1956 college entrance examination mathematics

Summary of test papers

This set of papers has nine questions , The scope of investigation is still the same as 1955 The annual range is similar , Focus on the knowledge points related to trigonometry and trigonometric functions , But the cosine theorem is examined too many times , There are two questions , There are still two problems in solving equations , One is to examine the discriminant , One is a system of binary equations , The solution of binary equations needs to use the whole idea and factorization , In fact, some topics can be used as the content of the connection between junior and senior . The specific distribution is shown in the table below ：

Topics of training value

Try to do it yourself ：

1． Use the logarithmic property to calculate .

4. The three sides of a triangle are 3 ruler ,4 Ruler and ruler , Find the degree of the maximum angle of this triangle

5. set up , It's the equation Two of them , verification ： =

8． There's a prism , The bottom is a diamond , = （ Pictured ） verification ： Plane Perpendicular to the bottom

9． 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

The text of the test paper

1． Use the logarithmic property to calculate .

【 Problem solving notes 】 Directly using the properties of logarithmic operation , You can get a complete square

• Similar problems

  • 1950 In the first 19 topic 19. if , , be =（ ）

A.0.5770
B.1.1038
C.6.1038
D.264.06
E.416.74

• 1951 In the first 18 topic It is known that , seek .
• 1953 In the first 4 topic seek
• 1954 In the first 2 topic Explain ：

2. set up Is the set of real Numbers , Prove the equation Both must be real numbers

【 Problem solving notes 】 Verify that the discriminant of quadratic equation is greater than 0 that will do

• Similar problems

  • 1951 In the first 7 topic If the equation The two are equal , be =？
  • 1953 In the first 2 topic The two are equal , seek k value

3. set up yes The edge of The midpoint of , too Make a straight line intersection On , too Make a straight line parallel to hand over On . verification : The area of is equal to Half the area of

【 Problem solving notes 】 drawing , As an auxiliary line

verification The area of is equal to that will do

4. The three sides of a triangle are 3 ruler ,4 Ruler and ruler , Find the degree of the maximum angle of this triangle

【 Problem solving notes 】 According to the big side to the big corner , The cosine theorem finds that the maximum angle is The right corner

• Similar problems

  • 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 In the first 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 .

5. set up , It's the equation Two of them , verification ： =

【 Problem solving notes 】 From the relationship between root and coefficient , You can know

What we need to prove is the relationship between strings , Consider converting strings to tangent , Both sides are divided by , That is to prove that will do , Then use the tangent formula of the sum of two angles .

• Similar problems

  • 1950 In the first 11 topic prove :

6． Solve equations

【 Problem solving notes 】 from You can get , take As a whole , Do factorization , obtain , Then we can know The relationship between , Then substitute another equation to solve it .

• Similar problems

  • 1953 In the first 10 topic 10． Explain

7． set up Is equilateral Circumscribed The last point , verification ： = +

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

according to And be similar , obtain ; According to And be similar obtain , Then add the two equations together

8． There's a prism , The bottom is a diamond , = （ Pictured ） verification ： Plane Perpendicular to the bottom

【 Problem solving notes 】 Here is to prove that the face is vertical , First of all, we have to find the line surface vertical , Connect Hand in point , Connect , The drawing is as follows ：

So easy to know , It's an isosceles triangle , And then come to , So we got it , Problem solving .

This is a relatively simple problem of proving that the surface is vertical , When answering, pay attention to the answer format .

9． 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

【 Problem solving notes 】 The sum theorem of internal angles of triangles and the mean term of equal deviation can tell that an angle is , The second question is carried out on the premise of the first question , It can be solved directly by cosine theorem . This is the second time that the cosine theorem has been used in this set of test papers .