Mathematics Test
Note to candidates: this question is ***28 small questions, with full score of 150 and test time of 120 minutes.
Multiple choice questions: (There is only one correct answer to each question. Please put the answer in brackets. There are *** 10 small questions in this big question, with 4 points for each small question and 40 points for * * *. )
1. equals ()
A.B. C. D。
2. In the function, the value range of the independent variable is ()
A.B. C. D。
3. In,,, and, it is ()
A.B. C. D。
4. The image of a linear function passes through point A and point B, as shown in the figure, and then inequality.
The solution set of is ().
A.B. C. D。
5. Our school participated in the physical education test in Grade Three, and a group of 10 people (girls) got the following results in standing long jump:
The score of standing long jump (M) is1.891.91.931.96.
Number of people 2 3 1 4
The mode and median of this group of students' standing long jump performance are () meters in turn.
A. 1.96 and1.91B.1.96 and1.92c.1.91and1.
6. The symmetry axis of parabola is a straight line, and it passes through point (3,2), then the value of is ().
A.0 B. 1 C.- 1 D.2
7. If an unary quadratic equation has two non-phases.
And so on, the value range of is ()
A.B.
C. and D.
8. The side length of the front view and the left view of the space geometry is 30㎝.
Regular triangle, the top view is a circle, then the transverse area of this geometric figure
Yes ()
A.2b . 2c . 2d . 2
9. As shown in Figure (a), there is a gray sector with an area of cm2 and a length of cm on the horizontal ground, which is perpendicular to the ground. If the sector in Figure (a) is scrolled to the right until it is perpendicular to the ground and does not slide, as shown in Figure (b), the distance that the point moves is ().
A.b . c . d
10. As shown in the figure, in △, the height of the side is a moving point on the side, and the intersection point and distance are set to, and the area of △ is, then the function image is roughly ().
Fill in the blanks: (Please fill in the answers on the lines. There are *** 10 small questions in this big question, with 3 points for each small question and 30 points for * * *. )
1 1. factorization
12. In the plane rectangular coordinates, the known point (,) is in the first quadrant, so the range of real numbers is.
13. The probability of drawing rational numbers and irrational numbers is.
14. If the vertex of the image of the quadratic function is on the axis, the value of is.
15. As shown in the figure, after a circular paper with a radius of ÷ is folded, the arc just passes through the center of the circle, and the length of the crease is.
16. As shown in the figure, in the grid diagram of 10×6 (the side length of each small square is 1 unit length), the radius ⊙A is 1, ⊙.
The radius of b is 2. To make ⊙A circumscribe static ⊙B, then ⊙A needs to turn right from the position shown in the figure.
Translate by one unit length.
17. As shown in the figure, the sum of two inverse proportional functions and the image in the first quadrant are
C 1 and C2, point p is on C 1, PC⊥x axis is on point c, C2 is on point a, and PD⊥y axis is on.
Points D and C2 intersect at point B, so the area of quadrilateral PAOB is.
18. As shown in the figure, in the right-angled trapezoid, ‖⊥⊥⊥⊥⊥⊥⊥⊥⊥⊥⊥⊥⊥⊥⊥⊥⊥⊥⊥⊥⊥ͥ8
19. As shown in the figure, in the diamond, the point starts from the point and moves to the point at the same speed along the edge. The following four conclusions are given: ①, ②, ③ When the point is the midpoint of the edge, ④ When the point is the midpoint of the edge, the area is the largest. The correct serial number in the above conclusion is. (What do you think?
20. Known straight line: (a non-zero natural number). When, straight line:
Let the area of △ (where O is the origin of the plane rectangular coordinate system) be: when, the straight line: intersects the axis and the axis at the point sum respectively, and let the area of △ be: ... and so on, the straight line intersects the axis and the axis respectively at the point sum, and the value is.
Three. Solution: (There are 6 small questions in this big question, and each small question scores 10, of which * * * * 60 points) When solving the following problems, the necessary calculus process or reasoning steps must be given.
2 1.( 1)(5 points)
(2)(5 points) Solve the equation.
22.( 10) Simplify first, then evaluate:, where.
23.( 10) Figure ① is a beautiful windmill pattern. Do you know how it was drawn? The windmill pattern can be drawn according to the following steps: in Figure ②, first draw a line segment OA, translate the line segment OA to CB to get the first blade F 1 of the windmill, then rotate the first blade OABC counterclockwise around the O point 180 to get the second blade F2, and then simultaneously rotate the F 1 and F2 counterclockwise around the O point by 90 to get the third and third blades.
(1) If the coordinate of point A is (4,0) and the coordinate of point C is (2, 1), write the coordinate of point B at this time;
(2) Please draw the second blade F2 in Figure ②;
(3) In the case of (1), connect OB and rotate the first blade counterclockwise by 180 to obtain the second blade.
What is the length of the path that point B passes through?
24.( 10 point) As shown in the figure, in the rectangular coordinate system, the image of the linear function and the image of the inverse proportional function intersect at a (1 4), at two points.
(1) Find the analytical formula of linear function;
(2) the area to be searched.
25.( 10) "Farmers can also reimburse medical expenses!" This is the result of implementing the new rural cooperative medical system in a city. As long as each villager pays 10 yuan every year, he can join the cooperative medical system, pay his own medical expenses first every year, and get a certain percentage of the refund at the end of the year. This measure has greatly enhanced farmers' ability to resist the risk of serious illness.
Obana, his classmates randomly surveyed some farmers in his hometown, and drew the following statistical chart according to the collected data.
According to the above information, answer the following questions:
(1) How many villagers were investigated this time, and how many villagers participated in the cooperative medical system and got a refund?
(2) If there are 10000 villagers in this township, please estimate how many people have participated in the cooperative medical system? Two years later, the number of people participating in the cooperative medical system increased to 9680. Assuming that the annual growth rate of these two years is the same, find this annual growth rate.
26.( 10 point) As shown in the figure, at the midpoint of,,, yes, it intersects with the point, intersects with the point, and the intersecting extension line is at the point.
(1) Verification:;
(2) If,, find the length of the line segment.
Four. Solution: (This big question is 2 small questions, each small question is 10 ***20 points) When solving the following problems, the necessary calculation process or reasoning steps must be given.
27.( 10) A factory plans to produce 600 sets of two types of student desks and chairs for a poor area to solve the learning problems of 1580 students. A set of tables and chairs (one table with two chairs) needs wood, and a set of tables and chairs (one table with three chairs) needs wood. This factory has wood in stock.
(1) How many production schemes are there?
(2) All the desks and chairs produced now should be transported to poverty-stricken areas. It is known that the production cost of each set of tables and chairs is 100 yuan, and the freight is 2 yuan; Manufacturing cost of each set of tables and chairs 120 yuan, freight 4 yuan. Find the relationship between the total cost (yuan) and the production of tables and chairs (sets), and determine the scheme with the least total cost and the least total cost. (Total cost, production cost and freight)
(3) According to the scheme in (2), is there any surplus wood? If so, please write directly and copy the above two kinds with extra wood.
Model tables and chairs can also provide tables and chairs for many students; If not, please explain why.
28.( 10) As shown in the figure, the parabola intersects with the shaft at two points, intersects with the positive semi-axis of the shaft at one point, and intersects with (0),.
(1) Find the analytical formula of parabola;
(2) As shown in Figure ①, make a rectangle and make an intersection point, which is a moving point on the side, and connect and make it intersect with this point. Let the length of the line segment be, and the length of the line segment be. When the point moves, sum the functional relationship and write the range of independent variables. What is the relationship between the image of this function and the part ≥0 in the parabola in Figure ① in the same rectangular coordinate system?
(3) As shown in Figure ②, in the parabola of Figure ①, a point is its vertex and a moving point on the parabola (non-coincidence). Take point (0) and sum (point, counterclockwise). When a point moves on a parabola, is there a certain positional relationship between a straight line and a parabola? If yes, write and prove your conclusion; If it does not exist, please explain why.
Proposer: Li Lan Examiner: Feng.
Answers to math test questions
First, multiple-choice questions:
The title is 1 23455 6789 10.
Answer d b b b b c c d
Second, fill in the blanks:
11.12.13.14.15.16.438+0 or 7.
17.4 18.4 19.①②③ 20.
Iii. Answer: (There are 6 small questions in this big question, and each small question is 10 ***60 points)
2 1.( 1) Original formula = ... 4 points.
= ... 5 points.
(2) The original equation is deformed as follows
................................. 1 min.
∴ ............................................... 3 points.
∴
Prove to be the root of the original equation.
The root of the original equation is 5 points.
22. Original formula =
=
=
=
When, the original formula =.
23.( 1)B(6， 1)； ........................................., two points.
(2) sketch; Six points.
(3) The figure swept by line segment OB is a semicircle. If you do BE⊥OA at point E through point B, we can get OE=6, BE= 1, OB2=37 from Pythagorean theorem.
∴ The motion path of point B is ............................................................................................................................... 10.
24. (1)∫a( 1, 4) on the inverse proportional function ∴ image,
The inverse proportional function of ∴ is when,, ∴B(3,) ............................................................................................................................................
∫A( 1, 4), B (3, 3) images about linear functions,
.........................................., 4 points.
∴ The analytical formula of linear function is ..................................................................................................................................................................
(2) Let the image of the linear function intersect the axis and the axis at points C and D respectively,
In the middle, do, then, do, then,
∴ c (4 4,0), d (0 0,0), ................................... 7 points.
∴
........................ 10.
25.( 1) 240+60 = 300 (person) ................................. 1 min.
240× 2.5% = 6 (person) 3 points.
(2) Because the percentage of participating in medical cooperation is 80%, therefore,
Therefore, it is estimated that there is 10000×80% = 8000 (person). ....................................................................................................................................
Let the annual growth rate be x, and the meaning of the question means 6 points.
8000× = 9680 ... 7 points.
Solution (excluding), that is, the annual growth rate is10%. ....................... scores 9 points.
A: * * * 300 people were investigated and 6 villagers received a refund. It is estimated that 8000 people participated in the cooperative medical system.
The annual growth rate is10% ...............10 minute.
26.( 1)∫d is the midpoint of AB, ∴AD=BD,
∴∠GAD=∠FBD BC,
∴ADG =∴BDF, 3 points.
∴△adg≌△bdf,∴ag=bf； Four points.
② connect eg.
It can be obtained from (1) △ adg △ BDF, GD=FD, and,
∴EG=EF。 Six points.
∵ ‖ , ,
∴∠ EAG+∠ ACB = 90, that is ∠ EAG = 90. .........................................................................................................................
∴ in the East Asian Games,
∴, and .................................................... scored 9 points.
∴ .....................................10.
27.( 1) If you provide a mass-produced desk and chair cover, then you can draw a mass-produced desk and chair cover from the meaning of the question.
2 points
3 points for the solution
Because it is an integer, there are 1 1 production schemes. Four points.
(2) 6 points
, decreases with the increase of.
When, at least 0.7 points.
When producing 220 sets of production tables and chairs and 380 sets of production tables and chairs, the total cost is the least.
At this point (yuan) 8 points.
(3) There is extra wood, which can solve the problem of desks and chairs for up to 8 students. 10.
28. The parabola intersects the shaft at two points, intersects the positive half shaft of the shaft at one point, and (,0),.
(1)∵, ∴ The symmetry axis of parabola is,
∵ (,0), ∴ (2, 0) ................................................................................................................................................
∴ ,∴ (0,4).
∴ ,∴ ,
So ... three points.
(2)∵ The quadrilateral is a rectangle, ∴ ∽.
That is,
∴, () ............................................ 5 points.
Say it again,
∴ of the parabola in Figure 1, when ≥0,
Translate the part ≥0 in the parabola by 4 units to the right to get (). .........................................................................................................................................................
(3), the reasons are as follows:
Connect and extend the intersection line to the point, set a straight line and intersect with the point.
A point is the vertex of a parabola,
∴ (0) and (0), (0),
, and
∴ ,
∵ ,∴ ,
∴∽, 9 points for ...........................................................................................................................
∴
∴, and then ∴ ...10.