A 2-input up/down synchrconous counter using two toggle flip-flops is shown in Fig.1. The counter's sequence is to be controlled by the input M as follows:
For M=1, sequence of Q₁, Q₀ is ..00, 01, 10, 11, 00, 01.......
For M=0, sequence of Q₁, Q₀ is ..00, 11, 10, 01, 00, 11......

(a)Design the necessary feedback logic for T₁ and T₀.

(b)Realize the feesback logic using inverters and 4-input multiplexers only. Use Q₁ and Q₀ as the control inputs of the multiplexer with Q₁ as the MSB.

The circuit diagram of a 2 bit A to D converter is shown in figure below. The combinational logic is to be disigned to provide a natural binary representation using $${D_1}$$ and $${D_0}$$ for the analog input $${V_i}$$. $${D_1}$$ is to be the most significant bit.
(a) Draw the Karnaugh maps for $${D_1}$$ and $${D_0}$$ in terms of $${C_2}$$, $${C_1}$$ and $${C_0}$$
(b) Obtain the minimal sum of products expressions for $${D_1}$$ and $${D_0}$$.
(c) Realize the logic for $${D_1}$$ and $${D_0}$$ using 2- input NAND gates only.
(d) Find the resolution of the Ato D converter.

The subtraction of a binary number Y from another binary number X, done by adding the 2's complement of Y to X, result in a binary number without overflow. This implies that the result is:

An electrostatic field is said to be conservative when