mathematical analysis
First, multiple-choice questions (3 points for each small question, 30 points for * * *)
1.(20 10 Hubei Ezhou 1, 3 points) In order to strengthen rural education, in 2009, the central government allocated 66.6 billion yuan for rural compulsory education.
A.6.66× 109 Yuan b. 66.6× 10 165438 Yuan c. 0 yuan D.6.66× 1065438 Yuan.
Analysis: 66.6 billion yuan = 666 million yuan = 6.66× 10 10 yuan. Therefore, D.
Answer d
It involves the scientific notation of knowledge points.
Comment on scientific notation is a necessary question in the examination paper of senior high school entrance examination every year. Write a number in the form of a× 10 (where 1 ≤ < 10, n is an integer, and this notation is called scientific notation). Its method is (1) to determine a, where a is a number with only one integer; (2) determine n; When the absolute value of the original number is ≥ 10, n is a positive integer, and n is equal to the integer digits of the original number minus1; When the absolute value of the original number is less than 1, n is a negative integer, and the absolute value of n is equal to the number of zeros before the first non-zero number from the left in the original number (including zeros on integer bits).
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2.(20 10 Hubei Ezhou, 2,3 points) The following data: 23,22,21,18,16,22 are () respectively.
2 1，22
The most commonly analyzed data is 22, that is, the mode is 22; The data are arranged from large to small as 23, 22, 22, 22, 2 1, 18, 16, with a median of 22.
Answer c
Representatives involved in knowledge point data
The two representative quantities of comment data-mode and median-belong to the basic questions of senior high school entrance examination, but they belong to the knowledge points commonly tested in statistics.
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3.(20 10 Hubei Ezhou, 3, 3 points) The front view of the geometry in the following figure is ().
The main view of the analysis is the same as the geometry we see ignoring the thickness. Choose B.
Answer b
Three views involving knowledge points
Comment on the three views of geometry in this question, which often appears in the senior high school entrance examination, is a low-level question.
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4.(20 10 Hubei Ezhou, 4,3 points) As shown in the figure, AD is the bisector of ∠BAC in △ABC, DE⊥AB intersects with AB at point E, and DF⊥AC intersects with AC at point F. If S △ ABC = 7, DE = 2, AB =
a4 b . 3 c . 6d . 5
Analysis ∵AD is the bisector of ∠BAC in △ABC, DE⊥AB, DF⊥AC, DE = DF = 2. ab = 4，∴ S △ Abd =× 4× 2 = 4。
Answer b
It involves the nature of the bisector of the knowledge point and the area of the triangle.
Comment on this problem to investigate the nature of bisector and the calculation of triangle area. It belongs to the low score of the senior high school entrance examination.
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5.(20 10 Ezhou, Hubei, 5,3 points) The images of the proportional function y = x and the inverse proportional function y = (k ≠ 0) are in the first quadrant.
Intersect at point A, OA =, then the value of k is ().
A.B. 1
The analysis is AB⊥x axis, vertical foot is b, ∫ point a is on y = x, ∴ ab = ob. ao =，∴ ab = ob = 1。 Y = passing point (1, 665.
Answer b
It involves knowledge points such as proportional function, inverse proportional function and Pythagorean theorem.
This topic belongs to the synthesis of linear function, inverse proportional function and Pythagorean theorem. The solution is to construct a right triangle from the point perpendicular to the X axis in the image, calculate the coordinates of this point from the Pythagorean theorem and known conditions, substitute it into the analytical formula, and find the value of the unknown coefficient.
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6.(20 10 Ezhou, Hubei, 6, 3 points) Celebrating May Day, the municipal labor union organized a basketball match in the form of single round robin (one game between every two teams). * * * played 45 games. This time, _ _ _ _ _ _ _ teams participated in the competition.
a . 12 b . 1 1 c . 9d . 10
According to analysis, there are X teams participating in the competition. According to the meaning of the question, the score is 45, and the solution is X 1 = 10, and X2 =-9 (irrelevant, so I choose D.
Answer d
One-variable quadratic equation involving knowledge points
Comment on this topic, it is a mid-range problem. The key to solve the problem is to clarify the calculation formula of single-cycle competition and list the quadratic equation of one variable.
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7.(20 10 Hubei Ezhou, 7,3 points) As shown in the figure, in the plane rectangular coordinate system, ∠ ABO = 90? Rotate △AOB clockwise around point O, so that point B falls at point B 1 on the X axis and point A falls at point A 1. If the coordinate of point B is (,), then the coordinate of point A/kloc-0 is ().
A.(3，-4) B.(4，-3) C.(5，-3) D.(3，-5)
The analysis is BC⊥x axis, and the vertical foot is C. According to the meaning of the question, OC =, BC =. ∴ OB = = 4。 ∫△ABO∽△bco, ∴ =, the solution is AB = 3. ∫△ABO rotation to get△ A6544.
Answer b
Involving knowledge point rotation, Pythagorean theorem, plane rectangular coordinate system
Comment on this topic, which is a comprehensive topic, mainly investigates the knowledge of rotation and pythagorean theorem through plane rectangular coordinate system. At the same time, Pythagorean Theorem is also an intermediate topic that involves many knowledge points in the senior high school entrance examination questions.
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8.(20 10 Ezhou, Hubei, 8,3 minutes) As shown in the figure, AB is the diameter ⊙O, C is the point ⊙O above, which connects AC. After passing through point C, make a straight line CD⊥AB to point D, E to point OB, and straight lines CE and ⊙O to point F, which are connected.
Then AG AF= = ()
a . 10 b . 12 c . 8d . 16
Analysis connection BC, ∫ab is the diameter, ∴ ∠ ACB = 90. ∫CD⊥ab, ∴∠ACG =∠b∠b and ∠F are exactly on the same arc.
Answer c
It involves the basic nature and similarity of knowledge circle.
The comment on this topic organically combines the basic nature and similarity of the circle and is comprehensive. In a circle, the angle between the circle and the diameter is equal to 90 and the angle between the circle and the same arc is equal to what is often involved in the senior high school entrance examination, and similarity is also one of the necessary contents. This topic belongs to the intermediate level.
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9.(20 10 Hubei Ezhou, 9,3 points) The image of quadratic function Y = AX2+BX+C (A ≠ 0) is shown in the figure, and the following conclusions are drawn: ① The signs of a and b are different; ② When x = 1 and x = 3, the function values are equal; ③4a+b = 0； ④ When y = 4, the value of x can only be 0. There are _ _ _ correct conclusions.
A. 1
The analysis shows that the symmetry axis is on the right side of Y axis, and A and B are different symbols, which are correct. From the abscissas of the intersection of the image and the X axis are -2 and 6, it is concluded that the symmetry axis is X = 2, and when X = 1 and X = 3, the function values are equal, and ② is correct; The symmetry axis is x = 2, that is, -= 2, ∴ 4a+b = 0, ③ correct; According to the symmetry of image and function, when y = 4, x = 0 or x = 4, ④ is wrong, so C. 。
Answer c
Images and properties of quadratic functions involving knowledge points
This question examines the relationship between the image of quadratic function and A, B and C. The key to solving the problem is to be familiar with the relationship between the image of the function, the open direction, the symmetry axis, the vertex coordinates, the intersection of the image with the X axis, the intersection with the Y axis and the image of the function when X = 1. This is a comprehensive topic.
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10.(20 10 Ezhou, Hubei Province, 10, 3 minutes) As shown in the figure, the quadrilateral OABC is a square with a side length of 6, with points A and C on the positive semi-axes of the X and Y axes respectively, point D on OA, the coordinate of point D is (2,0), and p on OB.
A.2 B. C.4 D.6
Analyzing connected CD, because point A and point C are symmetrical points about OB, the minimum value of ∴PA+PB is the length of CD. It is known that OC = 6, OD = 2, ∴ CD = = 2. Therefore, a was chosen.
Answer a
Axisymmetric and Pythagorean Theorems Involving Knowledge Points
A square is an axisymmetric figure, and the diagonal is one of the symmetry axes. Find the shortest distance between two points on the same side of the axis of symmetry, that is, find the distance from a symmetrical point to another point.
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Fill in the blanks (3 points for each small question, *** 18 points)
The arithmetic square root of 1 1.(20 10 Hubei Ezhou, 1 1, 3 points) 5 is.
Analysis because () 2 = 5 and > 0, the arithmetic square root of ∴5 is.
answer
Knowledge points involving arithmetic square root
Comment on the arithmetic square root is the positive square root of a positive number, and the arithmetic square root of 0 is 0. This question is the basic question in the senior high school entrance examination, which increases the credibility of the examination questions.
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12.(20 10 Ezhou, Hubei, 12, 3 minutes) If the diameter of the cone bottom is 2m and the length of the bus is 4m, the lateral area of the cone is m2.
The lateral area formula of analytic cone is πrl, where r is the radius of base circle and l is the length of generatrix. According to the meaning of the question, R = 1m, L = 4m, ∴ π RL = π× 1× 4 = 4π (m2).
Answer 4π
Relates to the lateral area of the knowledge pyramid.
Comment on the lateral area formula of cone, which is one of the frequently tested contents in the basic calculation of circle. As long as you memorize the formula and calculate it carefully, you can get the correct result. It is an intermediate topic.
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13.(20 10 Ezhou, Hubei Province, 13, 3 minutes) Given that α and β are two real roots of the equation x2-4x-3 = 0, then (α-3) (β-3) =.
According to the meaning of the question, α+β = 4 and α β =-3. ∴ (α-3) (β-3) = α β-3 (α+β)+9 =-3-3× 4+9 =-6.
Answer -6
It is related to the relationship between the roots and coefficients of a quadratic equation with one variable.
This topic examines the relationship between the roots and coefficients of quadratic equations. Firstly, the sum and product of two roots are obtained according to the relationship between roots and coefficients, and then they are substituted into the deformation of the formula for evaluation.
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14.(20 10 Ezhou, Hubei, 14, 3 minutes) A black bag contains three red balls and six white balls, except for the same color. The probability of randomly picking 1 ball is.
The analysis of * * * has 9 kinds of results, and the touched ball is white with 6 kinds of results, ∴P (the touched ball is white) = =.
answer
Probability involving knowledge points
Comment on this question. Finding classical probability by enumeration. Probability is one of the compulsory contents of the senior high school entrance examination, which is not very difficult and belongs to the middle and low-grade questions.
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15.(20 10 Hubei Ezhou 15, 3 minutes) The given radius ⊙O is 10, the length of chord AB is 10, point C is on ⊙O, and point C reaches chord AB.
Through the analysis of graphs, we can get three graphs: graph 1, graph 2 and graph 3. Let OD⊥AB be d, OA∶ob = 10, AB = 10, AD = BD = 5, OD = the area in Figure 3 is: 10×5×2 = 50.
Answer 25+25 or 50
It involves the vertical diameter theorem of knowledge points, Pythagorean theorem, discussion of sub-situations, and the area of graphs.
This question is a comprehensive question, which comprehensively examines the knowledge points such as vertical diameter theorem, pythagorean theorem, and sub-situation discussion thought.
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16.(20 10 Hubei Ezhou, 16,3 minutes) As shown in the figure, in the quadrilateral ABCD, AB = AC = AD, E is the midpoint of BC, AE = CE,
∠ BAC = 3∠ CBD, BD = 6+6, then AB =.
The analysis shows that DF⊥BA is in f, ab = ac, e is the midpoint of BC, AE ⊥ BC, be = ce. ∵ AE = ce, ∴△ABC, △ABE and △ACE are all isosceles right triangles, ∴ Abe = ∴∠ FAD = 30. Let DF = X, then AF = X, AB = AD = 2x. ∵ BD = 6+6, ∴ in Rt△BFD, x2+(x+2x) 2 = (6+6) 2, the solution.
Answer 12
Involving knowledge points isosceles triangle, Pythagorean theorem, unary quadratic equation
Comment on this topic is a comprehensive question that comprehensively examines the isosceles triangle, Pythagorean theorem, solving geometric problems with equations and other knowledge points. △ABC, △ABE and △ACE are isosceles right triangles, which is the key to solve the problem.
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Three. Problem solving (17~2 1 question, 8 points for each question, 22, 23 questions 10, 24 questions 12, * * 72 points).
17.(20 10 Hubei Ezhou 17,8 points) Solve the inequality group and write the integer solution of the inequality group.
The solution set of inequality ① and inequality ② is obtained through analysis, and then the solution set of inequality group is determined, so that the integer solution of this inequality group can be determined.
The answer to inequality -3 (x-2) ≥ 4-x is x ≤1; Solving inequality: x >-2; So the solution set of this inequality group is: -2 < x ≤ 1, so the integer solution of this inequality group is-1, 0, 1.
It involves solving inequalities, inequality groups, integer solutions and other knowledge points.
One-dimensional linear inequality group examination comments mainly highlight the foundation, the questions are generally not difficult, the coefficients are relatively simple, and the main examination methods are mastered.
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18.(20 10 Hubei Ezhou 18, 8 minutes) Simplify first, and then select a number from-1, 1, 2 as the value of x for evaluation.
Analysis first decomposes the factors to find the simplest common denominator, and then carries out mixed operation to turn it into the simplest fraction. Because the denominator of a fraction cannot be 0, you should pay attention to the range of letters when taking values.
The answer is original =, original = 2.
It involves simplifying the score of knowledge points and finding the value of the score.
Comment on this question, use fractional simplification and evaluation to solve problems, and examine students' ability to solve problems by comprehensively using fractional knowledge points. It is a moderately difficult problem with a certain degree of discrimination.
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19.(20 10 Ezhou, Hubei 19, 8 points) There was a football match between the No.4 senior high school and the No.6 senior high school in our city, and two PE teachers and two ninth-grade football fans were invited as referees. A ninth-grade football fan designed a kick-off method.
(1) Two PE teachers each toss a one-dollar coin. When both coins land, the head faces No.4 Middle School, otherwise No.6 Middle School kicks off. Please draw a tree diagram or list and find the probability of tee-off in No.4 middle school.
(2) Another ninth-grade football fan found that the kick-off method designed in front was unreasonable. He changed the rules: if two coins are heads up, the fourth high school will get 8 points, otherwise the sixth high school will get 4 points. According to the probability calculation, whoever scores the highest will kick off. Do you think the revised rules are fair? Please explain the reasons; If it is unfair, please design a fair way to kick off.
Analysis (1) Using the tree diagram or list method, list all possible situations in which two PE teachers each toss a dollar coin, and then find out how many situations are heads-up, so as to find out the probability of four high tee-offs.
(2) Calculate their respective probabilities first, and then calculate the score, so as to judge whether the design pair is fair.
Answer (1) list:
Xia Shang
Up, up, up, down.
Up and down, up and down.
As can be seen from the table, the probability of kick-off in senior high school.
(2) unfair. Because the four-high kick-off probability, the score is: the six-high kick-off probability is:, so it is unfair.
Modify the rule: if two coins face up, you will get 12 points in the fourth, otherwise you will get 4 points in the sixth. According to the probability calculation, whoever scores the highest will kick off.
When it comes to the probability of knowledge points, draw a tree diagram or list.
This topic reviews students' ability to apply probability and design rules fairly, which belongs to the middle level and has a certain degree of discrimination.
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20.(20 10, Ezhou, Hubei, 20, 8 minutes) During the period of Spring Festival travel rush, the passenger flow of a passenger station is increasing, and passengers often have to queue up for tickets for a long time. According to the investigation, about 400 people queue up to buy tickets every day, and new passengers keep entering the ticket office to queue up to buy tickets. When selling tickets, the ticket office adds 4 people every minute. There are three tickets for sale every minute at each ticket window. On a certain day, the relationship between the number of people in line at the ticket office y (people) and the time of ticket sales x (minutes) is shown in the figure. It is known that only two ticket windows are opened one minute before the ticket is sold (each person only needs to buy one ticket).
(1) Find the value of a 。
(2) The number of passengers queuing at the ticket office at the 60th minute.
(3) If all the passengers in the queue can buy tickets within half an hour after the ticket sales start, how many ticket sales windows should be opened at least at the same time so that passengers who arrive at the station later can buy them at any time?
Analysis (1) According to the image, there are still 320 people waiting in line for one minute, and the equation can be obtained:
400+ new queue number-ticket number =320.
(2) Find the BC resolution function and substitute it into the analytical formula to find the function value.
(3) the number of votes sold within half an hour is greater than or equal to the original 400 people and the number of votes needed for new people within half an hour.
The answer (1) is known from the image, so;
(2) If the analytical formula of BC is, then substitute (40,320) and (104,0), so that at that time, that is, the 60th minute after the ticket is sold, there are 220 passengers waiting in line at the ticket office;
(3) If six windows are opened at the same time, you can find out from the topic. Because it is an integer, at least six ticket sales windows should be opened at the same time.
It involves knowledge point equation and linear function.
Comment on this topic is to examine students' thinking methods of solving real-life problems with equations and functions. It is a basic skill test with universality and strong credibility.
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2 1.(20 10 Hubei Ezhou 2 1, 8 minutes) As shown in the figure, before the depression angle of 30 was measured at point A 500 meters below the sea surface, a black box signal was sent out at the bottom of seabed C. After sailing in a straight line for 4000 meters at the same depth, a black box signal is sent out at point B before the depression angle of 60.
Analyze and add auxiliary lines, construct a right triangle, and solve it by using the functional relationship between sides and angles.
Solution 1: let CF⊥AB be f, then ∴, ∵, ∴, ∴, ∴ The depth of submarine black box point C from the sea surface.
Solution 2: Make CF⊥AB at f, ∫, ∴, ∴, ∫, ∴, ∴, ∴, ∴ the depth from the submarine black box C to the sea surface.
It involves the solution of azimuth and right triangle of knowledge points.
Comment on the solution of right triangle is a compulsory knowledge point for senior high school entrance examination. It is generally not difficult to investigate the relationship between the angles of a right triangle and its application. This question mainly examines the ability of candidates to construct right-angled triangles to solve problems.
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22.(20 10 Hubei Ezhou 22, 10 minutes) The engineer has an iron plate, which is 12 decimeter long and 8 decimeters wide. The right triangle with the length AE=2 decimeters and AF=4 decimeters is cut off, and the remaining pentagons' rectangles MGCH and M must be cut off.
(1) If the area of the rectangular MGCH is 70 square decimeters, find the length and width of the rectangular MGCH.
(2) When the rectangular EM is what, the area of the rectangular MGCH is the largest? And find the perimeter of the rectangle at this time.
The analytic (1) rectangular MGCH has an area of 70 square decimeters, and the equation can be listed.
The proportional ratio can be obtained from PM∑AF, and another equation can be constructed.
(2)
The answer (1) is to extend HM to AB to p, GM to AD to r, let PM=x, RM=y, then,
, ∴ …① … ②
Simultaneous ① ② solution, ∴,.
Therefore, the length and width of rectangular MGCH are decimeter and decimeter respectively.
(2) EF=,,∵,∴，
The area of rectangular MGCH is =. At that time, the maximum area of rectangular MGCH was 72 square decimeters. At this time, EM=0, that is, point E and point M coincide. Find the perimeter of the rectangle at this time =2×(6+ 12)=36 decimeters.
It involves similar triangles, calculation of rectangular area and perimeter, equation, extreme value of quadratic function and other knowledge points.
The comment on this topic is to examine students' comprehensive ability to use knowledge, skillfully synthesize algebraic equations, functions, rectangles, similar triangles and other knowledge, and form comprehensive questions within the discipline, which has certain selection function, certain discrimination and reliability.
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23.(20 10 Hubei Ezhou 23, 10) As shown in the figure, a rectangular garden is surrounded by walls and fences, with an area of S square meters and one side parallel to the courtyard wall of X meters.
(1) If the maximum usable length of the courtyard wall is10m and the length of the fence is 24m, the middle of the garden is divided into two small rectangles by the fence, and the functional relationship between S and X is found.
(2) Under the condition of (1), if the enclosed garden area is 45 square meters, find the length of AB. Can you enclose a garden with an area of more than 45 square meters? If so, how should it be surrounded? If not, please explain why.
(3) When the maximum usable length of the courtyard wall is 40m and the length of the fence is 77m, N fences are built in the middle to form small rectangles. When these small rectangles are squares and X is a positive integer, please write a set of values of X and N that directly meet the conditions.
According to the area of rectangular garden, the functional relationship between (1)S and x is analyzed.
(2) In the functional relationship constructed by (1), when S=45, find the value of x. 。
(3) We can list relationships and find positive integer solutions within the range of values.
Answer (1) (0
(2) When S = 45, the solution is obtained, ∵ 0.
It doesn't matter. Give it up. ∴AB=5.
Yes Can form a garden with an area of more than 45 square meters.
At this time, the area is greater than 45, and AB= ab =.
(3)
It involves the quadratic function of knowledge points and the quadratic equation of one variable.
The comment on this topic is to examine students' ability to solve practical problems by using quadratic equation and quadratic function knowledge. By discussing the problem of solving quadratic equations and paying attention to the reality of life, it is helpful to stimulate students' enthusiasm for using mathematics and reflect the concept of the new curriculum.
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24.(20 10 Hubei Ezhou 24 12 minutes) As shown in the figure, in the rectangular coordinate system, A (- 1 0) and B (0 0,2), the moving point P moves along the ray BM passing through point B and perpendicular to AB, and the moving speed of point P is/kloc-.
(1) Find the coordinates of point C. 。
(2) Find the analytical formula of parabola passing through points A, B and C. 。
(3) If point P starts to move, point Q also starts from point C and moves to point A in the negative direction of X axis at the same speed as point P. After t seconds, the triangle with points P, Q and C as its vertices is an isosceles triangle. Find the value of t (point P stops moving to point C and point Q stops moving at the same time).
(4) Under the conditions of (2) and (3), when CQ=CP, find the coordinates of the intersection of the straight line OP and the parabola.
By analyzing (1) the similar properties of right triangle, it can be found that OC = 4；;
(2) The analytical expression of parabola can be set by three-point or two-root formula, and then the corresponding letter coefficient can be obtained by substituting coordinates;
(3) The triangle with vertices P, Q and C is an isosceles triangle, which can be discussed in three cases: CQ = PC, PQ = QC and PQ = PC to construct the equation.
The answer (1)C point coordinates are (4,0);
(2) Let the analytical formula of parabola passing through points A, B and C be y=ax2+bx+c(a≠0), and substitute the coordinates of points A, B and C to obtain:
The analytical formula of parabola is: y = x2+x+2.
(3) Let the movement time of P and Q be t seconds, then BP=t, CQ = T, and the triangle with P, Q and C as vertices is an isosceles triangle, which can be discussed in three cases.
① if CQ=PC, as shown in the figure, then PC = CQ = BP = T. ∴ There is 2t=BC=, ∴ t =.
(2) If PQ=QC, as shown in the figure, the intersection Q is DQ⊥BC and the intersection CB is at point D, then CD = PD. From △ABC∽△QDC, we can get PD=CD=, ∴, and get t =.
(3) If PQ=PC, as shown in the figure, the intersection point P is PE⊥AC and the intersection point E, then EC=QE=PC, ∴ T = (-t), and the solution is T =.
(4) When CQ=PC, we can know from (3) that t=, the coordinate of ∴ point P is (2 1), and the analytical formula of ∴ straight line OP is: y=x, so there is x =x2+x+2, that is, x2-2x-4 = 0, and the solution is obtained.
It involves isosceles triangle, right triangle, similar shape, quadratic function and equation (group).
Comment on this topic is a dynamic problem, and the key is to use the thinking method of classified discussion to study and explore flexibly. This topic is a rare and good topic, which is conducive to cultivating students' thinking ability, but it is difficult and obviously differentiated.
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