Mathematics (a)
Precautions:
1. Full volume *** 150, examination time 120 minutes.
Candidates must fill in (scribble) personal information such as school, name, admission ticket number, examination room and seat number on the answer sheet.
In the corresponding position.
Candidates must fill in (paint) the answers directly in the corresponding positions on the answer sheet.
1. Multiple choice questions (this question 15, 4 points for each question, ***60 points. Only one of the four options given in each question meets the requirements of the topic)
1. In the following figure, those that are axisymmetric but not centrosymmetric are
A B C D
2. It is known that the radii of two circles are 3cm and 2cm, respectively, and the center distance is 5cm, so the positional relationship between the two circles is as follows
A. externalization B. externalization C. intersection D. internalization
3. The top view of the geometric figure shown in Figure 1
The following statement is correct.
A. If the winning probability of a game is 10 times, it will definitely win the prize.
In order to understand the mental health status of middle school students in China, we should adopt the method of general survey.
C The mode and median of a set of data 6, 8, 7, 8, 8, 9, 10 are all 8.
D. if the variance of group a data and group b data, then group b data is more stable than group a data.
5. The value range of the independent variable X in the function y =+ is
A.x ≤ 2b.x = 3c.x < 2 and x≠3d.x ≤ 2 and x ≠ 3.
6. As shown in Figure 2, in the rectangular coordinate system, this point is a fixed point on the positive semi-axis of the shaft, which is
A moving point on the hyperbola (), when the abscissa of this point increases gradually,
The region will become
A. gradually increase B. unchanged C. gradually decrease D. increase first and then decrease.
7. The world financial crisis that broke out in 2008 is the most serious financial crisis in the world since 1930s. Affected by the financial crisis, the original price of a commodity in 200 yuan was 148 yuan after two consecutive price reductions. The following equation is correct.
A.B.
C.D.
8. As shown in Figure 3, a stone arch bridge in a park is circular arc (lower arc) with a span of 24 meters.
The radius of the arch is 13m, and the arch height is
A.5m B.8m C.7m D.5m.
9. In the same rectangular coordinate system, the image of the function and the function (which are constants and sums) may be
10. As shown in Figure 4, Ding Xuan walked from the street lamp to the street lamp at night. When he reached the finish line, he found his shadow behind him.
The top just touches the bottom of the street lamp. When he walked 20 meters forward to reach the point, he found that the top of his shadow just touched the bottom of the street lamp. Given that Ding Xuan's height is 1.5m, and the heights of two street lamps are 9m, the distance between the two street lamps is
24 meters long and 25 meters wide.
It is 28 meters long and 30 meters wide.
1 1. Translate the parabola to the left by 1 unit, and then translate it up by 3 units, then the analytical formula of the parabola after translation is
A.B.
C.D.
12. As shown in Figure 5, when planting trees on the flat land, it is required to have planting spacing (between two adjacent trees).
Horizontal distance) is 4m. If you plant trees on a hillside with a slope of 0.75,
It is also required that the plant spacing is 4m, so the slope spacing of two adjacent trees is
A.5m, B.6m, C.7m, D.8m
13. The image of quadratic function is shown in Figure 6, and the following relationship is incorrect.
A.< 0 B. >0
C.> 0 D. >0
14. As shown in Figure 7, a square piece of paper is folded in half twice, then three holes are punched in it, and the paper is unfolded.
15. As shown in Figure 8, points A, B, C and D are quartiles of circle O, and moving point P starts from the center of the circle O,
Do uniform motion along the route of O-C-D-O, and let the motion time be seconds ∠ degrees ∠APB.
For y degree, the most suitable image to show the functional relationship between y and t is.
Fill in the blanks (5 small questions in this question, 4 points for each small question, ***20 points)
16. As shown in Figure 9, in a grid composed of a small square with a side length of 1, the center o of ⊙O with a radius of 1 is at the grid point, then the tangent value of ∠AED is equal to.
17. As shown in figure 10, given the fan-shaped AOB area of 36㎡, the arc length AB is 9m, and the radius OA = m. 。
18. as shown in figure 1 1, if the vertex b of the square OABC and the vertex e of the square ADEF are both in the function ().
On the image, the coordinate of point E is (,).
19. Reading materials: Let two roots of the unary quadratic equation Ax2+BX+C = 0 (A ≠ 0) be x 1, x2, then these two roots have the following relations with the coefficient of the equation: x 1+X2 =-, X/kloc-. X2 =。 Fill in the blanks according to this material: it is known that x 1 and X2 are equations.
X2+6x+3 = 0, then the value of+is.
20. The quadratic function image is shown in figure 12, and the point is located at the coordinate origin.
Point,,,, On the positive semi-axis of the Y axis, point,,,
... on the image where the quadratic function is located in the first quadrant,
If △, △, △, …, △
Are equilateral triangles, then the side length delta =.
Third, answer the question (this question is 9 small questions, ***70 points. Write the necessary text description, proof process or performance when answering.
Calculation step)
2 1. (The full mark of this question is 10)
(1) (full mark for this small question is 5) Calculation:
(2) (The full score of this small question is 5) Solve the quadratic equation of one variable by matching method:
22. (The full mark of this question is 5) As shown in figure 13, it is necessary to form a right triangle.
(∠C is a right angle) To cut a semicircular iron sheet, you first need to
Draw a semicircle on this iron sheet so that its center is on the line segment AC.
It is tangent to AB and BC. Please draw it with a ruler and compasses (requirements
Draw with a ruler, keep the traces of the painting, and do not require writing methods)
23. (The full mark of this question is 7) This year, Lanzhou City has carried out educational activities with the theme of gratitude for life in primary and secondary schools in the city. All primary and secondary schools have carried out various forms of gratitude education activities in combination with students' reality. The following figures ① and ② are fan-shaped and bar-shaped statistical charts to investigate whether some students in a school know their mother's birthday. According to the information in the chart, answer the following questions: (1) Find out the number of students in this survey and fill in the bar statistics table.
(2) If there are 2,700 students in the school, how many students do you think know mom's birthday?
(3) What do you think of the analysis of the above data? (Answer in one sentence)
Eating zongzi on Dragon Boat Festival is a traditional custom of the Chinese nation. On the morning of the fifth day of May, my mother is going to Yang Yang.
I bought four zongzi: a sausage stuffing, a red jujube stuffing and two assorted stuffing. These four zongzi are different except for the internal fillings.
Everything is the same. Yang Yang likes to eat zongzi with assorted stuffing.
(1) Please use the tree diagram or list method to predict the probability that eating two zongzi is just assorted stuffing for Yang Yang;
(2) Before eating zongzi, Yang Yang is going to use the turntable as shown in the figure to carry out the simulation test of eating zongzi (this turntable is divided into equal parts
For four sectors, the position of the pointer is fixed. After rotating the turntable, it is allowed to stop freely, and one of the sectors will just stop at the position pointed by the pointer. If the pointer points to the intersection of two sectors, rotate the turntable again), which stipulates: continuous rotation.
Two turntables mean eating two zongzi at random, thus estimating the probability of eating two assorted dumplings. Do you think this simulation method is correct? Try to explain why.
25. (The full mark of this question is 7) As shown in figure 14, it is known to be the sum of images of a linear function.
Two intersections of inverse proportional function images.
(1) Find the analytical expressions of inverse proportional function and linear function;
(2) Find the coordinates of the intersection of the straight line and the axis and the area of △;
(3) Find the solution of the equation (please write the answer directly);
(4) Find the solution set of inequality (please write the answer directly).
26. (The full mark of this question is 7) As shown in figure 15, in quadrilateral ABCD, e is a point on AB, δ△ADE and δ△BCE are equilateral triangles, and the midpoints of AB, BC, CD and DA are P, Q, M and N respectively. Try to judge what a quadrilateral PQMN is and prove your conclusion.
27. (The full mark of this question is 9) As shown in Figure 16, in two concentric circles with O as the center, AB crosses the center O, intersects with the small circle at point A, intersects with the great circle at point B, and the tangent AC of the small circle intersects with the great circle at
Point d, CO shares ∠ ACB.
(1) Try to judge the position relationship between BC's straight line and small circle, and explain the reasons;
(2) Try to judge the quantitative relationship among AC, AD and BC, and explain the reasons;
(3) If, find the circle surrounded by the big circle and the small circle.
Area. (Results keep π)
28. (The full mark of this question is 9) As shown in Figure 17, the cross section of the highway tunnel is parabolic, with a maximum height of 6m and a bottom width of12m. Now, with the O point as the origin, the straight line where OM is established is the X axis.
cartesian coordinate system
(1) directly write the coordinates of point m and parabola vertex p;
(2) Find the analytical formula of this parabola;
(3) Establish a rectangular "supporting frame" AD- DC- CB,
Make points C and D on the parabola and points A and B on the ground OM.
What is the maximum total length of this "support frame"?
29. (The full mark of this question is 9) As shown in figure 1, in the square ABCD, the coordinates of point A and point B are (0, 10) and (8, 4) respectively.
Point c is in the first quadrant. The moving point p is on the side of the square ABCD, starting from point A, moving at a constant speed along A→B→C→D,
At the same time, the moving point Q moves on the positive semi-axis of the X axis at the same speed. When point P reaches point D, both points stop moving at the same time.
Let the movement time be t seconds.
(1) When the point P moves on the edge AB, the function image of the abscissa (length unit) of the point Q with respect to the moving time t (seconds) is shown in Figure ②. Please write down the coordinates when point Q starts to move and the moving speed of point P;
(2) Find the coordinates of the side length and vertex c of the square;
(3) When the value of t is (1), the area of △OPQ is the largest, and the coordinates of point P at this time are found;
(4) If the original velocities of point P and point Q remain unchanged, can OP and PQ be equal when point P moves at a uniform speed along A→B→C→D? If yes, write all qualified values of t; If not, please explain why.
Examination paper for junior high school graduates in Lanzhou in 2009.
Mathematics (1) Reference Answers and Grading Criteria
First, multiple-choice questions (this topic is entitled 15, with 4 points for each question and 60 points for * * *).
The title is12345678 91012131415.
Answer A B C C A C B B D D D A C D C
Two. Fill in the blanks (5 small questions in this big question, 4 points for each small question, ***20 points)
16.
17.8
18.( , )
19. 10
20.2008
Third, solve the problem (this big question is 9 small questions, with a score of ***70. Write the necessary text description, proof process or calculus steps when answering the question)
2 1. (The full mark of this question is 10)
(1) (the full mark of this question is 5 points)
Solution: The original formula = 3 points.
= 4 points
= 5 points
(Step 1: Calculate, give 1 point for each correct answer. )
(2) (The full score of this question is 5 points)
Solution: Move the term and get it.
1 point
The quadratic term is converted into 1, and it is obtained that
2 points
formula
4 points
From this, you can get
, 5 points
22. (The full score of this question is 5 points)
Make an angle bisector, get 2 points, make a semicircle, get 2 points, sum 1 point, ***5 points.
The above picture is the desired picture.
23. (The full mark of this question is 7 points)
Solution:
(1) (name),
This time, 90 students were investigated. 2 points
The complete bar chart is as follows:
(Note: If the as-built drawings are not shaded, no points will be deducted) 4 points.
(2) (name),
It is estimated that there are 1500 students in this school who know their mother's birthday. Six points.
(3) Omit (positive language expression, healthy and upward can score) .7 points.
24. (The full score of this question is 7 points)
Solution: (1) The tree diagram is as follows:
2 points
(I ate two zongzi, both of which were assorted stuffing) 3 points.
(2) The tree diagram of the simulation test is as follows:
5 points
(Two zongzi are mixed stuffing) 6 points.
This simulation test is incorrect. 7 points
25. (The full mark of this question is 7 points)
Solution: (1) is on the graph of the function.
.
The analytical formula of the inverse proportional function is:. 1.
Focus on the image of the function.
2 points
After that,
Get a solution
The analytical formula of linear function is: 3 points.
(2) is the intersection of a straight line and an axis.
When,
main points
4 points
5 points
(3) 6 points
(4) 7 points
26. (The full mark of this question is 7 points)
Proof: Connect AC and BD as shown in the figure.
∫PQ is the center line of △ABC,
∴ pqac. 1 point
Similarly, mnac.
∴ MN PQ, 2 points
∴ Quadrilateral PQMN is a parallelogram. Three points are in △AEC and △DEB,
AE=DE，EC=EB，∠AED=60 =∠CEB，
That is, ∠ AEC = ∠ Deb.
∴△AEC?△deb。 5 points ∴ AC = BD.
PQ = AC = BD = Pn6 Point PQMn is a diamond. Seven points
27. (The full mark of this question is 9 points)
Solution: The straight line of (1) is tangent to the small circle.
The reasons are as follows: if you work through the center of the circle, the vertical foot is,
Is the tangent of a small circle, passing through the center of the circle,
, 1 min
Divide equally.
.2 points
A straight line is the tangent of a small circle. 3 points
(2)AC+AD=BC
The reasons are as follows: connection.
Cut a small circle to the point, cut a small circle to the point,
.4 points
In and,
,
(HL)
.5 points
,
.6 points
(3). 7 points
, .8 points
Annular area
Again, 9 points.
Note: If the conclusions in questions (1) and (2) have been proved, but no judgment has been made before the proof, no points will be deducted.
28. (The full mark of this question is 9 points)
Solution: (1) m (12,0), p (6 6,6) .2 points.
(2) Let the analytical expression of parabola be: .3 minutes.
∫ Parabolic passing point (0,0),
∴, which is 4 points.
∴ The analytical formula of parabola is:
.5 points (3) Let A(m, 0), then
B( 12-m, 0), 0.6 points.
∴ Total length of "bracket" AD+DC+CB =
=.8 points
The image opening of this quadratic function is downward.
∴ When m = 3m, the maximum value of AD+DC+CB is 15m. Nine points
29. (The full mark of this question is 9 points)
Solution: (1) (1, 0) 1.
The moving speed of point P is 1 unit length per second. Two minutes.
(2) If the passing point is the BF⊥y axis at the point and the ⊥ axis at the point, then = 8.
∴ .
At Rt△AFB, 3 points.
The intersection point is the axis at this point, and the extension line intersects with this point.
* ∴△abf≌△bch.
∴ .
∴ .
The coordinate of point ∴c is (14, 12) .4.
(3) taking the crossing point P as the PM⊥y axis of point M and the PN⊥ axis of point N,
Then △ APM ∽△ ABF.
∴ .。
∴ .∴ .
Let the area of △OPQ be (square unit)
∴ (0≤ ≤ 10) 5 points
Note: If the range of independent variables is not indicated, no points will be deducted.
∵& lt； When ∴ is 0, the area of △OPQ is the largest. Six points.
At this point, the coordinate of p is (,) .7 points.
(4) When 4)or, OP and PQ are equal. Nine points
1 points plus one, no need to write the solution process.