GATE CS Digital Logic questions

Consider the $8$ -bit signed integers $X, Y$ and $Z$ represented using the sign-magnitude form. The binary representations of $X$ and $Y$ are as follows: $$X: 10110100 \quad Y: 01001100$$ Which of the following operations to compute $Z$ result(s) in an arithmetic overflow?

Consider the following Boolean expression of a function $F$ : $$F(P, Q)=(\bar{P}+Q) \oplus(\bar{P} Q)$$ Which of the following expressions is/are equivalent to $F$ ?

Consider the real valued variables $X, Y$ and $Z$ represented using the IEEE $754$ singleprecision floating-point format. The binary representations of $X$ and $Y$ in hexadecimal notation are as follows: $$X: 35 \mathrm{C} 00000 \quad Y: 34 \mathrm{A} 00000$$ Let $Z=X+Y$. Which one of the following is the binary representation of $Z$, in …

Consider a $2$-bit saturating up/down counter that performs the saturating up count when the input $P$ is $0$, and the saturating down count when $P$ is $1$. The Next State table of the counter is as shown. The counter is built as a synchronous sequential circuit using D flip-flops. InputCureent StateNext State …

Consider a Boolean function $F$ with the following minterm expression: $$F(P, Q, R, S)=\sum m(1,2,3,4,5,7,10,12,13,14)$$ Which of the following options is/are the minimal sum-of-products expression(s) of $F$ ?

Which one of the following options is not a property of Boolean Algebra? Note: ${+}$ is OR operation, $\cdot$ is AND operation, and ${\prime}$ is NOT operation

In a system, numbers are represented using $4$-bit two's complement form. Consider four numbers $N 1=1011, N 2=1101, N 3=1010$ and $N 4=1001$ in the system. Which of the following operations will result in arithmetic overflow?

The $32$-bit $\text{IEEE}$ $754$ single precision representation of a number is $0\text{xC}2710000$. The number in decimal representation is $\_\_\_\_$. (rounded off to two decimal places)

Consider the following $4$-variable Boolean function $$F(A, B, C, D)=\Sigma m(0,1,2,3,8,9,10,11)$$ Consider $A$ as MSB, $D$ as LSB. Which one of the following options represents the minimal sum of products form for the above function? Note: ${+}$ is OR operation, ${\cdot}$ is AND operation, ${\prime}$ is NOT operation

Consider the digital circuit shown below with two input lines $\text{A}$ and $\text{B}$, two select lines $\text{S0}$ and $\text{S1}$, and an output line $\text{Y}$. The blocks $\text{Q}$ and $\text{M}$ represent active high $2:4$ decoder and $4$-to-$1$ multiplexer, respectively. Out of $16$ possible input combinations, the number of …

Let $X$ be a $3$-variable Boolean function that produces output as $'1'$ when at least two of the input variables are $'1'$. Which of the following statement(s) is/are CORRECT, where $a, b, c, d, e$ are Boolean variables?

The number $-6$ can be represented as $1010$ in $4$-bit $2$'s complement representation. Which of the following is/are CORRECT $2$ 's complement representation(s) of $-6$?

Consider the following four variable Boolean function in sum-of-product form $$F\left(b_{3}, b_{2}, b_{1}, b_{0}\right)=\sum(0,2,4,8,10,11,12)$$ where the value of the function is computed by considering $b_{3} b_{2} b_{1} b_{0}$ as a $4$-bit binary number, where $b_{3}$ denotes the most significant bit and $b_{0}$ denotes the least significant bit. …

Consider the given sequential circuit designed using D-Flip-flops. The circuit is initialized with some value (initial state). The number of distinct states the circuit will go through before returning back to the initial state is ___________. (Answer in integer)

​Consider the following logic circuit diagram. Which is/are the CORRECT option(s) for the output function $F$ ?

The following two signed $2$ 's complement numbers (multiplicand M and multiplier Q) are being multiplied using Booth's algorithm: M: $1100110111101101$ and Q: $1010010010101010$ The total number of addition and subtraction operations to be performed is_____________. (Answer in integer)

In a $4$-bit ripple counter, if the period of the waveform at the last flip-flop is $64$ microseconds, then the frequency of the ripple counter in kHz is _________. (Answer in integer)

​​​​Given the following Karnaugh Map for a Boolean function $F(w, x, y, z)$ : Which one or more of the following Boolean expression(s) represent(s) $F$ ?

Three floating point numbers $X, Y$, and $Z$ are stored in three registers $\mathrm{R}_{\mathrm{X}}, \mathrm{R}_{\mathrm{Y}}$, and $\mathrm{R}_{\mathrm{Z}}$, respectively in $IEEE$ $754$ single precision format as given below in hexadecimal: …

Which of the following Boolean algebraic equation(s) is/are CORRECT?