Shandong province mathematics final exam over the years

58 (25 questions in Jiangxi Province in 2008) (this big question 10 points) is as shown in figure 1, the side lengths of squares and equilateral triangles are 1, the points slide on the line segments respectively, and the distance from the set point is, and the distance is, which is recorded as (when the points overlap).

(1) At that time (as shown in Figure 2), the value of (the result retains the root sign);

(2) What is the value of the point on the diagonal? Please state your reasons and find the value at this time (the result keeps the root sign);

(3) Please fill in the following table (accurate to 0.0 1):

0.03

0.29

0. 13

0.03

(4) If "points slide on line segments respectively" is changed to "points slide on the sides of squares respectively", please use the result of (3) to draw some points in Figure 4, and then draw an approximate figure formed by point movement.

(Reference data:. )

(Description: 1. (1) in the question, each correct one is worth 1 point;

2. Question (2), the correct answer is 1, the correct answer is 1, and the calculated value is1;

3. If four spaces are filled in, you will get 1 point;

3. If the figure is roughly drawn correctly, score 2 points.

59 (24 questions in Jinan, Shandong Province, 08) (this small question is full of 9 points)

Known: parabola (a≠0), vertex C.

(1,) intersects the X axis at points A and B,.

(1) Find the analytical expression of this parabola.

(2) As shown in the figure, take AB as the diameter, make a circle, intersect with parabola at point D, intersect with parabola symmetry axis at point E, and connect A, D, B and E in turn. Point P is a moving point on line AB (P does not coincide with points A and B), and the intersection point P is PM⊥AE at m and PN⊥DB at n, please judge whether it is a fixed value.

If yes, request the fixed value; If not, please explain why.

(3) Under the condition of (2), if the point S is a point on the line segment EP, then the point S is FG⊥EP.

, FG intersects edges AE and BE at points f and g respectively (f does not coincide with a and e, and g does not coincide with e and b). Please judge whether it is true. If it is true, please prove it. If not, please explain why.

Solution: (1) Let the analytical formula of parabola be

1 point

Substitute a (- 1, 0) into:

∴

2 points

∴

The analytical formula of parabola is:

3 points

(2) is a fixed value,

4 points

∵

AB is the diameter, ⅷ.

∠AEB=90 degrees

PM⊥AE,∴

PM∨BE

∴

△APM∽△ABE,∴

①

Similarly:

②

5 points

①

②:

6 points

(3)∵

The straight line EC is the parabola symmetry axis, ⅷ.

EC vertically bisects AB

∴

EA=EB

∵

∠AEB=90

∴

△AEB is an isosceles right triangle.

∴

EAB=∠EBA=45

7 points

As shown in the figure, point P is PH⊥BE at H,

As we all know, quadrilateral PHEM is rectangular,

∴PH=ME and ph∨ me

At △APM and △PBH.

∵∠AMP=∠PHB=90，

EAB=∠BPH=45

∴

PH=BH

And △APM∽△PBH

∴

①

8 points

In △MEP and △EGF,

∵

PE⊥FG，

∴

∠FGE+∠ seg =90

∠∠MEP+∠SEG = 90

∴

∠FGE =∠ European Parliament

∵

PME=∠FEG=90 degrees

∴△MEP∽△EGF

∴

②

From ① and ②:

9 points

If this question is classified, you can give full marks as long as it is reasonable. )

60.(08 Hangzhou, Zhejiang 24)

Set point A(0, t) and point Q(t, b) in the cartesian coordinate system xOy. The parabola f obtained by image translation of quadratic function satisfies two conditions: ① the vertex is q; (2) It intersects the X axis at two points (∣ ob ∣ < ∣OC∣), linking A and B.

(1) Is there such a parabola f? Please make a judgment and explain the reasons;

(2) If AQ∨BC, tan∠ABO=, find the analytical expression of the quadratic function corresponding to parabola F.

(08 Analysis of 24 Questions in Hangzhou, Zhejiang) √

The vertex of the parabola obtained from the translation image is,

∴

The analytical formula corresponding to parabola is:

2 points

∵

The parabola and the x-axis have two intersections.

1 point

Orders,

Get,,