if the involute is known show how to find the evolute ?

Prove via induction the following properties of Pascal’s Triangle:

•P(n,2)=(n(n-1))/2

• P(n+m+1,n) = P(n+m,n)+P(n+m−1,n−1)+P(n+m−2,n−2)+···+P(m,0) for all m ≥ 0

Theorem
two curves with the same intrinsic equation are necessary congruent
I need prove this theorem with details and thanks

Solve by variation of parameters:

A. y"−9y = 1/(1 − e^(3t))

B. y" +2y'+26y = e^-t/sin(5t)

Let P = (p1,...,pn) be a permutation of [n]. We say a number i is a fixed point of p, if pi = i.
(a) Determine the number of permutations of [6] with at most three fixed points.
(b) Determine the number of 9-derangements of [9] so that each even number is in an even position.
(c) Use the following relationship (not proven here, but relatively easy to see) for the Rencontre numbers:
Dn =(n-1)-(Dn-1 +Dn-2) (∗)
to perform an alternative proof of theorem 2.7. So, with the help of (∗), show that for all n ∈ N applies: n Dn =n! r=0 (-1)r r!
(Note: Of course, do not use Sentence 2.7 or Corollary 2.2, it is D0 = 1 and D1 = 0. Note that (∗) is also valid for n = 1 because of the factor (n - 1), no matter how we would define D-1. Then first look at the numbers An = Dn-nDn-1 (∗∗) and show that An = (-1)n is valid. Then divide both sides of (∗∗) by n! and deduce from this the assertion).

y'' - 8y = 4, y(1) = 9, y'(0) = 5

3) If you have a 40% probability of winning at a game of roulette, how many games can you expect to win after playing 30 games?

4) Calculate the variance of the problem above.

5) If Sarah rolls a 6-sided number cube, how many times can she expect to roll a 4 if she plays 18 games?

Find the general solution to the equation

?″+4?=1

this is abstract algebra

Let M be a Q[x]-module which is finite-dimensional as a vector space. What is its torsion submodule?

give an example to show it is false or argue why it is true.

∃!xP(x)⇒∃xP(x)

∃xP(x)⇒∃!xP(x)

∃!x¬P(x)⇒¬∀xP(x)

using the method of Characteristics, Explain the Process for solving the General Advection Equation

this is abstract algebra

Let M be a Q[x]-module which is finite-dimensional as a vector space. What is its torsion submodule?

Find the kernel (or nullspace) and the image of each linear map together with their basis.

(a) T :R4 →R3 given by f(x,y,z,w)=(3x+y−3z+3w , x+y+z+w , 2x+y−z+2w)

(b) T :R3 →R5 given by f(x,y,z)=(2x−y+6z , x−y−z , x+y−5z , z−y , −x+2z)

c) T :R4 →R3 given by f(x,y,z,w)=(x−y−3z+w , 2x−3y+z+2w , 3x+y−4z−w)