Given the following state table of an $$FSM$$ with two states $$A$$ and $$B,$$ one input and one output:

If the initial state is $$A = 0, B=0.$$ What is the minimum length of an input string which will take the machine to the state $$A=0, B=1$$ with Output$$=1?$$

The control signal functions of a $$4$$-bit binary counter are given below $$($$where $$X$$ “don’t care”$$):$$

The counter is connected as follows:

Assume that the counter and gate delays are negligible. If the counter starts at $$0,$$ then it cycles through the following sequence:

Consider the circuit in the diagram. The $$ \oplus $$ operator represents $$EX$$-$$OR.$$ The $$D$$ flip-flops are initialized to zeros (cleared).

The following data: $$100110000$$ is supplied to the ''data'' terminal in nine clock cycles. After that the values of $${q_2}{q_1}{q_0}$$ are

Consider the partial implementation of a $$2$$-bit counter using $$T$$ flip-flops following the sequence $$0$$-$$2$$-$$3$$-$$1$$-$$0,$$ as shown below.

To complete the circuit, the input $$X$$ should be