Math Test (Course Standard Edition)
(Examination time: 120 minutes, full mark: 120 minutes)
1. Multiple-choice questions (the full score of this question is 2 1, * * there are 7 small questions, with 3 points for each small question).
Each question below gives four conclusions numbered A, B, C and D, only one of which is correct. Don't choose, choose the wrong or multiple labels, do not score.
The arithmetic square root of 1.2 is ().
A.B.- C. D.2
2. The main idea of correct geometry is ().
A.B. C. D。
3. In order to know the health status of the elderly in this area, an extracurricular interest group conducted four different sampling surveys. What do you think is a reasonable sampling ().
A. The health status of 1000 elderly people was investigated.
The health status of 1000 elderly people was investigated in the hospital.
C. The health status of 10 elderly neighbors was investigated.
D. Using the household registration network of the police station to randomly investigate the health status of 10% elderly people in this area.
4. point P 1(x 1, y 1), point P2(x2, y2) are two points on the image of linear function y =-4x+3, and x 1 < x2, then the relationship between y 1 and y2 is
a . y 1 > y2 b . y 1 > y2 > 0 c . y 1 < y2 d . y 1 = y2
5. The position of △ABC in rectangular coordinate system is known as shown in the figure. If △A'B'C' and △ABC are symmetrical about y, then the coordinate of point A corresponding to point A' is ().
A.(-4，2) B.(-4，-2) C.(4，-2) D.(4，2)
6. As shown in the figure, when △ABC, BC = 4, ⊙A is centered at point A, and the radius is 2, it is tangent to BC at point D, AB at point E, AC at point F, P at point ⊙A, and ∠ EPF = 40, then the area of the shaded part in the figure is ().
a . 4πb . 4πc . 8πd . 8π
7. The owner of the store sells a commodity at a price not lower than 20% of the purchase price, but in order to get more profits, he marks it at a price higher than 80% of the purchase price. If you want to buy this 360 yuan-priced product, the store can only sell it at the highest price ().
A.80 yuan B. 100 yuan C. 120 yuan D. 160 yuan.
Fill in the blanks (the full mark of this question is 2 1, and there are 7 small questions in * *, with 3 points for each small question)
8. As shown in the figure, if the diameter ⊙O is AB =8cm, C is a point on ⊙O, and ∠ BAC = 30, then BC = _ _ _ _ _ cm.
9. Factorization: 4a3-4a2+A = _ _ _ _ _ _ _.
10. As shown in the figure, in △ABC, AB = AC, ∠ A = 50 and BD is the bisector of ∠ABC, then ∠ BDC = 0.
1 1. The voltage of a battery is constant. When using this power supply, the functional relationship between current I(A) and variable resistance R (ω) is shown in the figure. When the current of the electrical appliance is 10A, the variable resistance of the electrical appliance is _ _ _ _ _ Ω.
12. There are 12 white balls and several black balls in one pocket. In order to estimate the number of black balls in his pocket, Xiao Liang adopted the following methods: first, take out 10 balls from his pocket, find out the ratio of the number of white balls to 10, then put them back in his pocket and shake them evenly. The ratio of the number of white balls to 10 is 0.4, 0. 1, 0.2, 0. 1, 0.2, respectively. According to the above data, Xiao Liang can estimate that there are about black balls in his pocket.
13. As shown in the figure, P is a point in the regular triangle ABC, PA = 6, Pb = 8.
Pc = 10。 If △PAC rotates counterclockwise around point A to get △P'AB, then
The distance between point P and point P' is _ _ _ _, ∠ APB = _ _ _ _
14. As shown in the figure, the following geometric figure is composed of a small cube with a side length of 1.
It is laid on the ground according to certain rules. If all exposed surfaces are colored,
(the bottom surface is not drawn), only two surfaces in the nth geometry are drawn.
There is a small cube.
3. Painting question (the full mark of this question is 6 points)
Draw with compasses and rulers, instead of writing, but keep drawing traces.
15. The cylindrical water pipe in a residential area is broken. In order to replace the pipeline, the maintenance personnel need to determine the radius of the circular section of the pipeline. The following figure shows the cross section of a horizontal fractured pipeline with water.
(1) Please complete the circular part of this water pipe;
(2) If the water surface width of the water-filled part of the water pipeline is AB = 16 cm and the height of the deepest part of the water surface is 4cm, find the radius of this circular section.
4. Answer questions (this question is out of 72 points, and there are 9 small questions in * * *).
16. (Full score for this small question)
Solve the fractional equation: = 1.
17. (Full score for this small question)
During the Spring Housing Fair in Qingdao in 2006, a real estate company conducted a random questionnaire survey on consumers who participated in the Fair, and * * * distributed 1.2 million questionnaires, and actually recovered 1.0 million questionnaires. According to the questionnaire, the real estate company made the following statistics.
First, according to the annual income statistics of consumers surveyed:
Annual income (yuan) is less than 20,000, 20,000-40,000 (excluding 40,000), 40,000-60,000 (excluding 60,000), 60,000-80,000 (excluding 80,000) and above 80,000.
The number of consumers surveyed in each market segment accounts for 50% 26% 14% 7% 3% of the total number of consumers surveyed.
Two. A fan-shaped statistical chart made according to the number of consumers who intend to buy in different housing areas under investigation:
According to the above information, solve the following problems:
(1) The average annual income of consumers surveyed is 1 10,000 yuan. (Note: In the calculation, all below 20,000 yuan is regarded as 654.38+0,000 yuan, those between 20,000 and 40,000 yuan are regarded as 30,000 yuan, and so on, and those above 80,000 yuan are regarded as 90,000 yuan. )
(2) The number of consumers who intend to buy 80m2 ~100m2 is.
(3) If you are the developer of this real estate company, please talk about your future work plan from the construction area and other aspects (no more than 30 words).
18. (Full score for this small question)
Xiao Ming and Liang Xiao use the same turntable to play with purple. The rules of the game are as follows: Turn the turntable twice continuously. If two turntables become the same color or match purple (if one turntable becomes blue and the other becomes red, it can match purple), Xiaoming will get 1, otherwise Xiao Liang will get 1. Do you think this game is fair to both sides? Please explain the reasons; If it is unfair, please modify the rules to make the game fair to both sides.
19. (Full score for this small question)
In a math activity class, the teacher led the students to measure the width of a north-south river. As shown in the figure, a student observed point C on the other side of the river at point A on the east bank of the river, and measured that C was 3 1 west of A and 45/west of B. Please help the student to calculate the width of the river according to the above data.
20. (The full score for this short question is 8)
During the May Day Golden Week, a school plans to organize 385 teachers and students to rent a car. It is known that the rental company has 42 buses and 60 buses, 42 buses are rented in 320 yuan and 60 buses are rented in 460 yuan.
(1) How much does it cost for the school to rent these two cars separately?
(2) If the school rents eight buses of these two types at the same time (you can't get enough seats), and it will save more rent than renting a car alone. Please help the school choose the most economical car rental scheme.
2 1. (Full score for this small question)
As shown in the figure, in □ABCD, e and f are the midpoint of AB side and CD side respectively, BD is diagonal, and the extension lines of AG‖DB and CB are at g 。
(1) Verification: △ ade △ CBF;
(2) If the quadrangle BEDF is a diamond, what special quadrangle is the quadrangle AGBD? And prove your conclusion.
22. (Full score for this small question 10)
On the eve of the Cherry Festival in Beizhai, Laoshan, Qingdao in 2006, a fruit wholesale company conducted a survey and statistics on the market sales in previous years to guide the cherry sales this year, and obtained the following data:
Sales price x (yuan/kg) … 25 24 23 22 …
Annual sales volume (kg) … 2000 2500 3000 3500 …
(1) In the rectangular coordinate system as shown in the figure, make each set of ordered number pairs.
The point corresponding to (x, y). Connect these points and observe the result diagram.
Judge the functional relationship between y and x, and find the function between y and X.
Digital relationship;
(2) If the purchase price of cherries is 13 yuan/kg, try to find the sales profit.
The functional relationship between P (yuan) and selling price X (yuan/kg),
And find out when x takes what value, the value of p is the largest?
23. (Full score for this small question 10)
Hua, a famous mathematician in China, once said: "If there are few shapes, it is less intuitive, and if there are few shapes, it is difficult to be nuanced;" In mathematics, number and shape are the two most important research objects, and they are closely related. Under certain conditions, numbers and shapes can be transformed and infiltrated with each other.
The basic idea of the combination of numbers and shapes is that in the process of studying problems, we should pay attention to the combination of numbers and shapes, consider the specific situation of the problem, and turn the problem of graphic nature into the problem of quantitative relationship, or turn the problem of quantitative relationship into the problem of graphic nature, thus simplifying complex problems, concretizing abstract problems, making it difficult and easy, and obtaining a simple and successful scheme.
For example, find the value of 1+2+3+4+…+n, where n is a positive integer.
For this summation problem, if we adopt the method of pure algebra (adding two ends), the problem can be solved, but in the process of summation, we need to discuss the parity of n.
It is intuitive to use the method of combining numbers with shapes, that is, to use the properties of figures to illustrate the facts of quantitative relations. Now, the value of 1+2+3+4+…+n is found by using the properties of the graph. The scheme is as follows: As shown in the figure, the triangle pattern on the left side of the diagonal line is 1, 2, 3, …, and n small circles are arranged. The number of small circles that make up the whole triangle is exactly the value of the formula1+2+3+4+…+n. Find the value of the formula, and put the left triangle on the right side of the diagonal to form a parallelogram with the original triangle. At this time, there are n rows of small circles that make up the parallelogram * *, and each row has (n(n+ 1
(1) Imitate the above thinking method of combining numbers and shapes, design relevant graphs, and find the value of 1+3+5+7+…+(2n- 1), where n is a positive integer. (Requirements: Draw a chart and use the chart to make necessary inferences. )
(2) Try to design another graph and find the value of 1+3+5+7+…+(2n- 1), where n is a positive integer. (Requirements: Draw a graph and use it to make necessary inferences. )
24. (The full score of this short question is 12)
As shown in figure 1, there are two right triangles ABC and EFG(A with the same shape (point A coincides with point E). It is known that AC = 8 cm, BC = 6 cm, ∠ C = 90, EG = 4 cm, ∠ EGF = 90, and O is the midpoint on the hypotenuse of △EFG.
As shown in Figure ②, if the whole △EFG starts from the position in Figure ① and moves in the direction of ray AB at the speed of 1cm/s, and △EFG moves, then point P starts from the vertex G of △EFG and moves to point F on the right-angled side GF at the speed of 1cm/s, and when point P reaches point F, it stops moving.
(1) When is the value of x, OP‖AC?
(2) Find the functional relationship between Y and X, and determine the range of independent variable X. 。
(3) Is there a moment of 13∶24 for the ratio of quadrilateral OAHP area to △ABC area? If it exists, find the value of x; If it does not exist, explain why.
(Reference data:1142 =12996,1152 =13225,1162 =/.
Or 4.42 = 19.36, 4.52 = 20.25, 4.62 = 2 1. 16).
In 2006, the junior high school academic level examination in Qingdao, Shandong Province.
Reference answers and grading standards of mathematics test questions
Description:
1. If the candidate's solution is different from this solution, you can make corresponding scoring rules with reference to this scoring standard.
2. When the examinee gives an incorrect answer in a certain step, which affects the subsequent part, if the answer after this step does not change the content and difficulty of this question, the score of the latter part can be determined according to the degree of influence, but it shall not exceed half of the score of the latter part; If there is a serious mistake in the answer after this step, no extra points will be given.
3. In order to facilitate marking, the calculation steps in this solution are written in detail, but candidates are allowed to reasonably omit non-critical calculation steps in the process of solving problems.
4. Answer the score on the right, which means that the candidate should get the accumulated score if he does this step correctly.
1. Multiple-choice questions (the full score of this question is 2 1, * * there are 7 small questions, each with 3 points)
1.A 2。 C 3。 D 4。 A 5。 D 6。 B 7。 C
2. Fill in the blanks (the full score of this question is 2 1, * * there are 7 small questions, and each small question has 3 points)
8.4 9 . a(2a— 1)2 10.82 . 5 1 1.3 . 6 12.48 13.6 150
14.8n-4 or 4 (2n- 1)
Third, the painting question (the full score of this question is 6 points)
15.( 1) Make the correct graph and answer it. ............................................................................................................................................
(2) solution: o is OC⊥AB in d and arc AB in c,
∴bd= ∵oc⊥ab ab =× 16 = 8cm。
According to the meaning of the question, CD = 4cm ............................................................................................................................... 4.
Let the radius be x cm, then OD = (x-4) cm.
In Rt△BOD, it is concluded from Pythagorean theorem that:
OD2+BD2=OB2，∴( x-4)2+82=x2。 ………………………………5′
∴x= 10.
That is to say, the radius of this circular section is10cm. ........................................................................................................................................................
Fourth, answer the question (the full score of this question is 72 points, and there are 9 small questions in * * *)
16. (Full score for this small question)
Answer: = 1
2-x- 1=x-3，-2x=-4
∴x = 2……4′
Test: Substitute x = 2 into the original equation: left = 1 = right.
∴ x = 2 is the root ........................................ 6' of the original equation.
17. (Full score for this small question)
Solution: (1) 2.74 .................................... 2'
(2)360.……………………………………………………………………4′
(3) As long as the students' answers are reasonable, ............................................................................ 6'
18. (Full score for this small question)
Solution:
The second time, the first time, red, yellow and blue
Red (red, red) (red, yellow) (red, blue)
Yellow (yellow, red) (yellow, yellow) (yellow, blue)
Blue (blue, red) (blue, yellow) (blue, blue)
………………………………………………………………2′
You can get it from the table: p (Xiaoming wins) =, p (Liang Xiao wins) =.
Xiao Ming scored × 1 =, and Xiao Liang scored × 1 =.
This game is unfair. ...........................................................................................................................................................
Modification rules are not unique. If the color is the same or matches purple twice, Xiao Ming gets 4 points, otherwise Liang Xiao gets 5 points. ………………………………………………………………………………………………………………………………………………………………………………………………………………………'.
19. (Full score for this small question)
Solution: point c is CD⊥AB, and the vertical foot is d,
Let CD = x meters,
In Rt△BCD, ∠ CBD = 45,
BD = CD = x meters.
In Rt△ACD, ∠ DAC = 3 1,
Ad = ab+BD = (20+x) m，CD = x m，……………………………………………………………………………………………………………………
∫tan∠DAC =，
∴ =，∴x=30.
The width of this river is 30 meters. ....................................................................................................................................................................
20. (The full score for this short question is 8)
Solution: (1)385÷42≈9.2
∴ It costs 10 to rent a 42-seat bus, and the rent is 320×10 = 3,200 yuan. .................................................................................................................
385÷60≈6.4
∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴∴875
(2) If 42 buses are rented, 60 buses (8-x) are rented.
……………………………………………………5′
Solution: ≤ x ≤
∵x is an integer, ∴ x = 4,5. .....................................................................................................................................................
When x = 4, the rent is 320× 4+460× (8-4) = 3 120 yuan;
When x = 5, the rent is 320× 5+460× (8-5) = 2980 yuan.
Answer: When renting five 42-seat buses and three 60-seat buses, the rent will be the least. .................................................................................................................................................
Note: If students write "≤" as "
2 1. (Full score for this small question)
Solution: (1)∵ Quadrilateral ABCD is a parallelogram,
∴∠ 1=∠C,AD=CB,AB=CD。 …………………………………………………2′
Points e and f are the midpoint of AB and CD respectively.
∴AE= AB，CF= CD。
∴ae=cf 3 '
∴△ADE≌△CBF。 …………………………………………………………………4′
(2) When the quadrangle BEDF is rhombic,
The quadrilateral AGBD is a rectangle.
∵ quadrilateral ABCD is a parallelogram,
∴AD‖BC。
∫AG‖BD
∴ Quadrilateral AGBD is a parallelogram ................................................................................................................... 5'
∵ Quadrilateral BEDF is a diamond,
∴DE=BE。
AE = BE，
∴AE=BE=DE。
∴∠ 1=∠2,∠3=∠4.
∵∠ 1+∠2+∠3+∠4= 180 ,
∴2∠2+2∠3= 180 .
∴∠2+∠3=90 .
That is, ∠ ADB = 90 ............................................................... 7'
∴ Quadrilateral AGBD is a rectangle ..................................................................................................................................................................
22. (Full score for this small question 10)
Solution: (1) Draw dots and connect lines correctly. According to the image, y is a linear function of x .................... 1'
Let y = kx+b,
Points (25, 2000), (24, 2500) on the image,
∴
Solution:
∴ y =-500x+ 14500。 ………4′
P=(x- 13) y
=(x- 13)(-500 x+ 14500)
=-500 x 2+2 1000 x- 188500……
=-500(x-2 1)2+32000。
The functional relationship between p and x is p =-500x2+21000x-188500.
When the sales price is 2 1 yuan/kg, the maximum profit can be obtained. ....................................................................................................................................................
23. (Full score for this small question 10)
Solution: (1)
………………………………………………………3′
Because there are n rows of small circles * * * forming this parallelogram, each row has [(2n- 1)+ 1], that is, 2n, so there are (n×2n) small circles forming this parallelogram, that is, 2n2.
∴ 1+3+5+7+…+(2n- 1)= = N2。 ………………6′
(2)
…………………………………………………………………9′
Because there are n rows of small circles that make up this square, there is (n×n), which is n2.
∴ 1+3+5+7+…+(2n- 1)=n×n=n2。 ……………………………………… 10′
24. (The full score of this short question is 12)
Solution: (1)∫Rt△EFG∽Rt△ABC
∴ , .
∴fg= = 3 cm. …………………………………………………………………2′
When p is the midpoint of FG, OP‖EG, EG‖AC,
∴OP‖AC.
∴ x = = ×3= 1.5(s)。
∴, OP ‖ AC when x is1.5s. ...............................................................................................................................................
(2) In Rt△EFG, by Pythagorean theorem, EF =5cm. ..
∵EG‖ Ah,
∴△EFG∽△AFH。
∴ .
∴ .
∴ AH= ( x +5)，FH= (x+5)。 ……………………………………6′
O is OD⊥FP, and the vertical foot is D.
Point o is the midpoint of EF,
∴OD= EG=2cm。
∫FP = 3-x，
∴S quadrilateral oahp = s △ afh-s △ ofp
= AH FH- OD FP
= (x+5) (x+5)- ×2×(3-x)
= x2+x+3……7 '
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