Real math problems in 1958 college entrance examination

Summary of test papers

This set of papers has nine questions , Time interval 5 year , The binomial theorem is back . The triangular problems are 3 Avenue , The proof of identity 、 The trigonometric value of a special angle 、 And the quadratic equation of one variable constitute a very distinctive topic ; This paper investigates the knowledge point of line plane perpendicularity from several aspects ; There are three test questions in Ping several aspects , There are two related to the circle , The most distinctive thing is that there is a drawing problem , It's not just drawing .

Bright topics

1. The first 8 topic , A question you can't even think about , I actually took the drawing test , The propositioner is really thinking , At first, I looked very confused , Let's have a question like this , Although it's drawing , In fact, the core of the investigation is still in the calculation , Calculate the length of some line segments according to the known conditions , If the current college entrance examination questions change 、 Innovation can be made in the aspect of drawing , For example, put it in a triangle , The three heights of a triangle are known , Make the triangle and so on ;

2. The first 9 topic , The trigonometric function problem is perfectly combined with the univariate quadratic equation , stay 1957 There were such signs in the year of , It can be treated as a kind of problem .

The knowledge points involved are as follows ：

Topics of training value

Do it yourself

1. Find binomial In the expansion The coefficient of .

2. verification

4. verification ： Tetrahedron The two opposite edges in the （ That is, the two edges of different surfaces ） Perpendicular to each other

5. solve

9. It is known that the hypotenuse of a right triangle is , The height on the bevel is , It is proved that the two acute angles of this right triangle are trigonometric equations The root of the

The text of the test paper

1. Find binomial In the expansion The coefficient of .

【 Problem solving notes 】 binomial theorem , Using the general term formula

• Similar problems are ：

  • 1950 In the first 5 topic Expand binomial , The fifteenth item is （ ）

    A.
    B.
    C.
    D.
    E.

  • 1951 In the first 11 topic What is the constant term in the expansion ？

  • 1953 In the first 12 topic seek The constant term in the expansion of

  • 1954 In the first 3 topic Use binomial theorem to calculate , Make the error less than one thousandth

2. verification

【 Problem solving notes 】 Proof of triangular identity , Start on the right , Multiple use of sinusoidal angle doubling formula .

• Similar problems

  • 1950 In the first 11 topic prove :

  • 1956 In the first 5 topic set up , It's the equation Two of them , verification ： =

3. set up , Two strings of a circle , by The midpoint of , by The midpoint of , Make a straight line hand over On , hand over On , verification ：

【 Problem solving notes 】 To illustrate , It can be explained in in , , Then use the triangle outer angle theorem , So we need to make auxiliary lines , , The drawing is as follows ：

Title original drawing

After making auxiliary lines

4. verification ： Tetrahedron The two opposite edges in the （ That is, the two edges of different surfaces ） Perpendicular to each other

【 Problem solving notes 】 The subject , Many reference books contain , Even many courseware have , Proof of entry level questions . To prove the problem , It must be proved by the perpendicularity of the line and plane , Now let's briefly talk about the next group , The rest can be proved by the same reasoning , take The midpoint of , Connect , , Here's the picture ：

Because it's a tetrahedron , So every face is an equilateral triangle , then Is the midpoint , According to the three lines in one , You can get the vertical .

5. solve

【 Problem solving notes 】 You can get

• Similar problems are

  • 1951 In the first 12 topic What is the general solution of ？

  • 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 .

  • 1955 In the first 7 topic solve equations , seek Pass value of

6. Solve equations

【 Problem solving notes 】 Can be converted to ： , Then you can change yuan , Make , , be , Change the yuan , Get a simple equation .

• Similar problems

  • 1953 In the first 10 topic Explain

  • 1956 In the first 6 topic Solve equations

7. There are two concentric circles , The radius is , ( ）, Now from the center of the circle Make a circle with a radius greater than , Give a small circle to , from Make a straight line The diameter of the vertical circle , Hand in On ; from Make a straight line vertical , Hand in On , It is known that , seek Length

【 Problem solving notes 】 The acute angle trigonometric function is used to obtain ： , , , Then according to Pythagorean theorem, we can solve .

8. A triangle is known , Find a circle through And Midpoint , And with Tangent line . It is known that ： by Of The midpoint of . Seeking work ： One by one 、 Two points and with A circle tangent to a straight line

【 Problem solving notes 】 We can push it backwards , set up That is, the circle that meets the requirements （ Pictured ） because after 、 Two points and a straight line Tangent to a point , such , by The tangent of , by The secant of , therefore , due and , All are known , therefore , The length of can be made , From this, we can get some , So I passed 、 、 Three points can determine the circle

9. It is known that the hypotenuse of a right triangle is , The height on the bevel is , It is proved that the two acute angles of this right triangle are trigonometric equations The root of the

【 Problem solving notes 】 Let the two right sides of a right triangle be , From the equal area, we can know , Secondly, according to Pythagorean theorem, we get , We can work out , Then the sinusoidal values of two acute angles are calculated .

• Similar problems

  • 1957 In the first 7 topic 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 .