Prove that n|\phi(a^n-1) in "Topics in Algebra 2nd Edition" by

Using the integral test, find the positive values of p for which the series ∑∞ k=2 1/ (k(ln(k)))^p converges. Show your work and explain your reasoning.

Find k$k$ such that the following matrix M$M$ is singular.

M=⎡⎣⎢−225+k−5933−4114⎤⎦⎥

(1 point) A breathalyser test is used by police in an area to determine whether a driver has an excess of alcohol in their blood. The device is not totally reliable: 7 % of drivers who have not consumed an excess of alcohol give a reading from the breathalyser as being above the legal limit, while 10 % of drivers who are above the legal limit will give a reading below that level. Suppose that in fact 14 % of drivers are above the legal alcohol limit, and the police stop a driver at random. Give answers to the following to four decimal places.

Part a)
What is the probability that the driver is incorrectly classified as being over the limit?


Part b)
What is the probability that the driver is correctly classified as being over the limit?


Part c)
Find the probability that the driver gives a breathalyser test reading that is over the limit.


Part d)
Find the probability that the driver is under the legal limit, given the breathalyser reading is also below the limit.
 

Determine the average value of F(x, y, z) = xyz throughout the cubical region D
bounded by the coordinate planes and the planes x = 2, y − 2, z = 2 in the first octant.

A system for a random amount of time X (in units of months) is given by a density ;

$$\begin{gathered} f\left(x\right) = \frac{1}{4}x{e}^{-\frac{x}{2}} ; x > 0 \\ 0; x\le 0 \end{gathered}$$

(a) Find the moment generating function of X . Hence, compute the variance of X . 

(b) Deduce the expression for the k th moment. 

(c)Obtain the distribution function of X . Hence, compute that the probability that, 7 of such system, at least 4 will function for at least 6 units of months. State the assumptions that you make. 

Determine whether the given set S is a subspace of the Vector space V.

(there are multiple)

a−15=4a−3