# 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?
 S1 and S2 are both true S1 is true, S2 is false S1 is false, S2 is true 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?

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

 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?

 F1 unsatisfiable, F2 is satisfiable F1 and F2 are both satisfiable F1 is unsatisfiable, F2 is valid 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 (a) is correct.

 Question 4
Consider the following two statements:
 Only S1 is correct Only S2 is correct Both S1 and S2 are correct 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?
 If a language is context free it can always be accepted by a deterministic push-down automaton The union of two context free languages is context free The intersection of two context free languages is context free The complement of a context free language is context free

Question 5-Explanation:
 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
 N2 2N 2N N!

Question 6-Explanation:
 Question 7
More than one word are put in one cache block to
 exploit the temporal locality of reference in a program exploit the spatial locality of reference in a program reduce the miss penalty none of the above

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