True math problems in 1959 college entrance examination

Summary of test papers

The number of questions in this set of test papers is 1958 The annual number of questions has increased , Increased to 10 A title , After two years of logarithmic calculation, it returns , Not as diligent as the previous years ; The plural is an interval 9 Back to the college entrance examination ; The series appeared for the first time as a single topic in the college entrance examination ; The knowledge of triangle has never been broken , This set of papers combines cosine theorem and area formula to investigate , The current college entrance examination is still a point of investigation , Times are changing , The way of investigation has not changed much .

The distribution of topics and knowledge points is as follows ：

「 Bright questions 」

• The first 5 topic , The distance between parallel lines is equal everywhere , The lines and planes are parallel , Then the distance from any point on the straight line to the plane is equal everywhere ;

• The first 7 topic , The cosine theorem is combined with the area formula , The current college entrance examination is also a combination of ways to investigate , Now the college entrance examination has become a question of value range ;

• The first 8 topic , The sequence of numbers first appeared as a single test question , The front is basically an accessory of triangle related knowledge

Topics of training value and scope of application

Do it yourself

1． It is known that , , seek

2． seek Value .

3． Solving inequalities

4． seek Value

5． Three straight lines not in the same plane Parallel to each other , 、 by Up two points , Verify that the other two vertices are at And The volume of tetrahedron on is a constant value

6． The area of the top and bottom of the round table is , The diameter of the bottom is , The busbar is , Find the side area of the circular platform

7． It is known that in , , , Area is , seek and ．

8． Three numbers are known to form an equal difference sequence , The sum of the first and second numbers 3 Times the third number 2 times , If the second number subtracts 2, Then it becomes an equal ratio sequence , Find these three numbers

10． It is known that 、 、 Is a straight line Three o'clock , And ; by Outside a little , And , , seek

(1) Sine of 、 cosine 、 tangent ;

(2) Length ;

(3) Point to Distance of .

The text of the test paper

1． It is known that , , seek

【 Problem solving notes 】 ,

• The nature of logarithmic operation is similar to the problem

  • 1951 In the first 18 topic It is known that , seek .

  • 1953 In the first 4 topic seek

  • 1956 In the first 1 topic Use the logarithmic property to calculate .

2． seek Value .

【 Problem solving notes 】 The operational properties of complex numbers

• Similar problems

  • 1950 In the first 7 topic set up All are real numbers. , , And , be ( ), ( ).

  • 1950 In the first 12 topic set up 、 It's a real number , Known equation One of them is , seek 、 Value .

3． Solving inequalities

【 Problem solving notes 】 Solution of quadratic inequality in one variable

• Similar questions

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

  • 1957 In the first 2 topic Find the suitable inequality The real number x The scope of the

4． seek Value

【 Problem solving notes 】 Induction formula , The angle can be used again

• Similar questions

  • 1953 In the first 5 topic seek =?

  • 1957 In the first 3 topic verification

5． Three straight lines not in the same plane Parallel to each other , 、 by Up two points , Verify that the other two vertices are at And The volume of tetrahedron on is a constant value

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

spot Is a straight line The fixed point on the surface , A straight line And parallel , So don't argue In a straight line Which position on , The area of is always the same , also Parallel to each other , So there is no argument In a straight line Where is it , From the point of To plane The distance is a fixed value , So tetrahedral volume is a fixed value .

6． The area of the top and bottom of the round table is , The diameter of the bottom is , The busbar is , Find the side area of the circular platform

【 Problem solving notes 】 First find the radius of the upper and bottom circle , Then use the calculation formula of the side area of the circular platform

• Similar problems

  • 1951 In the first 17 There are cylinders and cones with the same bottom and height , The volume of the cylinder is known to be 18 Cubic feet , Find the volume of the cone

  • 1952 In the first 11 topic The radius of the bottom of the straight cone is 3 ruler , The inclined height is 5 ruler , How many cubic feet is its volume ？

  • 1954 In the first 5 topic We know that the radius of the ball is equal to r, Try to find the volume of the inscribed Cube

7． It is known that in , , , Area is , seek and ．

【 Problem solving notes 】 Using the area formula And the cosine theorem The equations of , Solve the equations

• The topic of cosine theorem

  • 1951 In the first 22 topic set up △ABC The three sides of , , , seek , Concurrent evidence by And The mean of the difference

  • 1952 In the first 23 topic Let the side length of the triangle be , , , The diagonal angles are , , seek , , , ask A triangle is an acute or obtuse angle ？

  • 1955 Year No 2 topic The length of one waist of an isosceles triangle is the length of the bottom 4 times , Find the cosine of each angle of this triangle .

  • 1956 In the first 4 topic The three sides of a triangle are 3 ruler ,4 Ruler and ruler , Find the degree of the maximum angle of this triangle

  • 1956 In the first 9 topic If the three angles of a triangle form an arithmetic sequence , Then one of the corners must be ; If three sides of the such a triangle form an equal ratio sequence , Then all three corners are , Try to prove

8． Three numbers are known to form an equal difference sequence , The sum of the first and second numbers 3 Times the third number 2 times , If the second number subtracts 2, Then it becomes an equal ratio sequence , Find these three numbers

【 Problem solving notes 】 Let the three numbers be , List about according to the meaning of the question The equations of , Solve the equations .

9． The circle is known The two strings of and The extension intersects at , too Point citation hand over The extension line of is on , too Point to circle The tangent of , verification ： .

【 Problem solving notes 】 According to the cutting line theorem ： , Second, prove

10． It is known that 、 、 Is a straight line Three o'clock , And ; by Outside a little , And , , seek

(1) Sine of 、 cosine 、 tangent ;

(2) Length ;

(3) Point to Distance of .

【 Problem solving notes 】

「 Law 1 」

set up , , spot The straight line The distance to .

You know

stay in , From the sine theorem ： ,

take Plug in in , You get about An equation of , Then we can find

「 Law two 」

Make auxiliary lines ： Over time do , hand over At point , Over time do hand over At point , The auxiliary lines are as follows ：

Easy to know ： , , And then you get , You can get The tangent of , Then according to the trigonometric function of the same angle , Then we can get the sine 、 Cosine value .

second 、 Just ask three questions and solve the right triangle .