There are 100 students in class A and class B. one sixth of class A is transferred to class B. at this time, the number of students in the two classes is just equal. How many students are there in class A and class B?

X-X/6=100/2
X=60
100-60=40
There are 60 and 40 each

There are 100 students in class A and class B, among which 29 students participate in the mathematics activity group. It is known that 1 / 3 of class A and 1 / 4 of class B have participated in mathematics activities. It is a one-dimensional equation to calculate the number of students in each class

Suppose there are x people in class A, then there are 100-x people in class B; 1 / 3x + 1 / 4 (100-x) = 29; 1 / 12x + 25 = 29, x = 48; class B has 100-48 = 52

Set A.B.C is a subset of I = {1.2.3.4.5.6.7.8.9.0}. If a ∪ B = I, find the number of ordered set pairs (A.B)

Because a ∪ B = I, there are only three possible assignments for each element in I, either only in a, only in B, or both
Therefore, these 10 elements produce a 3 ^ 10 (the 10th power of 3) distribution scheme
Because (A.B) is an ordered set pair, the above allocation does not produce repeated calculation
So the number is 3 ^ 10

As shown in the figure, points c, D, e divide line AB into four parts, and AC: CD: de: EB = 2:3:4:5. If CE = 14cm, find the length of line ab

Let AC = 2x, CD = 3x, de = 4x, EB = 5x
CE=CD+DE=3X+4X=14
X=2
Substituting AB = 28

|A | = 0, a is a matrix of order n, the existence of non-zero square matrix B, such that ab = Ba = 0
Detailed proof process, thank you~

Let a be a square matrix of n × n. firstly, there exists a nonzero column vector x (NX1) satisfying AX = 0 because a is not satisfied with rank. Secondly, there exists a nonzero column vector Y (n × 1) satisfying a (T) y = 0 because a (T) is also not satisfied with rank (T represents matrix transpose)

Judge the position relationship between the line and the circle, write the process line x + y = 2 and the circle X & # 178; + Y & # 178; = 2

y = 2-x
x^2+(2-x)^2 = 2
2x^2 -4x+2=0
(x-1)^2 = 0
x=1,y =1
There is only one intersection point between a line and a circle, which is tangent

Definite integral of xsinx from π to 0

(π,0) ∫ xsinx dx
=(π,0) ∫ -x dcosx
= -xcosx | (π,0) + (π,0) ∫cosxdx
= -(0-πcosπ) + sinx | (π,0)
= -π
Normally, it should be 0 to π. If it is, the result should be π

Who can tell me what a complex number is? It's like 10 divided into two numbers, the product of 40 can be (5 + √ - 15) (5 - √ - 15) = 25 - (15)

Complex number refers to the number a + bi which can be written in the following form, where a and B are real parts and I is an imaginary unit (i.e. - 1 open root)
The addition and subtraction of complex numbers are: addition and subtraction of real part and real part; addition and subtraction of imaginary part and imaginary part
Multiplication: (a + IB) * (c + ID) = AC + IAD + IBC BD = AC BD + I (AD + BC)
Division: first change the denominator into a real number. For example, if the denominator is a + IB, multiply it by its conjugate complex number a-ib (at the same time, the numerator should also be multiplied by (a-ib)) and finally change it into a ^ 2 + B ^ 2
Molecules become multiplication
Let z = a + IB, then the conjugate of Z is a-ib
（a+ib）*（a-ib）=a^2+b^2
|Z | = radical a ^ 2 + B ^ 2
Conjugation is the sign inversion of the coefficients of the imaginary part of a complex number

Solving Mathematical Olympiad problems by reducing research method
There are 1999 straight lines on the plane, no two of which are parallel to each other, and no three or more straight lines pass through the same point. Question: how many areas is the plane divided into by the 1999 straight lines?

Draw a line, the plane is divided into two regions, = 1 + 1 draw two lines, the plane is divided into four regions, = 1 + 1 + 2 draw three lines, the plane is divided into seven regions, = 1 + 1 + 2 + 3 draw four lines, the plane is divided into 11 regions, = 1 + 1 + 2 + 3 + 4... Draw 1999 lines, the plane is divided into regions

12345 stands for five numbers respectively, and after multiplying by four, it becomes 54321

40000a+4000b+400c+40d+4e=10000e+1000d+100c+10b+a
39999a+3990b+300c=960d+9996e
13333a+1330b+100c=320d+3332e
Because a

There are 100 students in class A and class B. one sixth of class A is transferred to class B. at this time, the number of students in the two classes is just equal. How many students are there in class A and class B?