Seek the final examination questions of the 2008 national junior high school mathematics competition (Shandong)

First, multiple-choice questions (this big question scored 50 points, and each small question scored 5 points)

Of the four alternative answers to the following questions, only one is correct. Please write the letters that you think are correct.

Fill in the code number in the box under the corresponding heading in the table below.

The title is 1 23455 6789 10.

Answer a question.

1. The result of calculating (-2)2005+(-2)2006 is

A. 2 BC to 2 BC 65438+AD 22005

2. Known, real number, if, then the size relationship between M and N is

A.m > nb.m = n.c.m < n.d. Not sure.

3. Let "●, ▲, ■" respectively represent three different objects, as shown in the following figure. The first two kinds of balance are balanced. If you want to balance the third balance, then "?" The number of "■" should be

A.5 B. 4 C. 3 D. 2

4. In order to promote a commodity with the same price, supermarket A reduced the price by 10% for two consecutive times, and supermarket B reduced the price by 20% at one time. Which supermarket has higher cost performance?

A.A.B.B.C Similarly, D is related to commodity prices.

5. According to the corresponding values in the table below, determine the value range of a solution of equation (≠0,,, is a constant).

3.23 3.24 3.25 3.26

-0.06 -0.02 0.03 0.07

A.3 < < b . 3.23 < < 3.24

C.3.24 < < 3.25 D. 3.25 < < 3.26

6. In the plane rectangular coordinate system, A(2, -2) is known, and point P is a point on the axis, so the point P that makes AOP an isosceles triangle is

1。

7. As shown in figure 1, fold the △ABC paper along the DE. When point A falls within the quadrilateral BCED, there is a constant quantitative relationship between ∠A and ∠ 1+∠2. Please try to find out this rule. The rule you find is

A.∠A =∠ 1+∠2 b . 2∠A =∠ 1+∠2

c . 3∠A = 2∠ 1+∠2d . 3∠A = 2(∠ 1+∠2)

8. As shown in Figure 2, the points A, D, G and M are on the semicircle O, and the quadrangles ABOC, DEOF and HMNO are all rectangles. Let BC =, EF = and NH =, then the following is correct.

A.> > B. > > C. > > D. = =

9. As shown in Figure 3, the rectangular window frame made of 8-meter-long aluminum alloy strips makes the window have the largest light transmission area, so the maximum light transmission area of the window is

A.m2b.m2c.m2d.4m2。

10. As shown in Figure 4, the side length of the square ABCD is 1, and E, F, G and H are points on each side, AE=BF=CG=DH. Let the area of the small square EFGH be S and AE be, then the function image about S is approximately

2. Fill in the blanks (7 small questions in this big question, 5 points for each small question, out of 35 points)

1 1. Forty students from Class Two, Grade Three of a school donated RMB 100 for Project Hope. Donations are as follows:

Donation (RMB) 1 2 3 4

No. 6

seven

The number of people in the table who donated money to 3 yuan, 2 yuan was accidentally contaminated by ink. If there are two students who donated money to 2 yuan and two students who donated money to 3 yuan, according to the meaning of the question, we can get the equations.

12. Known as two roots of quadratic equation, the value of algebraic expression is equal to.

13. As shown in Figure 5, points A, B, C, D and E are all on ⊙O, and ∠ A = 30 and ∠ O = 48, then ∠ E = 0.

14. As shown in Figure 6, an arc in rectangular coordinate system passes through grid points A, B and C, where the coordinate of point B is (4,4), then the center coordinate of the circle where the arc is located is _ _ _ _ _ _ _ _.

15. The images of proportional function and inverse proportional function intersect at point A and point C, with axis AB⊥ at point B and axis CD⊥ at point D (as shown in Figure 7), so the area of quadrilateral ABCD is.

16. As shown in Figure 8, the height of a beer bottle is 30cm, and the height of water in the bottle is 12cm. Cap the bottle and turn it upside down. At this time, the water level in the bottle is 20cm, so the ratio of the volume of water in the bottle to the volume of the bottle (excluding the thickness of the bottom of the bottle).

17. As shown in Figure 9, it is a graph formed by stacking small cubes with side length of. If we continue to arrange according to this rule, the nth layer needs a small cube (represented by an algebraic expression containing n).

Iii. Answer the questions (this big question is ***3 small questions, out of 35 points, of which 18 10,12, 20 questions 13).

18. A real estate development company plans to build 80 apartments A and B. The funds raised by the company are not less than 20.9 million yuan, but not more than 20.96 million yuan, all of which are used for building houses. The construction cost and selling price of these two apartments are as follows:

A b

Cost (ten thousand yuan/set) 25 28

Price (ten thousand yuan/set) 30 34

(1) What are the company's architectural plans for these two types of apartments?

(2) Which scheme does the company choose to build the house with the greatest profit?

(3) According to the market survey, the price of each B-type house remains unchanged, while the price of each A-type house increases by 10000 yuan (> 0), and both houses can be sold. How does the company get the maximum profit from building houses?

Note: Profit sales price cost

19. operation: put a triangular ruler on the square ABCD with the side length of 1 so that the vertex p of the right angle slides on the diagonal AC, and one side of the right angle always passes through the point B, and the other side intersects with the ray DC at the point Q (as shown in figures 10- 1 and10)

Inquiry: Let the distance between point A and point P be.

(1) When the point Q is on the edge CD, what is the size relationship between the line segment PQ and the line segment PB? Try to prove the conclusion you observed;

(2) Is it possible for △PCQ to become an isosceles triangle when point P slides on the line AC? If possible, point out the positions of all points Q that can make △PCQ an isosceles triangle, and find the corresponding values; If not, try to explain why.

20. it is known that A 1, A2 and A3 are three points on a parabola, A 1B 1, A2B2 and A3B3 are perpendicular to the axis, respectively, and the vertical feet are B 1, B2 and B3, and the intersection line segment of straight line A2B2 is at point C.

(1) as shown in figure 11,if the abscissas of A 1, A2, A3 are1,2, 3 in turn, find the length of line CA2;

(2) As shown in Figure 1 1-2, if the parabola is changed to parabola, the abscissas of A 1, A2 and A3 are continuous integers, and other conditions remain unchanged, the length of line segment CA2 is found;

(3) If the parabola is changed to a parabola, and the abscissas of points A 1, A2 and A3 are continuous integers, and other conditions remain unchanged, please guess the length of the line segment CA2 (represented by,, and write the answer directly).

Reference answer

I.1.d2.b3.a4.b5.c6.d7.b8.d9.c10.b.

Answer the prompt:

1. The original formula = (-2) 2005+(-2) 2005× (-2) = (-2) 2005× (1-2) = 22005. So I chose D.

2. So choose B.

3. Use the properties of the equation to solve. Choose a. For example,

4. If the original price of the commodity is a yuan, then the price reduction in supermarket A is a (1-kloc-0/0%) 2 = 0.81a, and that in supermarket B is a (1-20%) = 0.8a. Therefore, it is more economical to buy in supermarket B.

5. Using the solution of the equation as the intersection of the corresponding function image and the X-axis, the approximate range of a solution of the quadratic equation in one variable is estimated to be: 3.24 < < 3.25. Therefore, C.

6. Discuss it in three situations, namely: ① isosceles triangles with O as the vertex are △OP 1A, △ op2a; ② An isosceles triangle with one vertex is △ op3a; ③ The isosceles triangle with P as the vertex is △OP4A. So there are four points p that satisfy the condition.

7. the solution 1:∫△ADE and △A/DE are symmetric about the straight line DE,

∴∠AED=∠A/ED,∠ADE=∠A/DE，

∴∠ 1+∠2=2× 180-(∠AEA/+∠ada/)

= 360-2(∠AED+∠ AD)

=360 -2( 180 -∠A)

=2∠A。

Scheme 2: As shown in the figure, link AA/,

∫△ADE and△ a/DE are symmetric about the straight line de,

∴AE=A/E,AD=A/D，

∴∠EAA/=∠EA/A,∠DAA/=∠DA/A，

∴∠ 1+∠2=2∠eaa/+2∠daa/=2(∠eaa/+∠daa/)=2∠a.

8. Convert A, B and C into the other diagonal of the corresponding rectangle, and you will find that A, B and C are all equal to the radius of the circle, so choose D. 。

9. Set the length of the window rail as x meters. Then the vertical side length is m and the light transmission area is.

When, s is the maximum value = Choose C.

10 . s = 1-4×x( 1-x)= 2 x2-2x+ 1(0 < x < 1)。 So I chose B.

2.11.12.513.5414. (2,0) 15.2 16. 17.

Answer the prompt:

12.∫a and B are two roots of the equation x2-x- 1=0.

∴a2-a= 1，b2-b= 1。 ∴3a2+2b2-3a-2b=3(a2-a)+2(a2-a)=3+2=5.

13. If BO is connected, ∠ BOC = 2 ∠ A = 60,

∴∠e=∠BOD =(∠BOC+∠cod)=×(60+48)= 54。

14. Using the vertical relationship between grid lines, we can find the center line of chord AB, and using the diagonal lines of the square to bisect each other vertically, we can find the center line of BC, so we can get the center coordinate (2,0).

15. The coordinates of the image intersection points A and C of the function y=x are (1, 1) and (-1,-1) respectively, so the area of △AOB is equal, and the area of parallelogram ABCD can be obtained according to the inverse proportional function of the image being a central symmetric figure.

16. Although the shape of the beer bottle is irregular, the cylinder visible at the lower part of the bottle body can be replaced by the regular air volume V on the right side because the volume V of the bottle and the volume V of water in the bottle are unchanged. Let the bottom area of the bottle be Scm, then the left figure of water V =12smcm3, and the left figure of water v =

∫v bottle =V water +V empty =22Scm3, ∴ V water: V bottle =6: 1 1, so C should be chosen.

17. First floor:1; Second floor:1+2; Third floor:1+2+3; … …;

Nth floor: 1+2+3+…+n= pieces,

Three. 18.( 1) If there are X houses of type A, there will be (80-x) houses of type B. 。

According to the meaning of the question, 2090 ≤ 25x+28 (80-x) ≤ 209648 ≤ x ≤ 50: x is a non-negative integer, and ∴ x is 48, 49 and 50.

There are three housing plans:

48 sets of type A and 32 sets of type B; 49 sets of type A and 3 1 set of type B; 50 sets of type A and 30 sets of type B. 。

(2) The company made a profit from building (ten thousand yuan).

∴ When x=48, the maximum value W =432 (ten thousand yuan), that is, 48 A-type houses and 32 B-type houses get the most profits.

(3)w =(5+a)x+6(80-x)= 480+(a- 1)x。

∴ when 0 < a < 1, x = 48, w is the largest, that is, there are 48 type A houses and 32 type B houses.

When a= 1 and a- 1=0, the profits of the three housing schemes are equal.

When A > 1, x=50, w is the largest, that is, 50 buildings of type A and 30 buildings of type B. 。

19.( 1)PB=PQ。

It is proved that if P is MN‖BC, AB and DC intersect at M and N points respectively, quadrilateral AMND and quadrilateral BCNM are rectangles, and △AMP and △CNP are isosceles right triangles (as shown in figure 1).

∴NP=NC=MB.

∫∠BPQ = 90 degrees,

∴∠QPN+∠BPM=90。

And < pbm+< BPM = 90,

∴∠QPN=∠PBM.

∠∠QNP =∠PMB = 90，

∴△QNP≌△PMB.∴PB=PQ.

(2)△PCQ may become an isosceles triangle.

① When point P coincides with point A and point Q coincides with point D, then PQ=QC and △PCQ are isosceles triangles. X=0 at this time.

② Solution 1: When the Q point is on the extension line of the DC side and CP=CQ, △PCQ is an isosceles triangle (as shown in Figure 2).

At this time, QN=PM=, CP=, CN= CP= 1-.

∴CQ=QN-CN=. At this time, x= 1.