GATE CS Digital Logic questions

Which one of the following circuits implements the Boolean function given below? $f(x,y,z) = m_0+m_1+m_3+m_4+m_5+m_6$, where $m_i$ is the $i^{\text{th}}$ minterm.

If $x$ and $y$ are two decimal digits and $(0.1101)_2 = (0.8xy5)_{10}$, the decimal value of $x+y$ is ___________

If the numerical value of a $2$-byte unsigned integer on a little endian computer is $255$ more than that on a big endian computer, which of the following choices represent(s) the unsigned integer on a little endian computer?

Consider a Boolean function $f(w,x,y,z)$ such that $$\begin{array}{lll} f(w,0,0,z) & = & 1 \\ f(1,x,1,z) & =& x+z \\ f(w,1,y,z) & = & wz +y \end{array}$$The number of literals in the minimal sum-of-products expression of $f$ is _________

A multiplexer is placed between a group of $32$ registers and an accumulator to regulate data movement such that at any given point in time the content of only one register will move to the accumulator. The number of select lines needed for the multiplexer is ______.

If there are $m$ input lines and $n$ output lines for a decoder that is used to uniquely address a byte addressable $1$ KB RAM, then the minimum value of $m+n$ is ________ .

Consider the Boolean function $z(a,b,c)$. Which one of the following minterm lists represents the circuit given above?

Consider three registers $R1$, $R2$, and $R3$ that store numbers in $\textsf{IEEE-754}$ single precision floating point format. Assume that $R1$ and $R2$ contain the values (in hexadecimal notation) $\textsf{0x42200000}$ and $\textsf{0xC1200000},$ respectively. If $R3=\frac{R1}{R2}$, what is the value stored in $R3$?

In $16$-bit $2$’s complement representation, the decimal number $-28$ is:

Which one of the following is NOT a valid identity?

Consider $Z=X-Y$ where $X, Y$ and Z are all in sign-magnitude form. X and Y are each represented in $n$ bits. To avoid overflow, the representation of $Z$ would require a minimum of:

Two numbers are chosen independently and uniformly at random from the set $\{1,2,\ldots,13\}.$ The probability (rounded off to $3$ decimal places) that their $4\text{-bit}$ (unsigned) binary representations have the same most significant bit is ___________.

Consider three $4$-variable functions $f_1, f_2$, and $f_3$, which are expressed in sum-of-minterms as $f_1=\Sigma(0,2,5,8,14),$ $f_2=\Sigma(2,3,6,8,14,15),$ $f_3=\Sigma (2,7,11,14)$ For the following circuit with one AND gate and one XOR gate the output function $f$ can be expressed as:

What is the minimum number of $2$-input NOR gates required to implement a $4$ -variable function expressed in sum-of-minterms form as $f=\Sigma(0,2,5,7, 8, 10, 13, 15)?$ Assume that all the inputs and their complements are available. Answer: _______

Consider the minterm list form of a Boolean function $F$ given below. $$F(P, Q, R, S) = \Sigma m(0, 2, 5, 7, 9, 11) + d(3, 8, 10, 12, 14)$$ Here, $m$ denotes a minterm and $d$ denotes a don't care term. The number of essential prime implicants of the function $F$ is ___

Consider the unsigned 8-bit fixed point binary number representation, below, $$b_7 \: \: b_6 \: \: b_5 \: \: b_4 \: \: b_3 \: \: \cdot b_2 \: \: b_1 \: \: b_0$$ where the position of the primary point is between $b_3$ and $b_2$ . Assume $b_7$ is the most significant bit. Some of the decimal numbers listed below cannot be represented exactly in the …

Consider the sequential circuit shown in the figure, where both flip-flops used are positive edge-triggered $\text{D}$ flip-flops. The number of states in the state transition diagram of this circuit that have a transition back to the same state on some value of "in" is ____

Let $\oplus$ and $\odot$ denote the Exclusive OR and Exclusive NOR operations, respectively. Which one of the following is NOT CORRECT?

Consider a combination of $\text{T}$ and $\text{D}$ flip-flops connected as shown below. The output of the $\text{D}$ flip-flop is connected to the input of the $\text{T}$ flip-flop and the output of the $\text{T}$ flip-flop is connected to the input of the $\text{D}$ flip-flop. Initially, both $Q_{0}$ and $Q_{1}$ are set to $1$ (before the …

Consider the Karnaugh map given below, where $X$ represents "don't care" and blank represents $0$. Assume for all inputs $\left ( a,b,c,d \right )$, the respective complements $\left ( \bar{a}, \bar{b}, \bar{c}, \bar{d} \right )$ are also available. The above logic is implemented using $2$-input $\text{NOR}$ gates only. The minimum number of …