GATE EC Combinatorial Circuits questions

A Boolean function $f$ of two variables $x$ and $y$ is defined as follows: $$f(0,0)=f(0,1)=f(1,1)=1 ; \quad f(1,0)=0$$ Assuming complements of $x$ and $y$ are not available, a minimum cost solution for realizing $f$ using only $2$-input NOR gates and $2$-input OR gates (each having unit cost) would have a total cost of

The number of distinct Boolean expressions of $4$ variables is

The output of the $74$ series of $\text{TTL}$ gates is taken from a $\text{BJT}$ in

Without any additional circuitry, an $8: 1 \; \mathrm{MUX}$ can be used to obtain

The circuit shown in the figure has $4$ boxes each described by inputs $P, Q, R$ and outputs $Y, Z$ with $$\begin{array}{l} Y=P \oplus Q \oplus R \\ Z=R Q+\bar{P} R+Q \bar{P} \end{array}$$ The circuit acts as a

If the functions $W, X, Y$ and $Z$ are as follows $$\begin{array}{l} W=R+\bar{P} Q+\bar{R} S \\ X=P Q \bar{R} \bar{S}+\bar{P} \bar{Q} \bar{R} \bar{S}+P \bar{Q} \bar{R} \bar{S} \\ Y=R S+\overline{P R+P \bar{Q}+\bar{P} \cdot \bar{Q}} \\ Z=R+S+\overline{P Q+ \bar{P} . \bar{Q} . \bar{R} + P \bar{Q} . \bar{S}} \end{array}$$ Then

The DTL, TTL, ECL and CMOS families of digital $\mathrm{IC}_{\mathrm{s}}$ are compared in the following $4$ columns …

The circuit shown in the figure converts

If the input to the digital circuit the figure, consisting of a cascade of $20 \; \mathrm{XOR}$-gates is $\mathrm{X}$, then the output $Y$ is equal to

If the input $X_{3}, X_{2}, X_{1}, X_{0}$ to the ROM in the figure are $8-4-2-1$ BCD numbers, then the outputs $Y_{3} Y_{2} Y_{1} Y_{0}$ are

In the TTL circuit in the figure, $\mathrm{S}_{2}$ to $\mathrm{S}_{0}$ are select lines and $\text{X}_{7}$ to $\text{X}_{0}$ are input lines. $\text{S}_{0}$ and $\text{X}_{0}$ are LSBs. The output $Y$ is

For the logic circuit shown in the figure is the required input condition $(A, B, C)$ to make the output $(X)=1$ is

For the logic circuit shown in, the simplified Boolean expression for the output $Y$ is

Commercially available $\text{ECL}$ gears use two ground lines and one negative supply in order to

The logical expresion $y=\mathrm{A}+\overline{\mathrm{A}} \mathrm{B}$ is equivalent to

For a binary half-subtractor having two inputs $\text{A}$ and $\text{B}$, the correct set of logical expressions for the outputs $\text{D}$ ( = $\text{A}$ minus $\text{B}$ ) and $\text{X}(=$ borrow) are

The minimized form of the logical expression $(\bar{A} \bar{B} \bar{C}+\bar{A} B \bar{C}+\bar{A} B C+A B \bar{C})$ is

The $K$-map for a Boolean function is shown in the figure is The number of essential prime implicants for this function is

For the identity $\mathrm{AB}+\overline{\mathrm{A}} C+\mathrm{BC}=\mathrm{AB}+\overline{\mathrm{A}} \mathrm{C}$, the dual form is

Two $2^{\prime}$ 's complement number having sign bits $x$ and $y$ are added and the sign bit of the result is $z$. Then, the occurrence of overflow is indicated by the Boolean function