## Digital-Logic-Design

Question 1 |

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

1111 1111 1110 0100 | |

1111 1111 0001 1100 | |

0000 0000 1110 0100 | |

1000 0000 1110 0100 |

1’s complement = 1111 1111 1110 0011

2’s complement = 1’s complement + 1

2’s complement = 1111 1111 1110 0100 = (-28)

Question 2 |

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

0.502 | |

0.461 | |

0.402 | |

0.561 |

1 - 0001

2 - 0010

3 - 0011

4 - 0100

5 - 0101

6 - 0110

7 - 0111

8 - 1000

9 - 1001

10 - 1010

11 - 1011

12 - 1100

13 - 1101

The probability that their 4-bit binary representations have the same most significant bit is

= P(MSB is 0) + P(MSB is 1)

= (7×7)/(13×13) + (6×6)/(13×13)

= (49+36)/169

= 85/169

= 0.502

Question 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 + 2 bits | |

n - 1 bits | |

n + 1 bits |

To store overflow/carry bit there should be extra space to accommodate it.

Hence, Z should be n+1 bits.

Question 4 |

Which one of the following is NOT a valid identity?

(x + y) ⊕ z = x ⊕ (y + z) | |

(x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) | |

x ⊕ y = x + y, if xy = 0 | |

x ⊕ y = (xy + x'y')' |

(x+y) ⊕ z = (1+1)⊕ 0 = 1 ⊕ 0 = 1

x ⊕ (y+z) = 1⊕(1+0) = 1 ⊕ 1 = 0

So,

(x+y) ⊕ z ≠ x ⊕ (y+z)

Question 5 |

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

2 | |

4 | |

7 | |

1 | |

3(Option not given) |

Question 6 |

Consider three 4-variable functions f_{1}, f_{2} and f_{3}, which are expressed in sum-of-minterms as

f_{1}= Σ(0, 2, 5, 8, 14), f_{2}= Σ(2, 3, 6, 8, 14, 15), f_{3}= Σ(2, 7, 11, 14)

For the following circuit with one AND gate and one XOR gate, the output function f can be expressed as:

Σ (2, 14) | |

Σ (7, 8, 11) | |

Σ (2, 7, 8, 11, 14) | |

Σ (0, 2, 3, 5, 6, 7, 8, 11, 14, 15) |

f3 = ∑(2,7,11,14)

f1*f2 ⊕ f3 = ∑(2,8,14) ⊕ ∑(2,7,11,14)

= ∑(8,7,11)

(Note: Choose the terms which are not common)

Question 7 |

Let ⊕ and ⊙ denote the Exclusive OR and Exclusive NOR operations, respectively. Which one of the following is NOT CORRECT?

Question 8 |

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 ______

2 | |

3 | |

4 | |

5 |

Now lets draw characteristic table,

D

_{1}= Q

_{0}

D

_{0}= in

Question 9 |

b

_{7}b

_{6}b

_{5}b

_{4}b

_{3}b

_{2}b

_{1}b

_{0}

where the position of the binary 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 above representation:

(i) 31.500

(ii) 0.875

(iii) 12.100

(iv) 3.001

Which one of the following statements is true?

None of (i), (ii), (iii), (iv) can 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}= (11111.100)

_{2}= 2

^{4}+ 2

^{3}+ 2

^{2}+ 2

^{1}+ 2

^{0}+ 2

^{-1}

= 16 + 8 + 4 + 2 + 1 + 0.5

= (31.5)

_{10}

(ii) (0.875)

_{10}= (00000.111)

_{2}

= 2

^{-1}+ 2

^{-2}+ 2

^{-3}

= 0.5 + 0.25 + 0.125

= (0.875)

_{10}

(iii) (12.100)

_{10}

It is not possible to represent (12.100)

_{10}

(iv) (3.001)

_{10}It is not possible to represent (3.001)

_{10}

Question 10 |

Consider the minterm list form of a Boolean function F given below.

- F(P, Q, R, S) = Σm(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 _______.

3 | |

4 | |

5 |

There are 3 prime implicant i.e., P’QS, Q’S’ and PQ’ and all are essential.

Because 0 and 2 are correct by only Q’S’, 5 and 7 are covered by only P’QS and 8 and 9 are covered by only PQ’.

Question 11 |

The n-bit fixed-point representation of an unsigned real number X uses f bits for the fraction part. Let i = n-f. The range of decimal values for X in this representation is

2 ^{-f} to 2^{i} | |

2 ^{-f} to (2^{i} - 2^{-f}) | |

0 to 2 ^{i} | |

0 to (2 ^{i} - 2^{-f }) |

Number of bits in fraction part → f-bits

Number of bits in integer part → (n – f) bits

Minimum value:

000…0.000…0 = 0

Maximum value:

= (2

^{ n-f }- 1) + (1 - 2

^{-f}

= (2

^{n-f}- 2

^{-f})

= (2

^{i}- 2

^{ -f })

Question 12 |

When two 8-bit numbers A_{7}...A_{0} and B_{7}...B_{0} in 2’s complement representation (with A_{0} and B_{0} as the least significant bits) are added using a ripple-carry adder, the sum bits obtained are S_{7}...S_{0} and the carry bits are C_{7}...C_{0}. An overflow is said to have occurred if

the carry bit C _{7} is 1 | |

all the carry bits (C _{7},…,C_{0}) are 1 | |

i.e., A

_{7}= B

_{7}

⇾ Overflow can be detected by checking carry into the sign bits (C

_{in}) and carry out of the sign bits (C

_{out}).

⇾ Overflow occurs iff A

_{7}= B

_{7}and C

_{in}≠ C

_{out}

These conditions are equivalent to

Consider

Here A

_{7}= B

_{7}= 1 and S

_{7}= 0

This happens only if C

_{in}= 0

Carry out C

_{out}=1 when

Similarly, in case of

C

_{in}=1 and C

_{out}will be 0.

Question 13 |

Consider the Karnaugh map given below, where X represents *“don’t care”* and blank represents 0.

Assume for all inputs , the respective complements are also available. The above logic is implemented using 2-input NOR gates only. The minimum number of gates required is _________.

1 | |

2 | |

3 | |

4 |

As all variables and their complements are available we can implement the function with only one NOR Gate.

Question 14 |

Consider a combination of T and D flip-flops connected as shown below. The output of the D flip-flop is connected to the input of the T flip-flop and the output of the T flip-flop is connected to the input of the D flip-flop.

Initially, both Q_{0} and Q_{1} are set to 1 (before the 1^{st} clock cycle). The outputs

Q _{1}Q_{0} after the 3^{rd} cycle are 11 and after the 4^{th} cycle are 00 respectively | |

Q _{1}Q_{0} after the 3^{rd} cycle are 11 and after the 4^{th} cycle are 01 respectively | |

Q _{1}Q_{0} after the 3^{rd} cycle are 00 and after the 4^{th} cycle are 11 respectively | |

Q _{1}Q_{0} after the 3^{rd} cycle are 01 and after the 4^{th} cycle are 01 respectively |

Question 15 |

The representation of the value of a 16-bit unsigned integer X in hexadecimal number system is BCA9. The representation of the value of X in octal number system is

136251 | |

736251 | |

571247 | |

136252 |

_{16}

Each hexadecimal digit is equal to a 4-bit binary number. So convert

X = (BCA9)

_{16}to binary

Divide the binary data into groups 3 bits each because each octal digit is represented by 3-bit binary number.

X = (001 011 110 010 101 001)

_{2}

**Note**: Two zeroes added at host significant position to make number bits of a multiple of 3 (16 + 2 = 18)

X = (136251)

_{8}

Question 16 |

Given the following binary number in 32-bit (single precision) IEEE-754 format:

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} |

For single-precision floating-point representation decimal value is equal to (-1)

^{5}× 1.M × 2

^{(E-127)}

S = 0

E = (01111100)

_{2}= (124).

So E – 127 = - 3

1.M = 1.11011010…0

= 2

^{0}+ 2

^{(-1)}+ 2

^{(-1)}+ 2

^{(-4)}+ 2

^{(-5)}+ 2

^{(-7)}

= 1+0.5+0.25+0.06+0.03+0.007

≈ 1.847

(-1)

^{5}× 1.M × 2

^{(E-127)}

= -1

^{0}× 1.847 × 2

^{-3}

≈ 0.231

≈ 2.3 × 10

^{-1}

Question 17 |

Consider a quadratic equation *x ^{2} - 13x + 36 = 0* with coefficients in a base

*b*. The solutions of this equation in the same base b are

*x = 5*and

*x = 6*. Then

*b*=________.

8 | |

9 | |

10 | |

11 |

^{2}- 13x + 36 = 0 ⇾(1)

Generally if a, b are roots.

(x - a)(x - b) = 0

x

^{2}- (a + b)x + ab = 0

Given that x=5, x=6 are roots of (1)

So, a + b = 13

ab=36 (with same base ‘b’)

i.e., (5)

_{b}+ (6)

_{b}= (13)

_{b}

Convert them into decimal value

5

_{b}= 5

_{10}

6

_{10}= 6

_{10}

13

_{b}= b+3

11 = b+3

b = 8

Now check with ab = 36

5

_{b}× 6

_{b}= 36

_{b}

Convert them into decimals

5

_{b}× 6

_{b}= (b×3) + 6

_{10}

30 = b × 3 + 6

24 = b × 3

b = 8

∴ The required base = 8

Question 18 |

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 | |

(w + y)(wxy + wyz) = wxy + wyz |

__Option-A__:

wx + w(x + y) + x(x + y)

= (wx + wx) + wy + (x + xy)

= wx + wy + x(1 + y)

= wx + wy + x

= (w + 1)x + wy

= x + wy

__Option-B__:

__Option-C__:

__Option-D__:

(w + y)(wxy + wyz) = wxy + wyz + wxy + wyz = wxy + wyz

Question 19 |

Given f(w,x,y,z) = Σ_{m}(0,1,2,3,7,8,10) + Σ_{d}(5,6,11,15), where d represents the don’t-care condition in Karnaugh maps. Which of the following is a minimum product-of-sums (POS) form of f(w,x,y,z)?

_{m}(0,1,2,3,7,8,10) + Σ

_{d}(5,6,11,15)

K-Map for the function f is

Consider maxterms in K-map to represent function in product-of-sums (POS) form

f(w,x,y,z) = (w' + z')(x' + z)

Question 20 |

Consider a binary code that consists of only four valid code words as given below:

Let the minimum Hamming distance of the code be p and the maximum number of erroneous bits that can be corrected by the code be q. Then the values of p and q are

p=3 and q=1 | |

p=3 and q=2 | |

p=4 and q=1 | |

p=4 and q=2 |

Minimum Distance = p = 3

Error bits that can be corrected = (p-1)/2 = (3-1)/2 = 1

∴ p=3 and q=1

Question 21 |

The next state table of a 2-bit saturating up-counter is given below.

The counter is built as a synchronous sequential circuit using T flip-flops. The expressions for T_{1} and T_{0} are

By using above excitation table,

Question 22 |

Consider the Boolean operator with the following properties:

Then x#y is equivalent to

Ex-OR satisfies all the properties. Hence,

Question 23 |

The 16-bit 2’s complement representation of an integer is 1111 1111 1111 0101; its decimal representation is __________.

-11 | |

-12 | |

-13 | |

-14 |

It is a negative number because MSB is 1.

Magnitude of 1111 1111 1111 0101 is 2’s complement of 1111 1111 1111 0101.

1111 1111 1111 0101

0000 0000 0000 1010 : 1’s Complement

0000 0000 0000 1011 : 2’s complement

= (11)

_{10}

Hence, 1111 1111 1111 0101 = -11

Question 24 |

We want to design a synchronous counter that counts the sequence 0-1-0-2-0-3 and then repeats. The minimum number of J-K ﬂip-ﬂops required to implement this counter is __________.

4 | |

5 | |

6 | |

7 |

There are 3 transitions from 0.

Hence ⌈log

_{2}

^{3}⌉ = 2 bits have to be added to the existing 2 bits to represent 4 unique states.

Question 25 |

Consider the two cascaded 2-to-1 multiplexers as shown in the ﬁgure.

The minimal sum of products form of the output *X* is

^{st}MUX is

Now

Question 26 |

Consider a carry lookahead adder for adding two n-bit integers, built using gates of fan-in at most two. The time to perform addition using this adder is __________.

Θ(1) | |

Θ(log(n)) | |

Θ(√n) | |

Θ(n) |

_{k}(n))

Where n is number of bits added

and k is fan-in of the gates.

As we are adding n-bit numbers and fan-in is at most 2,

the solution is θ(log

_{2}(n)).

Question 27 |

Consider an eight-bit ripple-carry adder for computing the sum of A and B, where A and B are integers represented in 2’s complement form. If the decimal value of A is one, the decimal value of B that leads to the longest latency for the sum to stabilize is _________.

-1 | |

-2 | |

-3 | |

-4 |

If we do 2's complement of 1 = 0000 0001, we get -1 = "1111 1111"

So, if B = -1, every carry bit is 1.

Question 28 |

Let, x_{1}⊕x_{2}⊕x_{3}⊕x_{4} = 0 where x_{1}, x_{2}, x_{3}, x_{4} are Boolean variables, and ⊕ is the XOR operator. Which one of the following must always be **TRUE**?

x _{1}x_{2}x_{3}x_{4} = 0 | |

x _{1}x_{3}+x_{2} = 0 | |

x _{1} + x_{2} + x_{3} + x_{4} = 0 |

x

_{1}⊕ x

_{2}⊕ x

_{3}⊕ x

_{4}= 0 -----(1)

A) x

_{1}x

_{2}x

_{3}x

_{4}= 0

Put x

_{1}= 1, x

_{2}= 1, x

_{3}= 1, x

_{4}= 1

The given equation will be zero, i.e.,

1 ⊕ 1 ⊕ 1 ⊕ 1 = 0

But,

x

_{1}x

_{2}x

_{3}x

_{4}≠ 0

So, false.

B) x

_{1}x

_{3}+ x

_{2}= 0

Put x

_{1}= 1, x

_{2}= 1, x

_{3}= 0 , x

_{4}= 0

The given equation will be zero, i.e.,

1 ⊕ 1 ⊕ 0 ⊕ 0 = 0

But,

x

_{1}x

_{3}+ x

_{2}≠ 0

So, false.

D) x

_{1}+ x

_{2}+ x

_{3}+ x

_{4}= 0

Let x

_{1}=1, x

_{2}=1, x

_{3}=0, x

_{4}=0

The given equation will be zero, i.e.,

1 ⊕ 1 ⊕ 0 ⊕ 0 = 0

But,

x

_{1}+ x

_{2}+ x

_{3}+ x

_{4}≠ 0

So, false.

(i) True.

Question 29 |

Let *X* be the number of distinct 16-bit integers in 2’s complement representation. Let *Y* be the number of distinct 16-bit integers in sign magnitude representation.

Then *X-Y* is _________.

1 | |

2 | |

3 | |

4 |

^{16}

Since range is - 2

^{15}to 2

^{15}- 1

Y = 2

^{16}- 1

Here, +0 and -0 are represented separately.

X - Y = 2

^{16}- (2

^{16}- 1)

= 1

Question 30 |

Consider a 4-bit Johnson counter with an initial value of 0000. The counting sequence of this counter is

0, 1, 3, 7, 15, 14, 12, 8, 0 | |

0, 1, 3, 5, 7, 9, 11, 13, 15, 0 | |

0, 2, 4, 6, 8, 10, 12, 14, 0 | |

0, 8, 12, 14, 15, 7, 3, 1, 0 |

The state sequence is 0,8,12,14,15,7,3,1,0.

Question 31 |

The binary operator ≠ is defined by the following truth table

Which one of the following is true about the binary operator ≠?

Both commutative and associative | |

Commutative but not associative | |

Not commutative but associative | |

Neither commutative nor associative |

Question 32 |

A positive edge-triggered D flip-flop is connected to a positive edge-triggered JK flipflop as follows. The Q output of the D flip-flop is connected to both the J and K inputs of the JK flip-flop, while the Q output of the JK flip-flop is connected to the input of the D flip-flop. Initially, the output of the D flip-flop is set to logic one and the output of the JK flip-flop is cleared. Which one of the following is the bit sequence (including the initial state) generated at the Q output of the JK flip-flop when the flip-flops are connected to a free-running common clock? Assume that J = K = 1 is the toggle mode and J = K = 0 is the state-holding mode of the JK flip-flop. Both the flip-flops have non-zero propagation delays.

0110110... | |

0100100... | |

011101110... | |

011001100... |

The characteristic equations are

Q

_{DN}=D=Q

_{JK}

The state table and state transition diagram are as follows:

Consider Q

_{D}Q

_{JK}=10 as initial state because in the options Q

_{JK}=0 is the initial state of JK flip-flop.

The state sequence is

0 → 1 → 1 → 0 → 1 → 1

∴ Option (a) is the answer.

Question 33 |

The minimum number of JK flip-flops required to construct a synchronous counter with the count sequence (0, 0, 1, 1, 2, 2, 3, 3, 0, 0,……) is ___________.

2 | |

3 | |

4 | |

5 |

00

00

01

01

10

10

11

11

In the above sequence two flip-flop's will not be sufficient. Since we are confronted with repeated sequence, we may add another bit to the above sequence.

__0__00

__1__00

__0__01

__1__01

__0__10

__1__10

__0__11

__1__11

Now and every count is unique, occurring only once.

So finally 3-flip flops is required.

Question 34 |

The number of min-terms after minimizing the following Boolean expression is ______.

[D′ + AB′ + A′C + AC′D + A′C′D]′

1 | |

2 | |

3 | |

4 |

[D' + AB' + A'C + AC'D + A'C'D]'

[D' + AB' + A'C + C'D (A + A')']' (since A+A' = 1)

[AB' + A'C + (D' + C') (D' + D)]' (since D' + D =1)

[AB' + A'C + D' + C']'

[AB' + (A' + C') (C + C') + D']'

[AB' + A' + C' + D']'

[(A + A') (A' + B') + C' + D']'

[A' + B' + C' + D']'

Apply de-morgan's law,

ABCD

Question 35 |

A half adder is implemented with XOR and AND gates. A full adder is implemented with two half adders and one OR gate. The propagation delay of an XOR gate is twice that of an AND/OR gate. The propagation delay of an AND/OR gate is 1.2 microseconds. A 4-bit ripple-carry binary adder is implemented by using four full adders. The total propagation time of this 4-bit binary adder in microseconds is ____________.

19.1 | |

19.2 | |

18.1 | |

18.2 |

Here, each Full Adder is taking 4.8 microseconds. Given adder is a 4 Bit Ripple Carry Adder. So it takes 4*4.8 = 19.2 microseconds.

Question 36 |

The total number of prime implicants of the function f(w,x,y,z) = Σ(0, 2, 4, 5, 6, 10) is ______.

3 | |

4 | |

2 | |

1 |

Total 3 prime implicants are there.

Question 37 |

Consider the following Boolean expression for F:

F(P, Q, R, S) = PQ + P'QR + P'QR'S

The minimal sum-of-products form of F is

= Q(P+P’R) + P’QR’S

= Q(P+R) + P’QR’S

= QP + QR + P’QR’S

= QP + Q(R + P’R’S)

= QP + Q( R + P’S)

= QP + QR + QP’S

= Q(P+P’S) + QR

= Q(P+S)+ QR

= QP + QS + QR

Question 38 |

The base (or radix) of the number system such that the following equation holds is_________.

312/20 = 13.1

5 | |

6 | |

7 | |

8 |

(3r

^{2}+ r + 2) / 2r= (r+3+1/r)

(3r

^{2}+ r + 2) / 2r= (r

^{2}+3r+1) / r

(3r

^{2}+ r + 2) = (2r

^{2}+6r+2)

r

^{2}-5r = 0

Therefor r = 5

Question 39 |

Consider a 4-to-1 multiplexer with two select lines S1 and S0, given below

The minimal sum-of-products form of the Boolean expression for the output F of the multiplexer is

= P’Q + PQ’R + PQR’

= Q(P’ + P R’) + PQ’R

= Q(P’ + R’) + PQ’R

= P’Q + QR’ + PQ’R

Question 40 |

The dual of a Boolean function F(x_{1}, x_{2}, ..., x_{n}, +, ⋅, '), written as F^{D}, is the same expression as that of F with + and ⋅ swapped. F is said to be self-dual if F = F^{D}. The number of self-dual functions with n Boolean variables is

2 ^{n} | |

2 ^{(n-1)} | |

2 ^{(2n )} | |

2 ^{(2(n-1) )} |

^{n}.

Number of mutually exclusive pairs of minterms = 2

^{n-1}.

There are 2 choices for each pair i.e., we can choose one of the two minterms from each pair of minterms for the function.

Therefore number of functions = 2 x 2 x …. 2

^{n-1}times.

= 2

^{(2(n-1)) }

Question 41 |

Let k = 2^{n}. A circuit is built by giving the output of an n-bit binary counter as input to an n-to-2^{n} bit decoder. This circuit is equivalent to a

k-bit binary up counter. | |

k-bit binary down counter. | |

k-bit ring counter. | |

k-bit Johnson counter. |

A n x 2

^{n}decoder is a combinational circuit with only one output line has one and all others (2

^{n}-1) have zeros.

A n-bit binary Counter produces outputs from 0 to 2

^{n}i.e 000...00 to 111...11 and repeats.

The n x 2

^{n}Decoder gets the input (000..00 to 111...11 ) from the binary counter and only one output line has one and rest have zeros.

This circuit is equivalent to a 2

^{n}- bit ring counter.

Question 42 |

Consider the equation (123)_{5} = (x8)_{y} with x and y as unknown. The number of possible solutions is __________.

3 | |

5 | |

6 | |

7 |

(123)

_{5}= (x8)

_{y}

In R.H.S. since y is base so y should be greater than x and 8, i.e.,

y > x

y > 8

Now, to solve let's change all the above bases number into base 10 number,

5

^{2}× 1 +2 × 5 + 3 = y × x + 8

38 = xy + 8

xy = 30

⇒ yx = 30

So the possible combinations are

(1,30), (2,15), (3,10), (5,6)

But we will reject (5,6) because it violates the condition (y > 8).

So, total solutions possible is 3.

Question 43 |

Consider the following minterm expression for F:

F(P,Q,R,S) = Σ0,2,5,7,8,10,13,15

The minterms 2, 7, 8 and 13 are 'do not care' terms. The minimal sum-of-products form for F is:

Question 44 |

Consider the following combinational function block involving four Boolean variables x, y, a, b where x, a, b are inputs and y is the output.

f (x, y, a, b) { if (x is 1) y = a; else y = b; }

Which one of the following digital logic blocks is the most suitable for implementing this function?

Full adder | |

Priority encoder | |

Multiplexor | |

Flip-flop |

x is the select line, I

_{0}is 'b' and I

_{1}is a.

The output line, y = xa + x’b

Question 45 |

The above synchronous sequential circuit built using JK flip-flops is initialized with Q2Q1Q0 = 000. The state sequence for this circuit for the next 3 clock cycles is

001, 010, 011 | |

111, 110, 101 | |

100, 110, 111 | |

100, 011, 001 |

Question 46 |

Let ⊕ denote the Exclusive OR (XOR) operation. Let ‘1’ and ‘0’ denote the binary constants. Consider the following Boolean expression for F over two variables P and Q:

F(P,Q) = ((1⊕P)⊕(P⊕Q))⊕((P⊕Q)⊕(Q⊕0))

The equivalent expression for F is

P+Q | |

P⨁Q | |

⊕ is associative i.e P ⊕ (Q ⊕ R) = (P⊕Q) ⊕ R.

P ⊕ P = 0, 1 ⊕ P = P’ and 0 ⊕ Q = Q

(1 ⊕ P) ⊕ ((P ⊕ Q) ⊕ (P ⊕ Q)) ⊕ (Q ⊕ 0)

= P’⊕ (0) ⊕ Q

= P’ ⊕ Q

= (P ⊕ Q)’

Question 47 |

The smallest integer that can be represented by an 8-bit number in 2’s complement form is

-256 | |

-128 | |

-127 | |

0 |

^{n-1}to 2

^{n-1}-1.

The smallest 8-bit 2’s complement number is 1000 0000.

MSB is 1. So it is a negative number.

To know the magnitude again take 2’s complement of 1000 0000.

1000 0000

0111 1111 ← 1’s complement

1000 0000 ← 2’s complement (1’s complement +1)

= 128

-128 is 1000 0000 in 2’s complement representation.

Question 48 |

In the following truth table, V = 1 if and only if the input is valid.

What function does the truth table represent?

Priority encoder | |

Decoder | |

Multiplexer | |

Demultiplexer |

^{2}× 2 encoder. The inputs have priorities. So, it is a priority encoder.

Question 49 |

Which one of the following expressions does **NOT** represent exclusive NOR of x and y?

xy+x'y' | |

x⊕y' | |

x'⊕y | |

x'⊕y' |

x’ ⊕ y’ = xy’ + x’y = x⊕y. Hence option D is correct.

Question 50 |

The truth table

represents the Boolean function

X | |

X + Y | |

X ⊕ Y | |

Y |