GATE CS Digital Logic questions

Consider a system that uses $5$ bits for representing signed integers in $2$ 's complement format. In this system, two integers $A$ and $B$ are represented as $A$=$01010$ and $B$=$11010$. Which one of the following operations will result in either an arithmetic overflow or an arithmetic underflow?

​​Consider the circuit shown below where the gates may have propagation delays. Assume that all signal transitions occur instantaneously and that wires have no delays. Which of the following statements about the circuit is/are CORRECT?

Consider a Boolean expression given by $\text{F(X, Y, Z)}=\sum(3,5,6,7)$. Which of the following statements is/are CORRECT?

Consider a digital logic circuit consisting of three $2$-to-$1$ multiplexers $\text{M1, M2}$, and $\text{M3}$ as shown below. $\mathrm{X} 1$ and $\mathrm{X} 2$ are inputs of $\mathrm{M} 1$. $\text{X3}$ and $\text{X4}$ are inputs of $\text{M2}$. $\text{A, B}$, and $\text{C}$ are select lines of $\text{M1, M2}$, and $\text{M3}$, …

​​​​For a Boolean variable $x$, which of the following statements is/are FALSE?

Which of the following is/are EQUAL to $224$ in radix - $5$ (i.e., base - $5$) notation?

Consider $4$-variable functions $f 1, f 2, f 3, f 4$ expressed in sum-of-minterms form as given below. $$\begin{array}{l} f 1=\sum(0,2,3,5,7,8,11,13) \\ f 2=\sum(1,3,5,7,11,13,15) \\ f 3=\sum(0,1,4,11) \\ f 4=\sum(0,2,6,13) \end{array}$$ With respect to the circuit given above, which of the following options is/are CORRECT?

The output of a $2$-input multiplexer is connected back to one of its inputs as shown in the figure. Match the functional equivalence of this circuit to one of the following options.

A particular number is written as $132$ in radix-$4$ representation. The same number in radix-$5$ representation is _____________.

Consider a sequential digital circuit consisting of $\mathrm{T}$ flip-flops and $\mathrm{D}$ flip-flops as shown in the figure. $\text{CLKIN}$ is the clock input to the circuit. At the beginning, $\text{Q1, Q2}$ and $\text{Q3}$ have values $0,1$ and $1,$ respectively. Which one of the given values of $\text{(Q1, Q2, Q3)}$ can $\text{NEVER}$ …

A Boolean digital circuit is composed using two $4$-input multiplexers $\text{(M1 and M2)}$ and one $2$-input multiplexer $\text{(M3)}$ as shown in the figure. $\text{X0-X7}$ are the inputs of the multiplexers $\text{M1 and M2}$ and could be connected to either $0$ or $1.$ The select lines of the multiplexers are connected to Boolean …

Consider the $\textsf{IEEE-754}$ single precision floating point numbers $\text{P} = \textsf{0xC1800000}$ and $\text{Q} = \textsf{0x3F5C2EF4}.$ Which one of the following corresponds to the product of these numbers $(\text{i.e., P} \times \mathrm{Q} ),$ represented in the $\textsf{IEEE-754}$ single precision format?

Let $\text{R1}$ and $\text{R2}$ be two $4 – \text{bit}$ registers that store numbers in $2\text{’s}$ complement form. For the operation $\text{R1 + R2},$ which one of the following values of $\text{R1}$ and $\text{R2}$ gives an arithmetic overflow?

Consider a digital display system $\text{(DDS)}$ shown in the figure that displays the contents of register $\text{X}.$ A $16 – \text{bit}$ code word is used to load a word in $\text{X},$ either from $\text{S}$ or from $\text{R}.$ $\text{S}$ is a $1024-$word memory segment and $\text{R}$ is a $32-$word register file. Based on the value of …

Consider three floating point numbers $\textit{A, B}$ and $\textit{C}$ stored in registers $\text{R}_{\text{A}}, \text{R}_{\text{B}}$ and $\text{R}_{\text{C}},$ respectively as per $\textsf{IEEE-754}$ single precision floating point format. The $\text{32-bit}$ content stored in these registers $\text{(in hexadecimal form)}$ are as follows. …

Let the representation of a number in base $3$ be $210$. What is the hexadecimal representation of the number?

Consider the following representation of a number in $\text{IEEE 754}$ single-precision floating point format with a bias of $127$.$$S: 1\quad\quad E:\; 10000001\quad\quad F:\;11110000000000000000000$$ Here $S, \;E$ and $F$ denote the sign, exponent, and fraction components of the floating point representation. The decimal value corresponding to the …

Consider a $3$-bit counter, designed using $T$ flip-flops, as shown below: Assuming the initial state of the counter given by $\text{PQR}$ as $000$, what are the next three states?

Consider the following Boolean expression. $F=(X+Y+Z)(\overline X +Y)(\overline Y +Z)$ Which of the following Boolean expressions is/are equivalent to $\overline F$ (complement of $F$)?

The format of the single-precision floating point representation of a real number as per the $\text{IEEE 754}$ standard is as follows: $$\begin{array}{|c|c|c|} \hline \text{sign} & \text{exponent} & \text{mantissa} \\ \hline \end{array}$$ Which one of the following choices is correct with respect to the smallest normalized positive number …