Laplace transform involving step function \(\displaystyle{f{{\left({t}\right)}}}={\frac{{{\sin{{\left({2}{t}\right)}}}}}{{{e}^{{{2}{t}}}}}}+{t}{\left\lbrace\cdot\right\rbrace}{u}{\left({t}-{4}\right)}\)

For any vectors u, v and w, show that the vectors u+v, u+w and v+w form a linearly dependent set.

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.

Proving the double differential of $z = -z$ implies ${\displaystyle z=\sin x}$
${\displaystyle \frac{d^{2}z}{{ dx }^2}=-z}$
implies z is of the form ${\displaystyle a\sin x+b\cos x}$. Is there a proof for the same. I was trying to arrive at the desired function but couldn't understand how to get these trigonometric functions in the equations by integration. Does it require the use of taylor polynomial expansion of ${\displaystyle \sin x\text{or}\cos x}$?

Assume that 60% of the students at Remmington High studied for their Psychology test.  Of those that studied, 25% got an A, but only 8% of those who didn't study got an A.  What is the approximate probability that someone that gets an A actually studied for the Psychology test?

1. Suppose a fast-food restaurant can expect two customers every 3 minutes, on average. What is the probability that four or fewer customers will enter the restaurant in a 9-minute period?

find an equation in rectangular coordinates p+3=e and r-2 = 3cosΘ