Dr. Brajesh Kumar Jha

📘 Mathematics-I Interactive notes on mathematical functions, their domains, ranges, properties and animated graphs. Functions and Their Graphs A function describes a relationship between an input x and an output f(x) . Different functions produce characteristic curves. Click any function below to expand the section. The graph will then be drawn automatically. 1. Exponential Functions eˣ Exponential Function ⌄ f(x) = e x Domain ℝ Co-domain ℝ Range (0, ∞) The value of e x is always positive. The x-axis is a horizontal asymptote. ▶ Replay Graph 2ˣ Exponential Function ⌄ f(x) = 2 x Domain ℝ Co-domain ℝ Range (0, ∞) The graph passes through (0,1) and increases rapidly as x increases. ▶ Replay Graph 2. Trigonometric Functions sin Sine Function ⌄ f(x) = sin x Domain ℝ Co-domain ℝ Range ...

Bayes' Theorem Definition, Formula and 10 University-Level Solved Problems Definition of Bayes' Theorem Bayes' theorem is used to calculate the probability of an event after new information or evidence has become available. Let A and B be two events such that P(B) > 0. Then Bayes' theorem is P(A | B) = [P(B | A) P(A)] / P(B) Here: P(A | B) = posterior probability P(B | A) = likelihood P(A) = prior probability P(B) = probability of the evidence Prior Likelihood Posterior P(A) P(B | A) P(A | B) Figure 1: Bayes' theorem updates prior probability using new evidence. General Form If A 1 , A 2 , ..., A n form a partition of the sample space, then P(A i | B) = [P(A i ) P(B | A i )] / Σ P(A j ) P(B | A j ) Example 1: Two Machines A factory has two machines M 1 and M 2 . Machine M 1 produces 60% of the items and M 2 produces 40%. Their defective rates are 2% and 5%, respect...

Conditional Probability Definition, Formula and Solved Examples Definition of Conditional Probability Let A and B be two events in a sample space S , where P(B) > 0. The conditional probability of A given that B has already occurred is denoted by P(A | B) and is defined as P(A | B) = P(A ∩ B) / P(B) The occurrence of event B reduces the original sample space to B. We then determine what proportion of B also belongs to A. Sample Space S A B A ∩ B Figure 1: Conditional probability considers the part of A lying inside B. The multiplication rule follows directly: P(A ∩ B) = P(A | B) P(B) Similarly, P(A ∩ B) = P(B | A) P(A) Example 1: Drawing a Card A card is drawn randomly from a standard deck of 52 cards. Given that the selected card is a face card, find the probability that it is a king. K ♠ Q ♥ J ♦ 12 Face Cards Figure 2: Kings, Queens and Jacks form the 12 ...

Quiz on Sets and Relations Total Questions: 20 | 1 Mark Each Participant Information Name of Participant: School / College Name: 1. If A = {1,2,3,4}, then the number of elements in A is: A) 2 B) 3 C) 4 D) 5 2. Which of the following represents the empty set? A) {0} B) { } C) {1} D) {0,1} 3. If A = {1,2,3} and B = {3,4,5}, then A ∪ B is: A) {3} B) {1,2,4,5} C) {1,2,3,4,5} D) {1,2,3,3,4,5} 4. If A = {2,4,6,8} and B = {4,6,10}, then A ∩ B is: A) {2,8} B) {4,6} C) {4,6,10} D) {2,4,6,8,10} 5. If A = {1,2,3}, then the number of subsets of A is: A) 3 B) 6 C) 8 D) 9 6. If A ⊆ B, then: A) Every element of B belongs to A B) Every element of A belon...

Chapter 3: Application of Set Theory Set theory has many important applications in engineering, computer science, electronics, biotechnology, data science, databases and mathematical modelling. 1. Database and Information Systems Field: Computer Science and Software Engineering In database design, particularly in SQL, set theory is used to organise, combine and filter data. Operations such as union, intersection and difference have direct counterparts in database operations. Example: In an engineering inventory system, the set of available components can be intersected with the set of required components to determine which components are immediately available for manufacturing. 2. Digital Logic Design Field: Electronics and Communication Engineering Digital logic is closely related to set theory. Boolean operations such as AND, OR and NOT correspond respectively to intersection, union and complement of sets. For example, an AND gate produces an active ou...