A is a set containing $$n$$ elements. $$A$$ subset $$P$$ of $$A$$ is chosen at random. The set $$A$$ is reconstructed by replacing the elements of $$P.$$ $$A$$ subset $$Q$$ of $$A$$ is again chosen at random. Find the probability that $$P$$ and $$Q$$ have no common elements.

Let $$\overrightarrow A = 2\overrightarrow i + \overrightarrow k ,\,\overrightarrow B = \overrightarrow i + \overrightarrow j + \overrightarrow k ,$$ and $$\overrightarrow C = 4\overrightarrow i - 3\overrightarrow j + 7\overrightarrow k .$$ Determine a vector $$\overrightarrow R .$$ Satisfying $$\overrightarrow R \times \overrightarrow B = \overrightarrow C \times \overrightarrow B $$ and $$\overrightarrow R \,.\,\overrightarrow A = 0$$

Let $${z_1}$$ = 10 + 6i and $${z_2}$$ = 4 + 6i. If Z is any complex number such that the argument of $${{(z - {z_1})} \over {(z - {z_2})}}\,is{\pi \over 4}$$ , then prove that $$\left| {z - 7 - 9i} \right| = 3\sqrt 2 $$.

If $$\int {{{4{e^x} + 6{e^{ - x}}} \over {9{e^x} - 4{e^{ - x}}}}\,dx = Ax + B\,\,\log \left( {9{e^{2x}} - 4} \right) + C,} $$ then
$$A = .....,B = .....$$ and $$C = .....$$