Mathematics Week 2 Graded Assignment

1. Function Identification via Graph (Figure M1W8A-8.1) Question: A graph is shown in Figure M1W8A-8.1, ◦symbol signifies that the straight line does not touch the point and the - symbol signifies that the line touches the point. Choose the correct option. The graph cannot be a function, because it fails the vertical line test. The graph cannot be a function, because it passes the horizontal line test but fails the vertical line test. The graph can be a function, because it passes the vertical line test. The graph cannot be a function, because it passes the vertical line test but fails the horizontal line test. Solution: To check if the given graph represents a function, use the vertical line test. In Figure M1W8A-8.1, every vertical line crosses the graph only once (including both - and ◦ as per definition). Therefore, the graph represents a function.

Multiple Choice Questions (MCQ) 1. If $ 18^x - 12^x - (2 \times 8^x) = 0 $, then the value of $ x $ is: Options: ln 18 ln 2 ln 18 In 18 Answer: Option 1 (Note: The options are confusing and possibly mislabeled. The intended answer is likely a specific value or expression, but as shown, it is marked as Option 1, which is not a valid solution. However, the PDF marks Option 1 as correct. This may be an error in the PDF.)

1. Multiple Select Questions (MSQ) Question 1: Match the given functions in Column A with the equations of their tangents at the origin $(0,0)$ in column B and the plotted graphs and the tangents in Column C, given in Table M2W2G1. Function (Column A) Tangent at $(0,0)$ (Column B) Graph (Column C) i) $f(x)=x2^{x}$ a) $y=-4x$ 1) ii) $f(x)=x(x-2)(x+2)$ b) $y=x$ 2) iii) $f(x)=-x(x-2)(x+2)$ c) $y=4x$ 3) Options: Option 1: ii) $\rightarrow$ a) $\rightarrow 1$ Option 2: i) $\rightarrow$ b) $\rightarrow 3$ Option 3: iii) $\rightarrow$ b) $\rightarrow 1$ Option 4: iii) $\rightarrow$ c) $\rightarrow 2$ Option 5: i) $\rightarrow$ a) $\rightarrow 1$

1. Multiple Choice Questions (MCQ) Question 1: The maximum number of non-zero entries in an adjacency matrix of a simple graph having $ n $ vertices can be Options: (a) $ n^2 $ (b) $ n(n-1) $ (c) $ \frac{n(n-1)}{2} $ (d) $ n(n-1) $ Solution: The number of non-zero entries is equal to twice the number of edges for undirected graphs, but for directed simple graphs (no loops, no multiple edges), it is $ n(n-1) $. Since the question says “simple graph” (typically undirected, but the context here implies directed or maximum possible), and the solution says “sum of degrees” is $ n(n-1) $ if every vertex has degree $ n-1 $, which is only possible in a directed simple graph. Correct option: (d) $ n(n-1) $ Question 2: We have a graph $ G $ with 6 vertices. We write down the degrees of all vertices in $ G $ in descending order. Which of the following is a possible listing of the degrees? Options: (a) 6,5,4,3,2,1 (b) 5,5,2,2,1,1 (c) 5,3,3,2,2,1 (d) 2,1,1,1,1,1 Solution:

1. Multiple Choice Questions Question 1 An undirected graph $G$ has 20 vertices and the degree of each vertex is at least 3 and at most 5. Which of the following statements is true regarding the graph $G$? (a) The minimum number of edges that the graph $G$ can have is 60. (b) The maximum number of edges that the graph $G$ can have is 100. (c) The maximum number of edges that the graph $G$ can have is 60. (d) The minimum number of edges that the graph $G$ can have is 30.

1. Dijkstra’s Algorithm and Unique Shortest Path Question: An undirected weighted graph $ G $ is shown (not included here, but described in text). Find the set of all positive integer values of $ x $ such that if we use Dijkstra’s algorithm, there will be a unique shortest path from vertex $ a $ to vertex $ j $ that contains the edge $ (b, e) $.

1. Uniform Distribution Expectation and Variance Question: A random variable is uniformly distributed over $[a, b]$ with expectation $e$ and variance $v$. Find $ab$. Answer: $ab = e^2 - 3v$ Solution: