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: