Mathematics for Data Science I End Term: 8 August 2021 (May 2021 term)

Polynomial fit for the data given in the table recorded by a student is

$y = f(x) = e^{x(x^2-1)(x^2-4)} + c - 1$

Choose the value of $c$, so that SSE (sum squared error) will be minimum?

Let $f(x)$ and $g(x)$ be two functions defined from $\mathbb{R}$ to $\mathbb{R}$ such that

$f(x) = \begin{cases} 3 & \text{if } x \text{ is rational} \\ 4 & \text{if } x \text{ is irrational} \end{cases}$

,

$g(x) = \begin{cases} 5 & \text{if } x \text{ is rational} \\ 6 & \text{if } x \text{ is irrational} \end{cases}$

The value of $(f \circ g)(e) + (g \circ f)(\pi)$ is

Employees of the IITM Online Degree Program are assigned to visit the exam centers in Chennai. Vidya has been asked to arrange cabs for these employees to drop them at the centers. She has created a graph (shown below) and asked Sanaya to arrange the minimum number of cabs, but has forgotten to tell Sanaya what the nodes and the edges represent. Sanaya has assumed the following cases and found the answers accordingly.

Cases:
Case 1: Nodes represent the locations of employees and edges represent the roads connecting these locations. Each cab can pick up employees from only one location. Sanaya needs to arrange the minimum number of cabs under the assumption that each employee is willing to walk at most one road length from their current location to a neighboring location to be picked up. Case 2: Nodes represent the employees and if two employees share an edge, then they are assigned different centers. Now, Sanaya needs to arrange the minimum number of cabs such that no two employees assigned for different centers should come in the same cab. (Assume that any number of employees can come in a cab.)
Choose the correct option(s).

NOTE: Enter your answer to the nearest integer.

A company has branches in each of six cities $C_1, C_2, \ldots.., C_6$. Let a relation $R$ is defined as

1. $R = \{(a, b, c) \mid$ there is a direct flight between $a$ and $b, a \neq b,$ with a fare of ₹ $c\}$
2. If $(a, b, c) \in R$, then $(b, a, c) \in R$

An employee of that company wanted to travel from the city $C_2$ to the city $C_4$. If he traveled by the cheapest route possible, then find the total fare, he should pay. Given that in relation $R$,

1. $(C_1, C_2, 4005), (C_1, C_5, 5005), (C_1, C_6, 6005) \in R$
2. $(C_2, C_3, 5005) \in R$
3. $(C_3, C_4, 7005) \in R$
4. $(C_4, C_5, 4005) \in R$
5. $(C_5, C_6, 6005) \in R$

NOTE: Enter your answer to the nearest integer.

Srikanth has recently got married and he wants to maintain his monthly expenses. He has recorded his monthly expenses $E(n)$ (in thousands of ₹) for last year as $E(n) = \left(\frac{n^2}{1000}\right)(n^3 - 16n^2 + 55n) + 20$, where $n$ is the number of month and $n \in \{1, 2, \ldots, 11, 12\}$. He came to know that he should keep his monthly expense ₹20,000 per month for fulfilling his other plans. let $N_1$ be the set of month numbers when he has spent more than ₹20,000, $N_{-1}$ be the set of month numbers when he has spent less than ₹20,000, and $N_0$ be the set of month numbers when he has spent ₹20,000. Assuming last year expenses as the benchmark for his expenses, choose the set of correct options.

What is the weight of a minimum cost spanning tree of the graph given below?

NOTE: Enter your answer to the nearest integer.