GATE CS 1996

Question 1

Which of the following macros can put a micro assembler into an infinite loop? (i)

.MACRO M1 X
.IF EQ, X      ;if X=0 then
M1 X + 1
.ENDC
.IF NE X       ;IF X≠0 then
.WORD X        ;address (X) is stored here
.ENDC
.ENDM

(ii)

.MACRO M2 X
.IF EQ X
M2 X
.ENDC
.IF NE, X
.WORD X+1
.ENDC
.ENDM

• (ii) only
• (i) only
• Both (i) and (ii)
• None of the above

Question 2

Let F be the collection of all functions f: {1, 2,3} → {1, 2, 3}. If f and g ∈ F, define an equivalence relation ~  by f ~ g if and only if f(3) = g(3). a) Find the number of equivalence classes defined by ~. b) Find the number of elements in each equivalence class.

Question 3

Let A and B be sets and let A^c and B^c denote the complements of the sets A and B. The set (A−B) ∪ (B−A) ∪ (A∩B) is equal to a). A ∪ B b). A^c ∪ B^c c). A ∩ B d). A^c ∩ B^c

• a
• b
• c
• d

Question 4

Let X= {2, 3, 6, 12, 24}, Let ≤ be the partial order defined by X ≤ Y if x divides y. Number of edges in the Hasse diagram of (X,≤) is

• 3
• 4
• 9
• None of the above

Question 5

Suppose X and Y are sets and |X| and |Y| are their respective cardinalities. It is given that there are exactly 97 functions from X to Y. From this one can conclude that

• |X|=1,|Y|=97
• |X|=97,|Y|=1
• |X|=97,|Y|=97
• None of the above

Question 6

Which of the following statement is false?

• The set of rational numbers is an abelian group under addition
• The set of integers in an abelian group under addition
• The set of rational numbers form an abelian group under multiplication
• The set of real numbers excluding zero is an abelian group under multiplication

Question 7

Two dice are thrown simultaneously. The probability  that at least one of them will have 6 facing up is

• 1/36
• 1/3
• 25/36
• 11/36

Question 8

The formula used to compute an approximation for the second derivative of a function f at a point X₀ is   

[caption width="800"] [/caption]

• A

• B

• C

• D

Question 9

Let Ax=b be a system of linear equations where A is an m×n matrix and b is a m×1 column vector and X is an n×1 column vector of unknowns. Which of the following is false?

• The system has a solution if and only if, both A and the augmented matrix [Ab] have the same rank
• If m
• If m=n and b is a non-zero vector, then the system has a unique solution
• The system will have only a trivial solution when m=n, b is the zero vector and rank(A) = n

Question 10

Which two of the following four regular expressions are equivalent? (ε is the empty string). (i). (00)*(ε+0) (ii). (00)* (iii). 0* (iv). 0(00)*

• (i) and (ii)
• (ii) and (iii)
• (i) and (iii)
• (iii) and (iv)