An object moves in simple harmonic motion with

Can anyone help with paramaterization of conics?
Im struggling to wrap my head around an example. It considers the conic ${\displaystyle x^2+y^2-z^2=0}$ then proceeds:
Take ${\displaystyle A=[1,0,1]}$ and the line P(U) defined by ${\displaystyle x=0}$. Note that this conic and the point and line are defined over any field since the coefficients are 0 or 1. A point ${\displaystyle X\in P\left(U\right)}$ is of the form ${\displaystyle X=[0,1,t]}$ or [0, 0, 1] and the map ${\displaystyle \alpha }$ is

How do I evaluate B(v,v) or B(v,v)(a,b,c) like they have to go from the first line to the second?

describe maximum and minimum for the function of two variables x and y and saddle poin

Suppose that the series  of a sequence an in real converges to a real number. Show that limit to infinity summation k = n + 1 = 0

Sally has caught covid but doesn’t know it yet. She is testing herself with rapid antigen kits which have an 80% probability of returning a positive result for an infected person. For the purpose of this question you can assume that the results of repeated tests are independent. 

a) If sally uses 3 test kits what is the probability that at least one will return a positive result? 

b) In 3 tests, what is the expected number of positive results?

c) Sally has gotten her hands on more effective tests, these ones have a 90% probability of returning a positive result for an infected person. If she tested herself
twice with the new tests, how many positive results would she expect to see?

Calculate the laplace transform of
${\displaystyle t^{2}u(t-2)}$
I don't know how to manipulate $t^2$ in order for it to meet the form of the product between a function and a heaviside function.

Elliptic integrals The length of the ellipse

x=acost,y=bsint,0≤t≤2π$$x=a\cos t,y=b\sin t,0\le t\le 2\pi$$

turns out to be

=4a∫π/201−e2cos2t√dt$$=4a{\int }_0^{\pi /2}\sqrt{1-e^2{\cos }^{2}t}dt$$

where e$e$ is the ellipse's eccentricity. The integral in this formula, called an elliptic integral, is non elementary except when e=0$e=0$ or 1 a. Use the Trapezoidal Rule with n=10$n=10$ to estimate the length of the ellipse when a=1$a=1$ and e=1/2$e=1/2$ . b. Use the fact that the absolute value of the second derivative of f(t)=1−e2cos2t√$f(t)=\sqrt{1-e^2{\cos }^{2}t}$ is less than 1 to find an upper bound for the error in the estimate you obtained in part (a).

1. A strawberry jam company regularly receives large shipments of strawberries. For each shipment that is received, a supervisor takes a random sample of 500 strawberries to see what percent of the sample is bruised and performs a significance test. If the sample shows convincing evidence that more than 10%, percent of the entire shipment of strawberries is bruised, then they will request a new shipment of strawberries.

Let p represent the proportion of the strawberries in a shipment that are bruised.