are linked by a network, and each link has a maintenance cost. The graph represents how the computers are connected, where each node denotes a computer and each edge denotes a link between a pair of computers. The weights on the edges represent the maintenance cost (in hundreds of rupees). The objective is to select a subset of links such that the total maintenance cost is minimum while ensuring that all computers remain connected through the selected links.

Question 1

Suppose we obtain a Breadth-First Search (BFS) tree rooted at node A for an undirected graph with vertex set

V = \{A, B, C, D, E, F, G\}

Which of the following cannot be an edge in the original graph?

(A, D)
(C, F)
(D, G)
(B, F)

Question 2

Which of the following statements is/are true about the function

f(x) = x^2 + 2x - 8

f is one-one on its domain.
f has an inverse on its domain.
The vertex of this parabola is at $(-1, -9)$
The y-intercept of the given parabola is $-1$

Question 3

Consider the functions

f(x) = \sqrt{x + 4}

and

g(x) = \log(1 + x^2)

Which of the following options is/are true?

$(g \circ f)(x) = \log(2x + 5)$ on its domain of definition.
The domain of the function $(g \circ f)(x)$ is $[-4, \infty)$
The domain of the function $(g \circ f)(x)$ is $[-6, -1]$
$(g \circ f)(x) = \log(x + 5)$ on its domain of definition.

Question 4

Consider the following relations defined on the set of integers

\mathbb{Z}

R_1 = \{(x, y) \mid x, y \in \mathbb{Z} \text{ and } 7 \mid (x - y)\}

R_2 = \{(x, y) \mid x, y \in \mathbb{Z} \text{ and } x + y = 2\}

$R_1 \text{ is not transitive}$
$R_2 \text{ is symmetric}$
$R_1 \text{ is symmetric}$
$R_2 \text{ is transitive}$

Question 5

Consider the adjacency matrix of a graph G:

\begin{bmatrix} 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 \end{bmatrix}

Which of the following option(s) is/are true?

$\text{The graph has 5 nodes}$
$\text{There are 10 edges in the graph}$
$\text{There are 5 edges in the graph}$
$\text{The sum of the degrees of all the nodes is 20}$

Question 6

Consider the following directed graph:

Which of the following orderings can be a valid topological sort of the graph?

ABCDEFGH
ABCDEFHG
BACDEFGH
BACEDFGH

Question 7

Consider a quadratic function

q(x) = ax^2 + bx + c

where

a, b, c \in \mathbb{R},\ a \ne 0

with the following information: • The minimum value attained by

q(x)

is at

x = -1

•Discriminant value of

q(x)

8

•Slope of the function at

x = 1

8

Find:

q(1)

Your Answer: (Not answered)

Question 8

f(x) = \log_3\!\left(\log_{1/2}\!\left(\sqrt{5 - x - 1}\right) - 1\right)

has domain

(a, b) \subset \mathbb{R}

find the value of

a + b

Your Answer: (Not answered)

Question 9

Find

\lim_{n \to \infty} a_n

for the sequence {

a_n

}such that

a_n = \frac{n^5 - 3n^3 + \sin(n)}{2n^5 + \ln(n) + n^2}

Your Answer: (Not answered)

Question 10

If the function

f(x) = \begin{cases} Ax - B, & if x \le -1 \\ 2x^2 + 3Ax + B, & if -1 < x \le 1 \\ 4, & if x > 1 \end{cases}

is continuous for all

x \in \mathbb{R}

, find the value of

A + B

Your Answer: (Not answered)

Question 11

Consider the function

f(x) = x - \frac{4}{x}

on the interval

[2, 8]

Approximate the value of

\int_{2}^{8} 3f(x)\,dx

using the right-hand Riemann sum by taking 3 sub-intervals of equal length.

Your Answer: (Not answered)

Question 12

(Use the following information for Question 12): Two families have decided to enter into an alliance by marriage. The first family has

{S_1, S_2, S_3, S_4}

sons, and the second family has

{D_1, D_2, D_3, D_4}

daughters. To avoid impropriety, the families insist that each individual must marry someone who is either of the same age, one position younger, or one position older. The graph representing all such agreeable marriages is given (based on these conditions).

How many different acceptable marriage arrangements are possible?

Your Answer: (Not answered)

Question 13

What is the total maintenance cost (in hundreds of rupees) of the optimum subset of links?

Your Answer: (Not answered)

Question 14

Find the number of different ways of choosing an optimum subset of links for the given graph.

Your Answer: (Not answered)

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