$$P$$ is a $$16$$-bit signed integer. The $$2's$$ complement representtation of $$P$$ is $${\left( {F87B} \right)_{16}}$$ . The $$2's$$ complement representation of $$8{}^ * \,P$$ is

The Boolean expression for the output $$f$$ of the multiplexer shown below is

What is the probability that divisor of $${10^{99}}$$ is a multiple of $${10^{96}}$$ ?

Consider the set $$S = \left\{ {1,\,\omega ,\,{\omega ^2}} \right\},$$ where $$\omega $$ and $${{\omega ^2}}$$, are cube roots of unity. If $$ * $$ denotes the multiplication operation, the structure $$\left\{ {S,\, * } \right\}$$ forms