Hard math test

find the sum of the series ∞ n = 1 ne−3n correct to four decimal places.

Find the maximum area of a rectangle inscribed in a semicircle of radius r.

If a test for a disease is 95% accurate (true positive and true negative rates) and 5% of the population has the disease, what is the probability that a person has the disease given that they tested positive?

How many ways can 5 books be arranged on a shelf if 3 of the books are identical?

Find the volume of a cone with radius r and height h.

In a graph, find the shortest path from node A to node B using Dijkstra's algorithm.