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Boolean algebra. Combinational and sequential circuits. Minimization. Number representations and computer arithmetic (fixed and floating point)

$$\small{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline \textbf{Year}&\textbf{2019}&\textbf{2018}&\textbf{2017-1}&\textbf{2017-2}&\textbf{2016-1}&\textbf{2016-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum} \\\hline\textbf{1 Mark Count}&4&2&3&2&3&3&2&2.8&4 \\\hline\textbf{2 Marks Count}&2&2&0&4&2&0&0&1.7&4 \\\hline\textbf{Total Marks}&8&6&3&10&7&3&\bf{3}&\bf{6.2}&\bf{10}\\\hline \end{array}}}$$

# Previous GATE Questions in Digital Logic

1
In $16$-bit $2$’s complement representation, the decimal number $-28$ is: $1111 \: 1111 \: 0001 \: 1100$ $0000 \: 0000 \: 1110 \: 0100$ $1111 \: 1111 \: 1110 \: 0100$ $1000 \: 0000 \: 1110 \: 0100$
2
Which one of the following is NOT a valid identity? $(x \oplus y) \oplus z = x \oplus (y \oplus z)$ $(x + y) \oplus z = x \oplus (y+z)$ $x \oplus y = x+y, \text{ if } xy=0$ $x \oplus y = (xy+x’y’)’$
3
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: $n$ bits $n-1$ bits $n+1$ bits $n+2$ bits
4
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 $4-bit$ (unsigned) binary representations have the same most significant bit is _______________.
5
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: $\Sigma(7,8,11)$ $\Sigma (2,7,8,11,14)$ $\Sigma (2,14)$ $\Sigma (0,2,3,5,6,7,8,11,14,15)$
6
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: _______
–1 vote
7
Answer for Minimum no of nor gates question
8
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 ___
9
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 ... be exactly represented Only $ii$ cannot be exactly represented Only $iii$ and $iv$ cannot be exactly represented Only $i$ and $ii$ cannot be exactly represented
10
Consider the sequential circuit shown in the figure, where both flip-flops used are positive edge-triggered 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 ____
11
Let $\oplus$ and $\odot$ denote the Exclusive OR and Exclusive NOR operations, respectively. Which one of the following is NOT CORRECT? $\overline{P \oplus Q} = P \odot Q$ $\overline{P} \oplus Q = P \odot Q$ $\overline{P} \oplus \overline{Q} = P \oplus Q$ $P \oplus \overline{P} \oplus Q = ( P \odot \overline{P} \odot \overline{Q})$
12
What is the equivalent minimal Boolean expression (in sum of products form) for the Karnaugh map given below?
13
A 32-bit floating-point number is represented by a 7-bit signed exponent, and a 24-bit fractional mantissa. The base of the scale factor is 16, The range of the exponent is ___________, if the scale factor is represented in excess-64 format.
14
A sequential circuit takes an input stream of 0's and 1's and produces an output stream of 0's and 1's. Initially it replicates the input on its output until two consecutive 0's are encountered on the input. From then onward, it produces an output stream, which is ... to be used to design the circuit. Give the minimized sum-of-product expression for J and K inputs of one of its state flip-flops
15
Consider the synchronous sequential circuit in the below figure Given that the initial state of the circuit is S$_4$, identify the set of states, which are not reachable.
16
Consider the set X={a,b,c,d,e} under partial ordering R={(a,a),(a,b),(a,c),(a,d),(a,e),(b,b),(b,c),(b,e),(c,c),(c,e),(d,d),(d,e),(e,e)}. The Hasse diagram of the partial order (X,R) is shown below. The Hasse diagram of the partial ... i take the pair (b,d) it has LUB as e but no GLB since it is directed... may be the hasse diagram was undirected then it is already a lattice right correct me
17
Without any additional circuitry an $8:1$ MUX can be used to obtain Some but not all Boolean functions of $3$ variables All function of $3$ variables but none of $4$ variables All functions of $3$ variables and some but not all of $4$ variables All functions of $4$ variables
The next state table of a $2-$bit saturating up-counter is given below. $\begin{array}{cc|cc} Q_1 & Q_0 & Q_1^+ & Q_0^+ \\ \hline 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{array}$ The counter is built as a synchronous sequential circuit ... $T_1 = Q_1+Q_0, \quad T_0= \bar{Q_1} \bar{Q_0}$ $T_1 = \bar{Q_1}Q_0, \quad T_0= Q_1 + Q_0$
If $w, x, y, z$ are Boolean variables, then which one of the following is INCORRECT? $wx+w(x+y)+x(x +y) = x+wy$ $\overline{w \bar{x}(y+\bar{z})} + \bar{w}x = \bar{w} + x + \bar{y}z$ $(w \bar{x}(y+x\bar{z}) + \bar{w} \bar{x}) y = x \bar{y}$ $(w+y)(wxy+wyz) = wxy+wyz$
Given the following binary number in $32$-bit (single precision) $IEEE-754$ format : $\large 00111110011011010000000000000000$ The decimal value closest to this floating-point number is : $1.45*10^1$ $1.45*10^{-1}$ $2.27*10^{-1}$ $2.27*10^1$