I'll send you this test paper.
Analysis on the Selected Mathematics Finals of the National Senior High School Entrance Examination in 2008 (5)
50.(08 Yunnan Shuangbai) 25. (This little question (1) ~ (3) Ask *** 12 points; Questions (4) and (5) are extra points, with a score of 10, with 5 points for each small question, and extra points can be recorded in the total score; If the total score exceeds 120, it will be recorded as 120).
It is known that the parabola Y = AX2+BX+C intersects with the X axis at points A and B, and intersects with the Y axis at point C, where point B is on the positive semi-axis of the X axis, and point C is on the positive semi-axis of the Y axis, and the line segments OB and OC (OB
(1) Find the coordinates of points A, B and C;
(2) Find the expression of this parabola;
(3) Find the area of △ABC;
(4) If point E is a moving point on the line segment AB (not coincident with points A and B), then the intersection of point E is EF‖AC, and point BC intersects with point F, which connects CE. Let the length of AE be m and the area of △CEF be s, find the functional relationship between S and M, and write the value range of independent variable M;
(5) On the basis of (4), try to explain whether there is a maximum value of S, if so, request the maximum value of S, find the coordinates of point E at this time, and judge the shape of △BCE at this time; If it does not exist, please explain why.
(08 Analysis on 25 Questions of Yunnan Double cypresses) 25. (This small problem is 12) Solution: (1) Solve the equation X2- 10x+ 16 = 0 to get X 1 = 2, X2 = 8.
∵ point b is on the positive semi-axis of x axis, point c is on the positive semi-axis of y axis, and ob < oc.
∴ The coordinates of point B are (2,0), and the coordinates of point C are (0,8).
The symmetry axis of parabola Y = AX2+BX+C is straight line X =-2.
According to the symmetry of parabola, the coordinate of point A is (-6,0).
∴ The coordinates of points A, B and C are A (-6,0), B (2 2,0) and C (0 0,8) respectively.
(2)∵ point c (0 0,8) is on the image of parabola y = ax2+bx+c.
∴ c = 8. Substitute A (-6,0) and b (2 2,0) into the expression y = ax2+bx+8, and you get
0 = 36a-6b+80 = 4a+2b+8 gives a =-23b =-83.
The expression of parabola is y =-23x2-83x+8.
(3)∫AB = 8，OC=8
∴S△ABC = 12×8×8=32
(4) According to the meaning of the question, AE = m, then be = 8-m,
* OA = 6，OC=8，∴AC= 10
Ac ∴△BEF∽△BAC
∴ efac = beab means ef 10 = 8-M8 ∴ ef = 40-5m4.
If the intersection f is FG⊥AB and the vertical foot is g, then SIN ∠ FEG = SIN ∠ CAB = 45.
∴FGEF=45 ∴FG=45？ 40-5m2 = 8m
∴s=s△bce-s△bfe= 12(8-m)×8- 12(8-m)(8-m)
=12 (8m) (8-8+ m) =12 (8m) m =-12m+4m
The range of independent variable m is 0 < m < 8.
(5) existence. Reason:
∫ s =-12m2+4m =-12 (m-4) 2+8 and-12 < 0,
∴ When m = 4, S has a maximum value, and the maximum value of S = 8.
∵ m = 4, and the coordinate of ∴ point E is (-2,0).
△ BCE is an isosceles triangle.
51.(Chongqing Volume 08) (The answer to this question is missing) 28. (10) It is known that, as shown in the figure, the parabola intersects with the Y axis at point C (0,4), and intersects with the X axis at points A and B, and the coordinates of point A are (4,0).
(1) Find the analytical formula of parabola;
(2) Point Q is the moving point on line segment AB. Passing through point q is QE AC, and passing through BC at point e and connecting CQ. △ When the area of △CQE is the largest, find the coordinates of point Q;
(3) If the moving straight line parallel to the X axis intersects the parabola at point P and intersects the straight line AC at point F, the coordinate of point D is (2,0). Q: Is there such a straight line that △ODF is an isosceles triangle? If it exists, request the coordinates of point P; If it does not exist, please explain why.
52(08 Huzhou, Zhejiang) 24. (This little question is 12)
It is known that in a rectangle, a plane rectangular coordinate system as shown in the figure is established with a straight line as the axis and the axis respectively. It is a moving point (non-coincidence) on the edge, and the image of the inverse proportional function passing through this point intersects the edge at this point.
(1) Verification: the area is equal to;
(2) Remember, what is the maximum value? What is the maximum value?
(3) Please explore: Is there such a point? After the edge is folded in half, the point just falls on the floor? If it exists, find out the coordinates of the point; If it does not exist, please explain why.
Analysis of 24 questions in Huzhou, Zhejiang Province, 24. (This little question is 12)
(1) proves that the areas of,, and are,
Judging from the meaning of the question,
, .
, which is equal to the area of.
(2) According to the meaning of the question, the coordinates of the two points are,
,
.
When there is a maximum value.
.
(3) Solution: Suppose there is such a point. After the edge is folded in half, the point falls right on the edge, and the vertical foot is.
Judging from the meaning of the question:,,,
, .
Say it again,
.
, ,
.
,, solution.
.
There is a qualified point whose coordinates are.
53.(08 Huai 'an, Zhejiang) (The answer to this question is missing) 28. (This little question is 14)
As shown in the figure, in the plane rectangular coordinate system, the quadratic function y=a(x-2)2- 1. The vertex of the image is p, the intersection points with the X axis are A and B, and the intersection point with the Y axis is C. Connect BP and extend the intersection point with the Y axis to point D. 。
(1) Write the coordinates of point p;
(2) Connect AP, if △APB is an isosceles right triangle, find the value of A and the coordinates of points C and D;
(3) Under the condition of (2), connect BC, AC, AD, and point E(0, B) on the line segment CD (except end points C and D), and rotate the point E counterclockwise by △ BCD 90 to get a new triangle. Let the overlapping area of this triangle and △ACD be S, which is expressed by algebraic expression with B according to different situations. When b is a value, the area of the overlapping part is the largest. Write the maximum value.
54.(08 Jiaxing, Zhejiang) 24. As shown in the figure, in the rectangular coordinate system, it is known that two points are in the first quadrant and are regular triangles, the positive semi-axis of the intersecting axis of the circumscribed circle is at this point, and the tangent of the circle passing through this point is at this point.
(1) Find the coordinates of two points;
(2) Find the resolution function of the straight line;
(3) Let each be two moving points on the line segment and divide the perimeter of the quadrilateral equally.
Try to explore: the largest area?
(08 Zhejiang Jiaxing 24 questions analysis) 24. ( 1),.
Keep working,
Is a regular triangle,
, .
.
Lian,,,
.
.
(2) is the diameter of a circle,
The tangent of the circle again.
, .
.
Let the resolution function of the straight line be,
Then, solve
The resolution function of a straight line is.
(3) , , , ,
The perimeter of a quadrilateral.
Let's say that the area of is,
Then,.
.
When.
These points are on a line segment,
, the solution.
Satisfied,
The maximum area of is.
55(08 Jinhua, Zhejiang Province) (the answer to this question is temporarily missing) 24. (Subject 12 points) As shown in figure 1, in the plane rectangular coordinate system, it is known that AOB is an equilateral triangle, the coordinates of point A are (0,4), point B is in the first quadrant, and point P is the moving point on the X axis, which is connected with AP and put into it. (1) Find the analytical formula of straight line AB; (2) When point P moves to point (0), find the length of DP at this time and the coordinates of point D; (3) Whether there is a point P, so that the areas of Δ δOPD are equal, and if there is, the coordinates of the point P meeting the requirements are requested; If it does not exist, please explain why.
56(08 Lishui, Zhejiang) 24. As shown in the figure, in the plane rectangular coordinate system, the coordinates of a known point are (2,4), the straight line intersects with the axis at this point, the parabola moves in the direction from this point, intersects with the straight line at this point, and stops moving when the vertex reaches this point.
(1) Find the resolution function of the line where the line segment is located;
(2) Let the abscissa of the vertex of the parabola be,
(1) Use algebraic expressions to represent the coordinates of points;
② When the value is, the line segment is the shortest;
(3) When the line segment is the shortest, is there a point on the corresponding parabola that makes △
The area of is equal to the area of △, and if it exists, request the coordinates of this point; if
Does not exist, please explain why.
(08 Analysis of 24 Questions in Lishui, Zhejiang) 24. (This is entitled 14 points)
Solution: (1) Let the resolution function of a straight line be,
∵ (2,4),
∴ , ,
The resolution function on the straight line is ............................................ (3 points).
(2)①∫ The abscissa of vertex m is, moving on the line segment.
∴ (0≤ ≤2).
The coordinates of the vertex are (,).
The parabolic analytic function is.
When appropriate, (0≤ ≤2).
∴ The coordinate of the point is (2) ............................................ (3 points).
② ∵ = =, and ∵0≤ ≤2,
When appropriate, PB is the shortest .............................................. (3 points).
(3) When the line segment is the shortest, the analytical formula of parabola is ........................................................... (1 min).
Suppose there is a point on the parabola, then.
The coordinates of the set point are (,).
(1) When a point falls below a straight line, make a straight line//intersect the point.
∵ , ,
∴∴ The coordinate of the point is (0,).
The coordinate of the point is (2,3), and the resolution function of the straight line is.
The point falls on a straight line.
∴ = .
Solution, that is, point (2,3).
Point coincides with the second point.
At this time, there is no point on the parabola, so delta sum.
.......................... with equal triangle area (2 points)
② When the point falls on a straight line,
Do a symmetrical point about this point, cross a straight line//,and intersect with this point.
The coordinates of ∵, ∴ and ∴ are (0, 1) and (2, 5) respectively.
∴ The linear resolution function is.
The point falls on a straight line.
∴ = .
Solution:,.
Substitute, get,.
At this time, there is a point on the parabola.
Make the area of △ equal to that of △ ....................................... (2 points)
To sum up, there are some points on the parabola,
Make the area of delta equal to.
57(08 Quzhou, Zhejiang Province) 24. (This topic 14 points) It is known that the position of the right-angled trapezoidal paper OABC in the plane rectangular coordinate system is shown in the figure, and the coordinates of the four vertices are O (0 0,0), A (10/0,0), B (8 8,0) and C (0 0,0).
(1) Find the number of times ∠OAB, and find the functional relationship between S and T when point A' is on line AB;
(2) When the figure of the overlapping part of the paper is quadrilateral, find the value range of t;
(3) Is there a maximum value for S? If it exists, find this maximum value and find the value of t at this time; If it does not exist, please explain why.
(08 Zhejiang Quzhou 24 questions analysis) 24, (this question 14 points)
Solution: (1) ∫ The coordinates of point A and point B are A (10/0,0) and B (8 8,0) respectively.
∴ ,
∴
Be a point. When on line segment AB, ∫, TA=TA? ,
∴△A？ TA is an equilateral triangle,
∴ , ,
∴ ,
When a. When it coincides with b, AT=AB=,
So at this time.
(2) When is point A? When the extension line of line segment AB and point P is on line segment AB (not coincident with b),
The figure of the overlapping part of the paper is quadrilateral (as shown in figure (1), where e is TA? Intersecting with CB),
When point P and point B coincide, AT=2AB=8, and the coordinate of point T is (2,0).
And from (1) when a? When it coincides with B, the coordinate of T is (6,0).
So when the pattern of the overlapping part of the paper is quadrangular,
(3)s has a maximum value.
1 when,
On the left side of the symmetry axis t= 10, the value of s decreases with the increase of t,
When t=6, the maximum value of s is.
○2 When, from Figure 1, the area of the overlapping part.
∫△A？ The height of EB is,
∴
When t=2, the maximum value of s is;
○3 When, that is, when is point A? And point p is the extension of line AB (as shown in figure 2, where e is TA? Intersection with CB, f is the intersection of TP and CB),
∵, quadrilateral ETAB is isosceles, ∴EF=ET=AB=4,
∴
To sum up, the maximum value of s is, and the value of t at this time is.
58(08 Shaoxing, Zhejiang) 24. Put a rectangular piece of paper in a plane rectangular coordinate system. The moving point starts from the point and moves to the end point at the speed of 1 unit per second. When it moves for seconds, the moving point starts from this point and moves to the end point at the same speed. When one of the points reaches the end point, the other point stops moving. The moving time of a point is (seconds).
(1) is represented by the algebraic expression contained;
(2) When, as shown in figure 1, the edge is folded, and the point just falls on the edge, so as to find the coordinates of the point;
(3) Get the link by folding the edge, as shown in Figure 2. Q: Can sum be parallel? Can it be perpendicular to? If yes, find the corresponding value; If not, explain why.
(08 Analysis of 24 Questions in Shaoxing, Zhejiang) 24. (The full mark of this question is 14)
Solution: (1),.
(2) If it is in time, it will be overloaded and handed in, as shown in figure 1.
Then,,
, .
③ ① It can be parallel to.
As shown in fig. 2, if,
That is to say, at the same time,
.
② Not perpendicular to.
If it is unfolded, as shown in fig. 3,
Then.
.
.
Say it again,
,
And, moreover,
Does not exist.
59.(08 Suqian, Zhejiang) 27. (The full mark of this question is 12)
As shown in the figure, the radius ⊙ is, the square vertex coordinates are, and the vertex moves on ⊙.
(1) When the point moves to be on the same straight line with the point, try to prove that the straight line is tangent to ⊙;
(2) When the straight line is tangent to ⊙, find the functional relationship corresponding to the straight line;
(3) If the abscissa of a point is, the area of a square is, and the function relation of summation, the maximum value and the minimum value are obtained.
24. As shown in the figure, in a rectangle, the point, is the moving point on the edge (the point is different from the point, but the point coincides with it), the intersection point is a straight line, the edge intersects with the point, and then the point is folded in half along the moving line, the point corresponding to the point is a point, the length is set to, and the area of the overlapping part with the rectangle is.
(1);
(2) What is the value of the point on the edge of the rectangle?
(3)① The functional relationship between sum and;
② What is the value, the area of the overlapping part is equal to the area of the rectangle?
60(08 Wenzhou, Zhejiang) 24. (This question is 14)
As shown in the figure, the middle,, and are the midpoint of the edge, respectively. The point starts from the point, moves in the direction, makes the intersection, and makes the intersection.
When a point coincides with a point, the point stops moving.
(1) Find the length of the distance from the point to the point;
(2) Find the functional relationship about (the range of independent variables is not required);
(3) Is there a point that makes it an isosceles triangle? If it exists, request all the values that meet the requirements; If it does not exist, please explain why.
(08 Wenzhou, Zhejiang, 24 questions analysis) 24. (This question 14)
Solution: (1),,.
The point is the midpoint.
, .
,
, .
(2) , .
, ,
, ,
That is, the functional relationship about is:.
(3) Existence can be divided into three situations:
When the time is right, do too much, then.
, ,
.
, ,
, .
(2) When,
.
(3) When is a point on the middle vertical line,
So this point is the midpoint,
.
,
, .
To sum up, when it is or 6 or, it is an isosceles triangle.
6 1. (Yiwu, Zhejiang 08) (the answer to this question is missing) 24. As shown in figure 1, the vertices a and c of the right-angled trapezoidal OABC are on the positive and negative semi-axes of the Y-axis, respectively. Make a straight line after passing through point B and point C, and translate the straight line. The translated straight line intersects the axis of point D and the axis of point E. 。
(1) translate the straight line to the right, let the translation distance CD be (t 0), and the area swept by the straight line (the shaded part in the figure) be. The correlation function image is shown in Figure 2. OM is a line segment, MN is a part of a parabola, NQ is a ray, and the abscissa of n points is 4.
① Find the length of the trapezoid upper bottom AB and the area of the right-angled trapezoid OABC;
(2) When, find the resolution function of S;
(2) Under the condition of the problem (1), when the straight line moves to the left or right (including overlapping with the straight line BC), is there a point P on the straight line AB, which makes it an isosceles right triangle? If it exists, please directly write the coordinates of all points p that meet the conditions; If it does not exist, please explain why.