Find the math finale of the senior high school entrance examination in recent two years

1, known quadratic function

(1) At that time, the function value decreased with the increase of, and the value range of was found.

(2) With the vertex of the parabola as the vertex, the parabola is inscribed with a regular triangle (there are two points on the parabola). Is the area of △ an independent fixed value? If yes, request the fixed value; If not, please explain why.

(3) If the abscissa of the intersection of the parabola and the axis is an integer, find the value of the integer.

2. As shown in the figure, in the plane rectangular coordinate system, △ABC is a right triangle, ∠ ACB = 90, AC=BC, OA= 1,

OC=4, the parabola passes through point A and point B, and the vertex of the parabola is d 。

(1) Find the values of b and c;

(2) Point E is a moving point on the hypotenuse AB of the right triangle ABC (except point A and point B), and the intersection point E is the perpendicular of the X axis.

Parabolas intersect at point f, and when the length of line segment EF is the largest, find the coordinates of point e;

(3) Under the condition of (2): ① Find the area of quadrilateral with points E, B, F and D as vertices; ② Is there a point P on the parabola that makes △EFP a right triangle with EF on the right? If it exists, find the coordinates of all points p; If it does not exist, explain why.

3. (Full score of this question 12) As shown in the figure, in a square AMF⊥BCD with a side length of 2, P is the midpoint of AB, Q is a fixed point on the CD of the side, let DQ = T (0 ≤ T ≤ 2), the middle perpendicular of the straight line PQ intersects with the sides AD and BC at points M and N respectively, and Q is the point E and M.

(1) When t≠ 1, verification: △ peq △ nfm;

(2) Connect P, M, Q and N in turn, let the area of the quadrilateral PMQN be S, find the functional relationship between S and independent variable T, and find the minimum value of S. 。

4. As shown in the figure, the parabola intersects the axis at two points (,0) and (,0), and intersects the axis at one point. Here are the two roots of the equation.

(1) Find the analytical formula of parabola;

(2) A point is a moving point on a line segment. If it passes through a point, it will be ∨, intersect and connect with the point. When the area of the point is the largest, the coordinates of the point will be found.

(3) The point is on the parabola in (1), and the point is the moving point on the parabola. Is there a point on the axis that makes the quadrilateral of the vertex a parallelogram? If yes, find out the coordinates of all points that meet the conditions. If not, please explain why.

5. Situation observation

Cut the rectangular ABCD paper along the diagonal AC to get △ABC and △ a ′ c ′ d, as shown in figure 1. Overlap the vertex a ′ of △ a ′ c ′ d with point A, and rotate counterclockwise around point A, so that points D, A ′ (a ′) and B are on the same straight line, as shown in Figure 2.

Observing Figure 2, we can see that the line segment equal to BC is ▲, ∠ CAC ′ = ▲.

Problem inquiry

As shown in Figure 3, in △ABC, AG⊥BC is at point G, A is a right-angled vertex, AB and AC are right-angled sides respectively, isosceles Rt△ABE and isosceles Rt△ACF are outside △ABC, intersection points E and F are perpendicular to ray GA, and vertical feet are P and Q respectively. Try to explore the quantitative relationship between EP and FQ and prove your conclusion.

Extension extension

As shown in Figure 4, in △ABC, AG⊥BC is at G point, and a rectangle ABME and a rectangle ACNF are made by △ABC with AB and AC as one side respectively, and the ray GA intersects with EF at H point. If AB= k AE and AC= k AF, try to explore the quantitative relationship between he and HF and explain the reasons.

6. (The full mark of this question is 12) As shown in the figure, it is known that the images of linear function y =-x +7 and proportional function y = x intersect at point A, and intersect with X axis at point B. 。

(1) Find the coordinates of point A and point B;

(2) The intersection point A is the AC⊥y axis of point C, and the intersection point B is the l∑y axis. Moving point P starts from point O and moves to point A along the route of O-C-A at the speed of 1 unit per second; At the same time, the straight line L starts from point B and translates to the left at the same speed. In the process of translation, the straight line L intersects the X axis at point R, and the line segment BA or AO is at point Q. When point P reaches point A, both point P and straight line L stop moving. In the process of moving, let the moving point p move for t seconds.

(1) When t is what value, the area of a triangle with vertices A, P and R is 8?

② Is there an isosceles triangle with vertices A, P and Q? If it exists, find the value of t; If it does not exist, please explain why.

7.(20 1 1 Jining) As shown in the figure, ⊙C with a radius of 2 in the first quadrant is tangent to point A, and the tangent L of ⊙C passing through point D intersects with the X axis at point B, and P is the moving point on the straight line L. It is known that the analytical formula of the straight line PA is: y=kx+3.

(1) Let the ordinate of point P be P, and write the functional relationship between P and K. ..

(2) If ⊙C intersects with PA at point M and with AB at point N, there is △AMN∽△ABP wherever the moving point P is on the straight line L (except point B). Please prove the similarity of two triangles when point P is in the figure.

(3) Is there a k value that makes the area of △AMN equal? If yes, request a matching k value; If it does not exist, please explain why.

8. (Nanjing) (8 points) As shown in the figure, in Rt△ABC, ∠ ACB = 90, AC=6㎝, BC=8㎝, and P is the midpoint of BC. The moving point q starts from point p and moves at a speed of 2㎝/s along the direction of ray PC, p.

(1) When t= 1.2, judge the positional relationship between straight line AB and ⊙P, and explain the reasons;

⑵ O is known as the circumscribed circle of △ABC. If υp is tangent to υo, find the value of t 。

9.(9 points) As shown in figure 1, P is a point in △ABC, which connects PA, PB and PC. In △PAB, △PBC and △PAC, if there is a triangle similar to △ABC, then P is called the self-similarity point of △ABC.

(1) As shown in Figure ②, it is known that in Rt△ABC, ∠ ACB = 90, ∠ ACB > ∠ A, CD is the center line on AB, intersection B is ⊥ CD, and vertical foot is E. Try to explain that E is the self-similarity point of △ABC.

(2) In △ABC, ∠ A < ∠ B < ∠ C.

① As shown in Figure ③, use a ruler to make the self-similarity point P of △ABC (write out the practice and keep the drawing traces);

② If the internal P of △ABC is the self-similar point of the triangle, find the degrees of the three internal angles of the triangle.

10.( 1 1)

Problem situation

It is known that the area of a rectangle is a(a is a constant and a > 0). When the length of a rectangle is what, its circumference is the smallest? What is the minimum value?

mathematical model

Let the length of a rectangle be x and the circumference be y, then the functional relationship between y and x is.

exploratory research

⑴ We can learn from the previous experience in studying functions, and discuss the image properties of functions first.

Fill in the table below and draw an image of the function:

x…… 1 2 3 4……

y………

② Observe the image and write two different types of properties of the function;

③ when finding the maximum (minimum) value of the quadratic function y = ax2+bx+c (a ≠ 0), we can find the minimum value of the function (x > 0) by observing the image.

solve problems

⑵ Use the above methods to solve the problems in the "problem situation" and write the answers directly.

1 1, (this question 12 points)

It is known that two straight lines pass through point A (1, 0), point B,

And when two straight lines intersect at the same time at point C of the Y positive semi-axis, there happens to be

Parabolic symmetry axis and straight line passing through points a, b and c.

As shown in the figure, it intersects at point K.

(1) Find the coordinates of point C and the analytic function of parabola;

(2) Parabolic symmetry axis is defined by straight line, parabola, straight line and X axis.

Cut three line segments in turn. What is the quantitative relationship between these three line segments? Please provide a justification for the answer.

(3) When the straight line rotates around point C, the other intersection point with the parabola is M. Please find out the point M that makes △MCK an isosceles triangle, briefly describe the reason and write down the coordinates of the point M. ..

12. (Guangdong Province 20 1 1) As shown in Figures (1) and (2), the side length of rectangular ABCD is AB=6, BC=4, point F is on DC, and DF=2. Moving points M and N start from points D and B at the same time, respectively, and move in the direction of point A along ray DA and line segment BA (point M can be moved to the extension line of DA). When moving point N moves to point A, points M and N stop moving at the same time. Connecting FM and FN, when F, N and M are not in a straight line, we can get that the midpoint of △FMN and △FMN is △PQW. The speed of set points m and n is 1 unit/second, and the moving time of m and n is x seconds. Try to answer the following questions:

(1) describes △ fmn ∽△ qwp;

(2) Let 0≤x≤4 (that is, the time period when M moves from D to A). What is the value of x, and △PQW is a right triangle?

When x is in what range, △PQW is not a right triangle?

(3) What is the x value of the shortest line segment MN? Find the value of MN at this time.

13. (Guilin 20 1 1 year) The full score of this question is 12. ) The image of the known quadratic function is shown in the figure.

(1) Find the coordinates of the axis of symmetry and the axis intersection d;

(2) Translate the parabola upward along its symmetry axis, and set the translated parabola and axis, and the intersection points of the axes are A, B and C respectively. If < ACB = 90°, find the analytical formula of parabola at this time;

(3) Let the vertex of the translation parabola in (2) be m, the diameter of AB and the center of D ⊙D, try to judge the positional relationship between the straight line CM and D, and explain the reasons.

14, (10 minute) As shown in the figure, it is known that the parabola intersects the axis at two points A (1 0), B (0 0,0), and intersects the axis at point C (0 0,3). The vertex of the parabola is p, which connects AC.

(1) Find the analytical expression of this parabola;

(2) Find a point D on the parabola, make DC perpendicular to AC, and the straight line DC intersects the axis at point Q, and find the coordinates of point D;

(3) Whether there is a point m on the parabola axis of symmetry, so that S△MAP=2S△ACP, and if there is, find out the coordinates of the point m; If it does not exist, please explain why.

15. (The full mark of this question is 10) As shown in figure 1, put a square ABCD with a side length of 2 in a plane rectangular coordinate system, with point A at the coordinate origin and point C on the positive semi-axis of Y axis, and a parabola c 1 passing through points B, C and D at points M and N.

(1) Find the analytical formula of parabola c 1 and the coordinates of points m and n;

(2) As shown in Figure 2, the center G of another square with a side length of 2 is at point M, on the negative semi-axis of the X axis (left) and in the third quadrant. When point G moves from point M to point N along parabola c 1, the square moves with it and is always parallel to the X axis.

(1) Write directly the functional relationship between parabola c (C') and c (D') formed by the movement routes of points c' and d';

(2) As shown in Figure 3, when the square first moves to be on the same straight line with one side ABCD of the square,

Find the coordinates of G point.

16. (Full score of this question 12) As shown in the figure, the quadratic function intersects with the X axis at points A and B, and intersects with the Y axis at points C. Point P starts from point A and moves to point B at a speed of 1 unit per second. At the same time, point Q starts from point C and moves in the positive direction of Y axis at the same speed. The exercise time is t seconds. Let PQ intersect with straight line AC at G point.

(1) Find the analytical formula of straight line AC;

(2) Let the area of △PQC be S, and find the resolution function of S about t;

(3) Find a point m on the Y axis so that △MAC and △MBC are equal.

Waist triangle. Directly write the coordinates of all m points that meet the conditions;

(4) When the passing point P is PE⊥AC and the vertical foot is E, when the point P moves.

Whether the length of line segment eg has changed, please explain the reason.

17. As shown in figure 1, the coordinates of vertices A and B of square ABCD are (0, 10) and (8,4) respectively, and vertices C and D are in the first quadrant. Point P moves counterclockwise along the square from point A, and point Q starts from point E (4,0).

(1) Find the side length of a square ABCD.

(2) When the point P moves on the side of AB, the function image between the area s (square unit) of δOPQ and the time t(s) is a part of a parabola (as shown in Figure 2), and the moving speeds of the points P and Q are found.

(3) Find the resolution function of the area s (square unit) and the time t(s) in (2) and the coordinates of the point p where the area s takes the maximum value.

(4) If the velocity in (2) is kept constant at point P and point Q, the size of ∠OPQ increases with the increase of time t when point P moves along the AB side; When moving along the BC side, the size of ∠OPQ decreases with the increase of time T. When point P moves along these two sides, can it make ∠ OPQ = 90? If yes, directly write such points p; If not, write directly.