Digital Logic — GATE Previous Year Questions

Let \(A = 1111\,1010\) and \(B = 0000\,1010\) be two 8-bit 2's complement numbers. Their product in 2's complement representation is:

The minterm expansion of \(f(P,Q,R) = PQ + Q\overline{R} + P\overline{R}\) is:

The simplified sum-of-products (SOP) form of the Boolean expression

\((P + \overline{Q} + \overline{R}) \cdot (P + \overline{Q} + R) \cdot (P + Q + \overline{R})\)

is:

Consider the equation \((123)_5 = (x8)_y\) with \(x\) and \(y\) as unknown. The number of possible solutions is _______ .

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

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

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 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 _______ .

Consider the Boolean operator \(\#\) with the following properties:

\(x \# 0 = x\),   \(x \# 1 = \overline{x}\),   \(x \# x = 0\)   and   \(x \# \overline{x} = 1\).

Then \(x \# y\) is equivalent to:

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

\(F(P,Q,R,S) = \sum m(0,\,2,\,5,\,7,\,9,\,11) + d(3,\,8,\,10,\,12,\,14)\)

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 \(Z = X - Y\), 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:

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 minimum number of select lines needed for the multiplexer is _______ .