# GATE CS 2010

• Last Updated : 10 Nov, 2021

 Question 1
Let G = (V,E) be a graph. Define ξ(G) = Σd id x d, where id is the number of vertices of degree d in G. If S and T are two different trees with ξ(S) = ξ(T),then
 A |S| = 2|T| B |S| = |T|-1 C |S| = |T| D |S| = |T|+1
GATE CS 2010    Graph Theory
Discuss it

Question 1 Explanation:
The expression ξ(G) is basically sum of all degrees in a tree.   For example, in the following tree, the sum is 3 + 1 + 1 + 1.
    a
/ | \
b  c  d
Now the questions is, if sum of degrees in trees are same, then what is the relationship between number of vertices present in both trees? The answer is, ξ(G) and ξ(T) is same for two trees, then the trees have same number of vertices. It can be proved by induction. Let it be true for n vertices. If we add a vertex, then the new vertex (if it is not the first node) increases degree by 2, it doesn't matter where we add it. For example, try to add a new vertex say 'e' at different places in above example tee.
 Question 2

Newton-Raphson method is used to compute a root of the equation x2-13=0 with 3.5 as the initial value. The approximation after one iteration is

 A 3.607 B 3.667 C 3.676 D 3.575
GATE CS 2010    Numerical Methods and Calculus
Discuss it

Question 2 Explanation:

In Newton-Raphson's method, We use the following formula to get the next value of f(x). f'(x) is derivative of f(x). f(x)  =  x2-13
f'(x) =  2x

Applying the above formula, we get
Next x = 3.5 - (3.5*3.5 - 13)/2*3.5
Next x = 3.607

 Question 3

What is the possible number of reflexive relations on a set of 5 elements?

 A 225 B 220 C 215 D 210
GATE CS 2010    Combinatorics
Discuss it

Question 3 Explanation:

Number of reflexive relations is 2n2-n which is 220 for n = 5

 Question 4
Consider the set S = {1, ω, ω2}, where ω and w2 are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms
 A A group B A ring C An integral domain D A field
GATE CS 2010    Set Theory & Algebra
Discuss it

Question 4 Explanation:
A group is a set of elements together with an operation that combines any two of its elements to form a third element also in the set while satisfying four conditions called the group axioms, namely closureassociativityidentity and invertibility. (Source: http://en.wikipedia.org/wiki/Group_(mathematics) (S, *)  is a group with identity as 1
 Question 5

What is the value of Limn->∞(1-1/n)2n ?

 A e-1/2 B 1 C e-2 D 0
GATE CS 2010    Numerical Methods and Calculus
Discuss it

 Question 6
The minterm expansion of f(P, Q, R) = PQ + QR' + PR' is
 A m2 + m4 + m6 + m7 B m0 + m1 + m3 + m5 C m0 + m1 + m6 + m7 D m2 + m3 + m4 + m5
GATE CS 2010    Digital Logic & Number representation    Logic functions and Minimization
Discuss it

Question 6 Explanation:
K-map, = PQ + QR’ + PR’
= PQ(R+R’) + (P+P’)QR’ + P(Q+Q’)R’
= PQR + PQR’ +PQR’ +P’QR’ + PQR’ + PQ’R’
= PQR(m7) + PQR'(m6)+P’QR'(m2) +PQ’R'(m4)
= m2 + m4 + m6 + m7 
Option (A) is correct.
 Question 7
A main memory unit with a capacity of 4 megabytes is built using 1M × 1-bit DRAM chips. Each DRAM chip has 1K rows of cells with 1K cells in each row. The time taken for a single refresh operation is 100 nanoseconds. The time required to perform one refresh operation on all the cells in the memory unit is:-
A.100 nanoseconds
B.100×210 nanoseconds
C.100×220 nanoseconds
D.3200×220 nanoseconds
 A A B B C C D D
GATE CS 2010    Computer Organization and Architecture    Cache and main memory
Discuss it

Question 7 Explanation:
Number of chips required for 4MB MM = (4 * 2^20 * 8) / (1 * 2^20) = 32 chips In a refresh cycle, a whole row of a memory chip is refreshed at once. This implies the given time of 100 ns for one refresh operation refreshes one row of memory chip. Since there are 1K=2^10 such rows, time for refreshing a whole chip would be: 2^10 * 100 ns. Second question arises, how to arrange these chips as there can be many possible arrangements. There is a logical arrangement provided in the problem statement itself as "1M x 1 bit chip". This indicates that to make a "1M x 32 bits" MM, we need to arrange all 32 chips in a line. It is to be noted that a row in all chips in series can be refreshed in one refresh cycle. This makes the total time to refresh the 4MBytes of memory as same as that of one chip. Hence, time required to refresh MM = 100 * 2^10 ns. So, option (B) is correct.
 Question 8

P is a 16-bit signed integer. The 2's complement representation of P is (F87B)16.The 2's complement representation of 8*P

 A (C3D8)16 B (187B)16 C (F878)16 D (987B)16
GATE CS 2010    Number Representation
Discuss it

Question 8 Explanation:

P = (F87B)16 is -1111 1000 0111 1011 in binary

Note that most significant bit in the binary representation is 1, which implies that the number is negative. To get the value of the number performs the 2's complement of the number. We get P as -1925 and 8P as -15400

Since 8P is also negative, we need to find 2's complement of it (-15400)
Binary of 15400 = 0011 1100 0010 1000
2's Complement = 1100 0011 1101 1000 = (C3D8)16

 Question 9
 A (P(XOR)Q(XOR)R)' B P(XOR)Q(XOR)R C (P+Q+R)' D P+Q+R
GATE CS 2010    Digital Logic & Number representation
Discuss it

Question 9 Explanation:
For 4 to 1 mux truth table SEL                  INPUT                 O/P
 Q P R R’ R’ R F 0 0 X X X 1 1 0 1 X X 1 X 1 1 0 X 1 X X 1 1 1 1 X X X 1
p’q’r+p’qr’+pq’r’+pqr pXORqXORr
 Question 10

What does the following program print?

## C

#include
void f(int *p, int *q)
{
p = q;
*p = 2;
}
int i = 0, j = 1;
int main()
{
f(&i, &j);
printf("%d %d n", i, j);
getchar();
return 0;
}

 A 2 2 B 2 1 C 0 1 D 0 2
GATE CS 2010    C Pointer Basics    50 C Language MCQs with Answers
Discuss it

Question 10 Explanation:

/* p points to i and q points to j */
void f(int *p, int *q)
{
p = q;    /* p also points to j now */
*p = 2;   /* Value of j is changed to 2 now */
}

There are 64 questions to complete.
My Personal Notes arrow_drop_up