If u, v, w ∈ R n ,

Examples for when the quotient ring is necessarily / not necessarily an extension of the residue field.
Let R be an integral domain with unique maximal ideal ${\displaystyle \mathfrak{m}}$. Let $F:=Frac\left(R\right)$ be the field of fraction of R.

In a jet engine a diffuser decreases the kinetic energy of the air entering the engine.  The air enters a the diffuser at 132 kPa and 25°C with a velocity of 502 m/s and leaves at 229 kPa and 64°C.  The heat lost from the diffuser is estimated to be 6 kJ/kg.  Determine the exit velocity in m/s to 1 decimal place.  Take the specific heat at constant pressure of air to be 1.009 kJ/(kgK).

y''-2y'-3y=0 for y(0)=3, y'(0)=2

If 𝐿 [cos√𝑡/√𝑡] = √(𝜋𝑠)e^ℎ(𝑠)
then find ℎ(𝑠).

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?

1.     Is MBA major independent of the first degree? A random sample of 152 MBA students were surveyed and each person was asked to indicate their MBA major and their first degree. The data that resulted from the survey is summarized below.