Shanghai college entrance examination science mathematics volume

Formulas and conclusions commonly used in high school mathematics

1. Relationship between elements and sets

, .

2. De Morgan formula

3. Inclusion relation

4. Incompatibility principle

5. The number of subsets of the set is * * *; Proper subset has–1; There are–1nonempty space sets; There are -2 non-empty proper subset.

6. Three forms of the second resolution function.

(1) general formula;

(2) Vertex type;

(3) zero type.

7. Solving connected inequalities often has the following transformation forms.

8. An equation with only one real root is not equivalent to, and the former is a necessary but not sufficient condition for the latter. In particular, an equation with only one real root is equivalent to, or and, or and.

9. Maximum value of quadratic function on closed interval

The maximum value of the quadratic function in the closed interval can only be obtained at both ends of the interval, as shown below:

(1) When a >: 0, if, then

, , .

(2) When a < 0, if, then, if, then.

10. The distribution of real roots of quadratic equation with one variable

Basis: If, then the equation has at least one real root in the interval.

That is all right

The necessary and sufficient condition for the roots of the (1) equation to be in the interval is or;

(2) The necessary and sufficient condition for the equation to have a root on the interval is or or or;

(3) The necessary and sufficient condition for the equation to have a root on the interval is or.

The conditional basis of the quadratic inequality with 1 1. parameter in a certain interval.

The necessary and sufficient condition for (1) quadratic inequality with parameters (as parameters) to be constant in the subinterval of a given interval is.

(2) The necessary and sufficient conditions for the quadratic inequality with parameters (parameters) to be constant in the subinterval of a given interval are as follows.

(3) The necessary and sufficient condition of constancy is or.

12. Truth table

P q is not p p or q p and q.

True, false, true, true.

True or false, true or false.

True, true, true, false.

Fake, fake, fake, fake.

13. Negative form of common conclusion

The original conclusion is reversed, and the original conclusion is reversed.

Is there at least one without it?

Two, not all, at most one, at least two.

Greater than or not greater than at least

At most ()

Less than, not less than, at most

At least ()

For all of us,

Built somewhere,

wrong

and

For anyone,

It's untenable,

find

and

14. the relationship between the four propositions

Reciprocity proposition of original proposition

If p is q, if q is p.

Interaction

Mutual.

No, no.

Inverse inverse

No, no.

No proposition, no proposition

If it is not p, it is not the reciprocal of q. It's not q, it's not p.

15. Necessary and sufficient conditions

(1) Sufficient condition: If yes, it is a sufficient condition.

(2) Necessary condition: If yes, it is a necessary condition.

(3) Necessary and sufficient conditions: if, and, is a necessary and sufficient condition.

Note: If A is a sufficient condition for B, then B is a necessary condition for A; or vice versa, Dallas to the auditorium

16. Monotonicity of functions

(1) and then set it.

The world is increasing its functions;

The upper limit is a decreasing function.

(2) Let the function be differentiable in a certain interval, and if it is differentiable, it is increasing function; If there is, it is the subtraction function.

17. If the sum of functions is a subtraction function, then the sum function is also a subtraction function in the public domain; If the sum of functions is a decreasing function in its corresponding domain, the composite function is a increasing function.

18. Image characteristics of parity function

Odd function's image is symmetrical about the origin, and even function's image is symmetrical about y; On the other hand, if the image of a function is symmetrical about the origin, then this function is odd function; If the image of a function is symmetric about y, then the function is even.

19. If the function is even, then; If the function is even, then.

20. For the function (), if it holds, the symmetry axis of the function is the function; The image of the sum of two functions is symmetrical about a straight line.

2 1. If, the image of the function is symmetrical about this point; If so, the function is a periodic function with a period of.

22. Parity of Polynomial Functions

Polynomial functions are even terms (that is, odd terms) of odd function, and their coefficients are all zero.

Polynomial function is an even function with odd terms (even terms) whose coefficients are all zero.

23. Symmetry of function images

The image of the (1) function is symmetrical about a straight line.

(2) The image of the function is symmetrical about the straight line.

24. Symmetry of two function images

The (1) function and the image of the function are symmetrical about the straight line (axis).

(2) Functions and images of functions are symmetrical about a straight line.

(3) The image of function sum is symmetrical about the straight line y = x. 。

25. If the image of the function moves one unit to the right, the image of the function is obtained; If the image of the curve moves one unit to the upper right, the image of the curve is obtained.

26. The relationship between two mutually inverse functions

27. If a function has an inverse function, its inverse function is, no, and the function is.

28. Several common functional equations

(1) proportional function,.

(2) Exponential function.

(3) Logarithmic function.

(4) Power function.

(5) cosine function, sine function,

29. The period of several functional equations (convention a>0)

(1), then the period t = a；;

(2) ,

Or,

Or, then the period t = 2a.

(3), then the period t = 3a

(4) Then the period t = 4a.

(5)

Then the period t = 5a.

(6), then the period T=6a.

30. Power of fractional index

(1) (,and).

(2) (and).

Properties of 3 1. free radical

( 1) .

(2) when it is odd;

When it is an even number.

32. The operational properties of rational exponential power

( 1) .

(2) .

(3) .

Note: if a > 0 and p is an irrational number, then ap stands for a definite real number. The above operational properties of rational exponential power are applicable to the exponential power of irrational numbers.

33. Reciprocal formula of exponential and logarithmic expressions

34. Change the formula of logarithm base.

(,and, and,).

Inference (,and, and,).

35. Four operations of logarithm

If a > 0, a≠ 1, m > 0, n > 0, then

( 1) ;

(2) ;

(3) .

36. Set a function and remember it. If the domain of is, then, and; If the range of is, and. For the case of, it needs to be tested separately.

37. Logarithmic inversion inequality and its generalization

If,,, then the function

(1) when, and is increasing function.

(2) If is the decreasing function of sum.

Inference: Suppose,,, and, then

( 1) .

(2) .

38. The question of average growth rate.

If the base of the original output value is n and the average growth rate is 0, then the total output value of time is.

39. The relationship between the same series formula and the sum of the first n items

The sum of the first n items in the sequence is.

40. arithmetic progression's general formula

;

The first n terms and formulas are as follows

General formula of 4 1. geometric series

;

The summation formula of the first n terms is

Or ...

42. Equal ratio difference series: the general formula is

;

The first n terms and formulas are as follows

43. Installment payment (mortgage loan)

The repayment of each installment is RMB (the loan is RMB, which is paid off in installments with the interest rate of RMB per installment).

44. Common triangular inequalities

(1) If, then.

(2) If, then.

(3) .

45. The basic relationship of trigonometric functions with the same angle

, = , .

46. Sine and cosine induction formula

47. Sum angle and difference angle formula

;

(sine square formula);

The quadrant of the auxiliary angle is determined by the quadrant of the point.

48. Double angle formula

49. Triple angle formula

. .

50. Periodic formula of trigonometric function

Function, x∈R and the period of the function, x∈R(A, ω is a constant, and A≠0, ω > 0); The period of the function, (a, ω, constant, A≠0, ω > 0).

5 1. sine theorem

52. Cosine theorem

;

53. Area theorem

(1) (representing the heights of sides A, B and C respectively).

(2) .

(3) .

54. The theorem of triangle interior angle sum

In △ABC, there are

55. General solution of simple trigonometric equation

Specifically, there are

56. The simplest triangular inequality and its solution set

57. Algorithm of product of real number and vector

Let λ and μ be real numbers, then

(1) Binding Law: λ (μ a) = (λ μ a);

(2) The first distribution law: (λ+μ) a = λ a+μ a;

(3) The second distribution law: λ (a+b) = λ a+λ b 。

58. The number of vector product algorithm:

( 1) a？ b= b？ A (commutative law);

⑵(a)？ b= (a？ b)= a？ b= a？ (b) and:

(3)(a+b)？ c= a？ c +b？ c.

59. The basic theorem of plane vector

If e 1 and e 2 are two non-linear vectors on the same plane, then there are only a pair of real numbers λ 1 and λ2 for any vector on this plane, so that a = λ 1e 1+λ 2e2.

The vectors e 1 and e2 of non-* * lines are called a set of bases representing all vectors in the plane.

60. Vector parallel coordinate representation

Let a=, b=, and b 0, then a b(b 0).

53. the quantitative product (or inner product) of a and b.

Answer? b=|a||b|cosθ。

6 1. A? Geometric meaning of b

Quantity product a? B is equal to the product of the length of a |a| and the projection of b in the direction of a |b|cosθ.

62. Coordinate operation of plane vector

(1) Let A = and B =, then a+b=.

(2) Let A = and B =, then a-b=.

(3) Let A and B, then.

(4) Let a=, then a=.

(5) Let A = and B =, then A? b=。

63. The included angle formula of two vectors

(a=，b=)。

64. The distance formula between two points on the plane

(A, B).

65. Parallelism and verticality of vectors

Let a=, b=, b 0, then

A||b b=λa。

a b(a 0) a？ b=0。

66. Fixed fraction formula of line segment

Let,, be the equinox of a line segment, be a real number, and then

( ).

67. Coordinate formula of triangle center of gravity

The coordinates of the three vertices of △ABC are, respectively, and the coordinates of the center of gravity of △ABC are.

68. Point translation formula

Note: Any point P(x, y) on Figure F is translated to point and coordinate.

69. Several conclusions of "vector translation"

(1) points are translated by vector a= to get points.

(2) The image of the function is translated with vector a= to obtain an image, and the resolution function is.

(3) The image is translated by vector a=, and the resolution function of is.

(4) Curve: If an image is obtained after translation with vector a=, the equation is.

(5) Vector m= The vector obtained after translation according to vector a= is still m=.

70. Necessary and Sufficient Conditions of Five "Heart" Vector Forms of Triangle

Let it be a point on the plane and the opposite sides of the angle are respectively, then

(1) is the outer center.

(2) as the center of gravity.

(3) I care.

(4) for the heart.

(5) seek benefits for others.

7 1. Common inequalities:

(1) (take "=" if and only if A = B).

(2) (Take "=" if and only if A = B).

(3)

(4) Cauchy inequality

(5) .

72. Extreme value theorem

As we all know, all are positive numbers, so there is

(1) If the product is a constant value, then the sum of time has a minimum value;

(2) If the sum is a fixed value, the time product has a maximum value.

Publicity is well known, there are

(1) If the product is a constant value, it is maximum when it is maximum;

When it is the smallest, it is the smallest.

(2) If the sum is a fixed value, it is the maximum value and the minimum value;

When it is the smallest, it is the largest.

73. A quadratic inequality has two solution sets if it has the same symbol; If its sign is different, its solution is somewhere in between. In short, it exists between two identical symbols and two different symbols.

;

74. Absolute inequality

When a> is at 0, there is

Or ...

75. unreasonable inequality

( 1) .

(2) .

(3) .

76. Exponential inequality and logarithmic inequality

(1) When,

;

(2) When,

;

77. Slope formula

( 、 ).

78. Five equations of a straight line

(1) Point inclination (straight line passes through a point with a slope of).

(2) Oblique intercept (b is the intercept of a straight line on the Y axis).

(3) Two-point formula () (,()).

(4) Intercept type (horizontal intercept and vertical intercept of straight line respectively)

(5) General formula (where a and b are not 0 at the same time).

79. Parallelism and verticality of two straight lines

(1) If,

① ;

② .

(2) If and A 1, A2, B 1 and B2 are not zero,

① ;

② ;

80. Angle formula

( 1) .

( , , )

(2) .

( , , ).

When the straight line is a straight line, the angle between the straight line l 1 and l2 is.

8 1. Angle formula to

( 1) .

( , , )

(2) .

( , , ).

When it is a straight line, the angle between l 1 and l2 is.

82. Four common linear equations

(1) Fixed-point linear equations: The linear equations passing through the fixed point are (except straight lines), where the coefficients are undetermined; The equation of the straight line system passing through the fixed point is, where is the undetermined coefficient.

(2)*** point linear equations: the linear equations passing through the intersection of two straight lines are (division), where λ is the coefficient to be found.

(3) Parallel linear equations: When the slope k is constant and b changes along a straight line, it means parallel linear equations. The linear system equation parallel to the straight line is (), and λ is a parameter variable.

(4) Vertical linear system equation: The linear system equation perpendicular to the straight line (A≠0, B≠0) is, and λ is the parameter variable.

83. Distance from point to straight line

(Point, straight line:).

84. Or expressed plane area.

If a straight line is set, the plane area represented by or is:

If, when the same symbol is used, it means the area above the straight line; When the symbol is different from the symbol, it indicates the area below the straight line. In short, the sign is above and the sign is below.

If, when the same symbol is used, it represents the area to the right of the straight line; When the sign is different from the sign, it indicates the left area of the straight line. In short, the sign is on the right and the sign is on the left.

85. Or expressed plane area.

Set the curve (), and then

Or the plane area is:

The upper and lower parts of the plane area represented;

The upper and lower parts of the plane area represented.

86. Four kinds of equations of circle

Standard equation of (1) circle.

(2) General equation of circle (> 0).

(3) The parametric equation of the circle.

(4) Equation of the diameter of a circle (the end point of the diameter of a circle is,).

87. Equation of circle system

(1) The equation of the circle system passing through the point is

Where is the equation of a straight line and λ is the coefficient to be found.

(2) Line: The equation of the circle system intersecting with circle: is, and λ is the undetermined coefficient.

(3) Passing the circle: the equation of the circle system intersecting the circle: is, and λ is the undetermined coefficient.

88. The positional relationship between a point and a circle

There are three positional relationships between a point and a circle.

If, then

The point is outside the circle; A point on a circle; The point is in the circle.

89. The positional relationship between a straight line and a circle

There are three positional relationships between a straight line and a circle:

;

One of them is.

90. How to determine the position relationship between two circles

Let the center of two circles be O 1, O2, and the radius be r 1, r2.

;

Tangent equation of 9 1. circle

(1) known cycles.

(1) If the tangent point is known to be on the circle, there is only one tangent, and its equation is

When outside the circle, it represents the tangent equation passing through two tangent points.

② The tangent equation of a point outside the circle can be set as, and then k can be obtained by using the tangent condition. There must be two tangents at this time. Be careful not to miss the tangent parallel to the Y axis.

③ The tangent equation with a slope of k can be set as, and then using the tangent condition to find b, there must be two tangents.

(2) Known cycles.

The tangent equation of this point on the (1) circle is:

② The tangent equation of a circle with slope is.

92. The parameter equation of ellipse is.

93. Elliptic focal radius formula

, .

94. Inside and outside the ellipse

The (1) point is inside the ellipse.

(2) The point is outside the ellipse.

95. The tangent equation of ellipse

The tangent equation of a point on (1) ellipse is.

(2) The tangent equation of one point and two tangents outside the ellipse is

(3) The condition that an ellipse is tangent to a straight line is.

96. The focal radius formula of hyperbola

, .

97. The interior and exterior of hyperbola

The point (1) is in the hyperbola.

(2) The point is outside the hyperbola.

98. The relationship between hyperbolic equation and asymptote equation.

(1) If the hyperbolic equation is asymptote equation:

(2) If asymptote equation is a hyperbola, it can be set to.

(3) If the hyperbola has a common asymptote, it can be set to (,the focus is on the X axis and the focus is on the Y axis).

99. The tangent equation of hyperbola

The tangent equation of a point on the hyperbola (1) is.

(2) The tangent equation of one point and two tangents outside the hyperbola is

(3) The condition that a hyperbola is tangent to a straight line is.

100. formula of focal radius of parabola

Parabolic focal radius.

Over-focus chord length

The moving point on 10 1. parabola can be set to p or p, where.

102. The image of quadratic function is a parabola: (1) vertex coordinates are; (2) The coordinate of the focus is: (3) The equation of the alignment is.

Inside and outside the parabola.

The point (1) is in a parabola.

The point is outside the parabola.

(2) The point is in a parabola.

The point is outside the parabola.

(3) The point is in a parabola.

The point is outside the parabola.

(4) The point is in a parabola.

The point is outside the parabola.

104. tangent equation of parabola

The tangent equation of a point on the parabola (1) is.

(2) The tangent chord equation of two tangents passing through a point outside the parabola is.

(3) The condition that a parabola is tangent to a straight line is.

105. Two common equations of curve system

The equation of curve system at the intersection point of (1) intersecting curves is

(is a parameter).

(2)*** focal cone system equation, where when stands for ellipse; When means hyperbola.

106. Chord length formula of intersection of straight line and conic curve

The chord endpoint a is obtained by eliminating y from the equation, which is the inclination angle and slope of the straight line.

107. Two Symmetries of Conic Curves

(1) The curve whose center is symmetrical about a point is.

(2) The curve about line symmetry is

108. "four lines" equation

For the general quadratic curve, the equation is obtained by substitution, substitution, substitution, substitution and substitution.

The tangent, tangent chord, midpoint chord and midpoint equation of a curve are all derived from this equation.

109. Thinking method to prove the parallelism of straight lines

(1) is converted into judging that two straight lines on the * * * plane do not intersect;

(2) into two straight lines parallel to the third straight line;

(3) transforming into line-plane parallelism;

(4) in a vertical line;

(5) Conversion to face-to-face parallelism.

1 10. The thinking method to prove the parallelism between a straight line and a plane

(1) into a straight line and a plane with no common points;

(2) parallel lines;

(3) into a face-to-face parallel.

1 1 1. Think about the method of proving plane parallelism.

(1) is translated into judging that two planes have no common point;

(2) into parallel lines and planes;

(3) into vertical lines and planes.

1 12. Thinking method to prove verticality between straight lines

(1) is converted into intersecting perpendicularity;

(2) in a vertical line;

(3) transforming the projection of a straight line into a projection perpendicular to another straight line;

(4) The transformation line is perpendicular to the diagonal line forming the projection.

1 13. Thinking method of proving that a straight line is perpendicular to a plane

(1) is converted into that the straight line is perpendicular to any straight line in the plane;

(2) convert a straight line into two straight lines intersect on a plane;

(3) transforming a straight line into a vertical line parallel to the plane;

(4) transforming the straight line into another parallel plane;

(5) Transform the straight line into the intersection point perpendicular to two vertical planes.

1 14. Thinking method of proving plane verticality

(1) is converted into judging that the dihedral angle is a straight dihedral angle;

(2) into vertical lines and planes.

1 15. Algorithm of space vector addition and number multiplication vector operation

(1) additive commutative law: a+b = b+a.

(2) Additive associative law: (a+b)+c = a+(b+c).

(3) law of number multiplication distribution: λ (a+b) = λ a+λ b 。

1 16. the generalization of the parallelogram rule of plane vector addition in space

The sum of three vectors with the same starting point but not in the same plane is equal to the vector represented by the diagonal of the parallelepiped with the common starting point as the side.

1 17.*** line vector theorem

For any two vectors A and b(b≠0) in the space, there exists a real number λ for a‖b to make A = λ b. 。

Three-point line

* * * line instead of * * line instead of * * line.

1 18.*** Vector Theorem

There is a real number pair between the vector p of two non-* * lines and the vector a and B * * planes, so that.

It is inferred that there is an ordered real number pair, in which the point p in space is located in the plane MAB, therefore,

Or for any given point o in space, sort the real number pairs, and so on.

1 19. If () is satisfied for any point in space and three points A, B and C that are not * * * lines, then there are always four points * * * surfaces for any point in space; If, if the plane ABC, then the four points P, A, B and C are * * * planes; If the plane ABC, then P, A, B and C are not * * * planes.

Quadrant * * surface and * * * surface.

(plane ABC)

120. Fundamental Theorem of Space Vector

If the three vectors A, B and C are not * * * planes, then there is a unique ordered real array X, Y and Z for any vector P in the space, so that P = Xa+Yb+ZC.

It is deduced that O, A, B and C are four points in the non-* * plane, and any point P in space has only three ordered real numbers X, Y and Z..

12 1. projection formula

It is known that vector =a and axis, and e is the unit vector in the same direction. If point A is projected on a plane and point B is projected on a plane, then

?a，e÷= a？ e

122. Cartesian coordinate operation of vectors

Let a = and b = then

( 1)a+b =；

(2)a-b =；

(3)λa =(λ∈R)；

(4) a? b =；

123. Then let A and B.

= .

124. Lines in space are parallel or vertical.

Set,, and then

;