Indian Institute of Technology, Madras - BS in Data Science and Applications

Mathematics for Data Science 1. How many edges are there in a graph with 10 vertices each of degree 6? Solution: In any graph, the sum of degrees of all vertices is twice the number of edges. Sum of degrees = 10 × 6 = 60 So, number of edges = 60 / 2 = 301. 2. What is the minimum number of colours required to colour the given graph? Solution: The answer given is 3. This is a typical graph colouring problem, where the chromatic number is 31.

Mathematics for Data Science 1. How many edges are there in a graph with 10 vertices each of degree 6? Solution: Sum of degrees = 10 × 6 = 60. Each edge is counted twice, so number of edges = 60 ÷ 2 = 30[^1]. 2. What is the minimum number of colours required to colour the graph given below? Solution: The answer is 3. This is the chromatic number, meaning three colours are needed so that no two adjacent vertices share the same colour[^1].

Mathematics for Data Science 1️⃣ Polynomial Limits Question Question: Consider the polynomials $ p(x) = x^3 - 3x^2 + 100x - 1 $ and $ q(x) = x^3 + x + 5 $. Which of the following statements are correct? Correct Statement: $ p(x) - q(x) \to -\infty $ as $ x \to \infty $ ✅ Solution with Steps: Find the leading terms: Both $ p(x) $ and $ q(x) $ have $ x^3 $ as the highest degree term. Subtract the polynomials: $ p(x) - q(x) = (x^3 - x^3) + (-3x^2 - 0) + (100x - x) + (-1 - 5) = -3x^2 + 99x - 6 $ Analyze as $ x \to \infty $: The highest degree term is $ -3x^2 $. As $ x \to \infty $, $ -3x^2 \to -\infty $. Conclusion: So, $ p(x) - q(x) \to -\infty $ as $ x \to \infty $. 🎯 Final Answer: Option 2 is correct![^1] 2️⃣ Polynomial Graph Matching Question: Given $ p(x) = 0.3x^3(x^2 - 1)(x - 2)^2(x - 3) $, which figure represents the polynomial $ p(x) $?

Mathematics for Data Science Question 1 Question: Consider the polynomials $ p(x) = x^5 - 3x^4 + 100x - 1 $ and $ q(x) = x^5 + x + 5 $. Which of the following statements are correct? $ p(x) + q(x) \to \infty $ as $ x \to -\infty $ $ p(x) - q(x) \to -\infty $ as $ x \to \infty $ $ -5p(x) \to -\infty $ as $ x \to -\infty $ $ \frac{1}{2}q(x) \to \infty $ as $ x \to -\infty $ Step-by-Step Solution:

Given $ p(x) = 0.3x^3(x^2 - 1)(x - 2)^2(x - 3) $, which figure represents the polynomial $ p(x) $?

Question 1: Which of the following statements is/are true about the function $ f(x) = |x^2 - 4x + 3| + 17 $? Options: A. $ f $ is defined only for all $ x \in \mathbb{N} $. B. $ f $ is a bijective function. C. The range of $ f $ is $[0, \infty)$. D. The minimum value of $ f $ is 17. Solution 📝: Let’s analyze $ f(x) = |x^2 - 4x + 3| + 17 $.

Question 1: Choose the correct option from the following: Options: $ iv) - c) - 3) $ $ iv) - a) - 1) $ $ iv) - c) - 2) $ $ iv) - b) - 2) $ Solution 📝:

Given sets $ A $ (odd positive integers ≤ 20) and $ B $ (positive integers ≤ 30 divisible by 5), and relations?

If $ \alpha $ and $ \beta $ are the roots of $ x^2 + 4x + 1 = 0 $, then the equation whose roots are $ \alpha^2 $ and $ \beta^2 $ is?

Given the table of amount paid and distance (in km), and the best-fit line $ y = 4x + 1 $, what is the value of SSE?