Added Mathematics Practice Assignment Solutions

Author: aryan

content/exercises/practice-assignments/mathematics-1/w10pa1.md

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+---
+title: Week 10 Practice Assignment Solution
+weight: 10
+tags: 
+- Mathematics
+- Mathematics for Data Science - 1
+- Practice Assignment
+- Week-10
+categories:
+- Mathematics Practice Assignment
+series:
+- Mathematics Practice Assignment
+---
+
+Below are all the questions and their solutions extracted from the **Week-10-Practice-Assignment-Solution.pdf**[^1]. The PDF covers graph theory and discrete mathematics topics. The questions and solutions are organized by type and question number.
+
+---
+
+## **1. Multiple Choice Questions (MCQs)**
+
+**Question 1:**
+*Suppose we obtain the following DFS tree rooted at node 0 for an undirected graph with vertices $\{0,1,2,3,4,5,6,7,8,9,10\}$.*
+
+**Which of the following cannot be an edge in the original graph?**
+(a) $(1,4)$
+(b) $(0,4)$
+(c) $(7,10)$
+(d) $(2,9)$
+
+**Solution:**
+In a DFS tree for an undirected graph, edges between vertices in different branches cannot be present in the original graph if they would have already been visited.
+
+- **(a) $(1,4)$:** Both in the same branch—possible.
+- **(b) $(0,4)$:** Both in the same branch—possible.
+- **(c) $(7,10)$:** In different branches—**cannot be an edge**.
+- **(d) $(2,9)$:** Both in the same branch—possible.
+
+**Answer:** **(c) $(7,10)$**[^1][^2][^3]
+
+---
+
+**Questions 2–4** use the following context:
+Ten friends in a college decide to have a night party at one of their homes. On the day, the government closes many routes in the city. The graph $G = (V, E)$ represents their homes (nodes) and the open routes (edges).
+
+**Question 2:**
+**The possible place for the party is:**
+(a) Shiva's house
+(b) Abhi's house
+(c) Joe's house
+(d) Vishi's house
+
+**Solution:**
+Shiva's house is reachable by everyone, so it is the best possible place for the party.
+**Answer:** **(a) Shiva's house**[^1]
+
+---
+
+**Question 3:**
+Let $V_1 = \{\text{Kristi, Shiva, Nia}\}$ and $E_1 = E \cap (V_1 \times V_1)$, i.e., edges with both endpoints in $V_1$.
+**Choose the correct option:**
+(a) $G_1 = (V_1, E_1)$ is an undirected graph
+(b) $G_1 = (V_1, E_1)$ is a cyclic graph
+(c) $G_1 = (V_1, E_1)$ will not be a graph
+(d) $G_1 = (V_1, E_1)$ is a directed graph
+
+**Solution:**
+Given the edges in $E_1$ are directed, $G_1$ is a directed graph.
+**Answer:** **(d) $G_1 = (V_1, E_1)$ is a directed graph**[^1][^2][^3]
+
+---
+
+**Question 4:**
+**If Joe wants to have the party on his home, then at most how many members can join the party?**
+(a) 5
+(b) 6
+(c) 7
+(d) 8
+
+**Solution:**
+Joe's home is not reachable by Kristi and Shiva. With 10 members total, 8 can join.
+**Answer:** **(d) 8**[^1][^2][^3]
+
+---
+
+## **2. Multiple Select Questions (MSQs)**
+
+**Question 5:**
+*Suppose $G$ is a graph (shown in the figure). Let $V$ be the set of vertices of $G$, $V_i$ be the maximum independent set, and $V_c$ be the minimum vertex cover. Which of the following is/are true?*
+(a) Cardinality of $V_i$ is 4
+(b) Cardinality of $V_c$ is 3
+(c) Cardinality of $V_i$ is 5
+(d) Cardinality of $V_c$ is 4
+
+**Solution:**
+From the figure, the maximum independent set has 4 vertices and the minimum vertex cover has 4 vertices.
+**Answer:** **(a) and (d)**[^1]
+
+---
+
+**Question 6:**
+*Consider the graph below. Suppose we perform BFS/DFS so that when we visit a vertex, we explore its unvisited neighbors in a random order. Which of the following options are correct?*
+(a) If we perform Breadth First Search at node 0, then one of the possible orders in which the nodes will be visited is 01423567
+(b) If we perform Depth First Search at node 0, then one of the possible orders in which the nodes will be visited is 04123576
+(c) If we perform Breadth First Search at node 0, then one of the possible orders in which the nodes will be visited is 01423576
+(d) If we perform Depth First Search at node 0, then one of the possible orders in which the nodes will be visited is 04132567
+
+**Solution:**
+
+- **(a) and (c):** Possible BFS orders from node 0.
+- **(d):** Possible DFS order from node 0.
+- **(b):** Not a valid BFS or DFS order as per the graph structure.
+
+**Answer:** **(a), (c), and (d)**[^1][^2][^3]
+
+---
+
+**Question 7:**
+*Which of the following options are correct?*
+(a) In Depth First search of a directed graph, only back edges generate cycles.
+(b) If the maximum independent set of a graph $G$ contains only 1 element, then the graph must be connected.
+(c) If we add an edge to a tree $T$, then the resulting graph becomes cyclic.
+(d) In a connected graph $G$ having $n$ vertices, at least two vertices have the same degree.
+
+**Solution:**
+
+- **(a):** True for directed graphs—only back edges create cycles.
+- **(b):** True—if the maximum independent set is 1, the graph must be connected.
+- **(c):** True—adding any edge to a tree creates a cycle.
+- **(d):** True—in any connected graph with $n \geq 2$ vertices, at least two have the same degree.
+
+**Answer:** **(a), (b), (c), and (d)**[^1][^2][^3]
+
+---
+
+**Question 8:**
+*Consider the following directed graph. Suppose DFS is performed from node A, exploring unvisited neighbors in alphabetical order. Which options are correct?*
+(a) $AC$ is a forward edge
+(b) $CE$ is a backward edge
+(c) $BD$ is a forward edge
+(d) $FC$ is a backward edge
+
+**Solution:**
+
+- **(a):** $AC$ is a forward edge.
+- **(d):** $FC$ is a backward edge.
+
+**Answer:** **(a) and (d)**[^1][^2][^3]
+
+---
+
+## **3. Numerical Answer Type (NAT)**
+
+**Question 9:**
+*The cardinality of the maximum independent set of the graph given below is:*
+
+**Solution:**
+The maximum independent set has 4 vertices.
+**Answer:** **4**[^1][^2][^3]
+
+---
+
+**Question 10:**
+*The minimum coloring of the graph given below is:*
+
+**Solution:**
+The minimum number of colors required to color the graph such that no two adjacent vertices have the same color is 2.
+**Answer:** **2**[^1]
+
+---
+
+*All questions and solutions from the PDF have been extracted as above.*
+
+<div style="text-align: center">⁂</div>
+
+[^1]: Week-10-Practice-Assignment-Solution.pdf
+
+[^2]: https://www.studocu.com/in/document/indian-institute-of-technology-dharwad/mathematical-method/week-10-practice-assignment-solution/75879051
+
+[^3]: https://www.studocu.com/in/document/indian-institute-of-technology-madras/iitm-online-degree-data-science-and-programming/week-10-practice-assignment-solution/114209262
+
+[^4]: https://www.youtube.com/watch?v=WixAUZo9ycM
+
+[^5]: https://archive.nptel.ac.in/content/storage2/courses/downloads_new/106105171/noc20-cs06_Week_10_Assignment_03.pdf
+
+[^6]: https://www.youtube.com/watch?v=Win-qyjWuv8
+
+[^7]: https://www.scribd.com/document/759779857/Week-10-Practice-Assignment-Solution
+
+[^8]: https://archive.nptel.ac.in/content/storage2/courses/downloads_new/108106173/noc21_ee40_assignment_Week_10.pdf
+
+[^9]: https://www.youtube.com/watch?v=7tt_5fg9ACI
+
+[^10]: https://www.youtube.com/watch?v=4ylFoZ8Rp-8
+
+[^11]: https://gradedassignments.github.io/iit-madras-graded-assignments/
+