GATE-CS-2001

Last Updated : 22 Nov, 2021

Question 1
Consider the following statements:
S1: The sum of two singular n × n matrices may be non-singular
S2: The sum of two n × n non-singular matrices may be singular. 
Which of the following statements is correct?
Tick
S1 and S2 are both true
Cross
S1 is true, S2 is false
Cross
S1 is false, S2 is true
Cross
S1 and S2 are both false


Question 1-Explanation: 
Singular Matrix: A square matrix is singular if and only if its determinant value is 0. S1 is True: The sum of two singular n × n matrices may be non-singular It can be seen be taking following example. The following two matrices are singular, but their sum is non-singular.
M1 and M2 are singular
M1 =  1  1
      1  1
       
M2 =   1  -1
      -1   1

But M1+M2 is non-singular  
M1+M2 =  2  0
         0  2
S2 is True: The sum of two n × n non-singular matrices may be singular
M1 and M2 are non-singular
M1 =  1  0
      0  1
       
M2 =   -1  0
        0  -1

But M1+M2 is singular  
M1+M2 =  0  0
         0  0
Question 2

Consider the following relations:

R1(a,b) if (a+b) is even over the set of integers
R2(a,b) if (a+b) is odd over the set of integers
R3(a,b) if a.b > 0 over the set of non-zero rational numbers
R4(a,b) if |a - b| <= 2 over the set of natural numbers

Which of the following statements is correct?

Cross

R1 and R2 are equivalence relations, R3 and R4 are not

Tick

R1 and R3 are equivalence relations, R2 and R4 are not

Cross

R1 and R4 are equivalence relations, R2 and R3 are not

Cross

R1, R2, R3 and R4 are all equivalence relations



Question 2-Explanation: 

So basically, we have to tell whether these relations are equivalence or not.

  1. R1(a,b)
    • Reflexive : Yes, because (a+a) is even.
    • Symmetric : Yes, (a+b) is even ⟹ (b+a) is even.
    • Transitive : Yes, because (a+b) is even and (b+c) is even ⟹ (a+c) is even.
      So R1 is equivalence relation.
       
  2. R2(a,b)
    • Reflexive : No, because (a+a) is even.
      So R2 is not equivalence relation.
       
  3. R3(a,b)
    • Reflexive : Yes, because a.a > 0.
    • Symmetric : Yes, a.b > 0 ⟹ b.a > 0.
    • Transitive : Yes, because a.b > 0 and b.c > 0 ⟹ a.c > 0.
      So R3 is equivalence relation.
       
  4. R4(a,b)
    • Reflexive : Yes, because |a-a| ≤ 2.
    • Symmetric : Yes, |a-b| ≤ 2 ⟹ |b-a| ≤ 2.
    • Transitive : No, because |a-b| ≤ 2 and |b-c| ≤ 2 ⇏ (a-c) is even.
      So R4 is not equivalence relation.

So option (b) is correct.. 

Source: http://www.cse.iitd.ac.in/~mittal/gate/gate_math_2001.html

Question 3

Consider two well-formed formulas in prepositional logic. 

F1 : P ⇒ ¬P
F2 : ( P⇒¬P)∨(¬P⇒P)

Which of the following statements is correct? 
 

Cross

F1 unsatisfiable, F2 is satisfiable

Cross

F1 and F2 are both satisfiable

Cross

F1 is unsatisfiable, F2 is valid 

Tick

F1 is satisfiable, F2 is valid



Question 3-Explanation: 

The concept behind this solution is: 
a) Satisfiable 
If there is an assignment of truth values which makes that expression true. 
b) UnSatisfiable 
If there is no such assignment which makes the expression true 
c) Valid 
If the expression is Tautology 
Here, P => Q is nothing but –P v Q 
F1: P => -P = -P v –P = -P 
F1 will be true if P is false and F1 will be false when P is true so F1 is Satisfiable 
F2: (P => -P) v (-P => P) which is equals to (-P v-P) v (-(-P) v P) = (-P) v (P) = 
Tautology 
So, F1 is Satisfiable and F2 is valid 
Option (d) is correct. 
 

Question 4
Consider the following two statements: GATECS2000Q4
Tick
Only S1 is correct
Cross
Only S2 is correct
Cross
Both S1 and S2 are correct
Cross
None of S1 and S2 is correct


Question 4-Explanation: 
We can easily build a DFA for S1. All we need to check is whether input string has even number of 0\'s. Therefore S1 is regular. We can\'t make a DFA for S2. For S2, we need a stack. Therefore S2 is not regular.
Question 5
Which of the following statements is true?
Cross
If a language is context free it can always be accepted by a deterministic push-down automaton
Tick
The union of two context free languages is context free
Cross
The intersection of two context free languages is context free
Cross
The complement of a context free language is context free


Question 6
Given an arbitary non-deterministic finite automaton (NFA) with N states, the maximum number of states in an equivalent minimized DFA is at least
Cross
N2
Tick
2N
Cross
2N
Cross
N!


Question 7
More than one word are put in one cache block to
Cross
exploit the temporal locality of reference in a program
Tick
exploit the spatial locality of reference in a program
Cross
reduce the miss penalty
Cross
none of the above


Question 7-Explanation: 
Question 8
Which of the following statements is false?
Cross
Virtual memory implements the translation of a program‘s address space into physical memory address space
Cross
Virtual memory allows each program to exceed the size of the primary memory
Cross
Virtual memory increases the degree of multiprogramming
Tick
Virtual memory reduces the context switching overhead


Question 8-Explanation: 
See question 4 of http://www.geeksforgeeks.org/operating-systems-set-2/
Question 9
A low memory can be connected to 8085 by using
Cross
INTER
Cross
RESET IN
Cross
HOLD
Tick
READY


Question 9-Explanation: 
A low memory can be connected to 8085 by using READY signal, Communication is only possible when READY signal is set .So (D) is correct option
Question 10
Suppose a processor does not have any stack pointer register. Which of the following statements is true?
Tick
It cannot have subroutine call instruction
Cross
It can have subroutine call instruction, but no nested subroutine calls
Cross
Nested subroutine calls are possible, but interrupts are not
Cross
All sequences of subroutine calls and also interrupts are possible


Question 10-Explanation: 
Stack pointer register hold the address of top of stack, which is the location of memory at which CPU should resume its execution after servicing some interrupt or subroutine call. So if SP register is not available then no subroutine call instructions are possible. So (A) is correct option.
There are 50 questions to complete.

Share your thoughts in the comments

Similar Reads