The finale of the third grade of mathematics

Analysis on the Selected Mathematics Finals of the National Senior High School Entrance Examination in 2008 (Ⅱ)

14. (Changzhou, Jiangsu 08) (the answer to this question is missing) 28. As shown in the figure, parabola and X axis intersect at point B and point O respectively, and its vertex is A, which connects AB. Move AB's straight line along Y-axis, make it pass the origin O to get straight line L, and let P be the moving point on straight line L. 。

(1) Find the coordinates of point A;

(2) The quadrangles with points A, B, O, and P as vertices include rhombus, isosceles trapezoid, and right-angled trapezoid. Please write down the coordinates of the vertices p of these special quadrangles directly;

(3) Let the area of a quadrilateral with points A, B, O and P as vertices be S, and the abscissa of point P be X. If so, find the range of X. 。

13.(08 Huai 'an, Jiangsu) (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.

14.(08 Lianyungang, Jiangsu) 24. (The full score of this small question is 14)

As shown in the figure, there are two identical right-angled triangular cardboard I and II, and the lengths of their two right-angled sides are 1 and 2 respectively. Put them in a plane rectangular coordinate system, with the rectangular edge on the axis. Put a ruler next to the two cardboard above, so that the cardboard can move parallel to the edge of the ruler. When the board I moves to this position, let it intersect the point and the axis respectively.

(1) Find the functional relationship corresponding to the straight line;

(2) When a point is a moving point on a line segment (except the endpoint), try to explore:

① Is the distance from the point to the axis always equal to the length of the line segment? Please explain the reasons;

② Does the overlapping part (shaded part in the figure) of two pieces of cardboard have the largest area? If it exists, find out the maximum value and the time coordinate when taking the maximum value; If it does not exist, please explain why.

Analysis of 24 questions in Lianyungang, Jiangsu, 2008. Solution: (1) The lengths of two right angles of right-angled triangular paperboard are 1 and 2.

Know the coordinates of two points as follows.

Let the functional relationship corresponding to a straight line be .2 points.

There is a solution.

So the functional relationship corresponding to the straight line is .4 points.

(2)① The distance from the point to the axis is always equal to the length of the line segment.

Because the coordinates of the point are,

Therefore, the functional relationship corresponding to a straight line is.

Because the points are on a straight line,

So the coordinates of the point can be set to.

If the intersection point is the vertical line of the axis and the vertical foot is the point, there is.

Because the point is on a straight line, the score is 0.6.

Because the cardboard moves in parallel, yes, that is.

So ...

Method 1: Therefore,

So there it is.

Yes

So ...

Another 0.8.

So, yes, but,

So there is always. 10.

Method 2: Therefore, it is available.

Therefore.

So ...

Therefore, the point coordinates are.

Let the functional relationship corresponding to the straight line be,

Then there is a solution.

So the functional relationship of the straight line is .8 points.

Substitute in the coordinates of the point, and you can get.

However, there are always. 10 points.

② Starting from ①, the coordinates of the point are, and the coordinates of the point are.

. 12 point

At that time, there was a maximum value of.

The coordinate of the maximum point is. 14 point.

15.(08 Lianyungang, Jiangsu) 25. (The full score of this small question is 12)

We call the smallest circle that can completely cover a plane figure as the smallest covering circle of the plane figure. For example, the smallest covering circle of a line segment is the circle with the diameter of the line segment.

(1) Please draw the minimum covering circles of the two triangles in figure 1 respectively (it is required to draw with a ruler, keep the trace, and do not write);

(2) What is the rule of the minimum covering circle of a triangle? Please write your conclusion (without proof);

(3) There are four villages in a certain place (see Figure 2 for the location), and a TV signal relay station is planned. In order to make the residents of these four villages receive TV signals and make the transmission power required by the relay station minimum (the smaller the distance, the smaller the power required), where should this relay station be built? Please explain the reason.

25. Solution: (1) As shown in the figure: 4 points.

(Note: 1 Drawing is correct, score 2 points; If there is no trace or the trace is incorrect, you will not score. )

(2) If a triangle is an acute triangle, its minimum covering circle is its circumscribed circle; 6 points

If the triangle is a right-angled or obtuse triangle, its minimum covering circle is the circle with the diameter of the longest side of the triangle (the opposite side of the right-angled or obtuse angle) .8 minutes.

(3) This transfer station should be built in the center of the circumscribed circle (the intersection of the midline of the line segment and the midline of the line segment). 10.

The reason for this is the following：

By,

, ,

So this is an acute triangle,

So its minimum covering circle is,

Let this circumscribed circle be a straight line and an intersection point,

Then.

So the point is inside, which is also the smallest covering circle of quadrilateral.

Therefore, the transfer station is built in the center of the circumscribed circle, which can meet the requirements in the problem.

12 point

16 (Nanjing, Jiangsu, 08) 28. (10) The express train goes from A to B, and the local train goes from B to A. Both trains leave at the same time. Let the local travel time be, the distance between two trains be, and the dotted line in the figure shows the functional relationship with.

According to the pictures, make the following inquiries:

Information reading

(1) the distance between a and b is km;

(2) Please explain the actual meaning of the dots in the picture;

Image understanding

(3) Find the speed of the local train and the express train;

(4) Find the relationship between the line segment and the function it represents, and write the range of independent variables;

problem solving

(5) If the second express also goes from A to B, the speed is the same as that of the first express. Thirty minutes after the first express train meets the local train, the second express train meets the local train. How many hours does it leave after the first express?

(08 analysis of 28 questions in Nanjing, Jiangsu) 28. (This question 10)

Solution: (1) 900; 1 point

(2) The practical significance of the midpoint in the figure is that when the local train runs for 4 hours, the local train meets the express train. 2 points.

(3) According to the image, the traveling distance of the local train 12h is 900km.

So the speed of the local train is; 3 points

When the local train runs for 4 hours, the local train meets the express train, and the sum of the distances traveled by the two trains is 900km, so the sum of the speeds of the local train and the express train is 150 km/h.4 minutes.

(4) According to the meaning of the question, the express train travels 900km to the second place, so the express train travels to the second place. At this time, the distance between the two cars is, so the coordinates of the point are.

Let the functional relationship between the sum represented by the line segment be, substitution.

solve

Therefore, the functional relationship between the line segment and is .6 points.

The range of independent variables is .7 points.

(5) After 30 minutes, the local train meets the first express train and the second express train. At this point, the running time of the local train is 4.5h 。

Substitute and get.

At this time, the distance between the local train and the first express train is equal to the distance between the two express trains, which is 1 12.5km, so the interval between the two express trains is, that is, the second express train leaves 0.75 h later than the first express train10 minute.

17.(08 Nantong, Jiangsu) (Question 28 14) 28. It is known that hyperbola and straight line intersect at point A and point B, and point M(m, n) on the first quadrant (to the left of point A) is the moving point on hyperbola. Passing through point B is BD‖y axis and X axis intersect at point D (intersection point N).

(1) If the coordinates of point D are (-8,0), find the coordinates of point A and point B and the value of k. 。

(2) If b is the midpoint of CD and the area of quadrangle OBCE is 4, find the analytical formula of straight line CM.

(3) Let the straight lines AM and BM intersect with the Y-axis at two points P and Q, MA=pMP, MB=qMQ, and find the value of P-Q. 。

(08 Analysis of 28 Questions in Nantong, Jiangsu) 28. Solution: (1) ∫ d (-8,0),

The abscissa of point B is -8. If you substitute, you get y =-2.

The coordinate of point B is (-8, -2). And point A and point B are symmetrical about the origin ∴a(8 2).

So ..................... scored three points.

(2)∵N(0, -n), b is the midpoint of CD, and the four points A, B, M and E are all on hyperbola.

∴, b (-2m, -n), c (-2m, -n), e (-m, -n)...4 points.

S rectangular DCNO, S△DBO=, S△OEN =, ………………………………………….

∴S quadrilateral OBCE= S rectangle dcno-s △ dbo-s △ oen = k ................................................................ 8 points.

Get A (4, 1) and B (-4, 1) from the straight line and hyperbola.

∴ c (-4, -2), M(2, 2) ................................................................................................................ 9 points.

The analytical formula of the straight line CM is that from C and M on this straight line, we get

Solve.

The analytical formula of ∴ linear centimeter is ...................................................................................................................................................................

(3) As shown in the figure, the axes of AA 1⊥x, MM 1⊥x and vertical foot are A 1 and M 1 respectively.

Let the abscissa of point A be A and the abscissa of point B be -A, then

Similarly, 13 points.

∴ ............. 14 o'clock

18.(08 Suqian, Jiangsu) 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.

(08 Jiangsu Suqian 27 questions analysis) 27. Solution: (1) ∵ A quadrilateral is a square ∴.

On the same straight line, the straight line of ∴ ∴ is tangent to ∵;

(2) There are two situations when a straight line is tangent to ⊙:

(1) as shown in figure 1, when the point is set in the second quadrant, the axis is set to this point, and the side length of the square is set to, then the solution is OR (truncation).

able to pass

∴ ∴ ,

Therefore, the functional relationship of the straight line is:

(2) As shown in Figure 2, if the point is in the fourth quadrant and the axis is at this point, then the side length of the square is, then the solution is OR (truncation).

able to pass

∴∴, so the functional relationship of the straight line is.

(3) set, and then, by.

∴

∵

∴ .

19.(08 Taizhou, Jiangsu) 29. It is known that the image of quadratic function passes through three points (1, 0), (-3, 0), (0, 0).

(1) Find the analytic expression of quadratic function and make the image of this function in the given rectangular coordinate system; (5 points)

(2) If the image of the inverse proportional function and the image of the quadratic function intersect at point A (x0, y0) of the first quadrant, x0 falls between two adjacent positive integers. Please observe the image and write these two adjacent positive integers; (4 points)

(3) If the intersection of the image of the inverse proportional function and the image of the quadratic function in the first quadrant is A, then the abscissa of point A satisfies 2.

(08 Analysis of Question 29 in Taizhou, Jiangsu) IX. (The full mark of this question is 14) 29( 1) Let the analytical formula of parabola be y = a (x- 1) (x+3). ............................................................................

(As long as the analytical formula is correct, you will get 1 minute in any form.)

Substitute (0,-) to get a=.

The analytical formula of parabola is y = x2+x-............................................. 3 points.

(No matter what form the analytical formula is, as long as it is correct, score)

Draw a picture (sketch). (No points will be deducted if there is no list) 5 points.

(2) Draw the image of inverse proportion function in the first quadrant correctly. ............................................ scores 7 points.

As can be seen from the figure, the abscissa x0 of the intersection falls between 1 and 2, so that two adjacent positive integers are 1 and 2. Nine points

(3) According to the function image or function properties, when 2 < x < 3,

For y 1= x2+x-, y 1 increases with the increase of x, for y2 = (k > 0),

Y2 decreases with the increase of x, because A(X0, Y0) is the intersection of the quadratic function image and the inverse proportional function image, so when X0=2, y2 > y 1 is obtained from the inverse proportional function image above the quadratic function.

That is, >× 22+2-,and the solution is k > 5. ............................ 1 1 min.

Similarly, when X0=3, y 1 > y2 is obtained from the image of quadratic function number above the inverse scale.

That is × 32+3- >, and the solution is k < 18. ………………………………… 13

Therefore, the value range of k is 5 < k <18 .................................14 minute.

20.(08 Wuxi, Jiangsu) 27. (Full score for this small question 10)

As shown in the figure, it is known that the point moves in the positive direction of the axis at the speed of 1 unit length/second, and the vertex is rhombic, so that the point is in the first quadrant, and; Make a circle with the center and radius as the center. Set this point to two seconds and find:

The coordinates of (1) point (expressed by the contained algebraic expression);

(2) When a point is in motion, all the values that make it tangent to the straight line where the edge of the diamond is located.

(08 Jiangsu Wuxi 27 questions analysis) 27. Solution: (1) passes through the action axis,

, ,

The coordinates of this point are. (2 points)

(2)① When tangent (as shown in figure 1), the tangent point is, at this time,

, ,

. (4 points)

② When tangent to the axis (as shown in Figure 2), the tangent point is,

Excessive, then, (5 points)

,. (7 points)

(3) When it is tangent to a straight line (as shown in Figure 3), let the tangent point be,

Then,,

. (8 points)

If you go too far, then,

Simplify, acquire,

Solve,

The values are, and. (10)

2 1.(08 Wuxi, Jiangsu) 28. (Full score for this little question)

The transmitting diameter of telecommunication signal repeater is 365,438+0 km. Now it is required to select several installation points in a square urban area with a side length of 30km, and install a repeater at each point, so that the signals forwarded by these devices can completely cover the city. Q:

(1) Can you find such four installation points so that these points can meet the preset requirements after installing this forwarding device?

(2) After installing the forwarding equipment, how many installation points should be selected at least to make these points meet the preset requirements?

Requirements: Please draw the necessary schematic diagram when answering, and explain your reasons with necessary calculations, reasoning and words. (The following are some schematic diagrams of square urban areas with a side length of 30km for you to choose when solving problems. )

28. Solution: (1) Divide the square in figure 1 into four small squares as shown in the figure, and install these four forwarding devices at the intersection of the diagonals of these four small squares. At this time, the diagonal length of each small square is, each forwarding device can completely cover a small square area, and installing four such devices can meet the preset requirements.

(3 points) (The pattern design is not unique)

(2) Divide the original square into three rectangles as shown in Figure 2, so that each device is installed at the diagonal intersection of these rectangles, assuming that, then,

By, by,

, ,

Even if three such forwarding devices are installed in this way, the preset requirements can be met. (6 points)

Or: divide the original square into three rectangles as shown in Figure 2, so that it is the midpoint of the rectangle, and install each device at the diagonal intersection of these rectangles, so that three such forwarding devices can meet the preset requirements. (6 points)

To cover a square with two circles, a circle must pass through at least two adjacent vertices of the square. As shown in Figure 3, a square with a side length of 30 is covered by a circle with a diameter of 3 1, which means that the square cannot be completely covered by two circles with a diameter of 3 1.

Therefore, at least three such forwarding devices must be installed to meet the preset requirements. (8 points)

Scoring description: schematic diagram (figure 1, figure 2 and figure 3), each figure 1.

22.(08 Xuzhou, Jiangsu) (The answer to this question is missing) 28. As shown in figure 1, a pair of right triangles satisfy AB = BC, AC = DE, ∠ AB=BC = ∠ DEF = 90, ∠ EDF = 30.

In the operation, the right vertex E of the triangle DEF is placed on the hypotenuse AC of the triangle ABC, and then the triangle DEF is rotated around the point E, so that the edge DE and the edge AB intersect at the point P, and the edge EF and the edge BC are at the point Q..

To explore a way to make a difference,

(1) As shown in Figure 2, when, what is the quantitative relationship between EP and EQ? And give proof.

(2) As shown in Figure 3, what is the quantitative relationship between EP and EQ? , and explain the reasons.

(3) According to your query results of (1) and (2), try to write the quantitative relationship between EP and EQ as _ _ _ _ _ _ _ _, in which the range of values is _ _ _ _ _ (write the conclusion directly without proof).

Explore two ifs, AC = 30cm, continuous PQ, and let the area of △EPQ be S(cm2). During the rotation:

Is there a maximum or minimum value for (1) S? If it exists, find the maximum or minimum value, if it does not exist, explain the reason.

(2) With the different S values, what happened to the corresponding △EPQ number? The value range of the corresponding s value cannot be found.

23.(08 Yancheng, Jiangsu) (The answer to this question is missing) 28. (The full mark of this question is 12)

As shown in Figure A, in △ABC, ∠ACB is an acute angle. Point D is a moving point on ray BC, which connects AD and makes a square ADEF with AD as one side. On the right side of the advertisement.

Answer the following questions:

(1) If AB=AC, ∠BAC=90? .

① When point D is on BC line (not coincident with point B), as shown in Figure B, the positional relationship between line segment CF and BD is ▲, and the quantitative relationship is ▲.

② When point D is on the extension line of BC line, as shown in Figure C, does the conclusion in ① still hold? Why?

(2) If AB≠AC, ∠BAC≠90? Point d moves on line BC.

Try to explore: when △ABC meets certain conditions, CF ⊥ BC (except that point C coincides with point F)? Draw the corresponding picture and explain the reason. (draw without writing)

(3) If AC = and BC=3, under the condition of (2), let the side de of the square ADEF intersect with the line segment CF at point P, and find the maximum length of the line segment CP.

24.(08 Yangzhou, Jiangsu) (The answer to this question is missing) 26. (The full mark of this question is 14)

It is known that in right-angle ABCD, AB= 1, point M is on diagonal AC, straight line L passes through point M and is perpendicular to AC, and intersects with AD at point E. ..

(1) If the straight line L intersects the side BC at point H (as shown in figure 1), AM= AC, AD=A, find the length of AE; (represented by algebraic expression with a)

(2) In (1), divide the area ratio of the two parts of the straight line L by 2: 5, and find the value of a;

(3) If AM= AC and the straight line L passes through point B (as shown in Figure 2), find the length of AD;

(4) If the straight line L intersects the edges AD and AB at points E and F respectively, then AM= AC. Let the length of AD be X and the area of △AEF be Y, find the functional relationship between Y and X, and point out the value range of X (you don't need to write to find the value range of X).

In the past two years, under the guidance of the new curriculum reform concept, the questions are flexible? Novel design? Creative finale questions have mushroomed, one of which is axisymmetric? Translation? Spin? The test questions that combine graphic transformation such as folding with quadratic function become the protagonist of the finale of the senior high school entrance examination. Now, for example, the final evaluation of the 2006 senior high school entrance examination is as follows?

One? Combination of Graph Folding and Quadratic Function

[Comment] This topic studies the folding of triangle in coordinate system, and comprehensively investigates the properties of folding, the coordinates of points, the analytical formula of parabola, and the discrimination of right triangle. It is not only the organic combination of algebra and geometry, but also the unity of dynamic and static. It is gradient and difficult, which requires students to have solid basic skills and the ability to solve problems comprehensively by using mathematical knowledge? Item (3) should be able to make a reasonable guess according to the conditions and graphic characteristics, and use the reduction to absurdity to verify it reasonably and experience the concept of the new curriculum standard?

Two? Combination of Graph Rotation and Quadratic Function

Example 2. [Yichang] As shown in the figure, point O is the coordinate origin, and point A(n, 0) is the moving point on the X axis (n

(1) Find the value of k;

(2) Does the change of point A change the area ratio of △AMH to rectangular AOBC? Explain your reasons?

Analysis: (1) Get B(0, -2n) according to the meaning of the question.

When x=0, y = kx+m = m, and the ∴ f coordinate is (0, m).

And FB=-2n-m, in Rt△AOF,

[Comment] This topic examines the knowledge of rotation transformation, solving a right triangle, finding the coordinates of points, finding the resolution function by undetermined coefficient method, and finding the area of a graph by algebraic method. Geometric knowledge in the coordinate system, which combines conjecture and proof, not only allows students to appreciate the beauty of graphic transformation, but also realizes the dialectical thought of taking static by moving and changing strain in the process of mathematical inquiry?

Three? Combination of graphic translation and quadratic function

[Comment] After the curriculum reform, although the knowledge of circle was deleted, it lost its dominant position in the final exam, but the synthesis of circle and quadratic function is still one of the hot spots concerned by proposers? This question is based on several positional relationships between straight lines and circles, and takes the circle moving in translation as the carrier, skillfully putting the circle? The area of the quadrilateral? The content of triangle congruent geometry is related to the knowledge of quadratic function, which solves the maximum problem of kinematic geometry, permeates the idea of combining numbers and shapes and discusses them in different categories, which is very exploratory?

Four? Combining axisymmetric transformation with quadratic function

Example 4. [Yantai] As shown in the figure, it is known that the image of parabola L 1: Y = x2-4 intersects with X at a? At two o'clock,

(1) If parabola L 1 and L2 are symmetrical about X axis, find the analytical expression of L2;

(2) If point B is the moving point on parabola L 1 (B is different from A? C overlap), with AC as the diagonal, a? b？ Let the fourth vertex of the parallelogram with C as the vertex be D, and prove that point D is on L2;

(3) Exploration: When point B is located at L 1 on the X axis respectively? Is there a maximum and minimum area of the two parts of parallelogram ABCD in the image? If it exists, judge what kind of special parallelogram it is and find its area; If it does not exist, please explain why?

Analysis: Let the analytical formula of L2 be y = a (x-h) 2+k.

∵ The intersection of L2 and X axis A (-2,0), C (2,0), vertex coordinate (0,4), L 1 and L2 are symmetrical about X axis?

∴ L2 passes through a (-2,0), c (2,0), and the vertex coordinate is (0,4).

∴ y=ax2+4

∴ 0=4a+4 a=- 1

The analytical formula of ∴ L2 is y=-x2+4.

(2) Let B(x 1, y 1).

Point b is on L 1

∴ B(x 1，x 12-4)

∵ quadrilateral ABCD is a parallelogram, a? C is symmetric about o.

∴ bay D is symmetric about o.

∴ D(-x 1，-x 12+4)