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

Consider the Boolean operator $$ \ne $$ with the following properties:
$$x \ne 0 = x,\,\,x \ne 1 = \overline x ,\,\,x \ne x = 0$$ and $$x \ne \overline x = 1.$$ Then $$x \ne y$$ is equivalent to

Consider the recurrence relation $${a_1} = 8,\,{a_n} = 6{n^2} + 2n + {a_{n - 1}}.$$ Let $${a_{99}} = K \times {10^4}.$$ The value of $$K$$ is ____________.