If the equation (x-1) sin has four different roots x1, X2, X3 and X4 on (- 1,3), then the four roots are added If the equation (x-1) sin has four different roots x1, X2, X3 and X4 on (- 1,3), then what is the sum of the four roots?

Let y = X-1, then x = y + 1,
The equation is reduced to ysin [π (y + 1)] = 1,
It is reduced to ysin (π y) = - 1
Let f (y) = ysin (π y), then f (- y) = (- y) * sin [π (- y)] = ysin (π y) = f (y), so f (y) is an even function, and the image is symmetric about the Y axis,
By - 1

1. We know n positive integers x1, X2, X3 , xn satisfies X1 + x2 + X3 + +Xn = 2008, find the maximum value of the product of these n numbers
2. Let [x] denote the largest integer no more than x, and let {x} = x - [x], when a = [√ 19-91], B = [19 - √ 91], C = {√ 10}, d = √ {10}, find the value of AB + CD
3. [x] denotes the largest integer not exceeding X and solves the equation 3x + 5 [x] - 49 = 0

1、x1、x2、x3、… In xn, it is impossible to have a number greater than or equal to 5 because 5 is less than 2 × 3, 6 is less than 3 × 3 It is impossible to have three or more than three 2, because the product of three 2 is less than the product of two 3, so the maximum product of N numbers can only be composed of 668 3 and 2 2 products, the maximum value is 2 ^ 2 × 3 ^ 6682

The first n terms, the first 2n terms and the first 3N terms of the equal ratio sequence are known. Prove Sn ^ 2 + s2n ^ 2 = Sn (s2n + s3n)

Prove: ∵ the first n terms, the first 2n terms and the first 3N terms of the equal ratio sequence are known
∴S[n]=a[1](1-q^n)/(1-q)
S[2n]=a[1][1-q^(2n)]/(1-q)
S[3n]=a[1][1-q^(3n)]/(1-q)
∵S[n]^2+S[2n]^2
=[a[1](1-q^n)/(1-q)]^2+{a[1][1-q^(2n)]/(1-q)}^2
=a[1]^2{1-2q^n+q^(2n)+1-2q^(2n)+q^(4n)}/(1-q)^2
=a[1]^2{2-2q^n-q^(2n)+q^(4n)}/(1-q)^2
S [n] (s [2n] + s [3N])
=[a[1](1-q^n)/(1-q)]{a[1][1-q^(2n)]/(1-q)+a[1][1-q^(3n)]/(1-q)}
=a[1]^2{(1-q^n)[1-q^(2n)]+(1-q^n)[1-q^(3n)]}/(1-q)^2
=a[1]^2{1-q^n-q^(2n)+q^(3n)+1-q^n-q^(3n)+q^(4n)}/(1-q)^2
=a[1]^2{2-2q^n-q^(2n)+q^(4n)}/(1-q)^2
∴S[n]^2+S[2n]^2=S[n](S[2n]+S[3n])

Fill it in
4.3 cubic decimeter = () cubic decimeter () cubic centimeter
538 ml = () CC
2. Divide the four mooncakes equally among five people, and each person gets () pieces
3.12 and 16 least common multiple
4. In 7, 9, 10, 8, 11, 15, the mode is (), the average is (), and the median is ()
2. Cross the false and the true
The greatest divisor of a natural number (except 0) is equal to its least common multiple ()
A large space occupied by an object means that it is large ()
Prime numbers must be odd ()

4.3 cubic decimeter = (4) cubic decimeter (300) cubic centimeter
538 ml = (538) CC
2. Give 4 pieces of moon cakes to 5 people equally, and each person will get (4 / 5) pieces
3.12 and 16 least common multiple (48)
4. Among the numbers 7, 9, 10, 8, 11 and 15, the mode is (7, 9, 10, 8, 11 and 15), the average is (10), and the median is (9.5)
2. Cross the false and the true
The greatest divisor of a natural number (except 0) is equal to its least common multiple (pair)
A large space occupied by an object means that it has a large volume
Prime number must be odd (wrong)

As shown in the figure, in trapezoidal ABCD, angle a = 90 °, angle B = 120 ° and ad = root 3
As shown in the figure, in the right angle trapezoid ABCD, a = 90 °, B = 120 °, ad = root 3, ab = 6. Take point E on the bottom edge AB, and take point F on the ray DC, so that ∠ def = 120 °
Q: if the ray EF passes through point C, then the AE length is?

Because the ray EF passes through the point C, F and C coincide. From the vertical Bo of DC made by point B, OC = 1, BC = 2, DC = 7 ∠ a = 90 °, B = 120 ° so AB is parallel to CD. Because ∠ B = 120 °, def = 120 ° and ∠ BEC = ∠ ECD, triangle BEC is similar to triangle decbe / EC = EC / DC, so EC ^ 2 = be * DC, let be = x do eg

1 / 2 1 / 3 1 / 4 1 / 6 what symbol is equal to 1 / 8

1/2*(1/3-1/4+1/6)=1/8

Let a be a nonzero matrix of order n and | a | = 0. It is proved that there exists a nonzero matrix B of order n such that ab = 0
The problem requires the knowledge of the properties of matrix rank,

Because | a | = 0
So r (a)

Fill 0,1,2,3,7,8,9 in the box below to make the formula tenable: □ + □ = □ - □ = □. The number can not be used repeatedly

8+9=20-3=17

If a is a real matrix, it is proved that R (a '* a) = R (a)

For the linear equations a '* AX = 0 and Ax = 0, obviously, for Ax = 0, multiply a' on both sides left, then a '* AX = 0 can be known, so the solution vector satisfying AX = 0 is also the solution vector satisfying a' * AX = 0. For a '* AX = 0, multiply X' on both sides left, then x'a '* AX = 0 can be known, so the solution satisfying a' * AX = 0 can only be AX = 0

The difference between three times of a number minus 18 is 45.6!

Solution
Let this number be x, then
3X-18=45.6
3X=45.6+18
3X=63.6
X=21.2

If the equation (x-1) sin has four different roots x1, X2, X3 and X4 on (- 1,3), then the four roots are added If the equation (x-1) sin has four different roots x1, X2, X3 and X4 on (- 1,3), then what is the sum of the four roots?