The document provides formulas and concepts related to mathematics for MBA entrance exams. It covers topics such as number systems, arithmetic, algebra, geometry, modern math, averages, and more. Formulas are presented for laws of indices, factor theory, divisibility rules, algebraic formulas, and remainder/modular arithmetic. Examples are provided to illustrate how to apply the formulas and concepts. The goal is to compile important math formulas and fundamentals in one place to help students preparing for MBA entrance exams like CAT, XAT, and others.
Reading of the Introductions of William Jones and Buhler's ManusmritiAbhishek Sangavikar
This presentation is on the much maligned text in both India and in the Europe. Given the well known fact that Manusmriti was the 1st ever translated text into English Language, I'm focusing on the very reception of the text by European Indologists and Philisophers, What and How the text has been received, read, interpreted and translated? Is there any Epistemological violence involved in the reading of Manusmriti-- And if yes--how tto confront that forced violence? and what new insights can be seen in the text that leads us to the realm of heterogeneous culture of India which celebrates jagat against world & heaven; actional life against ideological life; nuanced jatis against secularism/ singularity/ universalism; dharma against religion/ god/ sovereignty.
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Reading of the Introductions of William Jones and Buhler's ManusmritiAbhishek Sangavikar
This presentation is on the much maligned text in both India and in the Europe. Given the well known fact that Manusmriti was the 1st ever translated text into English Language, I'm focusing on the very reception of the text by European Indologists and Philisophers, What and How the text has been received, read, interpreted and translated? Is there any Epistemological violence involved in the reading of Manusmriti-- And if yes--how tto confront that forced violence? and what new insights can be seen in the text that leads us to the realm of heterogeneous culture of India which celebrates jagat against world & heaven; actional life against ideological life; nuanced jatis against secularism/ singularity/ universalism; dharma against religion/ god/ sovereignty.
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
This article is all about what AI trends will emerge in the field of creative operations in 2024. All the marketers and brand builders should be aware of these trends for their further use and save themselves some time!
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
Unlocking the Power of ChatGPT and AI in Testing - A Real-World Look, present...Applitools
During this webinar, Anand Bagmar demonstrates how AI tools such as ChatGPT can be applied to various stages of the software development life cycle (SDLC) using an eCommerce application case study. Find the on-demand recording and more info at https://applitools.info/b59
Key takeaways:
• Learn how to use ChatGPT to add AI power to your testing and test automation
• Understand the limitations of the technology and where human expertise is crucial
• Gain insight into different AI-based tools
• Adopt AI-based tools to stay relevant and optimize work for developers and testers
* ChatGPT and OpenAI belong to OpenAI, L.L.C.
1. presents
MATH FORMULAS
& FUNDAS
For CAT, XAT & Other MBA Entrance Examinations
Ravi Handa
Avinash Maurya
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2. Presented by
Revolutionizing….
….Management Education
In association with
3. WHAT WE DO?
VIDEOS CONTENT DEVELOPMENT
The pool of 50 videos on our Articles / Books / Questions on Math,
website covers the entire Math General Knowledge, Group Discussion (GD)
syllabus for MBA Entrance Exams & Personal Interviews (PI)
(CAT/ XAT / MAT and others)
[Absolutely FREE !! ]
DOUBT CLEARING LIVE EVENTS
Get your doubts clarified on our Quizzes for corporates, schools, colleges
and clubs.
forum by expert Math, DI, LR
faculties.
[Absolutely FREE !!]
>>Click on the topics to visit their respective webpage <<
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4. Preface How to use the eBook effectively
On more than one occasion, students have come to me
Best View Details:-
asking for some sort of list of important formulas for CAT,
For Adobe Reader: Follow the path given below;
XAT and other MBA Entrance Examinations. The best
Toolbar>>View>>Full Screen Mode
solution that I could suggest to them was to look at any Or Ctrl+L
good MBA-prep book or material from a coaching Note: Click on the Topic Names in Contents to go to their
institute. More often than not, they have the formulas at respective pages. Contents are hyperlinked to different pages in
the beginning or the end or scattered throughout the the same document.
chapter. Till now, there exists no comprehensive list /
book of formulas and fundas. Bookmark Details:-
So, Avinash and I thought of making one. Once the idea Bookmarks (on left) are given to navigate between the topics
gathered some momentum various management training faster and more easily.
portals came on board to help us distribute it via their Note: Bookmarks won’t appear in Full Screen mode.
channels. We hope that our effort will not be in vain and
it would prove beneficial for students. Home page:-
We would love to hear what you think of the eBook. For Click on “Home” (at the bottom of every page) to go to the main
that you can use the feedback page or contact us directly. page titled HOME. All topics under HOME are hyperlinked to the
Our contact details are given at the end of the eBook. content page of that topic.
Ravi Handa & Avinash Maurya Suppose, one wants to go from Algebra to Geometry. Click
“Home” and then click Geometry. This will take one from any
page in Algebra to the content page of Geometry.
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5. HOME
Number System
Arithmetic
Arithmetic
Algebra
Geometry
Modern Math
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6. Glossary
Laws of Indices
Last digit of an
HCF and LCM
Factor Theory
Divisibility Rules
Algebraic Formulae
Remainder / Modular Arithmetic
Base System Concepts
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7. Glossary Absolute value: Absolute value of x (written as |x|) is the
distance of 'x' from 0 on the number line. |x| is always
Natural Numbers: 1, 2, 3, 4….. positive. |x| = x for x > 0 OR -x for x < 0
Whole Numbers: 0, 1, 2, 3, 4…..
Funda: The product of ‘n’ consecutive natural numbers
Integers: ….-2, -1, 0, 1, 2 ….. is always divisible by n!
Rational Numbers: Any number which can be expressed Funda: Square of any natural number can be written in
as a ratio of two integers for example a p/q format where the form of 3n or 3n+1. Also, square of any natural
‘p’ and ‘q’ are integers. Proper fraction will have (p<q) number can be written in the form of 4n or 4n+1.
and improper fraction will have (p>q)
Funda: Square of a natural number can only end in 0, 1,
Factors: A positive integer ‘f’ is said to be a factor of a 4, 5, 6 or 9. Second last digit of a square of a natural
given positive integer 'n' if f divides n without leaving a number is always even except when last digit is 6. If the
remainder. e.g. 1, 2, 3, 4, 6 and 12 are the factors of 12. last digit is 5, second last digit has to be 2.
Prime Numbers: A prime number is a positive number Funda: Any prime number greater than 3 can be written
which has no factors besides itself and unity. as 6k ±1.
Composite Numbers: A composite number is a number Funda: Any two digit number ‘pq’ can effectively be
which has other factors besides itself and unity. written as 10p+q and a three digit number ‘pqr’ can
effectively be written as 100p+10q+r.
Factorial: For a natural number 'n', its factorial is defined
as: n! = 1 x 2 x 3 x 4 x .... x n (Note: 0! = 1)
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8. Laws of Indices Last digit of an
n(Right)
1 2 3 4 Cyclicity
a(Down)
0 0 0 0 0 1
( ) 1 1 1 1 1 1
2 2 4 8 6 4
( ) 3 3 9 7 1 4
√
4 4 6 4 6 2
5 5 5 5 5 1
6 6 6 6 6 1
( ) 7 7 9 3 1 4
√ 8 8 4 2 6 4
9 9 1 9 1 2
Funda: If am = an, then m = n Funda: The fifth power of any number has the same
Funda: If a = b and m ≠ 0;
m m units place digit as the number itself.
Then a = b if m is Odd
Or a = ± b if m is Even
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9. HCF and LCM Factor Theory
For two numbers, HCF x LCM = product of the two. If N = xaybzc where x, y, z are prime factors. Then,
HCF of Fractions = Number of factors of N = P = (a + 1)(b + 1)(c + 1)
– – –
LCM of Fractions = Sum of factors of N =
Number of ways N can be written as product of two
Funda: If a, b and c give remainders p, q and r factors = P/2 or (P+1)/2 if P is even or odd respectively
respectively, when divided by the same number H,
then H is HCF of (a-p), (b-q), (c-r) The number of ways in which a composite number can be
resolved into two co-prime factors is 2m-1, where m is the
Funda: If the HCF of two numbers ‘a’ and ‘b’ is H, then, number of different prime factors of the number.
the numbers (a+b) and (a-b) are also divisible by H.
Number of numbers which are less than N and co-prime
Funda: If a number N always leaves a remainder R
to ( ) ( )( )( ) {Euler’s Totient}
when divided by the numbers a, b and c, then N = LCM
(or a multiple of LCM) of a, b and c + R.
Funda: If N = (2)a(y)b(z)c where x, y, z are prime factors
Relatively Prime or Co-Prime Numbers: Two positive Number of even factors of N = (a)(b+1)(c+1)
integers are said to be relatively prime to each other if Number of odd factors of N = (b+1)(c+1)
their highest common factor is 1.
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10. Divisibility Rules Algebraic Formulae
A number is divisible by:
a3 ± b3 = (a ± b)(a2 + ab + b2). Hence, a3 ± b3 is divisible by
2, 4 & 8 when the number formed by the last, last two, (a ± b) and (a2 ± ab + b2).
last three digits are divisible by 2,4 & 8 respectively. an - bn = (a – b)(an-1 + an-2b+ an-3b2 + ... + bn-1)[for all n].
3 & 9 when the sum of the digits of the number is Hence, an - bn is divisible by a - b for all n.
divisible by 3 & 9 respectively.
an - bn = (a + b)(an-1 – an-2b + an-3b2 ... – bn-1)[n-even]
11 when the difference between the sum of the digits in
Hence, an - b" is divisible by a + b for even n.
the odd places and of those in even places is 0 or a
multiple of 11. an + bn = (a + b)(an-1 – an-2b + an-3b2 + ... + bn-1)[n-odd]
6, 12 & 15 when it is divisible by 2 and 3, 3 and 4 & 3 and Hence, an + bn is divisible by a + b for odd n.
5 respectively. a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - ac - bc)
7, if the number of tens added to five times the number Hence, a3 + b3 + c = 3abc if a + b + c = 0
of units is divisible by 7.
13, if the number of tens added to four times the number
of units is divisible by 13.
19, if the number of tens added to twice the number of
units is divisible by 19.
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11. Remainder / Modular Arithmetic Example: * + * +
[ ] * + [ ] * + * + * + * +
[ ] * + [ ] * + * +
Case 1 – When the dividend (M) and divisor (N) have a {Such that 3x+5y=1}
factor in common (k) Valid values are x = -3 and y = 2
* + * + * + * +
Example: * + * + 0
Case 3 – Remainder when ( )
Case 2 – When the divisor can be broken down into is divided by ( ) the remainder is ( )
smaller co-prime factors.
* + * + {HCF (a,b) = 1}
Funda: If f(a) = 0, (x-a) is a factor of f(x)
Let * + * +
* + {Such that ax+by = 1}
Continued >>
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12. Remainder Related Theorems Fermat’s Theorem:
Euler’s Theorem: If N is a prime number and M and N are co-primes
Number of numbers which are less than N =
* +
and co-prime to it are
( ) ( )( )( ) * +
If M and N are co-prime ie HCF(M,N) = 1
Example: * + * +
( )
* +
Wilson’s Theorem
Example: * + If N is a prime number
( )
( 0) 0( )( )( ) * +
( )
( 0) 0 * +
* + * + Example: * + 0 * +
* + * + * +
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13. Base System Concepts Converting from base ‘n’ to decimal
Decimal (pqrst)n = pn4 + qn3 + rn2 + sn + t
Binary Hex
0 0000 0 Converting from decimal to base ‘n’
1 0001 1 # The example given below is converting from 156 to
2 0010 2 binary. For this we need to keep dividing by 2 till we get
3 0011 3 the quotient as 0.
4 0100 4 2)156 0
2)78 0
5 0101 5 2)39 1
2)19 1
6 0110 6 2)9 1
7 0111 7 2)4 0
2)2 0
8 1000 8 2)1 1
0
9 1001 9
10 1010 A Starting with the bottom remainder, we read the
11 1011 B sequence of remainders upwards to the top. By that, we
12 1100 C get 15610 = 100111002
13 1101 D
14 1110 E Funda: (pqrst)n x n2 = (pqrst00)n
15 1111 F
(pqrst)n x n3 = (pqrst000)n
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14. Arithmetic
Interest
Mixtures & Alligation
Time & Work
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15. Averages Funda: AM ≥ GM ≥ HM is always true. They will be
equal if all elements are equal to each other. If I have
u ts just two values then GM2 = AM x HM
Simple Average =
u r ts
Funda: The sum of deviation (D) of each element with
Weighted Average = respect to the average is 0
������ ������ ������������������������ ������ ������������������������
Arithmetic Mean = (a1 + a2 + a3 ….an) / n
������ ������������������������ ������ ������������������������ 0
Geometric Mean = ������������������������������������������������������
Funda: ������������������������ ������������������������������������������������ ������������������
������������ ������������ ������������������������������������������������
Harmonic Mean =
For two numbers a and b
Median of a finite list of numbers can be found by
AM = (a + b)/2 arranging all the observations from lowest value to
GM = √ highest value and picking the middle one.
HM =
Mode is the value that occurs most often
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16. Percentages
������������������������
Fractions and their percentage equivalents: Funda: r% change can be nullified by % change
������������������ ������
in another direction. Eg: An increase of 25% in prices
Fraction %age Fraction %age can be nullified by a reduction of [100x25/(100+25)] =
1/2 50% 1/9 11.11% 20% reduction in consumption.
Funda: If a number ‘x’ is successively changed by a%,
1/3 33.33% 1/10 10%
b%, c%...
1/4 25% 1/11 9.09% ������ ������ ������
Final value = ������ ( )( )( )
1/5 20% 1/12 8.33%
Funda: The net change after two successive changes of
1/6 16.66% 1/13 7.69% ������������
a% and b% is (������ ������
00
)%
1/7 14.28% 1/14 7.14%
1/8 12.5% 1/15 6.66%
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17. Interest Growth and Growth Rates
Amount = Principal + Interest Absolute Growth = Final Value – Initial Value
Simple Interest = PNR/100 Growth rate for one year period =
v u – t u
n x 100
Compound Interest = P(1+ ) –P t u
n v u – t u
Population formula P’ = P(1 ± ) SAGR or AAGR = x 100
t u
n
Depreciation formula = Initial Value x (1 – ) v u – t u
CAGR= ( ) –1
t u
Funda: SI and CI are same for a certain sum of money Funda: If the time period is more than a year, CAGR <
(P) at a certain rate (r) per annum for the first year. The
AAGR. This can be used for approximating the value of
difference after a period of two years is given by
CAGR instead of calculating it.
������������
∆
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18. Profit and Loss Mixtures and Alligation
r – st r Successive Replacement – Where a is the original
%Profit / Loss = 00 quantity, b is the quantity that is replaced and n is the
t u
number of times the replacement process is carried out,
In case false weights are used while selling, then
( )
% Profit = ( ) 00
Alligation – The ratio of the weights of the two items
r r – r mixed will be inversely proportional to the deviation of
Discount % = x 100 attributes of these two items from the average attribute
r
of the resultant mixture
u tt rst t
Funda: Effective Discount after successive discount of =
������������ u tt s t
a% and b% is (a + b – ). Effective Discount when
you buy x goods and get y goods free is x 100.
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19. Ratio and Proportion
Funda: If a/b = c/d = e/f = k
Compounded Ratio of two ratios a/b and c/d is ac/bd,
Duplicate ratio of a : b is a2 : b2 ������ ������ ������
=k
Triplicate ratio of a : b is a3 : b3
Sub-duplicate ratio of a : b is a : b ������������ ������������ ������������
Sub-triplicate ratio of a : b is ³ a : ³ b =k
Reciprocal ratio of a : b is b : a p q r
������������ ������ ������������ ������ ������ ������ ������
Componendo and Dividendo = kn
������������ ������������ ������ ������
If ≠
Four (non-zero) quantities of the same kind a,b,c,d are
said to be in proportion if a/b = c/d. Given two variables x and y, y is (directly) proportional to
x (x and y vary directly, or x and y are in direct variation)
The non-zero quantities of the same kind a, b, c, d.. are if there is a non-zero constant k such that y = kx. It is
said to be in continued proportion if a/b = b/c = c/d. denoted by
Proportion Two variables are inversely proportional (or varying
a, b, c, d are said to be in proportion if inversely, or in inverse variation, or in inverse
a, b, c, d are said to be in continued proportion if proportion or reciprocal proportion) if there exists a non-
zero constant k such that y = k/x.
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20. Time Speed and Distance Funda: Given that the distance between two points is
Speed = Distance / Time constant, then
1 kmph = 5/18 m/sec; 1 m/sec = 18/5 kmph If the speeds are in Arithmetic Progression, then
the times taken are in Harmonic Progression
SpeedAvg = = If the speeds are in Harmonic Progression, then
the times taken are in Arithmetic Progression
If the distance covered is constant then the average
speed is Harmonic Mean of the values (s1,s2,s3….sn) For Trains, time taken =
SpeedAvg = For Boats,
SpeedUpstream = SpeedBoat – SpeedRiver
SpeedAvg = (for two speeds) SpeedDownstream = SpeedBoat + SpeedRiver
If the time taken is constant then the average speed is SpeedBoat = (SpeedDownstream + SpeedUpstream) / 2
Arithmetic Mean of the values (s1,s2,s3….sn)
SpeedRiver = (SpeedDownstream – SpeedUpstream) / 2
SpeedAvg =
For Escalators,The difference between escalator
SpeedAvg = (for two speeds) problems and boat problems is that escalator can go
either up or down.
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21. Races & Clocks Two people are running on a circular track of length L
with speeds a and b in the opposite direction
Linear Races Time for 1st meeting =
Winner’s distance = Length of race Time for 1st meeting at the starting point =
LCM ( )
Loser’s distance = Winner’s distance – (beat distance +
start distance) Three people are running on a circular track of length L
with speeds a, b and c in the same direction
Winner’s time = Loser’s time – (beat time + start time) Time for 1st meeting = LCM ( )
st
Deadlock / dead heat occurs when beat time = 0 or beat Time for 1 meeting at the starting point =
distance = 0 LCM ( )
Circular Races
Clocks To solve questions on clocks, consider a circular
Two people are running on a circular track of length L track of length 360 . The minute hand moves at a speed
with speeds a and b in the same direction of 6 per min and the hour hand moves at a speed of ½
per minute.
Time for 1st meeting =
Time for 1st meeting at the starting point = Funda: Hands of a clock coincide (or make 180 ) 11
LCM ( ) times in every 12 hours. Any other angle is made 22
times in every 12 hours.
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22. Time and Work
If a person can do a certain task in t hours, then in 1 hour Funda: If A does a particular job in ‘a’ hours, B does the
he would do 1/t portion of the task. same job in ‘b’ hours and ABC together do the job in ‘t’
A does a particular job in ‘a’ hours and B does the same hours, then
job in ‘b’ hours, together they will take hours ������������������
C alone can do it in hours
������������ ������������ ������������
������������
A does a particular job in ‘a’ hours more than A and B A and C together can do it in hours
������ ������
combined whereas B does the same job in ‘b’ hours more ������������
B and C together can do it in hours
than A and B combined, then together they will take √ ������ ������
hours to finish the job.
Funda: If the objective is to fill the tank, then the Inlet
pipes do positive work whereas the Outlet pipes do
Funda: A does a particular job in ‘a’ hours, B does the
negative work. If the objective is to empty the tank,
same job in ‘b’ hours and C does the same job in ‘c’
������������������ then the Outlet pipes do positive work whereas the
hours, then together they will take hours.
������������ ������������ ������������ Inlet Pipes do negative work.
Funda: If A does a particular job in ‘a’ hours and A&B
together do the job in ‘t’ hours, the B alone will take
������������
hours.
������ ������
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23. Algebra Quadratic &
Other
Equations
Inequalities Logarithm
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24. Quadratic and Other Equations Funda: If c = a, then roots are reciprocal of each other
Funda: If b =0, then roots are equal in magnitude but
For a quadratic equation, ax2 + bx + c = 0, its roots opposite in sign.
±√
Funda: Provided a, b and c are rational
If one root is p + iq, other root will be p – iq
Sum of roots = If one root is p + ������, other root will be p – ������
Product of roots =
Cubic equation ax3+bx2+cx+d = 0
Discriminant ∆ Sum of the roots = - b/a
Sum of the product of the roots taken two at a
Condition Nature of Roots time = c/a
Product of the roots = -d/a
∆ 0 Complex Conjugate
Biquadratic equation ax4+bx3+cx2+dx+e = 0
∆ 0 Real and equal
Sum of the roots = - b/a
∆ > 0 and a perfect square Rational and unequal Sum of the product of the roots taken three at a
time = c/a
∆ > 0 and not a perfect square Irrational and unequal Sum of the product of the roots taken two at a
time = -d/a
Product of the roots = e/a
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25. Inequalities Quadratic Inequalities
If a > b and c > 0, (x – a) (x – b) > 0 {a < b}
a+c>b+c (x < a) U (x > b)
a-c>b-c
(x – a) (x – b) < 0 {a > b}
ac > bc
a/c > b/c a<x<b
If a, b ≥ 0, then an > bn and 1/an< 1/bn, where n is positive. For any set of positive numbers: AM≥GM≥HM
a < b and x > 0, then (a1+a2+ ….+an)/n ≥(a1.a2. …..an)1/n
If a and b are positive quantities, then
a > b and x > 0, then
≥ √
Modular Inequalities
|x- y| = |y - x| If a,b,c,d are positive quantities, then
|x. y| = |x| . |y|
|x+ y| < |x| + |y|
+ + + ≥
|x+ y| > |x| - |y| ≥
Continued >>
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26. If a,b,c …. k are n positive quantities and m is a natural Funda: If the sum of two or more positive quantities is
number, then constant, their product is greatest when they are equal
and if their product is constant then their sum is the
( ) least when the numbers are equal.
( ) If x + y = k, then xy is greatest when x = y
If xy = k, then x + y is least when x = y
������������ ������������ ������ ������ ������
Funda: >( ) [������ ≤ 0 ������������ ������ ≥ ]
������������ ������������ ������ ������ ������
( ) [0 ������ ]
������
Funda: For any positive integer n, 2≤ (
������
) ≤3
������ ������ ������
Funda: ambncp……..will be greatest when = =
������ ������ ������
Funda: If a > b and both are natural numbers, then
������������ ������ ������ {Except }
Funda: (n!)2 ≥ nn
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27. Logarithm
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( )
( )
( )
0
Ln x means
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29. Lines and Angles
Sum of the angles in a straight line is 180°
Vertically opposite angles are congruent (equal).
If any point is equidistant from the endpoints of a
segment, then it must lie on the perpendicular bisector
When two parallel lines are intersected by a transversal,
corresponding angles are equal, alternate angles are
equal and co-interior angles are supplementary. (All
acute angles formed are equal to each other and all
Funda: The ratio of intercepts formed by a transversal
obtuse angles are equal to each other)
intersecting three parallel lines is equal to the ratio of
corresponding intercepts formed by any other
transversal.
������ ������ ������
������ ������ ������
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30. Triangles Area of a triangle:
Sum of interior angles of a triangle is 180° and sum of
exterior angles is 360°.
Exterior Angle = Sum of remote interior angles.
Sum of two sides is always greater than the third side and
the difference of two sides is always lesser than the third
side.
Side opposite to the biggest angle is longest and the side
opposite to the smallest angle is the shortest.
= ½ x Base x Height
= ½ x Product of sides x Sine of included angle
= ( )( )( ); here s is the semi perimeter
[s = (a+b+c)/2 ]
=rxs [r is radius of incircle]
= [R is radius of circumcircle]
Continued >>
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31. A Median of a triangle is a line segment joining a vertex Funda: Centroid and Incenter will always lie inside the
to the midpoint of the opposing side. The three medians triangle.
intersect in a single point, called the Centroid of the
triangle. Centroid divides the median in the ratio of 2:1 - For an acute angled triangle, the Circumcenter
and the Orthocenter will lie inside the triangle.
An Altitude of a triangle is a straight line through a vertex - For an obtuse angled triangle, the Circumcenter
and perpendicular to the opposite side or an extension of and the Orthocenter will lie outside the triangle.
the opposite side. The three altitudes intersect in a single - For a right angled triangle the Circumcenter will
point, called the Orthocenter of the triangle. lie at the midpoint of the hypotenuse and the
A Perpendicular Bisector is a line that forms a right angle Orthocenter will lie at the vertex at which the
with one of the triangle's sides and intersects that side at angle is 90°.
its midpoint. The three perpendicular bisectors intersect Funda: The orthocenter, centroid, and circumcenter
in a single point, called the Circumcenter of the triangle. always lie on the same line known as Euler Line.
It is the center of the circumcircle which passes through
all the vertices of the triangle. - The orthocenter is twice as far from the
centroid as the circumcenter is.
An Angle Bisector is a line that divides the angle at one of - If the triangle is Isosceles then the incenter lies
the vertices in two equal parts. The three angle bisectors on the same line.
intersect in a single point, called the Incenter of the - If the triangle is equilateral, all four are the same
triangle. It is the center of the incircle which touches all point.
sides of a triangle.
Continued >>
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32. Theorems
Apollonius’ Theorem: AB2 + AC2 = 2 (AD2 + BD2)
Mid Point Theorem: The line joining the midpoint of any
two sides is parallel to the third side and is half the length
of the third side.
Interior Angle Bisector Theorem: AE/ED = BA/BD
Basic Proportionality Theorem: If DE || BC, then AD/DB
= AE/EC
Continued >>
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33. Special Triangles √
Height = x Side
Right Angled Triangle:
√
Area = x Side2
Inradius = 1/3 Height
Circumradius = 2/3 Height.
∆ABC ∆ ADB ∆ BDC Isosceles Triangle:
2
BD = AD x DC and AB x BC = BD X DC
Equilateral Triangle:
Angles equal to opposite sides are equal.
Area = √
All angles are equal to 60°. All sides are equal also. Continued >>
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34. 30°-60°-90° Triangle 30°-30°-120° Triangle
√
Area = * x2
√ 2
Area = * x Continued >>
45°-45°-90° Triangle
2
Area = x /2
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35. Similarity of Triangles
Two triangles are similar if their corresponding angles are
congruent and corresponding sides are in proportion.
Tests of similarity: (AA / SSS / SAS)
For similar triangles, if the sides are in the ratio of a:b
Corresponding heights are in the ratio of a:b
Corresponding medians are in the ratio of a:b
Circumradii are in the ratio of a:b
Inradii are in the ratio of a:b
Perimeters are in the ratio of a:b
Areas are in the ratio a2 : b2
Congruency of Triangles
Two triangles are congruent if their corresponding sides
and angles are congruent.
Tests of congruence: (SSS / SAS / AAS / ASA)
All ratios mentioned in similar triangle are now 1:1
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36. Polygons Quadrilaterals:
Sum of interior angles = (n-2) x 180° = (2n-4) x 90°
Sum of exterior angles = 360°
( )
Number of diagonals = nC2 – n =
Number of triangles which can be formed by the vertices
= nC3
Sum of the interior angles = Sum of the exterior angles =
Regular Polygon:
360°
If all sides and all angles are equal, it is a regular polygon. Area for a quadrilateral is given by ½ d1 d2 Sin .
All regular polygons can be inscribed in or circumscribed
about a circle. Cyclic Quadrilateral
Area = ½ x Perimeter x Inradius {Inradius is the
perpendicular from centre to any side}
( )
Each Interior Angle = ; Exterior = 360°/n
Continued >>
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37. If all vertices of a quadrilateral lie on the circumference of Parallelogram
a circle, it is known as a cyclic quadrilateral.
Opposite angles are supplementary
Area = ( )( )( )( ) where s is the
semi perimeter
Funda: Sum or product of opposite sides = Product of
diagonals
Opposite sides are parallel and congruent.
Opposite angles are congruent and consecutive angles
are supplementary.
Diagonals of a parallelogram bisect each other.
Perimeter = 2(Sum of adjacent sides);
Area = Base x Height = AD x BE
Funda: A parallelogram inscribed in a circle is always a
Funda: If a circle can be inscribed in a quadrilateral, its
Rectangle. A parallelogram circumscribed about a
area is given by = √������������������������
circle is always a Rhombus.
Continued >>
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38. A parallelogram with all sides equal is a Rhombus. Its
Funda: Each diagonal divides a parallelogram in two diagonals bisect at 90°.
triangles of equal area.
Funda: Sum of squares of diagonals = Sum of squares Perimeter = 4a; Area = ½ d1 d2 ; Area = d x √ ( )
of four sides
Rectangle
2 2 2 2 2 2
AC + BD = AB + BC + CD + DA
A parallelogram with all angles equal (90°) is a Rectangle.
Funda: A Rectangle is formed by intersection of the Its diagonals are congruent.
four angle bisectors of a parallelogram.
Perimeter = 2(l+b); Area = lb
Rhombus
Square
A parallelogram with sides equal and all angles equal is a
square. Its diagonals are congruent and bisect at 90°.
Perimeter = 4a; Area = a2; Diagonals = a√
Funda: From all quadrilaterals with a given area, the
square has the least perimeter. For all quadrilaterals
with a given perimeter, the square has the greatest
area.
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39. Kite Trapezium / Trapezoid
A quadrilateral with exactly one pair of sides parallel is
known as a Trapezoid. The parallel sides are known as
bases and the non-parallel sides are known as lateral
sides.
Two pairs of adjacent sides are congruent. Area = ½ x (Sum of parallel sides) x Height
The longer diagonal bisects the shorter diagonal at 90°. Median, the line joining the midpoints of lateral sides, is
half the sum of parallel sides.
Area = Product of Diagonals / 2
Funda: Sum of the squares of the length of the
diagonals = Sum of squares of lateral sides + 2 Product
of bases.
AC2 + BD2 = AD2 + BC2 + 2 x AB x CD
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40. Isosceles Trapezium
The non-parallel sides (lateral sides) are equal in length.
Angles made by each parallel side with the lateral sides
are equal.
Funda: If a trapezium is inscribed in a circle, it has to be
an isosceles trapezium. If a circle can be inscribed in a
trapezium, Sum of parallel sides = Sum of lateral sides.
Continued >>
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41. Hexagon (Regular) Number of diagonals = 9 {3 big and 6 small}
Length of big diagonals (3) = 2a
Length of small diagonals (6) = √ a
Funda: A regular hexagon can be considered as a
combination of six equilateral triangles. All regular
polygons can be considered as a combination of ‘n’
isosceles triangles.
2
Area of a Pentagon = 1.72 a
2
Area of an Octagon = 2(√ + 1) a
√ 2
Perimeter = 6a; Area = x a
Sum of Interior angles = 720°.
Each Interior Angle = 120°. Exterior = 60°
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42. Circles Chords / Arcs of equal lengths subtend equal angles.
Diameter = 2r; Circumference = 2 r; Area = r2
Chords equidistant from the centre of a circle are equal.
A line from the centre, perpendicular to a chord, bisects
the chord.
Equal chords subtend equal angles at the centre.
The diameter is the longest chord of a circle. Chord AB divides the circle into two parts: Minor Arc AXB
and Major Arc AYB
Measure of arc AXB = AOB =
Length (arc AXB) = x2 r
Area (sector OAXB) = = x r2
Area of Minor Segment = Shaded Area in above figure
Area of Sector OAXB - Area of ∆ OAB
A chord /arc subtends equal angle at any point on the
circumference and double of that at the centre. r2 [ – ]
0
Continued >>
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43. Properties of Tangents, Secants and Chords
PA x PB = PC x PD
= ½ [ m(Arc AC) – m(Arc BD) ]
The radius and tangent are perpendicular to each other.
There can only be two tangents from an external point,
which are equal in length PA = PB
PA x PB = PC x PD
= ½ [ m(Arc AC) + m(Arc BD) ]
Continued >>
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44. Properties (contd.) Alternate Segment Theorem
PA x PB = PC2
The angle made by the chord AB with the tangent at A
= ½ [ m(Arc AC) - m(Arc BC) ] (PQ) is equal to the angle that it subtends on the opposite
side of the circumference.
BAQ = ACB
Continued >>
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45. Common Tangents
No. of
Distance Between
Two Circles Common
Centers (d)
Tangents
One is
completely 0 < r1 - r2
inside other Length of the Direct Common Tangent (DCT)
Touch AD = BC = ( )
1 = r1 - r2
internally Length of the Transverse Common Tangent (TCT)
Intersect 2 r1 - r2 < d < r1 + r2 RT = SU = ( )
Touch Funda: The two centers(O and O’), point of intersection
3 = r1 + r2
externally
of DCTs (P)and point of intersection of TCTs (Q) are
One is
completely 4 > r1 + r2 collinear. Q divides OO’ in the ratio r1 : r2 internally
outside other whearea P divides OO’ in the ratio r1 : r2 externally.
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46. Solid Figures
Volume Total Surface Area Lateral / Curved Surface Area
Cube Side3 6 x Side2 4 x Side2
Cuboid LxBxH 2(LB + LH + BH) 2 (LH + BH)
Cylinder r2h 2 r (r + h) 2 rh
Cone (1/3) r2h r (r +L) rl {where L = √ }
Sphere (4/3) r3 r2 r2
3
Hemisphere (2/3) r r2 r2
Funda: There are 4 body diagonals in a cube / cuboid of length (√ x side) and √������ ������ respectively.
Continued >>
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47. Frustum / Truncated Cone Prism
It can be obtained by cutting a cone with a plane parallel
to the circular base.
It is a solid with rectangular vertical faces and bases as
congruent polygons (of n sides). It will have ‘2n’ Vertices;
‘n+2’ Faces and ‘3n’ Sides / Edges.
Volume = 1/3 h (R2 + r2 + Rr)
Lateral Surface Area = Perimeter x Height
Lateral Surface Area = (R+r) L
Total Surface Area = Perimeter x Height + 2 AreaBase
Total Surface Area = (R+r) L + (R2+r2)
Volume = AreaBase x Height
Continued >>
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48. Pyramid
Funda: If a sphere is inscribed in a cube of side a, the
radius of the sphere will be a/2. If a sphere is
circumscribed about a cube of side a, the radius of
the sphere will be √ a /2.
Funda: If a largest possible sphere is inscribed in a
cylinder of radius ‘a’ and height h, its radius r will be
r = h/2 {If 2a > h}
r=a {If 2a < h}
Funda: If a largest possible sphere is inscribed in a
cone of radius r and slant height equal to 2r, then the
It is a figure in which the outer surfaces are triangular and
radius of sphere = r/√
converge at a point known as the apex, which is aligned
directly above the centre of the base. Funda: If a cube is inscribed in a hemisphere of radius
Lateral Surface Area = ½ x Perimeter x Slant Height r, then the edge of the cube = r √
Total Surface Area = ½ x Perimeter x Slant Height +
AreaBase
Volume = ½ x AreaBase x Height
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49. Co-ordinate Geometry Slope(m) of a line is the tangent of the angle made by the
line with the positive direction of the X-Axis.
Distance between two points P(x1,y1) and Q(x2,y2) is given For a general equation ax + by + c = 0; slope (m) = -a/b.
by = ( ) ( ) For a line joining two points, P (x1,y1) and Q(x2,y2), the
slope(m) is =
If a point R (x,y) divides P(x1,y1) and Q(x2,y2) internally in
the ratio of m:n, the coordinates of R ie (x,y) are given by
Angle with X-
= , Slope(m) Type of line
Axis
If a point R (x,y) divides P(x1,y1) and Q(x2,y2) externally in > 0 (+ive) Rising Acute
the ratio of m:n, the coordinates of R ie (x,y) are given by 0 Parallel to X-Axis 0⁰
= , < 0 (-ive) Falling Obtuse
∞ Parallel to Y-Axis 90⁰
Equation of a line parallel to X-axis is y = a {Of X-Axis is y = 0}
Funda: The X axis divides the line joining P(x1,y1) and Equation of a line parallel to Y-Axis is x = a {Of Y-Axis is x = 0}
Q(x2,y2) in the ratio of y1 : y2
The intercept of a line is the distance between the point
Funda: The Y axis divides the line joining P(x1,y1) and where it cuts the X-Axis or Y-Axis and the origin. Y-
Q(x2,y2) in the ratio of x1 : x2 Intercept is often denoted with the letter ‘c’.
Continued >>
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50. Equation of a line For parallel lines, = 0°;
For parallel lines, = 90°;
General form: ax + by + c = 0
Distance of a point P (x1,y1) from a line ax + by + c = 0
Slope Intercept Form: Slope is m, y-intercept is c
y = mx + c d=| |
√
From origin, d = | |
Slope Point Form: Slope is m, point is x1,y1 √
y – y1 = m(x – x1) Distance between two parallel lines, ax + by + c1 = 0 and
ax + by + c2 = 0
Two Point Form: Two points are x1,y1 and x2,y2
d=| |
√
y – y1 =[ ](x – x1)
Two Intercept Form: X-intercept is a, Y-intercept is b. Funda: If we know three points A(x1,y1), B(x2,y2 ) and
C(x2,y2) of a parallelogram, the fourth point is given by
+ = 1 OR bx + ay = ab
(x1 + x3 – x2, y1 + y3 – y2)
Acute angle between two lines with slope m1 and m2 is
given by
Tan = | | Continued >>
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51. Triangle Circle
General Equation: x2 + y2 + 2gx + 2fy + c = 0
Centre is (-g, -f) and radius =
Centre is (h, k) and radius is r
( ) ( )
The vertices are P (x1,y1), Q(x2,y2 ) and R(x3,y3)
Centre is origin and radius is r
Incenter = { , } x 2 + y2 = r2
Centroid = { , }
Area = ½ [ x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2) ]
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52. Trigonometry Sin Cos Tan
00 0 1 0
sin = =
300 ⁄ √ ⁄ ⁄
√
cos = = 450 1
⁄ ⁄
√ √
tan = = 600 √ ⁄ ⁄ √
cosec = 900 1 0
s
Signs of T-ratios in Different Quadrants:
sec =
s
cot =
t
Some Basic Identities:
sin2 + cos2 = 1
1 + tan2 = sec2
1 + cot2 = cosec2
Continued >>
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53. Addition Formulae Trigonometric Rules
Sin (A + B) = sinA cos B + cosA sinB
Cos (A + B) = cos A cos B - sin A sin B
Tan (A + B) -
Subtraction Formulae
s
sin (A - B) = sin A cos B - cos A sin B Sine Rule: =s =
s
cos (A - B) = cos A cos B + sin A sin B
Cosine Rule: Cos A =
tan (A - B) =
Cos B =
Cos C =
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54. Set Binomial
Fundamentals Theorem
Modern Permutation
& Probability
Math Combination
Sequence
& Functions
Series
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55. Set Fundamentals Union of two sets is represented as A B and consists of
elements that are present in either Set A or Set B or both.
The number of elements in a set is called its cardinal Intersection of two sets is represented as A B and
number and is written as n(A). A set with cardinal number consists of elements that are present in both Set A and
0 is called a null set while that with cardinal number ∞ is Set B. n(A B) = n(A) + n(B) — n(A B)
called an infinite set.
Venn Diagram: A venn diagram is used to visually
Set A is said to be a subset of Set B if each and every represent the relationship between various sets. What do
element of Set A is also contained in Set B. Set A is said to each of the areas in the figure represent?
be a proper subset of Set B if Set B has at least one
element that is not contained in Set A. A set with ‘n’
elements will have 2n subsets (2n – 1 proper subsets)
The Universal set is defined as the set of all possible
objects under consideration.
Funda: Any set is a subset of itself, but not a proper
subset. The empty set, denoted by , is also a subset of
any given set X. The empty set is always a proper
subset, except of itself. Every other set is then a subset I – only A; II – A and B but not C; III – Only B; IV – A and C but not B;
V – A and B and C; VI – B and C but not A; VII – Only C
of the universal set.
n(A B C) = n(A) + n(B) + n(C) — n(A B) — n(A C) -
n(B C) + n(A B C)
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56. Binomial Theorem Some basic properties
For some basic values:
Funda: There is one more term than the power of the
( ) exponent, n. That is, there are terms in the expansion
( ) of (a + b)n.
( )
( ) Funda: In each term, the sum of the exponents is n,
( ) the power to which the binomial is raised.
( ) 0 0
Funda: The exponents of a start with n, the power of
Theorem the binomial, and decrease to 0. The last term has no
( ) factor of a. The first term has no factor of b, so
powers of b start with 0 and increase to n.
Funda: The coefficients start at 1 and increase
( )
through certain values about “half”-way and then
decrease through these same values back to 1.
Funda: To find the remainder when (x + y)n is divided
by x, find the remainder when yn is divided by x.
Funda: (1+x)n ≅ 1 + nx, when x<<1
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57. Permutation & Combination Linear arrangement of ‘r’ out of 'n' distinct items (nPr):
The first item in the line can be selected in 'n' ways AND
When two tasks are performed in succession, i.e., they the second in (n — 1) ways AND the third in (n — 2) ways
are connected by an 'AND', to find the total number of AND so on. So, the total number of ways of arranging 'r'
ways of performing the two tasks, you have to MULTIPLY items out of 'n' is
the individual number of ways. When only one of the two (n)(n - 1)(n — 2)...(n - r + 1) =
( )
tasks is performed, i.e. the tasks are connected by an
'OR', to find the total number of ways of performing the Circular arrangement of 'n' distinct items: Fix the first
two tasks you have to ADD the individual number of item and then arrange all the other items linearly with
ways. respect to the first item. This can be done in (n — 1)!
ways.
Eg: In a shop there are‘d’ doors and ‘w’ windows.
( )
Case1: If a thief wants to enter via a door or window, he Funda: In a necklace, it can be done in ways.
can do it in – (d+w) ways.
Case2: If a thief enters via a door and leaves via a Selection of r items out of 'n' distinct items (nCr): Arrange
of r items out of n = Select r items out of n and then
window, he can do it in – (d x w) ways.
arrange those r items on r linear positions.
n
Pr =nCr x r! n
Cr = = ( )
Continued >>
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58. Dearrangement If 'n' things are arranged in a row, the Partitioning
number of ways in which they can, be deranged so that
none of them occupies its original place is n+r-1
‘n’ similar items in No restrictions Cr-1
( ) ‘r’ distinct groups
( ) No group empty n-1
Cr-1
0
‘n’ distinct items in No restrictions rn
‘r’ distinct groups
Funda: Number of ways of arranging 'n' items out of Arrangement in a ( )
which `p' are alike, 'q' are alike, 'r' are alike in a line group important ( )
������
is given by = ‘n’ similar items in List the cases and then find out
������ ������ ������
‘r’ similar groups in how many ways is each case
possible
‘n’ similar items in List the cases and then find out
‘r’ similar groups in how many ways is each case
possible
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59. Probability
Funda: If the probability of an event occurring is P,
then the probability of that event occurring ‘r’ times in
P(A) = r n-r
‘n’ trials is = nCr x P x (1-P)
For Complimentary Events: P(A) + P(A’) = 1
For Exhaustive Events: P(A) + P(B) +P(C)… = 1
Odds
Addition Rule:
Odds in favor =
P (A U B) = P(A) + P(B) – P(A B)
For Mutually Exclusive Events P(A B) = 0 Odds against =
P (A U B) = P(A) + P(B)
Multiplication Rule:
P(A B) = P(A) P(B/A) = P(B) P(A/B)
For Independent Events P(A/B) = P(B) and P(B/A) = P(B)
P(A B) = P(A).P(B)
P (A U B) = P(A) + P(B) – P(A).P(B)
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60. Sequence, Series & Progression Sum of squares of first n natural numbers
( )( )
Arithmetic Progression
( ) Sum of cubes of first n natural numbers
( ) [ ( ) ] ( )
* +
������������ ������
Funda: Number of terms
������
Funda: Sum of first n odd numbers
( ������ ) ������
Geometric Progression
Funda: Sum of first n even numbers
������ ������(������ )
( )
Funda: If you have to consider 3 terms in an AP,
consider {a-d,a,a+d}. If you have to consider 4 terms,
Sum till infinite terms = (Valid only when r<1) consider {a-3d,a-d,a+d,a+3d}
Sum of first n natural numbers Funda: If all terms of an AP are multiplied with k or
divided with k, the resultant series will also be an AP
( )
with the common difference dk or d/k respectively.
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61. Functions One to One: Every element in the Domain has one and
only one image in the Co-Domain. Every element in Co-
Domain: Set of real and finite values that the Domain has one and only one pre-image in the Domain.
independent variable can take.
Many to One: If at least two elements in Domain have the
Range: Set of real and finite values that the dependent same image in the co-domain.
variable can have corresponding to the values of the
Onto Function: If for every element in the Co-Domain
independent variable
there is at least one pre-image in the Domain. In this case,
Co-Domain: Set of real and finite values that the Range = Co-Domain
dependent variable can have.
Into Function: If there is at least one element in the Co-
Domain which does not have a pre-image in the Domain.
Funda: Range is a subset of Co-Domain. Co-domain In this case, Range is a proper subset of Co-Domain.
may or may not have values which do not have a pre-
image in the domain. Even Function: f(x) is even if and only if f(-x) = f(x) for all
values of x. The graph of such a function is symmetric
Funda: It is not a function if for some value in the about the Y-Axis
domain, the relationship gives more than one value.
Eg: f(x) = √������ (At x = 4, f(x) could be both +2 and -2) Odd Function: f(x) is odd if and only if f(-x) = - f(x) for all
values of x. The graph is symmetric about the origin
Funda: Domain cannot have any extra value ie the
values at which the function does not exist. Funda: If f(x) is an odd function and f(0) exists
f(0) = 0
Continued >>
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62. Graphs If we consider f(x+2), it shifts left by 2 units
f(x) = |x|
If we consider –f(x), it gets mirrored in the X-Axis. If we consider f(x-2), it shifts right by 2 units.
Continued >>
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63. If we consider f(x) + 2, it shifts up by 2 units. If we consider f(2x) or 2f(x) ,the slope doubles and the
rise and fall become much sharper than earlier
If we consider f(x) – 2, it shifts down by 2 units. If we consider f(x/2) or ½ f(x), the slope halves and the
rise and fall become much flatter than earlier.
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