The conditions of comprehensive questions are hidden and changeable from topic setting to conclusion, from topic type to content, which determines the complexity of examination thinking and the diversity of problem-solving design. In the thinking of miscellaneous exams, we should grasp the ultimate goal of solving problems and share the goal in every step; Improve the accuracy of concept grasp and budget; Pay attention to the hint of the condition of the question. Don't be afraid of being slow in the first step of examination. In fact, fast and slow, clear problem-solving direction and reasonable problem-solving means are the premise and guarantee to improve problem-solving speed and accuracy. ?
Comprehensive questions have a large knowledge capacity, so we should consider a variety of problem-solving ideas, pay attention to the choice of ideas and calculation methods, and pay attention to the application of mathematical thinking methods. ?
Materialize abstract problems: abstract functions are represented by concrete functions with the same properties, and letters are represented by constants, that is, the concepts involved in the topic or the relationship between concepts are specific and clear, and sometimes tables or graphs can be drawn, which is convenient for applying general principles and laws to specific problem-solving processes. ?
Simplify complex problems: decompose comprehensive problems into simple problems related to various related knowledge, and transform complex forms into simple forms. ?
To solve mathematical comprehensive problems, you must have:?
Language conversion ability: Each mathematical comprehensive problem is composed of some specific written language, symbolic language and graphic language. Solving comprehensive problems often requires strong language conversion ability and the ability to convert ordinary language into mathematical language. ?
Conceptual transformation ability: the translation of comprehensive questions often requires strong mathematical concept transformation ability. ?
Ability to transform numbers and shapes: Solving problems by combining numbers and shapes refers to analyzing the algebraic and geometric meanings of the conditions and conclusions of the questions, trying to find out the idea of solving problems by combining algebra and geometry. ?
Mathematical ideas such as the combination of numbers and shapes, classified discussion and equation function are fully embodied in mathematical comprehensive problems and become the core of supporting problems in comprehensive problems. Make full use of the position, shape and size changes of geometric figures, and pay attention to the establishment of functional relationships between geometric elements; Put geometric figures into rectangular coordinates properly and answer related questions: pay attention to the relationship between almost graphic elements and equation roots. This exploration process is a solid foundation, innovation, and the embodiment of the vitality of mathematics review in senior high school entrance examination. ?
Here are two typical final exam questions to analyze and explain?
Example 1:?
Parabolic y? =? -x2？ +(m+2)x-3(m？ -1) X axis?
At point A and point B (A is to the right of B), the straight line y=(m+ 1)x-3?
After point a?
Find the analytical formula of parabola and straight line.
Y line =kx? (k<0) the intersection line y=(m+ 1)x-3 is at point p,? Turn it in
Parabolic y? =? -x2？ +(m+2)x-3(m？ -1) After point M?
Make the vertical line of the X axis, the vertical foot is D, and the intersection line y=(m+ 1)x-3 is at point N? . ?
Q: Can δδPMN become an isosceles triangle? If yes, find the value of k: if no, please explain the reason. ∵?
[Solution] (1)? Parabolic y? =? -x2？ +(m+2)x-3(m？ -1) the x axis is at point a and point b, when y=0, that is?
-x2？ +(m+2)x-3(m- 1)=0, and the solution is x 1=m- 1, x2=3,?
∴A(3,0),B(m- 1,0)？
∵ line y=(m+ 1)x-3 passes through point a,?
∴3(m+ 1)x-3=0,∴m=0？
∴ The analytical expression of parabola and straight line is y? =? -x2？ +2x+3 and y? =? x-3？
(2) Set the straight line y? =? X-3 intersects the y axis at point C.
∴C(0,-3),A(3,0)？
∴OC=OA？
∴∠OAC=∠NAD=45？
∵MN⊥x axis, ∴∠PMN? =45 ?
If △PMN is an isosceles triangle, K.
When PN=PM, then ∠PNM=∠PMN? =45 ?
∫∠ODM = 90？
∴OD=DM？ Let the coordinate of m be (m,-? m)？
∴-? m=k？ m？ , which is k? =? - 1?
When PN=MN,?
∫MN‖OC？
∴? PN/PC=MN/OC？
∠ACO=∠PNM？ =45 ?
∴PC=OC=3？
Passing point p makes PH perpendicular to y axis and h point?
∴PH=CP=sin45 =3×=？ √2/2?
CH=？ PH=，OH=3-？
∴P(3√2/2,3-(3√2)/2)？
And point p is on the straight line y=kx.
∴((3√2)/2)-3=3√2/k？
k=？ 1-√2?
To sum up, K? =? -1 or k= 1-√2?
Example 2?
In right-angled trapezoidal ABCD, AD‖BC, AB⊥BC, AD=6, BC=9,; P is the moving point on the side of BC (not coincident with point B), PQ⊥DP, the intersecting side AB is at point Q, and point Q is not coincident with point B?
(1) Find the length of AB; ?
(2) Let PC=x and BQ=y, and find the resolution function of Y and X, and?
The domain in which it is written; ?
(3) During the movement of point P, can it be tangent to ∠PDQ?
Value equals 2? If yes, request the value of BQ at this time; Like what?
If not, please explain why.
[Solution]: The intersection point D is DH⊥BC, and the vertical foot is H?
( 1)∵AB⊥BC,DH⊥BC,∴AB‖DH.？
In A.D. and B.C., the quadrangle ABHD was a rectangle.
∴BH=AD=6,AB=DH.？
∵BC=9,∴CH=3.？
∵,∴CD=5.？
∴AB=DH=4.？
(2)∵PQ⊥DP,∴∠BPQ+∠DPH=90。 ?
∠∠bpq+∠bqp = 90 ,∴∠dph=∠bqp.？
∴Rt△DPH∽Rt△PQB.？
That's it.
∴, that is, the resolution function of y and x is. ?
The domain name is 3
(3) Let tg∠PDQ=2, even if.
From Rt△DPH∽Rt△PQB, we can get.
DH=4，BP = BC-CP = 9-x
That is, the tangent of ∠PDQ cannot be equal to 2.
Come on, work together! I'm also trying ... _? )
The answer to the question is incomplete, please forgive me!