GATE CS 1998

Question 1

A die is rolled three times. The probability that exact one odd number turns up among the three outcomes is

• A. 1/6
• B. 3/8
• C. 1/8
• D. 1/2

Answer: (B)

GATE CS 1998 Probability
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The question is an example of Binomial Experiment, with two possibilities- Number is Even(E) or Number is odd(O).

Question 2

Consider the following set of equations

x+2y = 5
4x+8y = 12
3x+6y+3z = 15

This set-

• A. has a unique solution
• B. has no solutions
• C. has finite number of solutions
• D. has infinite number of solutions

Answer: (B)

GATE CS 1998 Linear Algebra
Discuss it

When the given set of equations are represented in matrix form, the coefficient matrix A is singular. Since the determinant value of A is 0, the system of equations is inconsistent. Therefore, option (B) is correct. This explanation is provided by Chirag Manwani.

Question 3

Which of the following statements applies to the bisection method used for finding roots of functions:

• A. converges within a few iteration
• B. guaranteed to work for all continuous functions
• C. is faster than the Newton-Raphson method
• D. requires that there be no error in determining the sign of the fuction

Answer: (B)

GATE CS 1998 Numerical Methods and Calculus
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This method is guaranteed to convert to the root off if f is a continuous function in space [a, b] and f (a) and f (b) have opposing symbols. The total error is limited to half of each step so that the path is changed sequentially, which is relatively slow. Ref: http://en.wikipedia.org/wiki/Bisection_method#Analysis

Question 4

Consider the function y = |x| in the interval [-1,1]. In this interval, the function is

• A. continuous and differentiable
• B. continuous but not differentiable
• C. differentiable but not continuous
• D. neither continuous nor differentiable

Answer: (B)

GATE CS 1998 Numerical Methods and Calculus
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The function y = |x| in the interval [-1,1] is |x| is continuous and differentiable everywhere except at x=0, where it is continuous but not differentiable. since [-1,1] contains 0 , in this interval it is continues but not differentiables. Hence, option (B) is correct.

Question 5

What is the converse of the following assertion?

I stay only if you go.

• A. I stay if you go
• B. If I stay then you go
• C. If you do not go then I do not stay
• D. If I do not stay then you go

Answer: (A)

GATE CS 1998 Propositional and First Order Logic.
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Question 6

Suppose A is a finite set with n elements. The number of elements in the largest equivalence relation of A is

• A. n
• B. n^2
• C. 1
• D. n+1

Answer: (B)

GATE CS 1998 Set Theory & Algebra
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Question 7

Let R₁ and R₂ be two equivalence relations on a set. Consider the following assertions: (i) R₁ ∪ R₂ is an equivalence relation (ii) R₁ ∩ R₂ is an equivalence relation Which of the following is correct?

• A. both assertions are true
• B. assertions (i) is true but assertions (ii) is not true
• C. assertions (ii) is true but assertions (i) is not true
• D. neither (i) nor (ii) is true

Answer: (C)

GATE CS 1998 Set Theory & Algebra
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Question 8

The number of functions from an m element set to an n element set is

• A. m+n
• B. m^n
• C. n^m
• D. m*n

Answer: (C)

GATE CS 1998 Set Theory & Algebra
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Question 9

If the regular set 'A' is represented by A= (01+1)* and the regular set 'B' is represented by B= ((01)* 1*)*, which of the following is true ?

• A. A ⊂ B
• B. B ⊂ A
• C. A and B are incomparable
• D. A = B

Answer: (D)

GATE CS 1998 Regular languages and finite automata
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Some of the regular expression always equivalent to (0+1)* such that

(0+1)* 
= (0*+1*)* 
= (01*)* 
= (0*+1)* 
= (0+1*)* 
= 0*(10*)* 
= 1*(01*)* 

Since,

(01+1)* = ((01)* 1* )*  

Therefore A = B.

Question 10

Which of the following sets can be recognized by a Deterministic Finite-state Automaton?

• A. The number 1, 2, 4, 8......,2^n,.......... written in binary.
• B. The number 1, 2, 4,....., 2^n,.......... written in unary.
• C. The set of binary strings in which the number of zeros is the same as the number of ones.
• D. The set {1, 101, 11011, 1110111,.......}

Answer: (A)

GATE CS 1998 Regular languages and finite automata
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If there is a infinite language and for that language if their is no any pattern exist then we can surely say that given language is not regular, but if pattern is exist for that language then it may or may not be regular language and for ensuring a given language is regular, if we are able to draw DFA for that language then surely it will be regular otherwise not regular, Therefore option (A) is regular language as it can written in binary i.e.,

L = {1, 10, 100, 1000, 10000, …} 

Regular expression is (10*), since, for this expression we can draw DFA. So, option (A) is correct.