GATE-CS-2003

Question 41

A piecewise linear function f(x) is plotted using thick solid lines in the figure below (the plot is drawn to scale). If we use the Newton-Raphson method to find the roots of f(x) = 0 using x0, x1 and x2 respectively as initial guesses, the roots obtained would be

• 1.3, 0.6, and 0.6 respectively
• 0.6, 0.6, and 1.3 respectively
• 1.3, 1.3, and 0.6 respectively
• 1.3, 0.6, and 1.3 respectively

Question 42

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

• 2-⁴⁰
• 2^-9
• 2²²
• 2³¹

Question 43

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?

• 5

• 6

• 7

• 8

Question 44

The literal count of a boolean expression is the 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"

• (11, 9)
• (9, 13)
• (9, 10)
• (11, 11)

Question 45

Consider the ALU shown below. If the operands are in 2\'s complement representation, which of the following operations can be performed by suitably setting the control lines K and C0 only (+ and - denote addition and subtraction respectively) ?

• A + B, and A - B, but not A + 1
• A + B, and A + 1, but not A - B
• A + B, but not A - B, or A + 1
• A + B, and A - B, and A + 1

Question 46

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

• 1
• 2
• 3
• 4

Question 47

Consider the following assembly language program for a hypothetical processor. A, B, and C are 8 bit registers. The meanings of various instructions are shown as comments.

	MOV B, # 0	;	B ← 0
	MOV C, # 8	;	C ← 8
Z :	CMP C, # 0	;	compare C with 0
	JZX	;	jump to X if zero flag is set
	SUB C, # 1	;	C ← C - 1
	RRC A, # 1	;	right rotate A through carry by one bit. Thus:
		;	if the initial values of A and the carry flag are a7...a0 and
		;	c0 respectively, their values after the execution of this
		;	instruction will be c0a7...a1 and a0 respectively.
	JC Y	;	jump to Y if carry flag is set
	JMP Z	;	jump to Z
Y :	ADD B, # 1	;	B ← B + 1
	JMP Z	;	jump to Z
X :

• the number of 0 bits in A0
• the number of 1 bits in A0
• A0
• 8

Question 48

Consider the following assembly language program for a hypothetical processor. A, B, and C are 8 bit registers. The meanings of various instructions are shown as comments.

	MOV B, # 0	;	B ← 0
	MOV C, # 8	;	C ← 8
Z :	CMP C, # 0	;	compare C with 0
	JZX	;	jump to X if zero flag is set
	SUB C, # 1	;	C ← C - 1
	RRC A, # 1	;	right rotate A through carry by one bit. Thus:
		;	if the initial values of A and the carry flag are a7...a0 and
		;	c0 respectively, their values after the execution of this
		;	instruction will be c0a7...a1 and a0 respectively.
	JC Y	;	jump to Y if carry flag is set
	JMP Z	;	jump to Z
Y :	ADD B, # 1	;	B ← B + 1
	JMP Z	;	jump to Z
X :

Which of the following instructions when inserted at location X will ensure that the value of register A after program execution is the same as its initial value ?

• RRC A, #
• NOP ; no operation
• LRC A, # 1 ; left rotate A through carry flag by one bit
• ADD A, # 1

Question 49

Consider the following deterministic finite state automaton M. Let S denote the set of seven bit binary strings in which the first, the fourth, and the last bits are 1. The number of strings in S that are accepted by M is

• 1
• 5
• 7
• 8

Question 50

Let G = ({S}, {a, b} R, S) be a context free grammar where the rule set R is S → a S b | SS | ε Which of the following statements is true?

• G is not ambiguous

• There exist x, y, ∈ L (G) such that xy ∉ L(G)

• There is a deterministic pushdown automaton that accepts L(G)

• We can find a deterministic finite state automaton that accepts L(G)