A, multiple-choice questions (this question * *10 small questions, each small question 3 points, ***30 points)
1.(3 points) The following numbers:,, 0. 10 1 ... (the middle 0 increases in turn),-π, and some are irrational numbers ().
A. 1 B. 2 C. 3 D.4
Test center: irrational number.
Analysis: According to the definition of irrational number (irrational number refers to infinite acyclic decimal).
Solution: There is an irrational number, 0.101… (the middle 0 increases in turn), ﹣ π, ***3,
So choose C.
Comments: Investigate the application of irrational numbers. Note: Irrational numbers refer to infinite cyclic decimals. Irrational numbers include three kinds of numbers: ① numbers with π, ② numbers with infinite roots, and ③ some regular numbers.
2.(3 points) (200 1? Beijing) It is known that if ACD and CE divide ∠ACD and ∠ A = 1 10, ∠ECD is equal to ().
A. 1 10
Test site: the nature of parallel lines; Definition of angular bisector.
Special topic: calculation problems.
Analysis: This problem mainly uses two parallel lines as internal angles, and then does it according to the concept of angular bisector.
Answer: solution: ∫AB∨CD,
According to the fact that two straight lines are parallel and are internal angles to each other, it is concluded that:
∴∠ACD= 180 ﹣∠A=70。
According to the definition of angle bisector, it is ∠ ECD = ∠ ACD = 35.
So choose D.
Comments: Investigate the properties of parallel lines and the concept of angular bisector.
3.(3 points) Among the following surveys, () is suitable for adopting the comprehensive survey method.
A. understand the air pollution in our city.
B. Understand the ratings of TV program Focus Interview
C. Know the time for each student in Class 7 (6) to do homework every day.
D. check the waterproof performance of a batch of watches produced by a factory.
Test sites: comprehensive survey and sampling survey.
Analysis: The results of the general survey are more accurate, but it needs more manpower, material resources and time, while the results of the sampling survey are similar.
Answer: Solution: A, we can't conduct a comprehensive investigation, but only conduct spot checks;
B, the TV station's survey of the ratings of a TV program being broadcast is suitable for sampling survey because of the heavy workload of census;
C, the quantity is small, which is convenient for investigation and suitable for comprehensive investigation;
D, large quantity, suitable for spot check.
So choose C.
Comments: This question examines the difference between sampling survey and comprehensive survey. The choice of census or sampling survey should be flexibly grasped according to the characteristics of the respondents. Generally speaking, sampling surveys should choose destructive surveys, surveys that cannot be conducted and surveys that are of little significance or value. For surveys with high precision, surveys with great importance often choose general surveys.
4.(3 points) The solution set of one-dimensional linear inequality group is expressed as () on the number axis.
A.B. C. D。
Test center: representing the solution set of inequality on the number axis; Solving linear inequalities.
Analysis: Find the solution set of each inequality separately, and then find its common solution set and express it on the number axis.
Solution: solution: obtained from ①, X.
Therefore, the solution set of the inequality group is: 0 ≤ x.
On the number axis, it is expressed as:
So choose B.
Comments: This question examines the solution set of inequality groups on the number axis and is familiar with "taking the same big; Take the small as the big; Small, large and medium search; The principle of "size cannot be found" is the key to solve this problem.
5.(3 points) The positive integer solution of binary linear equation 2x+y=8 has ()
A.2 B. 3 C. 4 D.5
Test center: Solve binary linear equation.
Special topic: calculation problems.
Analysis: Substitute x= 1, 2, 3, ... into the equation to find the value of positive integer y.
Solution: solution: when x= 1, you get 2+y=8, that is, y = 6；; When x = 2,4+y = 8, y = 4；; When x=3, 6+y=8, that is, y = 2；;
Then this equation has three positive integer solutions.
So choose B.
Comments: This topic examines and understands the binary linear equation. Note that x and y are positive integers.
6.(3 points) If point P(x, y) satisfies xy.
A. second quadrant B. third quadrant C. fourth quadrant D. second and fourth quadrants
Test center: coordinates of points.
Analysis: y > is obtained according to the properties of real numbers; 0, and then judge according to the coordinate characteristics of points in the second quadrant.
Solution: solution: ∫xy
∴y>； 0,
Point p is in the second quadrant.
So choose a.
Comments: This question examines the one-to-one correspondence between points on the coordinate plane of points and ordered real number pairs. Coordinate: The rectangular coordinate system divides the plane into four parts, which are called the first quadrant, the second quadrant, the third quadrant and the fourth quadrant. The points on the coordinate axis do not belong to any quadrant.
7.(3 points) As shown in the figure, if AB∨CD, ∠ A = 125, ∠ C = 145, the degree of ∠E is ().
A. 10 B. 20 C. 35 D. 55
Test site: the nature of parallel lines.
Analysis: If E is taken as EF∨AB, the degrees of ∠AEF and ∠CEF can be obtained according to the properties of parallel lines, and the degrees of ∠E can be obtained according to ∠ E = ∠ AEF ∠ CEF.
Answer: solution: e is EF∨AB,
∠∠A = 125，∠C= 145，
∴∠aef= 180 ﹣∠a= 180 ﹣ 125 = 55，
∠cef = 180 ﹣∠c= 180 ﹣ 145 = 35，
∴∠E=∠AEF﹣∠CEF=55 ﹣35 =20。
So choose B.
Comments: This question examines the nature of parallel lines. The key to solve this problem is to make auxiliary lines, which requires students to master the properties of parallel lines: two lines are parallel and the internal angles are complementary.
8.(3 points) If it is known that it is the solution of an equation group, then it is the solution of which of the following equations ().
a . 2x ~ y = 1b . 5x+2y = ~ 4c . 3x+2y = 5d。 None of the above.
Test site: the solution of binary linear equations; Solution of binary linear equation.
Special topic: calculation problems.
Analysis: Substitute x=2, y= 1 into the equations, and work out the values of A and B, then you can make a judgment.
Solution: solution: the equation is: a=2, b=3,
Substituting x=2 and y=3 into the left of 2x = 1, we get: 4 = 3 = 1, and the right is 1, so the left is equal to the right.
∴ is the solution of equation 2x ∴ y = 1
So choose a.
Comments: This question examines the solution of a binary linear equation. The solution of the equation is an unknown number that can make two equations in the equation hold.
9.(3 points) The following statement is not necessarily correct ()
A.B. C. D。
Test site: cube root; arithmetic square root
Analysis: According to the definition of cube root and square root, you can judge.
Solution: solution: when a and a are arbitrary numbers, the equations are true and correct, so the options are wrong;
When b and a are arbitrary numbers, the equation is correct, so this option is wrong;
C, the implicit condition a≥0 in the original formula, the equation is established and correct, so this option is wrong;
D, when a
So choose D.
Comments: This topic examines the application of cubic roots and square roots. Note: When a≥0 and =a, any number has a cubic root.
10.(3 points) If the inequality group has three integer solutions * * *, the range of a is ().
A.5 & lta & lt6 p = " " 5≤a≤6 & lt； = " " d . = " " 5≤a & lt； 6="" c.="" 5
Test site: integer solutions of unary linear inequalities.
Analysis: First, determine the solution set of the inequality group, and express it with a formula containing a.. According to the number of integer solutions, we can determine which integer solutions there are, and according to the situation of the solutions, we can get the inequality about A, so as to find the range of A. 。
Solution: Solution: 2
The inequality group has three integer solutions.
These three are 3, 4 and 5, so 5 ≤ A.
So choose C.
Comments: This question examines the integer solution of a group of linear inequalities. The key to solve this problem is to correctly solve the solution set of inequality groups and determine the value range of A. To solve the solution set of inequality groups, the following principles should be followed: the same size takes the big one, the same size takes the small one, and the middle of the small one cannot be solved.
Fill in the blanks (this topic is ***8 small questions, with 3 points for each small question and 24 points for * * *).
1 1.(3 points) (2009? The arithmetic square root of 9 is 3.
Test center: arithmetic square root.
Analysis: If the square of a nonnegative X is equal to A, then X is the arithmetic square root of A, and the result can be obtained according to this definition.
Answer: solution: 32 = 9,
The arithmetic square root of 9 is 3.
So the answer is: 3.
Comments: This topic mainly examines the equality of arithmetic square root, and the concept of arithmetic square root is easily confused with the concept of square root, leading to errors.
12.(3 points) Write the proposition "In the same plane, two straight lines perpendicular to the same line are parallel to each other" in the form of "If …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
Test sites: propositions and theorems.
Analysis: According to the proposition, two straight lines are perpendicular to the same straight line in the same plane; The conclusion is that these two straight lines are parallel to each other.
Solution: "In the same plane, two straight lines perpendicular to the same straight line are parallel to each other" is rewritten as "If two straight lines are perpendicular to the same straight line in the same plane, then these two straight lines are parallel to each other"
So the answer is: both straight lines are perpendicular to the same straight line, and these two straight lines are parallel to each other.
Comments: this topic examines propositions and theorems: the sentence that judges things is called a proposition, and the proposition consists of two parts: the topic and the conclusion; The correct proposition is called true proposition, and the wrong proposition is called false proposition; A reasoned true proposition is called a theorem.
13.(3 points) If the equation 2x+y=25 is written as an algebraic expression containing x, then y = 25 ~ 2x.
Test center: Solve binary linear equation.
Analysis: To write the equation 2x+y=25 as a formula containing X, it is necessary to move the term containing Y to the left of the equation and the other terms to the other side.
Solution: move the term and get y = 25 ~ 2x.
Comments: This question examines the basic operation skills of the equation, which means who should be placed on the left side of the equation and other items should be moved to the other side.
This problem can be moved directly.
14.(3 points) inequality x+4 >; The smallest integer solution of 0 is -3.
Test site: integer solution of one-dimensional linear inequality.
Analysis: First, the basic properties of inequality are used to solve inequality, and then positive integers suitable for conditions are found from the solution set of inequality.
Solution: Solution: X+4 > 0,
x & gt﹣4,
Then the solution set of inequality is x > 4,
Therefore, the inequality x+4 >; The smallest integer solution of 0 is -3.
So the answer is -3.
Comments: This question examines the integer solution of linear inequality in one variable. It is the key to solve the inequality correctly and find the solution set. The solution of inequality should be based on the basic properties of inequality.
15.(3 points) In the "Mathematics Papers" selection activity of a school, * * * collected 60 papers, evaluated and sorted them, and plotted the frequency distribution histogram in groups (as shown in the figure). It is known that the height ratio of five small rectangles from left to right is 1: 3: 7: 6: 3, so in this evaluation,
Test center: histogram of frequency (ratio) distribution.
Analysis: According to the ratio of the height of five small rectangles of 1: 3: 7: 6: 3 from left to right to the total number of articles, calculate the number of articles in each square respectively, and then get the answer according to the score of 80 or more and the score is an integer.
Solution: Solution: ∵ The height ratio of five small rectangles from left to right is 1: 3: 7: 6: 3, and * * * collected 60 sheets of paper.
∴ The number of items in the first box is: ×60=3 (items);
The number of articles in the second box is ×60=9 (articles);
The number of articles in the third box is: ×60=2 1 (articles);
The number of articles in the fourth box is: ×60= 18 (articles);
The number of articles in the fifth box is: ×60=9 (articles);
∴ The papers rated as excellent in this appraisal are: 9+ 18=27 (articles);
So the answer is: 27.
Comments: This question examines the ability to read the histogram of frequency distribution and the ability to obtain information by using statistical graphs; When using statistical charts to obtain information, we must carefully observe, analyze and study statistical charts in order to make correct judgments and solve problems.
16.(3 points) Two coal mines A and B in our city planned to produce 6 million tons of coal last year. Results A coal mine completed 1 15% of the planned coal mine last year, and B coal mine completed 120% of the planned coal mine last year. The two coal mines produced 7 10/00000 tons of coal, so as to seek for A and B coal mines last year. Two coal mines, A and B, originally planned to produce X million tons and Y million tons of coal respectively; Please list the equations.
Test center: abstract binary linear equations from practical problems.
Analysis: Using "Two coal mines A and B planned to produce 6 million tons of coal last year, as a result, A coal mine completed 1 15% of the planned output last year, B coal mine completed 120% of the planned output last year, and two coal mines * * * produced 765438+ 10,000 tons of coal", the binary can be solved once.
Solution: Assuming that Mine A originally planned to produce X million tons of coal and Mine B originally planned to produce Y million tons of coal, according to the meaning of the question:
,
So the answer is,
Comments: This question examines the knowledge of binary linear equations abstracted from practical problems. The key to solving the problem is to find two equivalent relationships from the topic, which is the basis of the equations.
17.(3 points) In the plane rectangular coordinate system, given the line segment ab∨x axis, the coordinate of endpoint A is (?14) and AB=4, then the coordinate of endpoint B is (? 5,4) or (3,4).
Test Center: Coordinate and Graphic Properties.
Analysis: According to the line segment AB∨x axis, the vertical coordinates of point A and point B are equal, and the answer can be obtained by using that point B may be on the right or left side of point A. 。
Solution: Solution: ∫ Line segment ab∨x axis, the coordinate of endpoint A is (1,4), AB=4.
Point b can be on the right or left of point a,
Then the coordinates of endpoint B are: (-5,4) or (3,4).
So the answer is: (-5,4) or (3,4).
Comments: This topic mainly examines the nature of coordinates and graphics, and classified discussion is the key to solving the problem.
18.(3 points) If the coordinates of point P(x, y) satisfy x+y=xy, then point P is called "harmony point", for example, harmony point (2,2) satisfies 2+2=2×2. Please write another coordinate of "harmonious point" (3,2).
Test center: coordinates of points.
Special topic: new definition.
Analysis: Let x=3, and calculate the corresponding value of y with x+y=xy, so as to get a coordinate of "harmonious point".
Solution: According to the meaning of the question, point (3,) satisfies 3+ =3×.
So the answer is (3,).
Comments: This question examines the one-to-one correspondence between points on the coordinate plane of points and ordered real number pairs. Coordinate: The rectangular coordinate system divides the plane into four parts, which are called the first quadrant, the second quadrant, the third quadrant and the fourth quadrant. The points on the coordinate axis do not belong to any quadrant.
Three, solve the problem (this big problem ***46 points)
19.(6 points) Solve the equation.
Test site: solving binary linear equations.
Analysis: first calculate the value of y according to the addition and subtraction elimination method, and then calculate the value of x according to the substitution elimination method.
Answer: Solution:,
①×5+②，2y=6，y=3，
Substitute y=3 into ① and x=6.
Therefore, the solution of the equation is.
Comments: This topic examines the solution of binary linear equation, and it is the key to understand the addition, subtraction, substitution and elimination of binary linear equation.
20.(6 points) Solve inequality: and judge whether it is the solution of this inequality.
Test center: solving one-dimensional linear inequality; Estimate the size of irrational numbers.
Analysis: First, the denominator, brackets and moving items are combined into similar items, and then the coefficient is converted into 1 to get the solution set of inequality, and then the judgment can be made.
Solution: remove the denominator and get 4 (2x+1) >12-3 (x-1).
If there are no brackets, you get: 8x+4 >; 12﹣3x+3，
Move project, get, 8x+3x & gt；; 12+3﹣4,
Combining with similar projects, we get:11x > 1 1,
The coefficient becomes 1, x >；; 1,
∵& gt； 1,
∴ is an unequal solution.
Comments: This question examines the ability to understand simple inequalities. Students who solve such problems often make mistakes because they don't pay attention to changing numbers when solving problems.
The solution of inequality should be based on the basic properties of inequality, and both sides of inequality should add or subtract the same number or algebraic expression of the direction of inequality. The direction of multiplication or division of the same positive number and unequal number on both sides of inequality remains unchanged; When both sides of inequality multiply or divide the same negative number at the same time, the direction of inequality changes.
2 1.(6 points) Learn to be reasonable and fill in the blanks:
As shown in the figure, if AD⊥BC is in D, EG⊥BC is in G, and E = ∠1,we can get AD bisection ∠BAC.
The reason for this is the following：
∵AD⊥BC in D, EG⊥BC in G, (known)
∴∠ADC =∠EGC = 90° (vertical resolution)
∴AD∥EG, (same angle is equal, two straight lines are parallel)
∴∠∠ 1 =∞∠2, (two straight lines are parallel and the internal dislocation angles are equal)
∠E=∠3, (two straight lines are parallel and have the same angle)
∫∠E =∠ 1 (known)
∴∠2 =∞∠3 (equivalent substitution)
∴AD bisector ∠BAC (definition of angular bisector)
Test center: determination and properties of parallel lines.
Topic: fill in the blanks by reasoning.
Analysis: This problem can be proved according to the definition of verticality and the nature and judgment theorem of parallel lines.
Answer: Solution: ∵AD⊥BC in D, EG⊥BC in G, (known)
∴∠ADC =∠EGC = 90° (vertical resolution)
∴AD∥EG, (same angle is equal, two straight lines are parallel)
∴∠∠ 1 =∞∠2, (two straight lines are parallel and the internal dislocation angles are equal)
∠E=∠3, (two straight lines are parallel and have the same angle)
∫∠E =∠ 1 (known)
∴∠2 =∞∠3 (equivalent substitution)
∴AD bisector ∠BAC (definition of angular bisector).
Comments: This question examines the judgment and nature of parallel lines and is a basic question. The key is to pay attention to the properties of parallel lines and the comprehensive application of judgment theorems.
22.(8 points) In the square grid as shown in the figure, the side length of each small square is 1, and the coordinates ABC of vertices A and C of a grid triangle (the triangle whose vertices are the intersections of grid lines) are (-4,5) and (-1, 3) respectively.
(1) Please make a plane rectangular coordinate system in the grid plane as shown in the figure;
(2) Please move △ABC to the right by 5 units, and then move down by 3 units to get △ A ′ B ′ C ′, and draw △ A ′ B ′ C ′ in the figure;
(3) Find the area of △ABC.
Test center: drawing-translation conversion.
Analysis: (1) Just translate the coordinate axis of point A to the origin according to the coordinate of point A;
(2) The coordinates of A,' B' and' C' can be obtained by using the coordinate translation property of points, and the answers can be obtained;
(3) It can be obtained by subtracting the area of the surrounding triangle from the area of the rectangle.
Solution: Solution: (1) ∫ The coordinate of point A is (4,5).
At point A ∴, move the Y axis 4 units to the right and move the X axis 5 units down. (2) As shown in the figure: △ A ′ B ′ C ′ is what you want; (3) The area of △ ABC is 3× 4× 3× 2× 1× 2× 2× 4 = 4.
Comments: This question mainly examines the translation transformation, the calculation method of triangle area and the determination method of coordinate axis. The correct translation of vertices is the key to solving the problem.
23.( 10) In the physical education test of the senior high school entrance examination in our city, 1 minute skipping rope is a multiple-choice question. There are several female students in the ninth grade of a middle school. 1 min chooses to skip rope. According to the test scoring standard, their scores are divided into four grades, such as A, B, C and D, and drawn into the following frequency distribution table (note
Frequency of grade score jump (times/1 min)
A12.5 ~15135 ~160m
b 10 ~ 12.5 1 10 ~ 135 30
C 5~ 10 60~ 1 10 n
D 0~5 0~60 1
The value of (1)m is 14, and the value of n is 30;
(2) The percentage of people with Grade C is10%;
(3) In this sample, please indicate which score has the most students?
(4) Ask the teacher to help calculate the passing rate of this 1 minute skipping test (10 or above means passing).
Test site: fan-shaped statistical chart; Frequency (frequency) distribution table.
Analysis: (1) First, divide the number of people in Grade B by its percentage to get the total number of people, then multiply it by 28% to get the value of m, and subtract the frequency of the other three groups to get the value of n;
(2) Divide the value of n by the total number of people to get the percentage;
(3) A conclusion can be drawn directly from the data in the statistical table;
(4) Calculate the number of people above 10, and then divide by 50 times 100% to draw a conclusion.
Answer: Solution: (1) 30 people, accounting for 60%, know the grade B by observing statistical charts and statistics tables.
∴ Total: 30÷60%=50,
∴m=50×28%= 14 people,
n=50﹣ 14﹣30﹣ 1=5； (2) The percentage of Grade C is ×100% =10%; (3) Class B has the largest number of people; (4) Pass rate × 100%=88%.
Comments: This question examines the application of frequency distribution table and fan-shaped statistical chart, and it is the key to understand the relationship between statistical table and statistical chart when answering.
24.( 10) (20 12? Yiyang) In response to the call of the municipal government to "create a national forest city", a residential district plans to purchase A and B seedlings 17, and it is known that A seedlings have one 80 yuan and B seedlings have one 60 yuan.
(1) If it happened that 1220 yuan bought two kinds of saplings, A and B, how many saplings would you buy?
(2) If the number of type B seedlings is less than the number of type A seedlings, please give a scheme with the least cost and calculate the cost of the scheme.
Test site: the application of one-dimensional linear inequality; Application of one-dimensional linear equation.
Topic: the finale.
Analysis: (1) If you buy X saplings of A species, then you need 1220 yuan to use two saplings of A and B. Combined with the unit price, you can get the equation and solve it.
(2) Combining the solution of (1) and the fact that the number of seedlings of B species is less than that of A species, we can find out the scheme.
Solution: Solution: (1) If you buy X seedlings of A species, you will buy (17-x) seedlings of B species. According to the meaning of the question:
80x+60( 17﹣x )= 1220，
Solution: x= 10,
∴ 17﹣x=7，
Answer: Buy a kind of seedlings 10 and 7 kinds of seedlings; (2) If X saplings of A species are purchased, X saplings of B species are purchased (17-x).
According to the meaning of the question:
17﹣x<； x，& ltp = " " & gt
Solution: x>,
The cost of purchasing A and B saplings is 80x+60 (17-x) = 20x+1020.
Then that minimum cost x is the minimum integer 9,
At this time, 17-x = 8,
At this time, the required cost is 20×9+ 1020= 1200 (yuan).
A: The most cost-saving plan is to buy 9 seedlings of A type and 8 seedlings of B type. At this time, the required cost is 1.200 yuan.
Comments: This topic mainly investigates the application of linear inequalities and linear equations. Increasing or decreasing according to the linear function is the key to solve the problem.
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