Calculate (m-3n) & sup3; · (3N-M) ^ 5=___ Multiplication of powers with the same base

Calculate (m-3n) & sup3; · (3N-M) ^ 5=___ Multiplication of powers with the same base

(M-3N)³·(3N-M)^5
=[-(3N-M)³]·(3N-M)^5
=-(3N-M)³·(3N-M)^5
=-(3N-M)^（3+5）
=-(3N-M)^8

If a tangent is drawn from a point on the line y = x + 1 to the circle (x-3) ^ 2 + y ^ 2 = 1, the minimum length of the tangent is
What a mess. Why is the distance and radius from a straight line to the center of a circle not equal? The tangent line is perpendicular to the center of a circle. Why do you still need to calculate D and use the Pythagorean theorem? What a mess,

Let the center of a given circle be g, the point on the line y = x + 1 satisfying the condition be a, and the tangent point be B. obviously, the coordinates of the center of the circle (x-3) ^ 2 + y ^ 2 = 1 are (3,0), and the radius is 1. The distance between the point (3,0) and the line y = x + 1 is 3-0 + 1 / √ (1 + 1) = 4 / √ 2 = 2 √ 2 ＞ 1, and the given line is in phase with the circle

Can the tolerance of arithmetic sequence be zero?

Yes
If a sequence starts from the second term, and the difference between each term and its previous term is equal to the same constant, the sequence is called the arithmetic sequence, and the constant is called the tolerance of arithmetic sequence. The tolerance is usually expressed by the letter D
The general formula of arithmetic sequence is: an = a1 + (n-1) d n is a positive integer
It can be seen from the above formula that an is a linear function (D ≠ 0) or a constant function (d = 0) of n

In the pyramid p-abcd, △ PBC is an equilateral triangle, ab ⊥ plane PBC, ab ∥ CD, ab = 12DC, DC = 3bC, e is the midpoint of PD. (1) prove: AE ∥ plane PBC; (2) prove: AE ⊥ plane PDC; (3) find the size of the sharp dihedral angle between planar pad and planar PBC

(1) It is proved that if we take the midpoint of PC as F and connect EF, then EF is the median line of △ PDC, that is, EF is parallel and equal to 12DC, ∵ ab ∥ CD, ∵ AB is parallel and equal to EF, ∥ AE ∥ BF, and ⊂ BF ⊂ plane PBC, ∩ quadrilateral aefb is rectangular, ∥ AE ∥ plane PBC. (3 points) (2) it is proved that ∵ PBC is an equilateral triangle, f is the midpoint of PC, ≁ BF ⊥ PC is EF ⊥ PC, EF ∩ BF = f, ≁ PC ⊥ is flat Let (1) know AE ⊥ EF, EF ∩ PC = f, ∩ AE ⊥ plane PDC. (7 points) (3) extend CB intersection Da to B /, connect Pb /, let BC = a, ∩ AB = 12DC, ∩ BB / = BP = a, take the midpoint of B / P as h, connect ah, BH, then BH ⊥ B / P, according to the three perpendicular theorem, ah ⊥ B / P, ∩ AHB is the plane angle of sharp dihedral angle formed by planar pad and planar PBC. (9 points) in RT △ AHB, ab = 32A The sharp dihedral angle between planar pad and planar PBC is π 3

It is known that the quadratic function f (x) = ax square + BX satisfies: ① f (1-x) = f (1 + x); ② the equation f (x) = x has two equal real roots
(2) Is there a real number m, n (M

（1）
f(1-x)=a(1-x)^2+b(1-x)=ax^2-(2a+b)x+(a+b)
f(1+x)=a(1+x)^2+b(1+x)=ax^2+(2a+b)x+(a+b)
∵f(1-x)=f(1+x)
∴2a+b=0
∵ f (x) = x, that is, ax ^ 2 + (B-1) x = 0, has two equal real roots
∴b-1=0
∴a=-1/2,b=1
∴f(x)=-1/2x^2+x
（2）
f(x)=-1/2x^2+x=-1/2(x-1)^2+1/2
Case 1: n ≤ 1
F (x) increases on [M, n]
That is: F (m) = - 1 / 2m ^ 2 + M = 3M
　　f(n)=-1/2n^2+n=3n
The solution is: M = - 4, n = 0
Case 2: n > 1 and the premise contradiction of M1
Case 3: m ≥ 1
F (x) decreases on [M, n]
That is: F (m) = - 1 / 2m ^ 2 + M = 3N
　　f(n)=-1/2n^2+n=3m
The solution is: M = n = 0, which contradicts the premise of M ≥ 1
So there are real numbers m, N, M = - 4, n = 0

The plane equation passing through point (3, - 1,3) and passing through line (X-2) / 3 = (y + 1) / 1 = (Z-1) / 2 is_____________ ..

The plane passes through (2, - 1,1) and (3, - 1,3), so the plane normal vector is perpendicular to (1,0,2) and (3,1,2), so the plane equation is 2x-4y-z-7 = 0 when the normal vector is (2, - 4, - 1) substituted into the point

4a²+9b²+（ ）=( )²

Complete square formula: M & # 178; + n & # 178; + 2Mn = (M + n) & # 178; 4A & # 178; = (2a) & # 178; 9b & # 178; = (3b) & # 178; 2a is equivalent to the upper m3b is equivalent to the upper n, so 2Mn is 2 (2a) (3b) = 12ab, so 4A & # 178; + 9b & # 178; + (12ab) = (2a + 3b) & # 178

In the cube abcd-a1b1c1d1 with known edge length of 1, e is the midpoint of A1B! To find the distance between the line AE and the plane abc1d1
In the cube abcd-a1b1c1d1, e is the midpoint of A1B1, and the angle between AE and plane abc1d1 is calculated

Take this picture instead
Let the side length be 2
The normal vector of plane abc1d1 is n = (1,0, - 1)
AE=(0,1,2)
cos<AE.n>=-2/(√2*√5)=-√10/5
The sine of the angle between AE and plane abc1d1 is √ 10 / 5

Arrange a natural number 1-100 as shown in the figure below. Use a rectangle to frame two lines of numbers. For example, the sum of the numbers in the box is 432. These six numbers are the most important
I'm in a hurry
Arrange a natural number 1-100 as shown in the figure below, and use a rectangle to frame out two lines of numbers. For example, the sum of the numbers in the box is 432. What is the minimum number of these six numbers
1 2 3 4 5 6 7 8
9 (10 11 12) 13 14 15 16
17 (18 19 20) 21 22 23 24
.......................................................
97 98 99 100

Let the minimum number be a, then the two numbers in the row with it are a + 1, a + 2, and the numbers in the second row are a + 8, a + 9, a + 10. Then the sum of the six numbers is 6A + 30 = 432, a = 67

Let a = {x} - 2

A = {x | - 2 ≤ x ≤ a}, so B = {y | - 1 ≤ y ≤ 2A + 3}, and set C cannot be determined, so this problem needs to be discussed. In addition, BUC = B means that B contains C
1. If - 2 ≤ a ≤ 0, then C = [A & sup2;, 4], so a & sup2; ≥ - 1 and 2A + 3 ≥ 4, a ≥ 1 / 2 is obtained;
2、0