Let ? = {?1, ?2, ?3, … … … … ?? } ?? ? = {?1, ?2, ?3, … … … … ?? } Prove by induction that the number of injective functions ? from ? ?? ? is ?!

Solve the initial value problem below using the method of Laplace transforms.

y''+7y'+12y=-7cost-9sint

y(pi/2)=1 y'(pi/2)=0

a computer retail store has 11 personal computers in stock. a buyer wants to purchase 4 of them. unknown to either the retail store or the buyer, 4 of the computers in stock have defective hard drives. assume that the computers are selected at random.
in how many different ways can the 4 computers be chosen
what is the probability that exactly one of the computers will be defective
what is the probability that at least one of the computers selected is defective

(a) Show that a group that has only a finite number of subgroups must be a finite group.

(b) Let G be a group that has exactly one nontrivial, proper subgroup. Show that G must be isomorphic to Zp2 for some prime number p. (Hint: use part (a) to conclude that G is finite. Let H

be the one nontrivial, proper subgroup of G. Start by showing that G and hence H must be cyclic.)

Q) if a fraction λ of the population susceptible to a disease that provides immunity against reinfection moves out of the region of an epidemic, the situation may be modeled by system
S'=-βSI-λS
I'=βSI-α I.
Show that both S and I approach zero as t→∞.

Where,
S=The susceptible population
I=infected individuals
β=infection rate

Find the original source where Fibonacci presented his puzzle about modeling rabbit populations. Discuss this problem and other problems posed by Fibonacci and give some information about Fibonacci himself.

prove that a ring R is a field if and only if (R-{0}, .) is an abelian group

Solve the initial value problem below using the method of Laplace transforms.

ty''-4ty'+4y=20, y(0)=5 y'(0)=-6

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. (Please show clear steps and explain them)

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with nitrous oxide. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 pounds/sq.in., the nitrous oxide chamber inside the rocket will explode. Tiff worked from a formula p=14.7e−h/10p=14.7e−h/10 pounds/sq.in. for the atmospheric pressure hh miles above sea level. Assume that the rocket is launched at an angle of αα above level ground at sea level with an initial speed of 1400 feet/sec. Also, assume the height (in feet) of the rocket at time tt seconds is given by the equation y(t)=−16t2+1400sin(α)ty(t)=−16t2+1400sin⁡(α)t. [UW]

a. At what altitude will the rocket explode?
b. If the angle of launch is αα = 12∘∘, determine the minimum atmospheric pressure exerted on the rocket during its flight. Will the rocket explode in midair?
c. If the angle of launch is αα = 82∘∘, determine the minimum atmospheric pressure exerted on the rocket during its flight. Will the rocket explode in midair?
d. Find the largest launch angle αα so that the rocket will not explode.

please show some shortcut tricks and technique formula for integration to solve out easy difficulties questions
I would give a postive rating..if you help me a little

Finding Surface Area In Exercises 43-46, find the area of the surface given by z = f(x, y) that lies above the region R.
f(x,y)=4-x^2 R: triangle with vertices (-2,2),(0,0),(2,2)