profit in 400 yuan? How much should each commodity be priced?
Lida dealership sells a kind of building materials for a factory (consignment here means that the manufacturer provides the goods for free first, and then settles the accounts after the goods are sold, and the unsold goods are handled by the manufacturer). When the price per ton is 260 yuan, the monthly sales volume is 45 tons. In order to improve the operating profit, the dealer intends to reduce the price to promote sales. According to market research, when the price per ton drops 10 yuan, the monthly sales will increase by 7.5 tons. Based on various factors, every ton of building materials sold needs to pay 100 yuan for manufacturers and other expenses.
(1) When the price per ton is 240 yuan, calculate the monthly sales at this time;
(2)(2) Under the principle of "small profits but quick turnover", ask the dealers how much they sell per ton of materials, and the monthly profit is 9000 yuan.
(3)(3) Xiao Jing said: "When the profit is the largest in the month, the monthly sales are also the largest." Do you think it's right? Please provide a justification for the answer.
In order to strengthen the macro-management of cigarette production and sales, the state implements an additional tax policy on cigarette sales. Now it is known that the market price of a certain brand of cigarettes is 70 yuan. If there is no tax increase, it will produce and sell 6.5438 million cigarettes a year. If the state levies additional tax, it will levy X yuan (called tax rate x%) for every 65.438+000 yuan sold. Then the annual production and sales will be reduced by10x00. If the additional tax levied on this business is 6.5438+0.68 million yuan, and the production and sales of cigarettes are under macro control, and the annual production and sales volume does not exceed 500,000 cigarettes, how to determine the tax rate?
For the first time, the buyer of a toy store spent 100 yuan on penguin toys, which were quickly sold out. When I bought it for the second time, I found that the wholesale price increased by 0.5 yuan, and I spent 150 yuan, and the number of toys I bought was more than that of the first time 10. The prices of these two batches of toys are both in 2.8 yuan. How many toys did you buy for the second time?
27. A shopping mall tried to sell a T-shirt at a cost of 60 yuan/piece. It is agreed that the unit price shall not be lower than the cost unit price and the profit shall not be higher than 40%. After the trial sale, it is found that the sales volume y (pieces) and the sales unit price x (yuan/piece) conform to the linear function bkxy+=, and when 70=x, 50 = Y；; When 80=x, 40 = y；;
(1) Write the value range of the sales unit price x;
(2)(2) Find the analytical formula of the linear function bkxy+=;
(3) If the profit of the shopping mall is W yuan, try to write the relationship between the profit W and the sales unit price X. What is the maximum profit that the shopping mall can get when the sales unit price is fixed?
Cut a 500px long wire into two sections and make a square around the length of each section. (1) If the sum of the areas of these two squares is equal to 425px2, what is the length of the wire cut into two sections? (2) The sum of the areas of two squares may be equal to 300px2.
Really? If yes, find out the length of two pieces of wire; If not, please explain why.
Travel problem: 1, A and B are 82km apart, A rides from A to B. After 9 minutes, B rides from B to A at a speed 2km faster than A, and they meet at a distance of 40km .. What are the speeds of A and B respectively?
30 A and b walk at the same speed from a and b, which are 20 kilometers apart. After meeting, they moved on, and B's speed remained unchanged. A walking more than before 1 km per hour. In this way, after A arrives at B, B needs 30 minutes to arrive at A, and how many kilometers does B walk per hour?
31railway distance between a and b 1600 km. After technical transformation, the train speed is increased by 20km/h, and the travel time from city A to city B is shortened by 4 hours. Under the existing safety conditions, the safe running speed of this railway should not exceed 140 km/h, please explain it with the mathematical knowledge you have learned.
32. Two people, A and B, ride in two directions, A and B respectively. 1 hour later, A set out, and 4 hours later, we met at C on the way. After meeting, they continued to move in the original direction. B stopped for 20 minutes on the way from C to A for some reason. As a result, B arrived at A 40 minutes earlier than A arrived at B from C. It is known that B travels 4 kilometers faster than A, so the speed of cycling can be calculated.
Engineering problem: 1. A company needs to complete the renovation project of the new office building within one month (3 1 day). If the two engineering teams of Party A and Party B work together, it can be completed within 12 days; If Team A and Team B complete the project alone, Team A will spend 65,438+00 days less than Team B, and (65,438+0) find the number of days required for Team A and Team B to complete the project alone. (2) If Team A is invited for construction, the company will pay 2000 yuan every day. If Team B is invited to carry out the construction, the company will pay 1400 yuan every day. Within the specified time: a. Please ask Group A to finish the project alone. B. ask group b to complete the project alone; Group a and group b are required to cooperate to complete the project. Which of the above three schemes has the lowest cost?
33. If the goods in a warehouse are empty, then Party A needs 65,438+00 hours, Party B needs 65,438+02 hours and Party C needs 65,438+05 hours.
Hours to complete, there are two warehouses A and B with goods in stock. A is in warehouse A and B is in warehouse ku. At the same time, C began to help A move, and then turned to help B halfway. When the goods in the last two warehouses move at the same time, how long does C help A and B respectively? (formula)
34 A and B both drive at a constant speed on the circular road, face to face, and meet every 2 minutes; Walking in the same direction, we meet every 6 minutes. It is known that A runs faster than B. How many laps do A and B run per minute?
35. The oil storage tank of an oil depot has two refueling pipes, A and B. It takes four hours to refuel by opening pipe A alone than by opening pipe B alone. After the two pipes were opened at the same time for 3 hours, the oil filling of pipe A stopped due to a fault, and the oil filling of pipe B continued for 9 hours. How many hours does it take to open the A pipe and the B pipe respectively to fill the fuel tank?
36 engineering problem: 1. A company needs to complete the renovation project of the new office building within one month (3 1 day). If the two engineering teams A and B jointly do it, it can be completed in 12 days; If Team A and Team B complete the project alone, Team A will spend 65,438+00 days less than Team B, and (65,438+0) find the number of days required for Team A and Team B to complete the project alone. (2) If Team A is invited for construction, the company will pay 2000 yuan every day. If Team B is invited to carry out the construction, the company will pay 1400 yuan every day. Within the specified time: a. Please ask Group A to finish the project alone. B. ask group b to complete the project alone; Group a and group b are required to cooperate to complete the project. Which of the above three schemes has the lowest cost?
37. If the goods in a warehouse are empty alone, Party A needs 65,438+00 hours, Party B needs 65,438+02 hours, and Party C needs 65,438+05 hours. For warehouse A and warehouse B with goods in stock, Party A is in warehouse A, and Party B starts to move the goods at the same time, and Party C starts to help Party A move, and then helps Party B halfway, and the last two. (formula)
38 A and b both drive at a constant speed on the circular road, face to face, and meet every 2 minutes; Walking in the same direction, we meet every 6 minutes. It is known that A runs faster than B. How many laps do A and B run per minute?
The oil storage tank of an oil depot has two refueling pipes, A and B. It takes 4 hours to refuel by opening pipe A alone than by opening pipe B alone. After the two pipes were opened at the same time for 3 hours, the oil filling of pipe A stopped due to a fault, and the oil filling of pipe B continued for 9 hours. How many hours does it take to open the A pipe and the B pipe respectively to fill the fuel tank?
Comment | 2 2
Report | 2013-06-1417: 44 Enthusiastic users can answer as soon as possible.
Review of the unit of quadratic equation in one variable
1. Multiple choice questions: (2 points for each small question, * * * 20 points)
1. The following equations are not necessarily quadratic equations ().
A.(a-3)x2=8(a≠0) B.ax2+bx+c=0
C.(x+3)(x-2)=x+5 D。
2. It is known that the unary quadratic equation ax2+c=0(a≠0). If the equation has a solution, there must be C = ().
A.-B.-1C.D. Not sure.
3. If the equation ax2+2(a-b)x+(b-a)=0 about x has two equal real roots, then A: B is equal to ().
A.- 1 or 2b. 1 or c .- or 1 d.-2 or 1.
4. If the unary quadratic equation ky2-4y-3=3y+4 about y has real roots, then the range of k is ().
A.k & gt-b. k ≥- and k ≠ 0 c.k ≥-d.k > and k ≠ 0.
5. If the two roots of the equation are known as a, the root of the equation is ().
A.B. C. D。
6. Equation x2+2(k+2)x+k2=0 Regarding X, if the sum of two real roots is greater than -4, then the value range of K is ().
A.k & gt- 1 b . k & lt； 0 degrees celsius-1< k & lt0d .- 1≤k & lt； 0
7. If the two real roots of the equation x2-kx+6=0 are respectively greater than the two real roots of the equation x2+kx+6=0 by 5, then the value of k is ().
5 BC
8. X that makes the score equal to zero is ()
A.6 b.- 1 or 6 c.- 1 d.-6
9. The solution of equation x2-4│x│+3=0 is ().
A.x = 1 or x = 3b.x = 1 and x = 3c.x =- 1 or x =-3d. There is no real root.
10. If the equations x2-k2- 16=0 and x2-3k+ 12=0 about x have the same real root, then the value of k is ().
A.-7b-7 or 4c-4d. 4
Fill in the blanks: (3 points for each small question, * * * 30 points)
1 1. As we all know, 3- is a root of the equation x2+mx+7=0, so m = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
12. given the equation 3ax2-bx- 1=0, ax2+2bx-5=0, and there are * * * same roots-1, then a = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
13. If the unary quadratic equation ax2+bx+c=0(a≠0) has the root of 1, then A+B A+B+C = _ _ _ _ _ _ _ _ _ _ _ _； If the root of A is-1, the relationship between B and A and C is _ _ _ _ _ _ _ _ _; If a root is zero, then c = _ _ _ _ _ _ _ _ _
14. If two of the equations 2x2-8x+7=0 are exactly the lengths of two right-angled sides of a right-angled triangle, then the length of the hypotenuse of this right-angled triangle is _ _ _ _ _ _ _.
15. The sum of all the real roots of the unary quadratic equation x2-3x- 1=0 and x2-x+3=0 is equal to _ _ _ _ _ _ _ _.
16. After two consecutive price increases 10%, the price of a food is one yuan, so the original price is _ _ _ _ _ _ _ _ _ _.
17. It is known that the product of two numbers is 12, the sum of the squares of these two numbers is 25, and the quadratic equation of one yuan based on these two numbers is _ _ _ _ _ _ _ _ _.
18. If the equation x2-2( 1-k)+k2=0 about x has real roots α, β, then the value range of α+β is _ _ _ _ _.
19. Let A be the sum of absolute values of all roots of the equation x2- x-520=0, then A2 = _ _ _ _ _ _ _
20. Cut a square with a side length of 5cm on each corner of the rectangular iron sheet, and then fold it into a box without a cover. The length of the iron sheet is twice the width, and the volume of the box is 1.5 cubic decimeter, so the length of the iron sheet is equal to _ _ _ _ _ _ _ _ _ _ _.
Three. Answer: (7 points for each question, ***2 1 point)
2 1. Let x 1, where x2 is the two real roots of the equation x2-(k+2)x+2k+ 1=0, and x12+x22 =11.
(1) Find the value of k; (2) Using the relationship between roots and coefficients, the quadratic equation of one variable is solved, so that one root is the sum of two roots of the original equation and the other root is the square of two differences of the original equation.
22. Let A, B and C be three sides of △ABC, the equation x2+2 x+2c-a=0 about X has two equal real roots, and the root of equation 3cx+2b=2a is 0.
(1) Verification: △ABC is an equilateral triangle;
(2) If A and B are two roots of the equation x2+mx-3m=0, find the value of m. 。
23. As shown in the figure, it is known that in △ABC, ∠ ACB = 90, the point C is CD⊥AB, the vertical foot is D, AD=m, BD= n, AC2: BC2 = 2: 1, and the equation X2-2 of X (n-65438.
Reference answer
1. 1.b dial: ax2+bx+c=0. Only when a≠0 is satisfied is it a quadratic equation.
2.d nudge: If the unary quadratic equation ax2+c=0(a≠0) has a solution, then ax2=-c, x2=, because x2≥0,
There are several solutions, so it is uncertain.
3.b nudge: According to the discriminant of the roots of a quadratic equation, the equation has two equal real roots.
Delta = 0, delta = [2 (a-b)] 2-4× a (b-a) = 4 (a-b) (2a-b), that is, 4(a-b)(2a-b)=0,
∴a=b or a=,
Namely a: b = 1 or a: b = 1: 2.
4.b nudge: k≠0 From the definition of a quadratic equation with one variable, it can be known that the equation has a real root and from the discriminant of the root of the quadratic equation with one variable.
Then △≥0, that is, k≥, so k≥ and k≠0, this problem is easy to miss two conditions: k≠0 and△ = 0.
5.d inching: from, to can be changed to, so its solution is x- 1=a- 1, that is, x=a or x- 1=, that is, X =. This question can easily be misunderstood as x=a or X =.
6. D. Hugging: The equation has two real roots, so △≥0, that is, [2(k+2)]2-4k2≥0, the solution is k≥- 1, and the sum of the two real roots is greater than -4, that is, -2(k+2)>-4, k &
∴- 1≤k<； 0. This problem is easy to ignore that there are two real roots, which need to meet the important condition of △≥0.
7. D. Guidance: Let two of x2-kx+b=0 be x 1, x2, then two of x2+kx+6=0 are x 1+5, x2+5, because x 1+x2=k, (x/.
8. inching: the condition that the score is zero: numerator =0, denominator ≠0, x2-5x-6=0, x=6 or-1, x+ 1, x≦- 1, so x.
9. A cue: ∵x2≥0, │x│0, ∴x2-4│x│+3=0 is the solution of the equation │x│2-4│x│+3=0, (│ x │-65433).
10.d nudge: if two equations have the same real root, then x2+k2-16 = x2-3k+12, and the solution is k=-7 or 4.
When k=- 7, the equation has no real root, ∴ k = 4.
Second,
1 1.m =-6 and the other is 3+.
Pointing: according to the relationship between the root and coefficient of a quadratic equation, let the other root of the equation be x 1,
Then (3- )x 1=7, x 1=3+, (3+ )+(3- )=-m, then m =-6.
12.a= 1，B =-2。 Cuddle:-1 is the root of two equations, then 3A+b=-2. 1 = 0, a-2b-5=0, the solution is A = 1, and B =-.
13.a+b+c=0，b=a+c，c=0。
14.3 inching: let two roots be x 1, x2, and according to the relationship between roots and coefficients, x 1+x2=4, x 1x2=,
According to Pythagorean Theorem, the square of the hypotenuse length = (x1+x2) 2-2x1x2 =16-2x9, and the hypotenuse length is 3.
15.3 inching: x2-3x-1= 0 △ =13 > δ is 0, x2-x+3=0 =- 1 1