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

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 ______.

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:

Which one of the following is NOT a valid identity?

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.

Consider three 4-variable functions f₁, f₂ and f₃, which are expressed in sum-of-minterms as

f1 = Σ(0, 2, 5, 8, 14),  f2 = Σ(2, 3, 6, 8, 14, 15),  f3 = 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:

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

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 ______

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 _______.

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

When two 8-bit numbers A₇…A₀ and B₇…B₀ in 2’s complement representation (with A₀ and B₀ as the least significant bits) are added using a ripple-carry adder, the sum bits obtained are S₇…S₀ and the carry bits are C₇…C₀. An overflow is said to have occurred if

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 _________.

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 Q₀ and Q₁ are set to 1 (before the 1^st clock cycle). The outputs

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

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

The decimal value closest to this floating-point number is

Consider a quadratic equation x² – 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=________.

If w, x, y, z are Boolean variables, then which one of the following is INCORRECT?

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)?

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

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 T₁ and T₀ are

Consider the Boolean operator with the following properties:

Then x#y is equivalent to

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

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 __________.

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

The minimal sum of products form of the output X is

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 __________.

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 _________.

Let, x₁⊕x₂⊕x₃⊕x₄ = 0 where x₁, x₂, x₃, x₄ are Boolean variables, and ⊕ is the XOR operator. Which one of the following must always be TRUE?

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 _________.

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

The binary operator ≠ is defined by the following truth table

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

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.

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 ___________.

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

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

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 ____________.

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

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

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

        312/20 = 13.1     

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

The dual of a Boolean function F(x₁, x₂, …, 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

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

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

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:

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?

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

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

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

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

What function does the truth table represent?

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

The truth table

represents the Boolean function