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Test 3 Digital Logic | Computer Science(CS)

GATE Previous year Topic Wise Questions and Answers | Digital Logic

26 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

Assuming all numbers are in 2's complement representation, which of the following numbers is divisible by 11111011?

  1. 11100111
  2. 11100100
  3. 11010111
  4. 11011011
Question 2 Multiple Choice (Single Answer)

he sum of the number of times each literal appears in the expression. For example, the literal count of (xy + xz') is 4. What are the minimum possible literal counts of the product-of-sum and sum-of product representations respectively of the function given by the following Karnaugh map? Here, X denotes “don't care”

  1. (11, 9)
  2. (9, 13)
  3. (9, 10)
  4. (11, 11)
Question 3 Multiple Choice (Single Answer)

Consider the following circuit composed of XOR gates and non-inverting buffers

enter image description here

The non-inverting buffers have delays d1 = 2 ns and d2 = 4 ns as shown in the figure. Both XOR gates and all wires have zero delay. Assume that all gate inputs, outputs and wires are stable at logic level 0 at time
0. If the following waveform is applied at input A, how many transition(s) (change of logic levels)
Occur (s) at B during the interval from 0 to 10 ns?

enter image description here

  1. 1
  2. 2
  3. 3
  4. 4
Question 4 Multiple Choice (Single Answer)

You are given a free running clock with a duty cycle of 50% and a digital waveform f which changes only at the negative edge of the clock. Which one of the following circuits (using clocked D flip-flops) will delay the phase of f by 180°?

Question 5 Multiple Choice (Single Answer)

Consider the circuit above. Which one of the following options correctly represents f (x, y, z)?

  1. $x\bar{z}+xy+\bar{y}z$$
  2. $x\bar{z}+xy+\overline{yz}$
  3. $xz+xy+\overline{yz}$
  4. $xz+x\bar{y}+\bar{y}z$
Question 6 Multiple Choice (Single Answer)

We consider the addition of two $2's$ complement numbers $ b_{n-1}b_{n-2}\dots b_{0}$ and $a_{n-1}a_{n-2}\dots a_{0}$. A binary adder for adding unsigned binary numbers is used to add the two numbers. The sum is denoted by $ c_{n-1}c_{n-2}\dots c_{0}$ and the carry-out by $ c_{out}$. Which one of the following options correctly identifies the overflow condition?

  1. $ c_{out}\left ( \overline{a_{n-1}\oplus b_{n-1}} \right )$
  2. $ a_{n-1}b_{n-1}\overline{c_{n-1}}+\overline{a_{n-1}b_{n-1}}c_{n-1}$
  3. $ c_{out}\oplus c_{n-1}$
  4. $ a_{n-1}\oplus b_{n-1}\oplus c_{n-1}$
Question 7 Multiple Choice (Single Answer)

Consider a Boolean function $ f(w,x,y,z)$. Suppose that exactly one of its inputs is allowed to change at a time. If the function happens to be true for two input vectors $ i_{1}=\left \langle w_{1}, x_{1}, y_{1},z_{1}\right \rangle $ and $ i_{2}=\left \langle w_{2}, x_{2}, y_{2},z_{2}\right \rangle $ , we would like the function to remain true as the input changes from $ i_{1}$ to $ i_{2}$ ($ i_{1}$ and $ i_{2}$ differ in exactly one bit position) without becoming false momentarily. Let $ f(w,x,y,z)=\sum (5,7,11,12,13,15)$ . Which of the following cube covers of $f$ will ensure that the required property is satisfied?

  1. $ \overline{w}xz,wx\overline{y},x\overline{y}z,xyz,wyz$
  2. $ wxy, \overline{w}xz,wyz$
  3. $ wx\overline{y} \overline{z}, xz, w\overline{x}yz$
  4. $ wx\overline{y}, wyz, wxz, \overline{w}xz, x\overline{y}z, xyz$
Question 8 Multiple Choice (Single Answer)

Consider numbers represented in 4-bit gray code. Let h3 h2 h1 h0 be the gray code representation of a number n and let g3 g2 g1 g0 be the gray code of (n + 1) (modulo 16) value of the number. Which one of the following functions is correct?

  1. g0 (h3 h2 h1 h0) = $\sum$1, 2, 3, 6, 10, 13, 14, 15
  2. g0 (h3 h2 h1 h0) = $\sum$4,9,10,11,12,13, 14,15
  3. g0 (h3 h2 h1 h0) = <$\sum$2, 4,5, 6, 7,12,13,15
  4. g0 (h3 h2 h1 h0) = $\sum$0,1,6, 7,10,11,12,13
Question 9 Multiple Choice (Single Answer)

Given two three bit numbers a2 a1 a0 and b2 b1 b0 and c, the carry in, the function that represents the carry generate function when these two numbers are added is:

  1. a2 b2 + a2 a1 b1 + a2 a1 a0 b0 + a2 a0 b1 b0 + a1 b2 b1 + a1 a0 b2 b0 + a0 b2 b1 b0
  2. a2 b2 + a2 a1 b0 + a2 a1 b1 b0 + a1 a0 b2 b1 + a1 a0 b2 + a1 a0 b2 b0 + a2 a0 b1 b0
  3. a2 + b2 + (a2$\oplus$b2) (a1 + b1 +(a1$\oplus$b1) (a0 + b0))
Question 10 Multiple Choice (Single Answer)

What is the maximum number of different Boolean functions involving n Boolean variables?

  1. n2
  2. 2n
  3. 22n
  4. $2^{n^2}$
Question 11 Multiple Choice (Single Answer)

How many 3-to-8 line decoders with an enable input are needed to construct a 6-to-64 line decoder without using any other logic gates?

  1. 7
  2. 8
  3. 9
  4. 10
Question 12 Multiple Choice (Single Answer)

Consider the following Boolean function of four variables:
f (w, x, y, z) = $\sum$(1,3,4,6,9,11,12,14)
The function is:

  1. independent of one variables.
  2. independent of two variables.
  3. independent of three variables.
  4. dependent on all the variables.
Question 13 Multiple Choice (Single Answer)

The control signal functions of a 4-bit binary counter are given below (where X is “don't care”):
Clear Clock Load Count Function

The counter is connected as follows:

Assume that the counter and gate delays are negligible. If the counter starts at 0, then it cycles through the following sequence:

  1. 0, 3, 4
  2. 0, 3, 4, 5
  3. 0, 1, 2, 3, 4
  4. 0, 1, 2, 3, 4, 5
Question 14 Multiple Choice (Single Answer)

Define the connective * for the Boolean variables X and Y as: X * Y = XY + X'Y'. Let Z = X *Y. Consider the following expressions P, Q and R.

$\pi P$ : X = Y * Z Q : Y = X * Z R : X *Y * Z = 1

Which of the following is TRUE?

  1. Only P and Q are valid.
  2. Only Q and R are valid.
  3. Only P and R are valid.
  4. All P, Q, R are valid.
Question 15 Multiple Choice (Single Answer)

Suppose only one multiplexer and one inverter are allowed to be used to implement any Boolean function of n variables. What is the minimum size of the multiplexer needed?

  1. 2n line to 1 line
  2. 2n+1 line to 1 line
  3. 2n−1 line to 1 line
  4. 2n−2 line to 1 line
Question 16 Multiple Choice (Single Answer)

Let f (w, x, y, z) = $\sum$(0,4,5,7,8,9,13,15). Which of the following expressions is/are NOT equivalent to f?

P. x’y’z’ + w’xy’ + wy’z + xz
Q. w’y’z’ + wx’y’ + xz
R. w’y’z’ + wx’y’ + xyz + xy’z
S. x’y’z’ + wx’y’ + w’y

  1. P only
  2. Q and S
  3. R and S
  4. S only
Question 17 Multiple Choice (Single Answer)

In a look-ahead carry generator, the carry generate function Gi and the carry propagate function Pi for
inputs Ai and Bi are given by:
Pi = Ai $\oplus$ Bi and Gi = Ai Bi
The expressions for the sum bit Si and the carry bit Ci+1 of the look-ahead carry adder are given by:
Si = Pi Ci and Ci+1 = Gi + Pi Ci, where Co is the input carry.
Consider a two-level logic implementation of the look-ahead carry generator.
Assume that all Pi and Gi are available for the carry generator circuit and that the AND and OR gates
can have any number of inputs. The number of AND gates and OR gates needed to implement the
look-ahead carry generator for a 4-bit adder with 3 2 1 0 4 S ,S ,S ,S and C as its outputs are
respectively:

  1. 6, 3
  2. 10, 4
  3. 6, 4
  4. 10, 5
Question 18 Multiple Choice (Single Answer)

Given f1, f3 and f in canonical sum of products form (in decimal) for the circuit.


$f_1 = \Sigma m(4, 5, 6, 7, 8)$
$f_3 = \Sigma m(1, 6, 15)$
$f = \Sigma m(1, 6, 8, 15)$

Then f2 is

  1. $\Sigma m(4, 6)$
  2. $\Sigma m(4, 8)$
  3. $\Sigma m(6, 8)$
  4. $\Sigma m(4, 6, 8)$
Question 19 Multiple Choice (Single Answer)

In the IEEE floating point representation, the hexadecimal value 0x00000000 corresponds to

  1. the normalized value 2-127
  2. the normalized value +0
  3. the normalized value 2-126
  4. the special value +0
Question 20 Multiple Choice (Single Answer)

Let $r$ denote number system radix. The only value(s) of $r$ that satisfy the equation $\sqrt{121_r}={11}_r$, is/are

  1. decimal 11
  2. any value >2
  3. decimal 10
  4. decimal 10 and 11
Question 21 Multiple Choice (Single Answer)

If $P, Q, R$ are Boolean variables, then

$(P + \bar{Q}) (P.\bar{Q} + P.R) (\bar{P}.\bar{R} + \bar{Q})$ simplifies to

  1. $P.\bar{Q}$
  2. $P.\bar{R}$
  3. $P.\bar{Q} + R$
  4. $P.\bar{R} + Q$
Question 22 Multiple Choice (Single Answer)

In the Karnaugh map shown below, X denotes a don't care term. What is the minimal form of the function represented by the Karnaugh map?

  1. $\bar{b}.\bar{d} + \bar{a}.\bar{d}$
  2. $\bar{a}.\bar{b} + \bar{b}.\bar{d} + \bar{a}.b.\bar{d}$
  3. $\bar{b}.\bar{d} + \bar{a}.b.\bar{d}$
  4. $\bar{a}.\bar{b} + \bar{b}.\bar{d} + \bar{a}.\bar{d}$
Question 23 Multiple Choice (Single Answer)

What is the minimum number of gates required to implement the Boolean function (AB + C) if we have to use only 2-input NOR gates?

  1. 2
  2. 3
  3. 4
  4. 5
Question 24 Multiple Choice (Single Answer)

(1217)8 is equivalent to

  1. (1217)16
  2. (028F)16
  3. (2297)10
  4. (0B17)16
Question 25 Multiple Choice (Single Answer)

The following is a scheme for floating point number representation using 16 bits.

Bit Position 15 14 .... 9 8 ...... 0
s e m
Sign Exponent Mantissa
Let s, e, and m be the numbers represented in binary in the sign, exponent, and mantissa fields respectively. Then the floating point number represented is:

$$\begin{cases}(-1)^s \left(1+m \times 2^{-9}\right) 2^{e-31}, & \text{ if the exponent } \neq 111111 \ 0, & \text{ otherwise} \end{cases}$$

What is the maximum difference between two successive real numbers representable in this system?

  1. 2-40
  2. 2-9
  3. 222
  4. 231
Question 26 Multiple Choice (Single Answer)

A 1-input, 2-output synchronous sequential circuit behaves as follows:
Let zk, nk denote the number of 0's and 1's respectively in initial k bits of the input
(zk+nk=k). The circuit outputs 00 until one of the following conditions holds.
zk - nk=2. In this case, the output at the k-th and all subsequent clock ticks is 10.
nk - zk = 2. In this case, the output at the k-th and all subsequent clock ticks is 01.

What is the minimum number of states required in the state transition graph of the above circuit?

  1. 5
  2. 6
  3. 7
  4. 8
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