Consider a carry lookahead adder for adding two $$n$$-bit integers, built using gates of fan-in at most two. The time to perform addition using this adder is

Consider the two cascaded $$2$$-to-$$1$$ multiplexers as shown in the figure.

The minimal sum of products form of the output $$X$$ is

Two eigenvalues of a $$3 \times 3$$ real matrix $$P$$ are $$\left( {2 + \sqrt { - 1} } \right)$$ and $$3.$$ The determinant of $$P$$ is _______.

$$\mathop {\lim }\limits_{x \to 4} {{\sin \left( {x - 4} \right)} \over {x - 4}} = \_\_\_\_\_\_\_.$$