A first order reaction A $$\to$$ B, requires activation energy of 70 kJ mol^-1. When 20% solution of A was kept at 25^oC for 20 minutes, 25% decomposition tooks place. What will be the percentage decomposition in the same time in a 30% solution maintained at 40^oC? Assume that activation energy remains constant in this range of temperature.

Numbers are selected at random, one at a time, from the two- digit numbers $$00, 01, 02 ......, 99$$ with replacement. An event $$E$$ occurs if only if the product of the two digits of a selected number is $$18$$. If four numbers are selected, find probability that the event $$E$$ occurs at least $$3$$ times.

Let $$\vec a = 2\hat i - \hat j + \hat k,\vec b = \hat i + 2\hat j - \hat k$$ and $$\overrightarrow c = \widehat i + \widehat j - 2\widehat k - 2\widehat k$$ be three vectors. A vector in the plane of $${\overrightarrow b }$$ and $${\overrightarrow c }$$, whose projection on $${\overrightarrow a }$$ is of magnitude $$\sqrt {2/3,} $$ is :

In a triangle $$ABC, D$$ and $$E$$ are points on $$BC$$ and $$AC$$ respectively, such that $$BD=2DC$$ and $$AE=3EC.$$ Let $$P$$ be the point of intersection of $$AD$$ and $$BE.$$ Find $$BP/PE$$ using vector methods.