#class 12 maths relations and functions

[edusquaremaths]

Q5 Exercise1.1 I Class 12 Maths NCERT Chapter 1 Relations and Functions | NCERT solutions

NCERT Class 12Chapter: Relations and FunctionsExercise 1.1 Question 5: Check whether the relation R in R defined by R = {(a, b) : a ≤ b3} is reflexive, symmetric or transitive.

[class12maths]

Relations and Functions: Class 12 Maths Video Tutorials — Mathyug

Are you a Class 12 student gearing up for your Maths exams? Do the concepts of relations and functions seem daunting? Fret not! Ashish Sir from MathYug has designed an exceptional self-study course that covers everything you need to master these topics. This comprehensive course includes video tutorials, detailed notes, and assignments that ensure thorough understanding and practice.

What You’ll Learn

Ashish Sir’s course on MathYug is meticulously structured to provide a deep understanding of relations and functions, a crucial component of Class 12 Mathematics. The course is divided into two main parts:

1. Relations

In this segment, you will explore various types of relations. Here’s a brief overview of what you’ll learn:

Empty Relations: Understanding the concept of relations where no element of a set is related to any element of another set.

Universal Relations: Learning about relations where every element of a set is related to every element of another set.

Trivial Relations: Delving into the simplest form of relations that only relate to the identical pairs.

Reflexive Relations: Understanding relations where every element is related to itself.

Symmetric Relations: Exploring relations where if one element is related to another, the second element is also related to the first.

Transitive Relations: Learning about relations where if one element is related to a second, and the second to a third, then the first is related to the third.

Equivalence Relations: Discovering relations that are reflexive, symmetric, and transitive.

Equivalence Classes: Understanding the partition of a set into disjoint subsets where each subset is an equivalence class.

To ensure comprehensive understanding, the course provides practice questions from diverse sources such as NCERT Textbook exercises, NCERT Examples, Board’s Question Bank, RD Sharma, and NCERT Exemplar. You can download the PDF of assignments within the course for additional practice.

2. Functions

The second part of the course focuses on functions, including:

One to One and Onto Functions: Understanding functions where each element of one set is paired with a unique element of another set, and functions that map elements from one set onto every element of another set.

Composite Functions: Learning how to combine two functions to form a new function.

Inverse of a Function: Exploring how to find the function that reverses the effect of the original function.

Similar to the relations segment, this part also includes practice questions from various reputable sources. The assignments are designed to reinforce your understanding and are available for download within the course.

Sample Videos

To give you a glimpse of the quality of instruction, Ashish Sir has shared sample videos covering some of these essential topics. Watching these will help you understand his teaching methodology and the depth of content provided.

If you’re aiming to excel in your Class 12 Maths exams, Ashish Sir’s course on MathYug is your go-to resource. With detailed video tutorials, comprehensive notes, and extensive assignments, you will gain a solid understanding of relations and functions. Start your journey towards mastering these crucial topics today!

For more information and to access the course, visit MathYug. Happy learning!

[ashishkumarletslearn]

Class 12 Maths Exercise 1.1 | Relations and Functions - MathYug

Delve into the realm of Relations and Functions in Class 12 Maths with MathYug's expert teacher Ashish Kumar, fondly known as Agam Sir. In this comprehensive video lecture, we unravel the complexities of NCERT Exercise 1.1, exploring empty relations, universal relations, and more.

Timestamps:

00:00:05 Introduction and How Ashish Sir is going to teach you Class 12 Maths on MathYug

00:06:37 Empty Relations and Universal Relations (Trivial Relations) Lecture 1 (NCERT Exercise 1.1, NCERT Exemplar, Board’s Question Bank)

00:12:27 Question 1 Let R be the relation in the set N given by R = {(a, b) : a = b – 2, b ≥ 6}. Choose the correct answer. A. (2, 4) ∈ R B. (3, 8) ∈ R C. (6, 8) ∈ R D. (8, 7) ∈ R (N)

00:15:17 Question 2 Let A= {1, 2, 3,} and define R = {(a, b): a – b = 12}. Show that R is an empty relation on Set A. (B)

00:17:47 Question 3 Let A be the set of all students of a boy’s school. Show that the relation R in A given by R = {(a, b) : a is the sister of b} is the empty relation and R′ = {(a, b) : the difference between the heights of a and b is less than 3 meters} is the universal relation. (N)

00:28:17 Question 4 If A is the set of students of a school then write, which of the following relations are Universal, Empty, or neither of the two. R1 = {(a, b) : a, b are ages of students and |a – b| > 0} R2 = {(a, b) : a, b are weights of students, and |a – b| > 0} R3 = {(a, b) : a, b are students studying in the same class} (B)

00:40:47 Question 5 Let A = {1, 2, 3,} and define R = {(a, b): a + b < 0}. Show that R is a universal relation on set A. (B)

Join us as we embark on a journey to master Class 12 Maths, guided by Agam Sir's expertise.

[mathyug]

Unlock a deeper understanding of Reflexive, Symmetric, Transitive, and Equivalence Relations, and master Exercise 1.1 in Relations and Functions Class 12 Maths with expert instructor Ashish Sir on MathYug.Explore additional video lectures on our website: https://mathyug.com/class-12-maths

[quietimtryingtostudy]

9. Let W denote the set of words in the English dictionary. Define the relation R by R= {(x, y) ∈ W x W such that x and y have at least one letter in common}. Show that this relation R is reflexive and symmetric, but not transitive.

(i vaguely remember my teacher saying that we don't have word problems from this chapter or smtg. but just in case, ya know?)

Took help from one of my friends for this one- but I understood the entire procedure by the end!

(Yeah, I understand. Thanks for sending it though!!)

[divorcetual]

theres multiple grandmasters??? well. it makes sense that its a title. news to me

Yeah !! "Grandmaster" is just the name for the "owner"/leader of Koro at the time (technically Koro [the city/settlement] is run by Koro [the AI], but since it isnt 100% functional rn, the de facto leader is The Grandmaster- they're the one who manages project Koro and oversees any settlements incorporated into Koro [the city]). Its mostly a familial line (being the Grandmaster is suuuuper reliant on the person having this One Specific Fear or else it doesnt work, so if the persons kid doesnt develop/they dont have a kid, the title is passed down to a non-related heir, who is then adopted into the family). Yoki and the first Grandmaster dated for a bit wayyyy back like. even before he got his cellres bug.

Sooo. If the grandmaster title has been around since b4 Yoki got a bug ,,, 400 years post-bomb and he got his bug 80 years b4 that, but the first guy probably didnt take on the alias until he was in his 30s-40s. Yoki n him dated as teens n Yoki was in his 20s when he got the bug. Lets say 15 years after that the Grandmaster appeared, so 80-15 is 65. So ~465 years of grandmasters. The title is only transferred upon death, n lets average an age of death of 60 since a lot of them probably died younger than usual after the bomb. I'll say each grandmaster has a kid at ~20 then dies at 60, so the new grandmaster gets it at age 40,, i havent had a math class in years oh my god. 465/40 is ~11.6

So theres been around 12 grandmasters total so far !!

[sculptural-renaissance]

SGU Week Day 2 (I promise I can count; I'm just behind!!): Favorite Character

Leave it to me to get fixated on a character who only shows up in 4 episodes of a show that got cancelled 12 years ago. 😫 I thoroughly enjoyed SGU from its inception, but Amanda Perry is what took the show from something I really enjoyed to my current hyperfixation (yes, it's her fault I'm here, this fictional lady). She lives rent-free in my head. 💖

(Also fair warning this is gonna be long as hell)

I'm aware that her primary function is that of "love interest," but her character resonated with me on a much deeper level. Maybe I didn't travel a billion light-years to be with the person I love, but I did move 1,000 miles away from everything I knew to make a life with the man who became my husband. I know what it's like to feel rejected because the person you love shows their love differently (though I've been married for almost 6 years now after 3.5 years of long-distance, and then a lengthy engagement, so I know from experience that loving differently doesn't mean loving less). I know what it's like to feel like I'm not valued as a person, though in my case, I was mistreated because I'm autistic as opposed to being paralyzed, and I was regularly used and taken advantage of for my musical abilities, with little respect for my health or needs. I wasn't a person, I was a thing. I've been targeted by people I couldn't fight back against, even for things I didn't do. I know what it's like to be left behind for a lot of social milestones (driving, dating, etc.) I know what it's like to finally find that one safe person who sees you for you.

Amanda is relatable as hell to me. I do want to focus on her specifically in this post, though obviously her romantic endeavors are a factor. Tbh one of the main reasons I like her so much is, in addition to all the other stuff I just mentioned, is that she was the one to initiate things with Rush. I have little dating experience, myself, but I was the instigator in most of my relationships, especially with my husband. Gotta love those ladies who are active instead of waiting around for a guy to make up his mind!

Anyway, I make a lot of SGU memes and dumb lil comics that I share on Facebook, and a decent chunk of them are about Amanda (she even has her own multi-episode arc!!), so I'll share some of them here. I guess if people like them, I can put them on tumblr. I'll add that the vast majority of my content takes place post-stasis, so this Amanda inhabits her own meat body, cloned from DNA the SGC had on file, and is not a computer ghost or a body-swapper with questionable ethics. 😂

I have generated A LOT of memes lmao

Well that didn't turn out to be a metaphor for anything... 😏 I didn't catch this the first time, but I ugly-laughed on my second watch-through.

When the lockdowns were first imposed in my town, I had no idea what that meant, and I was literally about to get in my car and commit a crime because I thought my husband was going to be trapped at work for 2 weeks. Obviously, that wasn't the case, but I felt this scene.

Since my brain has chosen to attach itself to a character with minimal backstory, I'm obligated to fill in the blanks. One thing I find disappointing in media is that even the nerdy women tend not to have nerdy interests, which absolutely is not what I've experienced in real life. My thought is that Amanda was already something of a prodigy ("Little Miss Brilliant" probably isn't a nickname you get for being a regular genius lmao) before her accident, probably skipped a grade or two in school, private tutors in math and science, that kind of thing, and was later homeschooled before starting college as a teen. She was probably pretty lonely because of that. I'm not quite a "genius" (I have a Master's in music and an Etsy shop, and I uhhhhh make memes about an old TV show for fun), but I am a former gifted kid. Even though I took gifted classes from 4th-12th grade (academics, not that pull-out enrichment crap), I felt very lonely even among my gifted classmates, and I struggled with making friends throughout school because I was just too different. I was weird. I liked weird stuff. The girls were mean. The boys that were my friends didn't like the girly stuff I enjoyed, so I had to lock that part of myself away to fit in. The kids a grade ahead of me obviously left, and the kids a grade below me didn't want to keep in touch when I moved on to high school or college. It was lonely. I see similar things in Amanda's brief appearances. She's on the Destiny for 3 weeks, and she pretty much just sticks to Rush because he's "safe." She tries to be friendly with Eli, but he ends up making fun of her, and he's not very nice to her later on, either (sometimes for good reason; she's my fave but she's no saint). TJ isn't exactly in a position to be socializing during a complicated surgery, so she gets a pass. 😉

This was rambly, but my point is that a lot of lonely nerds are drawn to sci-fi, even the girls. I know for a fact that I'm projecting, but come on, she designs hyperdrives. You really gonna sit there and tell me she never watched Star Trek? 😂

As a side note (oh boy even more rambling), in my fan content, she and Eli have a very sweet friendship based on their shared nerd interests. I really don't think they'd get along in canon as things were left.

Speaking of sci-fi, I like to think that, after getting recruited for the Stargate program, Amanda watched all of Wormhole X-treme to prepare herself. I also like to think that no one else on Destiny's crew has seen it except Eli, who saw a handful of episodes randomly. I just really need him to see a stargate for the first time and be all like WOW JUST LIKE WORMHOLE X-TREME 😂 These are two separate excerpts, one mentioning the show and tying it back to actual events in SG-1, and the other showing a snippet of an episode. Replicators are kinda like Legos, so of course the humanoid Wormhole X-treme versions would be Lego people (I'm not creative, I'm just silly).

Going back to feeling lonely, I do think Amanda would be pretty imaginative, even if she's kinda one-note. In a fic I'm working on, she confesses that all of her fantasies involve Rush and having working legs (disclaimer that there's obviously nothing wrong with using a wheelchair and that disabled people can and do have fulfilling romantic relationships and sex lives, but it seems clear in the show that she would like to be able to walk again, so I have stuck to that for this particular story), and the narrative quality of her fantasies is very much on par with dollar store romance novels. She reminds me a little of Tina from Bob's Burgers and her "erotic friendfiction," so I made a few crossover memes. That's Tina's copy of "Buttloose" in the 4th panel. Panel 3 is also a borrowed quote from Tina. I do think Amanda might struggle with tidiness since she spent a good 25 years not being able to pick up after herself at all. I also struggle with this due to executive dysfunction and growing up in a borderline hoarder environment, so I guess I'm projecting again RIP 😅

No, really, Camile, what do you mean????

I guess in Amanda's defense, she doesn't find Rush off-putting in any way since he's not unnecessarily mean to her. 😂

I'm not sure if I'll ever complete my fic, but one thing I'm exploring, both there and, to a less complicated degree, in my memes, is Amanda's use of the neural link while she's a computer ghost. In Seizure, she seemed to use it with ease (apart from that one colossal fuckup lmaooooo), and I'd really like to see what she could do with it after her file is no longer quarantined. I've done a lot of cringe rambling and infodumping (congrats to anyone who's actually made it this far), so I won't go into detail, but she does work on expanding use of the link so she and Ginn can use it at the same time and connect to multiple people so they can socialize and function more as crew members instead of something like a guardian angel or familiar spirit. I'd also like, as mentioned in a previous post for this event, if she could help Lisa "see" again via the neural link. Another possibility, as alluded to in the above photo, is making "holograms," which aren't really holograms, but are more like skins for the ship's AI. What's cooler, accessing a medical database in the infirmary, or asking the EMH from Voyager to give you the info you need? If you're a loser like me, it's absolutely the second one.

More backstory!! My hc is that Amanda's parents were killed in the accident that paralyzed her (on the way to some math tournament with the high school mathletes or something idk I'm a musician I don't know what STEM people do for fun), and she was raised by her grandparents. We all love to make our faves suffer, and I'm no exception. This is just a silly little comic, but there is a certain sadness of being intellectually advanced but emotionally immature, something common among gifted kids and neurodivergent people (and I'm both, wheeeee). Things are thankfully getting better now. I love seeing more acceptance. But it was difficult in the early 00s when I was a teen, and I'm sure it would have been harder for Amanda in the '80s and '90s. It's also difficult when you have to rely on someone else for transportation, but they are unwilling to take you places (I went nowhere in college except school and church and my parents were still always mad at me, which is weird of them because they actively discouraged me from driving and would not teach me or help me get over my fears, so uhhhh yeah).

Last one~

So yes, I'm definitely salty that my top 2 SGU ladies got fridged twice (Ginn's my second fave because of course she is). I do think it opened up a lot of interesting possibilities for both characters, but as the show was tragically cut short, we have to rely on fan content. Anyway, the point I'm trying to make here is that Amanda and Ginn would have been fine had Amanda simply studied Bobby Hill's Guide to Self-Defense. Yes, it's bad, but I am a bad person and I generate irreverent things.

If you made it this far, then congratulations, you're an obsessed weirdo like me!! I know Amanda isn't exactly popular, nor is SGU, especially not so long after the fact, so I don't really expect anyone to read all this. If you do, I hope I was able to convey why she is so special to me. If nothing else, I hope you enjoyed the memes.

Also this took me like 5 hours to write, Jesus Christ I need to sleep

[ddaleishere45]

Exploring S.T.E.M

By Dale V. Aldea (Grade 11-Perfection)

As a senior high school student, you've probably heard the term "STEM" being talked around countless times. Maybe you have heard that this strand is for students who wants to be a Doctor, an Engineer, even a Software Genius! But what exactly is STEM? Is it just more than just a senior high school strand? In this blog, we'll dive into the fascinating world of the STEM strand.

S.T.E.M and its origins

STEM is one of many strands you can take up in Senior High School. It stands for Science, Technology, Engineering, and Mathematics. The S.T.E.M acronym was first introduced in 2001 by a a group of scientific administrators at the U.S. National Science Foundation (Hallinen, 2023). The organization previously used the acronym SMET when referring to the career fields in those disciplines or a curriculum that integrated knowledge and skills from those fields but was later changed to STEM and since then, STEM-focused curriculum has been extended to many countries like the United States, Australia, China, France, South Korea, Taiwan, United Kingdom, and was later introduced here in the Philippines by the K-12 program.

Decoding the Acronym

S.T.E.M - SCIENCE Science is all about understanding the natural world, from the smallest particles to the vastness of the space. It involves asking questions, making hypotheses, conducting experiments, and analyzing data to discover the basic principles that govern the universe. If you aim to take science related courses or medicine in college, well you're in luck because this strand offers primary education for the said courses.

S.T.E.M - TECHNOLOGY

Technology is everything from smartphones and laptops to life-saving medical devices and cutting-edge space exploration equipment. It's the practical application of scientific knowledge to create tools, systems, and solutions that make our lives more efficient, convenient, and safe. If you're interested in software development, computer engineering, or even app designs in college, this strand is for you.

S.T.E.M - ENGINEERING

Engineering is the art of using scientific and technical knowledge and using it to design and make solutions. From bridges and skyscrapers to renewable energy systems and electric cars, engineers turn ideas into real, functional realities. The field of engineering is big, with specialties like civil engineering, mechanical engineering, electrical engineering, and aerospace engineering, to name just a few. Well in this strand you can learn the basics of those courses to prepare yourselves in college.

S.T.E.M - MATHEMATICS

Mathematics serves as the language of STEM. It is the foundation upon which all the other subjects are based. Our high school math classes help us develop critical thinking skills, problem-solving abilities, and logical reasoning qualities that are highly sought after in any profession. Whether you are planning on taking a career directly related to mathematics or use it as a tool to excel in another S.T.E.M field, its importance cannot be overstated.

Is STEM important?

The S.T.E.M strand in senior high school very important for students like me. The strand prepares students who are planning on taking on courses that are related to science, technology things, engineering, architecture and many more. The strand also focuses on enhancing the students problem-solving skills, critical thinking and creativity. This is a big help, giving us students a big boost in knowledge in the fields of science and math for college.

So keep this in mind, STEM isn't just a some random acronym made for Senior High School, it is a gateway of understanding the world around us. The strand equips students with knowledge and curiosity for the fields in science, technology, engineering and mathematics, creating a path for a bright future. STEM is more than just a strand, it is a journey of discovery and learning to limitless possibilities.

References:

[mbti-notes]

Anon wrote: Hello, I am 16 years old. I have taken many tests, and read your blogs and descriptions about the Cognitive Functions and processes. I have narrowed my decisions down to two types. INTP AND ISTP. If you are able to find it, I could your advice as a resource to help me find my official type.

Dominant Ti- INTP and ISTP argument.

Between the ages of 6-12 and before some steess happened to me, I was a very curious child, watching many videos about how things worked, many History and Science Vids just to learn for fun. I tend to notice inconsistencies with data, and real life systems such as schools. Additionally, I also tend to have my own framework on how the world works. Such as me thinking that relationships are like an abstract system were you have to stabilize it at all angles to keep it running. I also think that many concepts like Newtons Laws can be applicable to many things like relationships . Like the 3rd Law where for every reaction, there is an equal and opposite reaction. A good example of this is that the more you love the person in a relationship, the more it will hurt when you break up. My one family member, who is an ISTJ btw told me that my logic makes sense and that it is kind of weird. I also have the thinking that School is just a chore, and that School is not real knowledge to me. I also tend to like to analyze things I am interested in such as Personality Theory, History, Science, and Math and knowing how they were formed and how it happened so I can expand my logical framework into a rich innerworld full of knowledge.

Inferior Fe-

One time when I got stressed, mainly due to me not reading social ques well which caused me to get used. I started getting so clingy to people, which caused people to become distant from me, which I didnt know was bad. I also had an emotional breakdown in front of my friend because I failed to admit that my crush didnt like me, which caused me to lose that friend. Recently, after a friend and I started growing distant, I started becoming so paranoid about how other people percieve me, which caused me to have a fear that people will laugh at me and betray me. During that time, I suddenly got stressed, and then went outside to be agressively social to someone I didnt know by cussing and emphasising logic to an extreme. I also noticed that it was harder for me to think, mainly due to the fact that I am so paranoid about others. But before that, I had an expierence where I got jealous, which caused me to suddenly want to get involved in a family friends personal business, by asking for their phone number via their parents, just so I can get invasive and make them feel uncomfy. I tend to want to have socially interactions, but I tend to fail at them due to how awkward I was.

INTP Argument Aux Ne and Tert Si argument:

I also noticed, that not only did I learn about many topics like history and science by watching videos. I also tend to make up my fictional World Wars, drawing maps based on those in the videos, and even coming up with new ways I believe the worlds problems could be fix. I tend to also find bizarre connections to things, such as me connecting Physics and Relationships, two things I somehow connected, even though they are not related to each other. I also tend to easily jump from topic to topic when talking to people about my ideas and even having normal conversations. Additionally, I seem to get really energized when brainstorming about all the possibilities, which to others can seem kinda quirky. I also tend to have a problem sticking to one possibility sometimes, due to the fact I can easily accept other possible theorys, even though it is not likely to happen. My Si shows up as me having vivid memories of things like dates. However my Si can be somewhat inconsistent due to the fact that somedays, I can be great with memories while on other days, I cant even remember where I sit in class. My Si kinda somewhat developed, thanks to my family member who is an ISTJ. She forced me to somewhat develop it because she tends to punish me for forgetting habits, routines, and chores by taking away my phone for 2 hours.

ISTP Argument Aux Se and Tert Ni

At the Age of 13, I suddenly was more active. I started playing tennis as a sport, and wanted to explore the world more by trying out new things around me. I also got less picky with food, and suddenly wanted to start trying things with a lot of topics despite me not liking those things in the past. I also started acting before thinking just for me to experience things. One time, I suddenly had a hunch that caused me to accuretly guess my friends birthday. I felt like I was in a Ti-Ni loop one time, where I was so obsessed about the future, that I slowed got less interested in staying present, and suddenly started worring more about the future.

-----------------------

Several problems with your submission:

- Overall, my impression is you don't have very deep knowledge of yourself. While some of this can be chalked up to youth, you also didn't follow the submission instructions properly, so you haven't done enough self-reflection. Remember, I'm only able to see what you're able to show, and you haven't shown enough, barely scratched the surface.

- Your descriptions of T and N overlap too much for comfort. I don't know if this is due to not understanding the difference conceptually, not being able to spot the difference in yourself, or a combination of both. Until you are able to disentangle the two cognitive processes, the dominant function remains unclear, which means it will be very difficult for the rest of the stack to snap into place.

- A lot of people, especially introverts, are socially awkward/inept but it doesn't mean they have inferior Fe - do not confuse the two. Until you are able to clarify the reasons for and/or explain the psychological mechanisms underlying your socializing issues, inferior Fe remains unproven. Additionally, there's a lot more to Fe than socializing. Until you have a better understanding of what Fe is and what exact role it plays in your mind in relation to the other functions, its stack position remains up in the air.

- The auxiliary and tertiary function descriptions were unhelpful due to lacking detailed examples and not providing exactly the information I asked for in the guide. ISTP seems unlikely, though.

Generally speaking, I need to reach >95% confidence before I'm willing to confirm or disconfirm a possibility. Unfortunately, since none of the stack positions are clear, your submission does not meet my threshold as sufficient proof of INTP, nor does it allow me to rule it out.

[3dprintinginedushaikhsamreen]

Application of 3D Printing in STEM Education

3D printing has revolutionized various fields, and its application in STEM (Science, Technology, Engineering, and Mathematics) education is particularly significant. By providing hands-on learning experiences, 3D printing enhances students’ understanding of complex concepts and encourages creativity and innovation. Here are some examples of how 3D printing is being utilized in STEM education.

 1.Engineering Prototyping

In engineering courses, students can design and print prototypes of their projects. mechanical engineering students can create working models of machines or mechanisms, allowing them to test and examine their designs quickly. This hands-on experience solidifies theoretical knowledge and develop problem-solving skills.

2. Biology and Anatomy Models

3D printing is extensively used in biology classes to create models of cells, organs, and organisms. Students can print detailed models of the human heart or brain, which helps them understand anatomy in a easier way.

3.Mathematics Visualization

Students can create geometric shapes and complex mathematical surfaces, allowing them to explore properties such as volume and surface area. This practical application helps fill the gap between theoretical math and real-world applications.4

4. Robotics and Coding

Students can design and print custom parts for their robots, coding them to specific functions. This process integrates coding with mechanical design, providing a comprehensive learning experience that enhances both programming and engineering skills.

5.Environmental Science Projects

Students can use 3D printing to create models that simulate ecological systems or showcase the impact of human activities on environments. For instance, they might design models of habitats to study biodiversity or print pollution models to visualize and analyze data. These projects foster awareness and understanding of environmental issues.

 6.Art and Design Integration

In STEM courses, students can design artistic sculptures or functional items, . This integration encourages innovation and creative problem-solving, essential skills in both STEM and the arts.

7. Space Exploration Stimulation 

With the growing interest in space exploration, students can engage in projects that involve 3D printing to design and create models of spacecraft or Martian habitats. This not only enhances their understanding of aerospace engineering but also inspires interest in STEM fields related to space science.

8. Accessibility and Customization

3D printing also allows for the customization of educational tools and resources. For example, students with disabilities can have assistive devices tailored to their needs, fostering inclusivity in STEM education. This adaptability makes learning more accessible and engaging for all students.

9. Chemistry Molecular Models

In chemistry classes, students can use 3D printing to create models of molecules and chemical compounds. By printing these structures, students gain a better understanding of molecular geometry, bonding, and chemical reactions, which can be challenging to visualize through diagrams alone.

10. Physics Experiments

Physics students can design and print custom apparatus for experiments. For example, they might create resonance or simple machines to study forces, motion, and energy transfer. This hands-on approach helps to understand theoretical principles through practical application.

 11.Historical Artifacts in Social Studies

In social studies or history classes, students can print replicas of historical artifacts, such as ancient tools or architectural features. This experience enhance their understanding of historical contexts making learning more engaging.

 12.Renewable Energy Models

Students studying renewable energy technologies can design and print models of solar panels, wind turbines, or hydroelectric systems. These projects can be used to simulate energy production and understand the principles of sustainability, engineering, and environmental science.

13. Customized Tools and Gadgets

In technology and design courses, students can create customized tools or gadgets tailored to specific tasks or problems. For example, they could design ergonomic tools for specific user needs, promoting innovation and practical application of design thinking.

14. Architectural Models

Architecture students can use 3D printing to create detailed scale models of their designs. This allows them to visualize relationships and aesthetics, providing insights into both structural integrity and design principles.

15.Game Design and Development

In computer science and game design programs, students can print 3D models of characters or environments they create. This integration of physical and digital design fosters creativity and helps students understand the complexities of game development.

 16. Assistive Technology Projects

Students can engage in projects that focus on designing assistive technologies, such as prosthetic limbs or adaptive devices. By creating functional prototypes, students learn about engineering design, user-centered design, and the impact of technology on accessibility.

17. Personalized Learning Tools

Teachers can use 3D printing to create customized learning tools, such as tactile math manipulatives for younger students or specialized equipment for science labs. This tailored approach supports diverse learning styles and needs within the classroom.

 18. Bioengineering Simulations

In bioengineering programs, students can print models of biological systems or tissues to study concepts like tissue engineering and regenerative medicine. This application enhances understanding of complex biological processes and fosters innovation in medical technology.

 19. Virtual Reality Integration

Combining 3D printing with virtual reality (VR) allows students to create physical models that can be used alongside VR simulations. This integration provides a multidimensional learning experience, enabling students to explore concepts in both physical and digital spaces.

20.Student Competitions and Challenges

Many educational institutions host competitions that encourage students to solve real-world problems using 3D printing. For instance, challenges might involve designing and building sustainable housing models or innovative transportation solutions, fostering collaboration and creativity.

 Conclusion

These examples illustrate the diverse applications of 3D printing in STEM education, highlighting its potential to enhance learning experiences, encourage innovation, and prepare students for future careers in technology and engineering fields. As educators continue to explore and implement 3D printing in their curriculum.

[shiva10010]

JEE Main 2025! Prep Your Math Strategy with the Latest Syllabus

The Joint Entrance Examination (JEE) Main is a critical gateway for aspiring engineers in India. While the official syllabus for JEE Main 2025 will likely be released by the National Testing Agency (NTA) in November 2024, you can get a head start on your preparation using the previous year's syllabus as a guide.

This article focuses on the Mathematics section of the JEE Main 2025 exam.

Key Elements of the JEE Main 2025 Maths Syllabus

The JEE Main Maths syllabus covers a wide range of topics from Class 11 and 12 NCERT textbooks. Here's a quick glimpse of the major areas you'll encounter:

Algebra: Sets, relations and functions, complex numbers, matrices and determinants, mathematical induction, binomial theorem, sequence and series.

Calculus: Limit, continuity and differentiability, integral calculus, differential equations.

Geometry: Coordinate geometry, three-dimensional geometry, vector algebra.

Other Quantitative Techniques: Statistics and probability, trigonometry, mathematical reasoning.

Why eSaral is Your Perfect JEE Main 2025 Maths Companion

While the syllabus provides a roadmap, in-depth preparation is essential for success. This is where eSaral steps in:

Complete Syllabus Coverage: eSaral ensures you cover all the essential JEE Main Maths topics.

Detailed Learning Material: Access a treasure trove of resources, including crisp explanations, illustrative examples, and topic-wise revision notes [refer to JEE Main 2025 Maths Revision Notes on eSaral for details].

Catered Learning Styles: Whether you prefer short and focused content or a deep dive into concepts, eSaral offers options to match your learning style.

Practice Makes Perfect: Sharpen your skills with ample practice problems available on the platform.

Additional Tips to Conquer JEE Main 2025 Maths

Focus on NCERT Textbooks: They form the foundation for the entire syllabus.

Regular Practice: Solve problems consistently to build speed, accuracy, and exam temperament.

Mock Tests: Take regular mock tests to simulate the exam environment and identify areas for improvement.

Clear Your Doubts: Don't hesitate to seek clarification from teachers or online forums.

Conclusion

JEE Main 2025 Maths can be aced with the right approach and resources. By leveraging the detailed syllabus and comprehensive learning materials offered by eSaral, you can approach the exam with confidence and a strategic plan. Remember, consistent effort and focused practice are key to achieving your engineering dreams.

[edusquaremaths]

Q4 Exercise1.1 I Class 12 Maths NCERT Chapter 1 Relations and Functions | NCERT solutions

NCERT Class 12Chapter: Relations and FunctionsExercise 1.1 Question 4: Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

[ashishkumarletslearn]

Training Teachers to Teach Class 12 Maths — MathYug

In the ever-evolving landscape of education, having access to high-quality learning resources can make a world of difference. For Class 12 students, mastering mathematics is crucial, not only for academic success but also for future endeavors in various fields. At MathYug, we understand this importance and have dedicated ourselves to providing top-tier educational materials. Today, we’re excited to introduce our training program designed specifically for teachers who aspire to elevate their teaching skills and make a significant impact on their students’ learning journey.

Sample Videos: A Glimpse into Quality Education

To give you a taste of what MathYug has to offer, we are sharing a selection of sample videos from our Class 12 Maths tutorials. These videos exemplify the high-quality, in-depth teaching style that Ashish Kumar, affectionately known as Agam Sir, is known for.

Relations and Functions

2. One to One and Onto Functions

3. Inverse Trigonometric Functions

4. Continuity and Differentiability

5. Applications of Derivatives

6. Integrals

7. Inverse Trigonometric Functions

8. Basics of Logarithms, Log Table and Antilog Table

Why Choose MathYug for Teacher Training?

MathYug stands out as a premier platform for learning and teaching mathematics, especially for Class 12 students. Our resources are meticulously designed to cater to the diverse needs of both learners and educators, ensuring that each student can grasp complex mathematical concepts with ease. Here’s why MathYug is the best choice for your teaching journey:

Expert Guidance

All our content is created by Ashish Kumar (Agam Sir), a renowned educator with years of experience in teaching mathematics. His unique teaching style simplifies complex topics, making them easily understandable. By training with MathYug, teachers can learn how to effectively convey these methods to their students.

Comprehensive Coverage

Our tutorials cover the entire Class 12 Maths syllabus, aligned with the NCERT guidelines. This ensures that students are well-prepared for their board exams and other competitive exams. Teachers trained with MathYug will be equipped to offer their students a thorough and well-rounded mathematical education.

Interactive Learning

We believe in making learning engaging and interactive. Our video lessons are complemented by practical exercises, assignments, and downloadable PDFs to reinforce learning. Teachers will be trained to use these resources to create a dynamic and interactive classroom environment.

Elevate Your Teaching Experience with MathYug

At MathYug, we are committed to providing the best possible educational resources to help both students and teachers excel in mathematics. Our Class 12 Maths tutorials are designed to build confidence, enhance understanding, and foster a love for learning. By sharing these sample videos, we hope to give teachers a glimpse of the quality education that awaits them on our platform.

Join MathYug Today!

Experience the difference with MathYug and take your teaching skills to new heights. Subscribe to our Class 12 Maths membership and gain access to a comprehensive collection of video lessons, study materials, and expert guidance from Ashish Sir. Let’s embark on this journey of academic excellence together!

Visit MathYug now and start your journey towards mastering Class 12 Maths with ease and confidence.

[mathyug]

Sample Videos of Class 12 Maths Tutorials by Ashish Sir on MathYug

A Glimpse into Quality Education

To give you a taste of what MathYug has to offer, we are sharing a selection of sample videos from our Class 12 Maths tutorials. These videos exemplify the high-quality, in-depth teaching style that Ashish Sir is known for.

Relations and Functions

2. One to One and Onto Functions

3. Inverse Trigonometric Functions

4. Continuity and Differentiability

5. Applications of Derivatives

6. Integrals

7. Inverse Trigonometric Functions

8. Basics of Logarithms, Log Table and Antilog Table

Why Choose MathYug?

MathYug stands out as a premier platform for learning mathematics, especially for Class 12 students. Our resources are meticulously designed to cater to the diverse needs of learners, ensuring that each student can grasp complex mathematical concepts with ease. Here’s why MathYug is the best choice for your maths journey:

Expert Guidance: All our content is created by Ashish Kumar (Agam Sir), a renowned educator with years of experience in teaching mathematics. His unique teaching style simplifies complex topics, making them easily understandable.

Comprehensive Coverage: Our tutorials cover the entire Class 12 Maths syllabus, aligned with the NCERT guidelines. This ensures that students are well-prepared for their board exams and other competitive exams.

Interactive Learning: We believe in making learning engaging and interactive. Our video lessons are complemented by practical exercises, assignments, and downloadable PDFs to reinforce learning.

Elevate Your Learning Experience with MathYug

At MathYug, we are committed to providing the best possible educational resources to help students excel in mathematics. Our Class 12 Maths tutorials are designed to build confidence, enhance understanding, and foster a love for learning. By sharing these sample videos, we hope to give you a glimpse of the quality education that awaits you on our platform.

Join MathYug Today!

Experience the difference with MathYug and take your mathematical skills to new heights. Subscribe to our Class 12 Maths membership and gain access to a comprehensive collection of video lessons, study materials, and expert guidance from Ashish Sir. Let’s embark on this journey of academic excellence together!

Visit MathYug now and start your journey towards mastering Class 12 Maths with ease and confidence.

[mpboardsolutions-blog]

MP Board Class 12th Maths Book Solutions in English Medium

MP Board Class 12th Maths Chapter 1 Relations and Functions

Chapter 1 Relations and Functions Ex 1.1

Chapter 1 Relations and Functions Ex 1.2

Chapter 1 Relations and Functions Ex 1.3

Chapter 1 Relations and Functions Ex 1.4

Chapter 1 Relations and Functions Miscellaneous Exercise

MP Board Class 12th Maths Chapter 2 Inverse Trigonometric Functions

Chapter 2 Inverse Trigonometric Functions Ex 2.1

Chapter 2 Inverse Trigonometric Functions Ex 2.2

Chapter 2 Inverse Trigonometric Functions Miscellaneous Exercise

MP Board Class 12th Maths Chapter 3 Matrices

Chapter 3 Matrices Ex 3.1

Chapter 3 Matrices Ex 3.2

Chapter 3 Matrices Ex 3.3

Chapter 3 Matrices Ex 3.4

Chapter 3 Matrices Miscellaneous Exercise

MP Board Class 12th Maths Chapter 4 Determinants

Chapter 4 Determinants Ex 4.1

Chapter 4 Determinants Ex 4.2

Chapter 4 Determinants Ex 4.3

Chapter 4 Determinants Ex 4.4

Chapter 4 Determinants Ex 4.5

Chapter 4 Determinants Ex 4.6

Chapter 4 Determinants Miscellaneous Exercise

MP Board Class 12th Maths Chapter 5 Continuity and Differentiability

Chapter 5 Continuity and Differentiability Ex 5.1

Chapter 5 Continuity and Differentiability Ex 5.2

Chapter 5 Continuity and Differentiability Ex 5.3

Chapter 5 Continuity and Differentiability Ex 5.4

Chapter 5 Continuity and Differentiability Ex 5.5

Chapter 5 Continuity and Differentiability Ex 5.6

Chapter 5 Continuity and Differentiability Ex 5.7

Chapter 5 Continuity and Differentiability Ex 5.8

Chapter 5 Continuity and Differentiability Miscellaneous Exercise

MP Board Class 12th Maths Chapter 6 Application of Derivatives

Chapter 6 Application of Derivatives Ex 6.1

Chapter 6 Application of Derivatives Ex 6.2

Chapter 6 Application of Derivatives Ex 6.3

Chapter 6 Application of Derivatives Ex 6.4

Chapter 6 Application of Derivatives Ex 6.5

Chapter 6 Application of Derivatives Miscellaneous Exercise

MP Board Class 12th Maths Chapter 7 Integrals

Chapter 7 Integrals Ex 7.1

Chapter 7 Integrals Ex 7.2

Chapter 7 Integrals Ex 7.3

Chapter 7 Integrals Ex 7.4

Chapter 7 Integrals Ex 7.5

Chapter 7 Integrals Ex 7.6

Chapter 7 Integrals Ex 7.7

Chapter 7 Integrals Ex 7.8

Chapter 7 Integrals Ex 7.9

Chapter 7 Integrals Ex 7.10

Chapter 7 Integrals Ex 7.11

Chapter 7 Integrals Miscellaneous Exercise

MP Board Class 12th Maths Chapter 8 Application of Integrals

Chapter 8 Application of Integrals Ex 8.1

Chapter 8 Application of Integrals Ex 8.2

Chapter 8 Application of Integrals Miscellaneous Exercise

MP Board Class 12th Maths Chapter 9 Differential Equations

Chapter 9 Differential Equations Ex 9.1

Chapter 9 Differential Equations Ex 9.2

Chapter 9 Differential Equations Ex 9.3

Chapter 9 Differential Equations Ex 9.4

Chapter 9 Differential Equations Ex 9.5

Chapter 9 Differential Equations Ex 9.6

Chapter 9 Differential Equations Miscellaneous Exercise

MP Board Class 12th Maths Chapter 10 Vector Algebra

Chapter 10 Vector Algebra Ex 10.1

Chapter 10 Vector Algebra Ex 10.2

Chapter 10 Vector Algebra Ex 10.3

Chapter 10 Vector Algebra Ex 10.4

Chapter 10 Vector Algebra Miscellaneous Exercise

MP Board Class 12th Maths Chapter 11 Three Dimensional Geometry

Chapter 11 Three Dimensional Geometry Ex 11.1

Chapter 11 Three Dimensional Geometry Ex 11.2

Chapter 11 Three Dimensional Geometry Ex 11.3

Chapter 11 Three Dimensional Geometry Miscellaneous Exercise

MP Board Class 12th Maths Chapter 12 Linear Programming

Chapter 12 Linear Programming Ex 12.1

Chapter 12 Linear Programming Ex 12.2

Chapter 12 Linear Programming Miscellaneous Exercise

MP Board Class 12th Maths Chapter 13 Probability

Chapter 13 Probability Ex 13.1

Chapter 13 Probability Ex 13.2

Chapter 13 Probability Ex 13.3

Chapter 13 Probability Ex 13.4

Chapter 13 Probability Ex 13.5

Chapter 13 Probability Miscellaneous Exercise

[ncertmcq]

Relations and Functions Class 12 Notes Maths Chapter 1