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.