mathematics

Volume one multiple-choice question (***24 points)

A, multiple-choice questions (this topic has ***l2 small questions, each small question 2 points, * * * 24 points. Only one of the four options given in each small question meets the requirements of the topic, please select it and black it on the answer sheet)

The reciprocal of 1 |-6| Yes (D)

A. 6th century BC

2. the quadrant where the point (a 2. 1) is located is (b)

A. first quadrant B. second quadrant C. third quadrant D. fourth quadrant

3. The following operation is correct (A)

A.B. C. D。

4.20 1 1 in the first quarter, the investment in fixed assets in our province was 47.56 billion yuan. This data can be expressed as (c) by scientific notation.

Yuan RMB

5. Both sides of ∠AOB are shown in the figure. OA and OB are plane reflectors, ∠ AOB = 35, and there is a little E on OB. After a beam of light from point E is reflected by point D on OA, the reflected light DC is just parallel to OB, so the degree of ∠DEB is (b).

1 10 D. 120

6. Fold a rectangular piece of paper in half according to Figure (1) and Figure (2) in turn, then cut it along the dotted line in Figure (3), and finally spread the paper in Figure (4) to get the pattern (a).

7. Every regular polygon with an outer angle equal to 45 is (c).

A. Regular hexagon regular heptagon regular octagon regular nonagon

8. The figure is three views of the workpiece, and the dimensions are marked in the figure, so the volume of the workpiece is (B).

a . 13πb . 17πc . 66πd . 68π

The solution of the fractional equation is (b)

A.B. C. D。

10. During the "May Day" festival, the cost price of an electrical appliance was marked up by 30%, and then it was sold for 2080 yuan at a 20% discount (80% of the marked price). The cost price of the electric appliance is X yuan. According to the meaning, the following equation is correct (A).

A.B.

C.D.

1 1. As shown in the figure, in △ABC, AB=AC, points D and E are the midpoint of AB and AC, respectively, points G and F are on BC, and the quadrilateral DEFG is a square. If DE=2cm, the length of AC is (d).

A. cm diameter 4 cm

12. Given the quadratic function image as shown in the figure, the symmetry axis is a straight line x= 1, then the following conclusion is correct (b).

The two roots of the equation are

When x>0, y decreases with the increase of x 。

Volume 2 Non-multiple choice questions (***96 points)

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

13. Calculation: _ _ _ _ _ _ _ ()

14. As shown in the figure, the quadrilateral ABCD is a parallelogram. Adding | and a condition _ _ _ _ _ _ _ _ _ can make it a rectangle. (∠ ABC = 90 or AC=BD)

15. During the "Twelfth Five-Year Plan" period, Shanxi will be built into a strong tourist province in the central and western regions, and the service industry led by tourism will become a rich driving force for Shanxi's economic development. In 20 10, the total tourism revenue of the whole province is about 1000 billion yuan. If the total tourism revenue of the whole province is to reach144 billion yuan by 20 12, the average annual growth rate should be _ (20%).

16. The picture shows a set of regular patterns with the same length of sticks. The pattern (1) needs four sticks, and the pattern (2) needs 10 sticks ... If you put it down according to this rule, the first pattern needs a wooden stick _ _ _ _ _ _ _. (6n-2)

17. As shown in the figure, △ABC is an isosceles right triangle, ∠ ACB = 90, and AB=AC. △ AB ′ c ′ is obtained by rotating △ABC clockwise by 45 degrees around point A.. If AB=2, the area of the swept part (shadow part) of the line segment BC during the above rotation is _ _ _ _. ( )

18. As shown in the figure, AB =12 is known; AB⊥BC in B, AB⊥AD in A, AD=5, BC = 10. Point e is the midpoint of the CD, so the length of AE is _ _ _ _ _ _ _ _. ( )

Third, the solution (this big question is ***8 small questions, ***78 points. Answer to write about friends, proof process or calculus steps)

19. (There are two small problems with this problem. 1 item 8, item 2 6, *** 14)

(1) Simplify first. Re-evaluate:

, among them.

Solution: original formula =, when, original formula =

(2) Solving the inequality group: express its solution set on the number axis.

Solution: Obtained from ①,

From 2,

∴ 。

Represents abbreviations on the number axis.

20. (7 points) As shown in the figure, in the plane rectangular coordinate system, the image of a linear function intersects with the X axis and the Y axis at points A and B, with the image of an inverse proportional function at points C and D, and with the DE⊥x axis at point E. It is known that the coordinates of point C are (6,) and DE = 3.

(1) Find the analytic expressions of inverse proportional function and linear function.

(2) Answer directly according to the diagram: When x is what value, the value of the linear function is greater than the value of the inverse proportional function?

Solution: The analytical formula of (1) proportional function is

Analytical formula of linear function

(2) When or when. The value of the linear function is greater than the value of the inverse proportional function,

2 1. (8 points for this question) Xiaoming and Xiao Liang play games. They washed three playing cards with brand numbers 2, 3 and 4, and then put them on the table with their backs up. The rules of the game are as follows: first, randomly draw a card from it, take the card number as the tenth digit, then put it back and wash it again, then randomly draw a card from it and put the card number on the table. If the composition of the two digits is exactly a multiple of 3, Xiao Liang wins.

Do you think the rules of the game are fair to both sides? Please use a diagram or list to explain the reason.

Solution: The rules of the game are unfair to both sides.

The reasons are as follows. Draw a tree according to the meaning of the question:

Points for attention: If candidates write "two teachers" directly in the form, they will be given 4 points as long as they answer correctly.

It can be seen from the tree diagram (or table) that there are nine possible results * * *, namely: 22, 23, 24, 32.33, 34, 42, 43, 44, and the possibility of each result is the same, and there are six possible results * * *.

∴P (Xiao Mingsheng) =,

∴P (Xiao Liangsheng) =

(small) > p (sheng), this game rule is unfair to both sides.

22. (9 points in this question) As shown in the figure, △ABC is a right triangle, ∠ ACB = 90.

Exercise and Operation (1) Draw with a ruler according to the following requirements, and indicate the corresponding letters in the drawing (keep drawing traces and don't write).

(1) is the circumscribed circle of △ABC with the center of o;

② Take line segment AC as one side, and do equilateral △ ACD on the right side of AC;

③ Connect BD, cross ⊙O at point F, and connect AE.

Scoring description: ① Item 2, ② Item 2, ③ Item 1, as shown in the figure.

If the candidate is a vertical line with two sides or three sides, no points will be deducted.

(2) Synthesis and application in your drawings, if AB=4 and BC=2, then:

(1) the positional relationship between ad and ⊙O is _ _ _ _ _. (two points) (tangency)

② The length of line segment AE is _ _ _ _ _ _ _. (2 points) (or)

23.( 10) A class implements a quantitative assessment system. In order to understand the students' learning situation, Mr. Wang made statistics on the comprehensive evaluation scores of students in Group A and Group B for six weeks in a row, and made the obtained data into the following statistics:

Comprehensive evaluation score statistics (unit: points)

(1) Please complete the following table according to the data in the table (note: the variance calculation result is accurate to 0. 1).

Solution:

average number

median

discrepancy

Group a

14

14

1.7

Group b

14

15

1 1.7

(2) According to the data in the comprehensive evaluation score statistics table, please draw the broken-line statistical chart of the comprehensive evaluation score of Group B in the figure below.

Solution: The picture on the right is a line chart.

(3) Please make a brief evaluation of the learning situation of Group A and Group B for six consecutive weeks according to the information in the dotted statistical chart.

Solution: As can be seen from the line chart, the performance of Group A is relatively stable, but the progress is not great, and there is a slight downward trend.

The results of group B are not stable enough, but they are improving rapidly, showing an upward trend.

Scoring description: The answer is not unique, as long as it meets the meaning of the question, it can be scored.

24. (Question 7) As shown in the picture, students in the comprehensive practice activity group of a school want to measure the height of a tree DE in the park. On the steps in front of a pavilion in front of the tree, they measured the elevation of the tree top D at point A at an angle of 30, and walked to point C under the steps in the direction of the tree, and measured the elevation of the tree top D at an angle of 60. It is known that the height AB of point A is 2m, and the slope AC of the step is (namely AB: BC). Please calculate the height of the tree DE according to the above items (ignore the height of the inclinometer).

Solution: The tree DE is 6 meters high.

25. (9 points for this question) As shown in figure (1), in Rt△ABC, ∠ ACB =-90, CD⊥AB, the vertical foot is D, AF divides equally ∠CAB, CD is at point E, and CB is at point F.

(1) verification: CE = CF

Proof: ellipsis

(2) Translate △ADE( 1) in the figure to the right along AB to the position of △ a ′ d ′ e ′, so that the point e ′ falls on the edge of BC, and other bars.

As shown in Figure (2), what is the quantitative relationship between BE' and CF? Please prove your conclusion.

Solution: Equality

Proof: As shown in the figure, point E is EG⊥AC in G.

And ∵ AF ∠CAB, ED⊥AB, ∴ ed = eg

From the nature of translation, we can know that D'E'=DE, ∴ d' e' = ge.

∠∠ACB = 90 degrees. ∴∠ACD+∠DCB=90

∵CD⊥AB is in D. ∴∠ B+∠ DCB = 90.

∴ ∠ACD=∠B

In Rt△CEG and Rt△BE'D',

∠∠GCE =∠B，∠CGE=∠BD'E '，CE=D'E '

∴△CEG≌△BE'D'

∴CE=BE'

According to (1), CE=CF,

(Please refer to the score for other proofs. ) 。

26.( 14) As shown in the figure, in the plane rectangular coordinate system, quadrilateral A→B→C is a parallelogram. The straight line passes through two points, O and C. The coordinates of point A are (8, O), and the coordinates of point B are (1 1.4). The moving point P starts from point O on the line segment OA at a speed of 65438+ per second. When one of the two points P and Q reaches the end point, the other point also stops moving, and the moving time of P and Q is set as t seconds (). The area of △ MPQ is S.

(1) The coordinate of point C is _ _ _ _ _ _ _ _ _, and the analytical formula of the straight line is _ _ _ _ _ _ _ _ _. (Each space 1 point, ***2 points)

(3,4);

(2) Before Q and M meet, try to find the functional relationship between S and T, and write the corresponding range of T. ..

Solution: according to the meaning of the question, OP=t, AQ = 2t. It is discussed in three situations:

① As shown in Figure L, the coordinate of point M is ().

If the intersection point C is the CD⊥x axis in D and the intersection point Q is the QE⊥ x axis in E, you can get △AEO∽△ODC.

∴ ,∴ ,∴ ,

∴ The coordinate of point Q is (), ∴PE=

∴S=

② As shown in Figure 2, when the intersection Q is qf ⊥ and the X-axis IS is f,

* ,∴of=

∴ The coordinate of point Q is (), ∴PF=

∴S=

(3) When point Q intersects with point M, solve it.

③ When, as shown in Figure 3, MQ=, MP=4.

S=

(1) (2) (3) The range of the three independent variables T .......................................................................................................................................................... (8 (8)

Scoring description: in ① and ②, every L analytical expressions will get 2 points, and in ③, every L analytical expressions will get L points. In ①, ② and ③, the range of the three independent variables T is correct.

You can get 1 minute.

(3) Try to find the value of T in question (2) and find the maximum value of S. ..

.

Solution: ① In time,

∵, the parabolic opening is upward, and the symmetry axis is a straight line.

∴ When appropriate, S increases with the increase of T. ..

At that time, s had a maximum value of.

(2) when, ∵, parabolic opening downward.

At that time, s had a maximum value of.

(3) when, ∵. ∴ s decreases with the increase of t 。

When again, S = 14. When S = 0.

To sum up, when s has a maximum value, the maximum value is.

Points for attention in grading: ① ② ③ each 1, conclusion1; If there is only one calculation error in the values of S and T in ②, resulting in an error in the corresponding S or T in the final conclusion, ② and the conclusion will be deducted intermittently, and only 1 point will be deducted; Candidates can score as long as they answer that S decreases with the increase of T 。

(4) With the movement of P and Q, when point M moves on line segment CB, let the extension line of PM intersect with the straight line of point N. Try to explore: When t is what value, △QMN is an isosceles triangle? Please write the value of t directly.

Solution: When △QMN is an isosceles triangle.