1957 college entrance examination mathematics problems

Summary of test papers

The total of this set of papers is 9 A title , Calculation with index 、 Solution of quadratic inequality in one variable 、 There are three problems in triangle 、 Solid geometry has 2 Problem , The characteristic topic is ：

1. The first 4 Avenue , This question is in the ordinary high school textbook （2019） taught A There's something in version 2 of compulsory , This question can be adapted and extended , It will be mentioned in later articles ;

2. The first 5 Avenue , Although we have examined the relevant knowledge of different planes and straight lines , But its essence is the embodiment of the nature of parallel faces , The method of making auxiliary lines is very representative , You can solve more than one problem , Represents different ideas ;

3. The first 9 Avenue , Combine trigonometry with cubic equation to investigate , It's novel , It can be used as a question for the second round review of senior three .

The details are as follows ：

Topics of training value and scope of application

Do it yourself before you see ：

1. Simplification

2. Find the suitable inequality The real number x The scope of the

3. verification

4. In tetrahedron in , , 、 、 、 The order is edge 、 、 、 The midpoint of , verification ： For a diamond

5. set up , It is a straight line with different planes , by , The common vertical line of , Over The midpoint of and with , Parallel planes , by A little , by A little Verify the line segment By plane Bisection .

6. verification : equation One root of is 1. Let the three roots of this equation be The sine of the three internal angles of , seek 、 、 The degree of and Value .

Text

1． Simplification

【 Problem solving notes 】 The operation of exponent , All written in the form of prime factor exponential power , Then calculate according to the exponential power algorithm .

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2． Find the suitable inequality The real number x The scope of the

【 Problem solving notes 】 The new version of Renjiao A In the textbook edition , Put the quadratic inequality of one variable into the second chapter of compulsory one , The solving steps of quadratic inequality of one variable can be summarized as ： Hua Zheng 、 solve 、 Value

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  • 1951 In the first 9 topic When when ,x What is the range of values for ？

3． verification

【 Problem solving notes 】 Use the angle doubling tangent formula to solve

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  • 1952 In the first 14 topic equation A general understanding of x=？

  • 1953 In the first 6 topic if , seek Value .

  • 1954 In the first 9 topic Try by , seek Pass value of .

4． In tetrahedron in , , 、 、 、 The order is edge 、 、 、 The midpoint of , verification ： For a diamond

【 Problem solving notes 】 The original title of the textbook , Classic title , The new version of Renjiao A Compulsory Edition II 134 Face example 1, Next to the question section , The new teaching of straight lines parallel to straight lines can use

This topic can be further adapted , Adapted as follows ：

• Variant 1 In tetrahedron in , , 、 、 、 The order is edge 、 、 、 The midpoint of , verification ： Is a rectangle .

• Variant 2 In tetrahedron in , And , 、 、 、 The order is edge 、 、 、 The midpoint of , verification ： Is a square

5． set up , It is a straight line with different planes , by , The common vertical line of , Over The midpoint of and with , Parallel planes , by A little , by A little Verify the line segment By plane Bisection .

【 Problem solving notes 】

Law 1 ： A straight line Make a plane , Over time do , And plane 、 Hand in the points respectively , , set up , And plane Hand in the points respectively , The picture is as follows ：

From the property theorem of plane parallel, we can know the line segment By plane Bisection .

Law two ： Connect Intersection plane At point , set up , And plane Hand in the points respectively At two o 'clock , From the property theorem of parallel lines and planes, we can know , , The picture is as follows ：

Then according to the proportional theorem of parallel line segments, it can be proved that .

6． Solve equations

【 Problem solving notes 】 Using the properties of logarithmic operation 、 Properties of exponential power operation , Get the formal equations , Solve the integral equations .

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7． set up The radius of the inscribed circle is , verification : The height on the edge

【 Problem solving notes 】 Draw the following figures according to the meaning of the title ：

Connect , set up The tangent point with the inscribed circle is , Connect , The picture is as follows :

According to the meaning of the question, we can know , According to the angle doubling formula, we can know , then Use an angle to represent , , , Then cut it into strings .

8． set up It is an acute triangle , With Circle the diameter , And from Make the tangent of this circle Tangent to a circle spot , By in Take... From the side , And I've been do Perpendicular to The extension line of the edge intersects , verification ：

(1) .

(2) The area of = The area of .

【 Problem solving notes 】 Draw the following pictures according to the meaning of the title ：

set up And Intersect at a point , coupling , , , Here's the picture ：

You know ： , Then we can know from the cutting line theorem ： , According to the meaning You can prove that the first question ;

Second questions ： The key is to explain

9． verification : equation One root of is 1. Let the three roots of this equation be The sine of the three internal angles of , seek 、 、 The degree of and Value .

【 Problem solving notes 】 I believe that when you come, you will bring Substitute into the equation to find Value , However, we can't find out , What should I do ？ Using polynomial division , hold Divide obtain ：

Here we might as well set , that It's the equation The root of the , According to the relationship between root and coefficient and the sum theorem of triangle inner angle , You can get a lot of equations , Then the problem is solved

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