Dr. Brajesh Kumar Jha

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...

B.Tech Probability Questions with Detailed Solutions Topic: Simple / Classical Probability Level: B.Tech Engineering Mathematics Marks: 6-10 Marks Each Question 1: Probability Based on Playing Cards A card is drawn at random from a well-shuffled standard deck of 52 playing cards. Find the probability that the card drawn is: a king or a queen, a red face card, neither an ace nor a king, either a heart or a face card. Show Detailed Solution Total number of cards in a standard deck: \( n(S) = 52 \) The classical definition of probability is: \[ P(E)=\frac{\text{Number of favourable outcomes}} {\text{Total number of equally likely outcomes}} \] (a) Probability of drawing a King or Queen Number of kings = 4 Number of queens = 4 Therefore, ...

1. Standard Integrals \(\displaystyle \int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\qquad n\ne -1\) \(\displaystyle \int \frac{1}{x}\,dx=\ln|x|+C\) \(\displaystyle \int (ax+b)^n\,dx=\frac{(ax+b)^{n+1}}{a(n+1)}+C,\qquad n\ne -1\) \(\displaystyle \int e^x\,dx=e^x+C\) \(\displaystyle \int a^x\,dx=\frac{a^x}{\ln a}+C,\qquad a>0,\ a\ne1\) \(\displaystyle \int \sin x\,dx=-\cos x+C\) \(\displaystyle \int \cos x\,dx=\sin x+C\) \(\displaystyle \int \tan x\,dx=-\ln|\cos x|+C\) \(\displaystyle \int \cot x\,dx=\ln|\sin x|+C\) \(\displaystyle \int \sec^2x\,dx=\tan x+C\) \(\displaystyle \int \csc^2x\,dx=-\cot x+C\) \(\displaystyle \int \sec x\tan x\,dx=\sec x+C\) \(\displaystyle \int \csc x\cot x\,dx=-\csc x+C\) \(\displaystyle \int \sec x\,dx=\ln|\sec x+\tan x|+C\) \(\displaystyle \int \csc x\,dx=\ln|\csc x-\cot x|+C\) \(\displaystyle \int \sin(ax)\,dx=-\frac1a\cos(ax)+C\) \(\displaystyle \int \cos(ax)\,dx=\frac1a\sin(ax)+C\) \(\displaystyle \int \tan(ax)\,dx=-\frac1a\ln|\cos...

Sticky Ad Probability Problems with Detailed Solutions Click each question to expand the detailed interpretation and solution. Question 1: Two unbiased dice are thrown Two fair dice are rolled together. Find the probability that: (i) the sum is even (ii) the sum is less than 5 (iii) the sum is at most 12 Since each die has 6 possible outcomes, the total number of outcomes is: $$6\times6=36$$ We calculate the number of favorable outcomes for each condition. (i) Sum is even The sum is even when: both numbers are even both numbers are odd Number of even numbers on a die = 3 Number of odd numbers on a die = 3 $$3\times3+3\times3=18$$ Therefore, $$P(\text{sum is even})=\frac{18}{36}=\frac12$$ (ii) Sum is less than 5 Possible sums are 2, 3 and 4. Number of favorable outcomes: $$1+2+3=6$$ H...

Differential Equations: Detailed Step-by-Step Solutions In this article, we solve three important differential equations step by step using standard methods such as complementary function and particular integral , integrating factor , and separation of variables . 1) Solve \(y''+4y=\sin(3x)\) This is a linear differential equation with constant coefficients: \[ y''+4y=\sin(3x) \] The general solution is: \[ y=y_c+y_p \] where \(y_c\) is the complementary function and \(y_p\) is the particular integral. Step 1: Complementary Function The auxiliary equation is: \[ m^2+4=0 \] \[ m^2=-4 \] \[ m=\pm 2i \] Therefore, the complementary function is: \[ y_c=C_1\cos 2x+C_2\sin 2x \] Step 2: Particular Integral \[ (D^2+4)y=\sin 3x \] \[ y_p=\frac{1}{D^2+4}\sin 3x \] Using: \[ f(D)\sin ax=f(-a^2)\sin ax \] \[ y_p=\frac{1}{-9+4}\sin 3x \] \[ y_p=-\frac15\sin 3x \] Step 3: General Solution \[ y=C_1\cos 2x+C_2\sin 2x-\frac15\sin 3x \] Fi...

First Order Homogeneous Differential Equations (Solved Examples) A differential equation of the form \[ \frac{dy}{dx}=f\left(\frac{y}{x}\right) \] is called a homogeneous differential equation . We use the substitution \[ y=vx \quad \text{or} \quad v=\frac{y}{x} \] so that \[ \frac{dy}{dx}=v+x\frac{dv}{dx} \] Example 1: Solve \( \frac{dy}{dx}=\frac{x+y}{x} \) View Solution \[ \frac{dy}{dx}=1+\frac{y}{x} \] Let \[ v=\frac{y}{x}, \quad y=vx \] \[ \frac{dy}{dx}=v+x\frac{dv}{dx} \] Substitute \[ v+x\frac{dv}{dx}=1+v \] \[ x\frac{dv}{dx}=1 \] \[ \frac{dv}{dx}=\frac{1}{x} \] Integrate \[ v=\ln|x|+C \] Since \(v=\frac{y}{x}\) \[ \frac{y}{x}=\ln|x|+C \] \[ y=x(\ln|x|+C) \] Example 2: Solve \( \frac{dy}{dx}=\frac{x+y}{x-y} \) View Solution Let \[ v=\frac{y}{x}, \quad y=vx \] \[ \frac{dy}{dx}=v+x\frac{dv}{dx} \] Substitute \[ v+x\frac{dv}{dx}=\frac{1+v}{1-v} \] \[ x\frac{dv}{dx}=\frac{1+v}{1-v}-v \] \[ x\frac{dv}{dx}=\frac{1+v^2}{...