ExercisesPYQPYQSum of Squared Errors (SSE) for Best-Fit LineSum of Squared Errors (SSE) for Best-Fit LineIITM Quiz 11️⃣ Sum of Squared Errors (SSE) for Best-Fit Line (From Image 1)Question: Given the table of amount paid and distance (in km), and the best-fit line $ y = 4x + 1 $, what is the value of SSE?Step-by-Step Explanation: Calculating SSE for the Best-Fit Line 📊 Let’s break down how to find the Sum of Squared Errors (SSE) for the given data and the best-fit line $ y = 4x + 1 $, step by step!1️⃣ List the Data Points Amount Paid ($ y $)Distance ($ x $)8020601560161002558142️⃣ Calculate Predicted Values ($ y_{pred} $) Use the best-fit line equation $ y = 4x + 1 $:For $ x = 20 $: $ y_{pred} = 4 \times 20 + 1 = 81 $For $ x = 15 $: $ y_{pred} = 4 \times 15 + 1 = 61 $For $ x = 16 $: $ y_{pred} = 4 \times 16 + 1 = 65 $For $ x = 25 $: $ y_{pred} = 4 \times 25 + 1 = 101 $For $ x = 14 $: $ y_{pred} = 4 \times 14 + 1 = 57 $3️⃣ Find the Error for Each Point Error is the difference between the actual and predicted value:$$ \text{Error} = y_{\text{actual}} - y_{\text{pred}} $$$ y_{actual} $$ y_{pred} $Error8081$ -1 $6061$ -1 $6065$ -5 $100101$ -1 $5857$ +1 $4️⃣ Square Each Error $$ \text{Squared Error} = (\text{Error})^2 $$ErrorSquared Error-11-11-525-11+115️⃣ Sum All Squared Errors (SSE) $$ \text{SSE} = 1 + 1 + 25 + 1 + 1 = \boxed{29} $$🏁 Final Answer The Sum of Squared Errors (SSE) for the given data and the best-fit line $ y = 4x + 1 $ is:$$ \boxed{29} $$What Does SSE Mean? SSE measures how well the line fits the data.A smaller SSE means the line fits the data points more closely.Here, each squared error shows how far the actual value is from the predicted value, and their sum gives the total error for all points.That’s how you find SSE, step by step, with formulas and calculations! 🚀Roots of a Quadratic EquationMore QuestionsIIT M FOUNDATION ET1 EXAM QPF1 S2 30 Previous Year QuestionsMathematics 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.IIT M FOUNDATION ET1 EXAM QPF2 S2 30 Previous Year QuestionsMathematics 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].IIT M QUALIFIER AN EXAM QDQ1 27 Oct 27 Oct 2024Mathematics 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) $?IIT M QUALIFIER AN EXAM QDQ2 27 Oct 27 Oct 2024Mathematics 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:Polynomial Graph Matching IITM Quiz 1Given $ p(x) = 0.3x^3(x^2 - 1)(x - 2)^2(x - 3) $, which figure represents the polynomial $ p(x) $?QUIZ 2 FOUNDATIONAL 01 DEC 2024Question 1: Choose the correct option from the following: Options: $ iv) - c) - 3) $ $ iv) - a) - 1) $ $ iv) - c) - 2) $ $ iv) - b) - 2) $ Solution 📝:QUIZ 2 Foundational 4 Aug 2024Question 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 $.Relations and Sets IITM Quiz 1Given sets $ A $ (odd positive integers ≤ 20) and $ B $ (positive integers ≤ 30 divisible by 5), and relations?
