Mathematics Test
* Examination time 120 minutes, full mark on the paper 150.
Scoring reviewer
1. Multiple choice questions (Only one of the alternative answers to the following questions is correct. Fill in the number of the correct answer in the brackets after the question. 3 points for each small question, ***24 points)
The value range of the independent variable x in the function is ()
A.B. C. D。
In ⊙O with a radius of 1, 120? The arc length of the central angle is ()
A.B. C. D。
. The straight line is known, when the straight line does not pass ().
A. first quadrant B. second quadrant C. third quadrant D. fourth quadrant
Solving fractional equation by substitution method. If set, the original equation can be reduced to the whole equation of ().
A.B. C. D。
The vertex coordinate of parabola is ()
A.(，)b .(，)c .(，)d .(，)
. As shown in Figure 1, the slope of trapezoidal revetment stone dam is 1:3, and the dam height is 2m, so the slope length is ().
A.B.
C.D.
Given that the radii of two circles are 2 and 3 respectively, and the distance between the centers of two circles is 4, then the positional relationship between the two circles is ().
A. externalization B. externalization C. intersection D. internalization
It is often windy in spring in Shenyang, which brings a lot of inconvenience to people's travel. Xiao Ming observed the continuous wind change of 12 hours on April 6, and drew the image of the wind change with time (as shown in Figure 2), then the following statement is correct ().
From 08: 00 to 14: 00, the wind continued to increase.
0800 to 12, and the maximum wind force is 7.
The wind is minimum at 8 o'clock, and the wind is minimum at D.20
Scoring reviewer
Fill in the blanks (3 points for each small question, 24 points for * * *)
The coordinates of the point (,) symmetrical about the origin are.
The root of the unary quadratic equation is.
A set of data, the variance of 0, 1, 2, 3 is.
In △,,, 30, then the degree of ∠ is.
As shown in Figure 3, PB is the tangent of ⊙O, A is the tangent point, and D is the upper point. If ∠ BAC = 70, the degree of ∠ADC is.
Given that the radius of the bottom of the cone is 2, the length of the bus is 4, and the measured area of the cone is.
It is known that the side length of a circle inscribed with a regular hexagon is 1, then the side length of a circle inscribed with a square is.
As shown in Figure 4, ⊙M intersects the X axis at points A (2 2,0) and B (8 8,0), and is tangent to the Y axis at point C, then the coordinate of the center m is.
Scoring reviewer
Three. (6 points for 17, 8 points for 18, 8 points for 19, 20 points for 10, and 32 points for * *).
. Calculation:
Solve the equation:
Read the following problem solving process:
Title: It is known that the two real roots of the equation are P and Q. Is there an M value that satisfies P and Q? If it exists, find the value of m; If it does not exist, please explain why.
Solution: There is an m value that satisfies the meaning of the question, which is obtained from the relationship between the roots and coefficients of a quadratic equation.
p+q=m，pq= 1。 ∴ .* ,∴m= 1.
Answer the following questions after reading: Is the above problem-solving process correct? If it is not correct, write the correct problem-solving process.
As shown in fig. 5, it is known that a straight line intersects the axis and the axis at points a and b, respectively, and intersects the hyperbola (
(1) The analytical expressions of straight line AB and hyperbola are obtained respectively;
(2) Find the coordinates of point D;
(3) write directly with images: when the value of x is in what range, >; .
Scoring reviewer
Iv. (For each small question 10, ***20)
In a factory, it is known that ∠ ACB = 90, AC = 3° and BC = 4° from several right-angled triangular iron plates with the same shape (as shown on the right). Now we are going to cut the remaining materials of two iron plates, and the scheme is as follows:
Scheme 1: As shown in Figure 6, use AB to cut out a sector with point C as the center and tangent to point D;
Scheme 2: As shown in Figure 7, cut a semicircle with the center of O on BC, which is tangent to AB and AC at points D and C respectively;
(1) Calculate the area of the cutting pattern of the above two schemes respectively, and fill in the calculation results directly on the horizontal line.
According to the scheme 1, the cut graphic area is.
According to scheme 2, the graphic area cut out is.
⑵ Write the calculation process of semi-circular area cutting according to Scheme 2.
As shown in Figure 8, A and B are two villages, AB, BC and CD are roads, BD is fields, AD is river width, and CD and AD are perpendicular to each other. Now laying a cable from E to Village A and Village B, there are two laying schemes:
Option 1: Option 2:.
Measured km, km, km, ∠ BDC = 45, ∠ Abd = 15.
It is known that the construction cost of underground cable is 20,000 yuan/km, and that of underwater cable is 40,000 yuan/km.
(1) Find out the river width AD (as a result, the root sign is retained);
⑵ Find out the length of highway CD;
(3) Which scheme has low cost for laying cables? Please explain your reasons.
Scoring reviewer
V. (12)
During the Spring Housing Fair in Shenyang in 2005, a company conducted a random questionnaire survey on consumers who participated in the housing fair, and distributed 1000 questionnaires, all of which were recovered. According to the questionnaire, sort out the annual income of consumers and make a table1; After sorting out the areas where consumers intend to buy a house, make Table 2 and make a histogram of bias frequency distribution (as shown in Figure 9).
Table 1 (annual income of consumers surveyed)
Annual income (ten thousand yuan)1.21.83510
Number of consumers surveyed (persons) 200 500 200 70 30
Table 2 (Information on the housing area that the surveyed consumers intend to purchase, note: the housing area is rounded off)
Grouping (square meter) frequency
40.5~60.5 0.04
60.5~80.5 0. 12
80.5~ 100.5 0.36
100.5~ 120.5
120.5~ 140.5 0.20
140.5~ 160.5 0.04
Total 1000 1.00
Please answer the following questions based on the above information:
(1) According to the table 1, the average annual income of consumers surveyed is 1 ten thousand yuan; The median annual income of consumers surveyed is
Ten thousand yuan; Average and median can better reflect the general level of consumers' annual income.
(2) According to Table 2, the number of people who intend to buy a house of 100.5 ~ 120.5 m2 is; The proportion of consumers who intend to buy a house with an area not exceeding 100 square meter is.
(3) Complete the frequency distribution histogram in Figure 9.
Scoring reviewer
Six, (12 points)
As shown in figure 10, △ABC is inscribed in ⊙O, and AD divides ∠BAC, and the intersection BC is at point E ⊙O is at point D. 。
(1) Let point D be MN‖BC, and prove that MN⊙O is tangent;
(2) Verification:
(3) As shown in figure 1 1, AE bisects the outer corner of ∠BAC ∠FAC, the extension line intersecting BC is at point E, and the extension line intersecting EA ⊙O is at point D. Is the conclusion still valid? If yes, please write out the proof process; If not, please explain why.
Scoring reviewer
Seven, (12 points)
In order to realize the goal of building Shenyang into a forest city, in the greening work this spring, the Greening Office plans to purchase and plant 400 saplings for a residential area. A sapling company provides the following information:
Message 1: There are three kinds of saplings to choose from: poplar, lilac, willow, and the number of poplar and lilac trees should be equal.
Message 2: The following table:
Wholesale price of each seedling (yuan) Air purification index of each seedling after two years.
Poplar 3 0.4
Lilac tree 2 0. 1
Willow p 0.2
Suppose poplar and willow are purchased as X and Y plants respectively.
(1) Write the functional relationship between y and x (the range of independent variables is not required);
⑵ When the wholesale price p of each willow tree is equal to 3 yuan, how to arrange the purchase quantity of these three kinds of saplings, so that the air purification index of these 400 saplings will not be lower than 90 after two years, so as to minimize the total cost of purchasing saplings? What is the lowest total cost?
(3) When there is a relationship between the wholesale price p (yuan) and the purchase quantity y (plants) of each willow, find the functional relationship between the total cost w (yuan) and the purchase quantity x (plants) (the range of independent variables is not required).
Scoring reviewer
Eight, (14 points)
As shown in figure 12, the straight line intersects with the X axis at point A, intersects with the Y axis at point B, and point C (m, n) is any point in the second quadrant. The circle with the center at point C is tangent to X axis at point E and straight line AB at point F. 。
(1) When the quadrilateral OBCE is a right angle, find the coordinates of point C;
(2) As shown in figure 13, if ⊙C is tangent to the Y axis at point D, find the radius r of ⊙C;
⑶ Find out the functional relationship between m and n;
(4) Can △OEF be an equilateral triangle during the movement of ⊙C (only "yes" or "no")?