Show that : $$\mathop {\lim }\limits_{n \to \infty } \left( {{1 \over {n + 1}} + {1 \over {n + 2}} + .... + {1 \over {6n}}} \right) = \log 6$$

For a biased die the probabilities for the different faces to turn up are given below :

This die tossed and you are told that either face $$1$$ or face $$2$$ has turned up. Then the probability that it is face $$1$$ is ...............

An anti-aircraft gun can take a maximum of four shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shot are $$0.4, 0.3, 0.2$$ and $$0.1$$ respectively. What is the probability that the gun hits the plane?

Let $$\overrightarrow A ,\overrightarrow B ,\overrightarrow C $$ be vectors of length $$3, 4, 5$$ respectively. Let $$\overrightarrow A $$ be perpendicular to $$\overrightarrow B + \overrightarrow C ,\overrightarrow B $$ to $$\overrightarrow C + \overrightarrow A $$ to $$\overrightarrow A + \overrightarrow B .$$ Then the length of vector $$\overrightarrow A + \overrightarrow B + \overrightarrow C $$ is ..........