Solution for IIT Madras Course Mathematics I Garaded Assignments

Q: If the slope of parabola y = ax² + bx + c, where a, b, c ∈ ℝ{0} at points (3, 2) and (2, 3) are 31 and 14 respectively, then find the value of a. Solution Step by Step 📝 Step 1: Find the derivative 🔍 For parabola y = ax² + bx + c, the slope at any point is: $ \frac{dy}{dx} = 2ax + b $

Q1. Which of the following are irrational numbers❓ (a) $3^{1/3}$ (b) $(\sqrt{8}+\sqrt{2})(\sqrt{12}-\sqrt{3})$ (c) $\frac{\sqrt{18}-3}{\sqrt{2}-1}$ (d) $\frac{\sqrt{8}+\sqrt{2}}{\sqrt{8}-\sqrt{2}}$

1. Incident and Reflected Ray through Points (with Figure M1W2Q6) Question: A incident ray is passing through the point (2, 3) makes an angle α with horizontal. The ray gets reflected at point M and passes through the point (5, 2) as shown in figure below. Fig: M1W2Q6 Choose the set of correct option(s). The equation of incident ray is −5x −3y + 19 = 0 The equation of incident ray is 3x + 2y −12 = 0 The equation of reflected ray is 5x −3y −19 = 0 The equation of reflected ray is 2x + y −12 = 0 Answer: Option a and c

1. Consider three Airports A, B, and C. Two friends Ananya and Madhuri want to meet at Airport C. Ananya Boarded Flight 1 from Point A to C which is 1200 km, due to bad weather, Flight 1 slowed down, and the average speed was reduced by 200 km/h and the time increased by 30 minutes. Madhuri boarded Flight 2 from Point B to C which is 1800 km, the average speed of Flight 2 is 720 km/h. What is the waiting time, and who will be waiting at the airport? (Given Ananya and Madhuri boarded at the same time) Waiting Time is 1 hr and Ananya is waiting.

Multiple Select Question (MSQ) 1. Consider the line $\left(L_{x}\right)$ and parabola $\left(P_{x}\right)$ as shown in below figure. Which among the following represents the graph of $\frac{P_{x}}{L_{x}}$ ? Answer: Option b Solution:

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 Choice/Statement Analysis Statements about sequences: Statement: If ${a_n}$ and ${b_n}$ are two sequences of real numbers, then ${a_n + b_n}$ is a convergent sequence. Counterexample: Let $a_n = 1$ for all $n$, $b_n = -1$ for all $n$. Both converge, but $a_n + b_n = 0$ for all $n$, which is convergent. However, the PDF says “option 1 is not correct,” which may refer to a different statement or a misinterpretation. The given explanation is not clear, but the PDF concludes: “Hence option 1 is not correct.” Statement: If ${a_n}$ is an increasing sequence, then ${(-1)^n a_n}$ is a decreasing sequence. Counterexample: $a_n = n$ for all $n$. Then ${(-1)^n a_n}$ is not decreasing. Solution: Hence option 2 is not correct. Statement: If ${a_n} \to a$, ${b_n} \to b$, and both $a, b$ are non-zero, then ${a_n b_n} \to ab$ must be non-zero. Solution: This is correct. Conclusion: Option 3 is correct. Statement: If ${a_n} \to a$ and ${b_n} \to a$, then ${a_n - b_n} \to 0$. Solution: This is correct. Conclusion: Option 4 is correct. Statement: If a sequence is divergent, then any subsequence is also divergent. Counterexample: Let $a_n = n$ if $n$ is odd, $a_n = 1$ if $n$ is even. ${a_n}$ is divergent, but ${a_{2n}}$ is constant and hence convergent. Conclusion: Option 5 is not correct. 2. Function Type Matching Match the following functions to their types:

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. Match Functions with Graphs and Area Question: Match the given functions with their graphs and the area under the curve over $[-1,1]$: Function Graph Area under curve $[-1,1]$ (i) $f(x) = x$ (b) $\int_{-1}^{1} x,dx = 0$ (ii) $f(x) = x $ (iii) $f(x) = x^2$ (a) $\int_{-1}^{1} x^2,dx = \frac{2}{3}$ Solution: (i) $\rightarrow$ (b) $\rightarrow$ (3) (Note: The “3” here is likely a typo or mislabel; area is 0, not “3”. The correct area solution is as above.) Corrected: (i) $\rightarrow$ (b) $\rightarrow$ area $= 0$ **(ii) $\rightarrow$ (c) $\rightarrow$ area $= 1$ **(iii) $\rightarrow$ (a) $\rightarrow$ area $= \frac{2}{3}$ 2. Curves Enclosing Negative Area Question: Which of the following curves enclose a negative area on the x-axis in the interval $1$? Area above the x-axis is positive, below is negative. If the area below the x-axis is more than above, the net area is negative.

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:

15 Formula Calculation Formula: n * m Results: 10, 30, 60 Formula Calculation Formula: n + m Results: 11, 8 Math Calculator m1: m2: n1: n2: x: formula1: formula2: formula3: Solve a: b: Formula: Solve Result: x: y: z: Formula: Solve Result: Add Now Add Now

Solution for IIT Madras Course Mathematics I Practice Assignments

1. Venn Diagram Analysis Question: Given below is a Venn diagram for sets of students who take Maths, Physics, and Statistics. Which of the option(s) is(are) correct? [Notation: For sets P and Q, P \ Q denotes the set of elements in P which are not in Q.] Maths Physics A D B F G E C Statistics D is the set of students who take both Maths and Statistics. D∪E∪F∪G is the set of all students who take at least two subjects. E is a subset of the set of the students who have not taken Maths. Maths \ D is the set of all students who have taken only Maths. Physics \ (D∪G∪E) is the set of all students who have taken only Physics. Solution: According to Figure 1, D is the set of students who take both Maths and Physics. Hence the first statement is not valid.

1. Multiple Choice Question (MCQ) Question: If $ R $ is the set of all points which are 5 units away from the origin and are on the axes, then $ R $ is: $ R = {(5,5), (-5, 5), (-5, -5), (5, -5)} $ $ R = {(5,0), (5, -5), (5, 5), (-5,0)} $ $ R = {(5,0), (0,5), (5,5), (0, -5)} $ $ R = {(5,0), (0,5), (-5,0), (0, -5)} $ $ R = {(5,0), (0,5), (-5,0), (-5,5)} $ There is no such set. Solution: The points on the axes and 5 units from the origin are $(5,0)$, $(0,5)$, $(-5,0)$, and $(0,-5)$. Correct option: $ R = {(5,0), (0,5), (-5,0), (0, -5)} $

1. Multiple Choice Questions (MCQ) 1. What will be the equation of the tangent to the curve $ f(x) = 2x^2 + 9x + 20 $ at point $(-3, 11)$? Options: $ y = 3x $ $ y = -3x + 2 $ $ y = -3x + 20 $ $ y = -\frac{x}{3} + 2 $ $ y = \frac{x}{3} + 20 $ $ y = -\frac{x}{3} $ Solution: Slope at $ x = -3 $: $ m = 2 \times 2 \times (-3) + 9 = -3 $ Equation of tangent: $ y = -3x + c $ Passes through $(-3, 11)$: $ 11 = -3 \times (-3) + c \implies c = 2 $ Answer: $ y = -3x + 2 $123

Multiple Choice Questions (MCQ) Question 1: Which of the following polynomial functions represents the profit from selling Tamil books? ○ 2x³ + 4x² - 2x - 4 ○ x³ - 2x² - x + 2 ○ x³ + 2x² - x - 2 ○ 2x³ - 4x² - 2x + 4 Solution: The profit from selling Tamil books is $x^3 + 2x^2 - x - 2$.

Multiple Choice Questions (MCQ) Question 1: Choose the correct option. ○ f₃ is not a function. ○ f₆ is not a function. ○ f₅ is not a function. ○ All of the above are functions. Solution: Vertical line test fails only for f₆ and therefore f₆(x) is not a function. Question 2: Choose the correct option. ○ f₁ and f₃ are one-one functions in the given domain. ○ f₂ and f₄ are one-one functions in the given domain. ○ f₃ and f₅ are one-one functions in the given domain. ○ f₅ is one-one function in the given domain. Solution: The functions f₂ and f₄ are strictly decreasing functions in the domain [-4, 4], therefore these are one-to-one functions.

Question 1 If $ b > 0 $ and $ 4\log_x b + 9\log_{b^5x} b = 1 $, then the possible value(s) of $ x $ is (are) (a) $ b^{10} $ ✓ (b) $ b^9 $ (c) $ b^{-2} $ ✓ (d) $ b^5 $ (e) $ b^4 $ Solution: Let $ p = \log_b x $. Equation: $ \frac{4}{p} + \frac{9}{5 + p} = 1 $ Solve: $ p^2 - 8p - 20 = 0 $, roots $ p = -2, 10 $. So, $ x = b^{-2} $ or $ x = b^{10} $.

Question 1: Function Behavior from Graph Statement: From the graph, clearly the values of the function tend to 0 as $ x $ tends to $ \infty $. The graph is smooth (no sharp edges or sharp turns) and so the graph has a unique tangent at all points (at $ x = 1 $ and $ x = -1 $ too). The function values are decreasing (slope of the curve is negative) in $[-0.5, 0]$ (Option 7).

Multiple Select Questions (MSQ) and Matching Question 1 (Matching) Given: Column A: Functions (i) $ f(x) = x e^x $, (ii) $ f(x) = e^{-2x} - 1 $, (iii) $ f(x) = e^x - 1 $ Column B: Tangent equations (a) $ y = -2x $, (b) $ y = x $, (c) $ y = 0 $ Column C: Figure numbers (1, 2, 3) Solution:

Multiple Select Questions (MSQ) and Matching Question 1: Matching Function, Area, and Graph Question: Match the given functions in Column A with the (signed) area between its graph and the interval $[-1,1]$ on the X-axis in Column B and the pictures of their graphs and the highlighted region corresponding to the area computation in Column C. Functions (Column A) Area under the curve (Column B) Graphs (Column C) i) $ f(x) = 5x - 1 $ a) $\pi/2$ 1) ii) $ f(x) = x^3 $ b) 0 2) iii) $ f(x) = \frac{1}{x^2 + 1} $ c) -2 3) Options:

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.