Tuesday, July 18, 2017

Best Books to Learn about Stocks Trading and Investing





Best Books to Learn about Stocks Trading and Investing


  1. Trading Psychology 2.0: From Best Practices to Best Processes Hardcover
  2. How to Make Money Trading the Ichimoku System: Guide to Candlestick Cloud Charts Paperback  – Nov 2015 +++++++
  3. Following the Trend: Diversified Managed Futures Trading
  4. Trading in the Zone by far my preferred choice.
  5. Reminiscences of a Stock Operator (Wiley Investment Classics) -  Jesse Livermore
  6. Market Wizards, Updated: Interviews With Top Traders
  7. Trend Following (Updated Edition): Learn to Make Millions in Up or Down Markets
  8. Security Analysis: Sixth Edition, Foreword by Warren Buffett (Security Analysis Prior Editions)
  9. Buffettology: The Previously Unexplained Techniques That Have Made Warren Buffett The Worlds
  10. Trading System and Methods
  11. Trade Your Way to Financial Freedom
 
 
Here are ten of the top trading books ever written summarized in one sentence:

#1 How to Make Money in Stocks: A Winning System in Good Times and Bad by William O'Neil: Buy only the best innovative growth stocks at the proper buy point out of price bases and let them run as far as they will go.


#2 Reminiscences of a Stock Operator by Edwin Lefevre: It’s a bull market you know and the big money is made in holding good positions over the long term and not trying to trade inside the day to day noise.


#3 Market Wizards, Interviews With Top Traders by Jack D. Schwager: Here are how the best money managers in the world make money, will you listen to them?


#4 How I Made $2,000,000 in the Stock Market by Nicolas Darvas: Find the best stocks that are being accumulated in high volume near their all time highs once you find them add to the winners and cut the losers short and let them run as far as they will go.


#5 Trend Following: Learn to Make Millions in Up or Down Markets by Michael Covel: Give up your opinions, forecasts, and fundamentals and use a robust system to trade the market trends by following the actual price action.


#6: How to Trade In Stocks by Jesse Livermore: Trade the pure price action of the leading stocks as they go higher and higher understand that they do have normal pull back reactions but learn how to not be stopped out until they have ran their full course.


#7 Trade Your Way to Financial Freedom by Van Tharp: Quit looking for the Holy Grail of trading and instead focus on developing a trading system that fits you and trade it while managing risk.


#8 Trading for a Living: Psychology, Trading Tactics, Money Management by Alexander Elder: You must manage the three M’s of trading to be successful: manage the money, the mind, and the trading method to be profitable.


#9 Trading in the Zone: Master the Market with Confidence, Discipline and a Winning Attitude by Mark Douglas: Trading is more of a mental game than a mathematical one and it takes faith in your system and yourself to give you the confidence to trade with discipline and mental control.


#10 The Complete TurtleTrader: How 23 Novice Investors Became Overnight Millionaires by Michael Covel : Richard Dennis showed some everyday people how to become millionaires by trading his firm’s capital with a simple trend following methodology based on breakouts with a few filters while using risk management.


#11 Trade Like a Stock Market Wizard: How to Achieve Super Performance in Stocks in Any Market by Mark Minvervini
U.S. Investing Champion Mark Minervini reveals the proven, time-tested trading system he used to achieve triple-digit returns for five consecutive years, averaging 220% per year for a 33,500% compounded total return.


Technical Analysis Books


1# The Art and Science of Technical Analysis:


Market Structure, Price Action and Trading Strategies by Adam Grimes
Modern day book which explains how technical analysis can be used to capture statistically validated patterns in certain types of market conditions.
Supported by extensive statistical analysis of the markets it gives you a realistic sense of how markets behave, when and how technical analysis works, and what it really takes to trade successfully.

2# Technical Analysis of the Financial Markets by John Murphy:


A Comprehensive Guide to Trading Methods and Applications
If you want to get started on technical analysis, this is the book to read.


3# Mind over Markets:


Power Trading with Market Generated Information by James F. Dalton
Market Profile is used widely by professional scalpers. If you aspire to be one, you need to read this.


4# Stan Weinstein's Secrets For Profiting in Bull and Bear Markets by Stan Weinstein


A simple book to read, packed with practical trading knowledge you can use.  You will learn the exact time to buy, sell and exit your trades.


5# Trading for a living


6# Profitable Candlestick Trading: Pinpointing Market Opportunities to Maximize Profits


Worthy mentions


1# One Good Trade: Inside the Highly Competitive World of Proprietary Trading by Mike Bellafiore
If you have interest in working for a proprietary trading firm or want to be an intraday trader, this book will tell you what it takes.  Also it tells you how professional traders read the depth of market.


2# Trading in the Zone: Master the Market with Confidence, Discipline and a Winning Attitude by Mark Dougals

An excellent book on trading psychology.

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1. The Little Book That Still Beats The Market by Joel Greenblatt


I  usually recommend this book to anyone who wants to learn stock  investing. However, I strongly believe that both new and old investors  will benefit from the wisdom shared in the book. Greenblatt explains the  logic and the irrationality of the markets using such a simple language  that even a school going kid should be able to understand it.




2. Beating The Street by Peter Lynch

The  celebrated Peter Lynch of the Fidelity Magellan fund demonstrates how a  basic layman using common sense can actually beat the experts from Wall  Street. Just like most other books I like it is written in very simple  and easy to understand language. I plan on reading this book again very  soon. Usually this is the second book I recommend to people.




 3. The Little Book That Builds Wealth by Pat Dorsey

This  is the type of book that I fall in love with. Easy to read, small and  has a world of information. Don’t be fooled by the gimmicky title of the  book as I rate it within the 3 best books on investing I ever read. The  book explains the concept of sustainable competitive advantage (aka  economic moat) that brings superior returns




4. The Little Book of Value Investing by Christopher H. Browne


Another  brilliant book on value investing from the little book, big profits  series. Explains the concept of value investing crisply. A great book by  a great investor.




5. Market Wizard Series by Jack D. Schwager


There  are four books in this series. Each of them are filled with interviews  with some of the best traders and investors. From Ray Dalio to Joel  Greenblatt to Paul Tudor Jones, Mr Schwager has interviewed them all.  And these are not the typical interviews that we read. Each of them have  very insightful and deep questions and answers.
 6. Common Stocks and Uncommon Profits by Philip A. Fisher
Phil  Fisher can be considered as one of the gurus of fundamental investing.  Many great investors in the world have mentioned him as an influence.  His strategy of a holistic approach to fundamental due diligence by not  only looking at a company but also studying the suppliers, customers and  competitors has become the gold standard in the equity analysis world.




7. You Can Be a Stock Market Genius by Joel Greenblatt


The  ultimate guide to special situation (mergers, acquisitions, spin-offs,  divestitures) investing. Greenblatt outdoes himself with this book and  proves clearly that the efficient market hypothesis is a myth even in  the highly developed markets. Special situations very often lead to  irrationality in valuation and these can be exploited by the clever  investor.




8. The Essays of Warren Buffet by Warren Buffet

This  is a compilation of the letters Warren Buffet wrote to his  shareholders. The letters have been organized according to category. The  reader will clearly understand why this man is one of the richest  people in the world.




9. Best practices for Equity Research Analysts by James Valentine


Hands  down the best book on equity research. Short, compact and highly  practical guide for both buy and sell side analysts.  A must read for  all investors in my opinion. It might be a bit hard to apply in real  life but the book title clearly mentions that these are the ‘best  practices’.




10. Margin of safety by Seth A. Klarman


Another  excellent book on value investing. The author shows with theory, logic  and practical examples why investment fads are pretty much always bogus.  The best way to make money is to have a strong disciplined fundamental  analysis approach.




11.The intelligent Investor by Benjamin Graham




12. Encyclopedia of Chart Patterns - Thomas Bulkowski




13. MASTERING THE TRADE - by John F. Carter.




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https://www.trading212.com/en/Video-tutorials
http://www.swing-trade-stocks.com/stock-market-books.html
http://www.traderplanet.com/articles/view/165435-top-ten-trading-books/
http://www.traderplanet.com/options/
https://www.tradingwithrayner.com/best-trading-books-of-all-time/#ta

Nepolion Hill secret to Financial Freedom

Nepolion Hill secret to Financial Freedom


https://www.youtube.com/watch?v=kj9Ny3kv0zA

Monday, July 17, 2017

How to Open Zerodha Demat Account





How to open Zerodha demat account


Click the below link and submit your details -




https://zerodha.com/open-account?c=ZW9465

Friday, July 14, 2017

CBSE Class 11 and 12 Maths Notes

CBSE Class 11 and 12 Maths Notes : Sets, Relations and Binary Operations
April 27, 2014 by Anuj William Leave a Comment
Set
Set is a collection of well defined objects which are distinct from each other. Sets are usually denoted by capital letters A, B,C,… and elements are usually denoted by small letters a, b,c,… .
If ‘a’ is an element of a set A, then we write a ∈ A and say ‘a’ belongs to A or ‘a’ is in A or ‘a’ is a member of A. If ‘a’ does not belongs to A, we write a ∉ A.
Standard Notations
• N : A set of natural numbers.
• W : A set of whole numbers.
• Z : A set of integers.
• Z+/Z– : A set of all positive/negative integers.
• Q : A set of all rational numbers.
• Q+/Q– : A set of all positive/ negative rational numbers.
• R : A set of real numbers.
• R+/R–: A set of all positive/negative real numbers.
• C : A set of all complex numbers.
Methods for Describing a Set
(i) Roster/Listing Method/Tabular Form In this method, a set is described by listing element, separated by commas, within braces.
e.g., A = {a, e, i, o, u}
(ii) Set Builder/Rule Method In this method, we write down a property or rule which gives us all the elements of the set by that rule.
e.g.,A = {x : x is a vowel of English alphabets}
Types of Sets
1. Finite Set A set containing finite number of elements or no element.
2. Cardinal Number of a Finite Set The number of elements in a given finite set is called cardinal number of finite set, denoted by n (A).
3. Infinite Set A set containing infinite number of elements.
4. Empty/Null/Void Set A set containing no element, it is denoted by (φ) or { }.
5. Singleton Set A set containing a single element.
6. Equal Sets Two sets A and B are said to be equal, if every element of A is a member of B and every element of B is a member of A and we write A = B.
7. Equivalent Sets Two sets are said to be equivalent, if they have same number of elements.
If n(A) = n(B), then A and B are equivalent sets. But converse is not true.
8. Subset and Superset Let A and B be two sets. If every element of A is an element of B, then A is called subset of B and B is called superset of A. Written as
A ⊆ B or B ⊇ A
9. Proper Subset If A is a subset of B and A ≠ B, then A is called proper subset of B and we write A ⊂ B.
10. Universal Set (U) A set consisting of all possible elements which occurs under consideration is called a universal set.
11. Comparable Sets Two sets A and Bare comparable, if A ⊆ B or B ⊆ A.
12. Non-Comparable Sets For two sets A and B, if neither A ⊆ B nor B ⊆ A, then A and B are called non-comparable sets.
13. Power Set (P) The set formed by all the subsets of a given set A, is called power set of A, denoted by P(A).
14. Disjoint Sets Two sets A and B are called disjoint, if, A ∩ B = (φ).
Venn Diagram
In a Venn diagram, the universal set is represented by a rectangular region and a set is represented by circle or a closed geometrical figure inside the universal set.

Operations on Sets
1. Union of Sets
The union of two sets A and B, denoted by A ∪ B is the set of all those elements, each one of which is either in A or in B or both in A and B.

2. Intersection of Sets
The intersection of two sets A and B, denoted by A ∩ B, is the set of all those elements which are common to both A and B.

If A1, A2,… , An is a finite family of sets, then their intersection is denoted by

3. Complement of a Set
If A is a set with U as universal set, then complement of a set, denoted by A’ or Ac is the set U – A .

4. Difference of Sets
For two sets A and B, the difference A – B is the set of all those elements of A which do not belong to B.

5. Symmetric Difference
For two sets A and B, symmetric difference is the set (A – B) ∪ (B – A) denoted by A Δ B.

Laws of Algebra of Sets
For three sets A, B and C
(i) Commutative Laws
A ∩ B = B ∩ A
A ∪ B = B ∪ A
(ii) Associative Laws
(A ∩ B) ∩ C = A ∩ (B ∩ C)
(A ∪ B) ∪ C = A ∪ (B ∪ C)
(iii) Distributive Laws
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(iv) Idempotent Laws
A ∩ A = A
A ∪ A = A
(v) Identity Laws
A ∪ Φ = A
A ∩ U = A
(vi) De Morgan’s Laws
(a) (A ∩ B) ′ = A ′ ∪ B ′
(b) (A ∪ B) ′ = A ′ ∩ B ′
(c) A – (B ∩ C) = (A – B) ∩ (A- C)
(d) A – (B ∪ C) = (A – B) ∪ ( A – C)
(vii) (a) A – B = A ∩ B’
(b) B – A = B ∩ A’
(c) A – B = A ⇔A ∩ B= (Φ)
(d) (A – B) ∪ B= A ∪ B
(e) (A – B) ∩ B = (Φ)
(f) A ∩ B ⊆ A and A ∩ B ⊆ B
(g) A ∪ (A ∩ B)= A
(h) A ∩ (A ∪ B)= A
(viii) (a) (A – B) ∪ (B – A) = (A ∪ B) – (A ∩ B)
(b) A ∩ (B – C) = (A ∩ B) – (A ∩ C)
(c) A ∩ (B Δ C) = (A ∩ B) A (A ∩ C)
(d) (A ∩ B) ∪ (A – B) = A
(e) A ∪ (B – A) = (A ∪ B)
(ix) (a) U’ = (Φ)
(b) Φ’ = U
(c) (A’ )’ = A
(d) A ∩ A’ = (Φ)
(e) A ∪ A’ = U
(f) A ⊆ B ⇔ B’ ⊆ A’
Important Points to be Remembered
• Every set is a subset of itself i.e., A ⊆ A, for any set A.
• Empty set Φ is a subset of every set i.e., Φ ⊂ A, for any set A.
• For any set A and its universal set U, A ⊆ U
• If A = Φ, then power set has only one element i.e., n(P(A)) = 1
• Power set of any set is always a non-empty set.
Suppose A = {1, 2}, thenP(A) = {{1}, {2}, {1, 2}, Φ}.(a) A ∉ P(A)
(b) {A} ∈ P(A)
• (vii) If a set A has n elements, then P(A) or subset of A has 2n elements.
• (viii) Equal sets are always equivalent but equivalent sets may not be equal.
The set {Φ} is not a null set. It is a set containing one element Φ.
Results on Number of Elements in Sets
• n (A ∪ B) = n(A) + (B)- n(A ∩ B)
• n(A ∪ B) = n(A)+ n(B), if A and B are disjoint.
• n(A – B) = n(A) – n(A ∩ B)
• n(A Δ B) = n(A) + n(B)- 2n(A ∩ B)
• n(A ∪ B ∪ C)= n(A)+ n(B)+ n(C)- n(A ∩ B) – n(B ∩ C)- n(A ∩ C)+ n(A ∩ B ∩ C)
• n (number of elements in exactly two of the sets A, B, C) = n(A ∩ B) + n(B ∩ C) + n (C ∩ A)- 3n(A ∩ B ∩ C)
• n (number of elements in exactly one of the sets A, B, C) = n(A) + n(B) + n(C) – 2n(A ∩ B) – 2n(B ∩ C) – 2n(A ∩ C) + 3n(A ∩ B ∩ C)
• n(A’ ∪ B’)= n(A ∩ B)’ = n(U) – n(A ∩ B)
• n(A’ ∩ B’ ) = n(A ∪ B)’ = n(U) – n(A ∪ B)
• n(B – A) = n(B)- n(A ∩ B)
Ordered Pair
An ordered pair consists of two objects or elements in a given fixed order.
Equality of Ordered Pairs Two ordered pairs (a1, b1) and (a2, b2) are equal iff a1 = a2 and b1 = b2.
Cartesian Product of Sets
For two sets A and B (non-empty sets), the set of all ordered pairs (a, b) such that a ∈ A and b ∈ B is called Cartesian product of the sets A and’ B, denoted by A x B.
A x B={(a,b):a ∈ A and b ∈ B}
If there are three sets A, B, C and a ∈ A, be B and c ∈ C, then we form, an ordered triplet (a, b, c). The set of all ordered triplets (a, b, c) is called the cartesian product of these sets A, B and C.
i.e., A x B x C = {(a,b,c):a ∈ A,b ∈ B,c ∈ C}
Properties of Cartesian Product
For three sets A, B and C
•  n (A x B)= n(A) n(B)
• A x B = Φ, if either A or B is an empty set.
• A x (B ∪ C)= (A x B) ∪ (A x C)
• A x (B ∩ C) = (A x B) ∩ (A x C)
• A x (B — C)= (A x B) — (A x C)
• (A x B) ∩ (C x D)= (A ∩ C) x (B ∩ D)
• If A ⊆ B and C ⊆ D, then (A x C) ⊂ (B x D)
• If A ⊆ B, then A x A ⊆ (A x B) ∩ (B x A)
• A x B = B x A ⇔ A = B
• If either A or B is an infinite set, then A x B is an infinite set.
• A x (B’ ∪ C’ )’ = (A x B) ∩ (A x C)
• A x (B’ ∩ C’ )’ = (A x B) ∪ (A x C)
• If A and B be any two non-empty sets having n elements in common, then A x B and B x A have n2 elements in common.
• If ≠ B, then A x B ≠ B x A
• If A = B, then A x B= B x A
• If A ⊆ B, then A x C = B x C for any set C.
Relation
If A and B are two non-empty sets, then a relation R from A to B is a subset of A x B.
If R ⊆ A x B and (a, b) ∈ R, then we say that a is related to b by the relation R, written as aRb.
Domain and Range of a Relation
Let R be a relation from a set A to set B. Then, set of all first components or coordinates of the ordered pairs belonging to R is called : the domain of R, while the set of all second components or coordinates = of the ordered pairs belonging to R is called the range of R.
Thus, domain of R = {a : (a , b) ∈ R} and range of R = {b : (a, b) ∈ R}
Types of Relations
(i) Void Relation As Φ ⊂ A x A, for any set A, so Φ is a relation on A, called the empty or void relation.
(ii) Universal Relation Since, A x A ⊆ A x A, so A x A is a relation on A, called the universal relation.
(iii) Identity Relation The relation IA = {(a, a) : a ∈ A} is called the identity relation on A.
(iv) Reflexive Relation A relation R is said to be reflexive relation, if every element of A is related to itself.
Thus, (a, a) ∈ R, ∀ a ∈ A = R is reflexive.
(v) Symmetric Relation A relation R is said to be symmetric relation, iff
(a, b) ∈ R (b, a) ∈ R,∀ a, b ∈ A
i.e., a R b ⇒ b R a,∀ a, b ∈ A
⇒ R is symmetric.
(vi) Anti-Symmetric Relation A relation R is said to be anti-symmetric relation, iff
(a, b) ∈ R and (b, a) ∈ R ⇒ a = b,∀ a, b ∈ A
(vii) Transitive Relation A relation R is said to be transitive relation, iff (a, b) ∈ R and (b, c) ∈ R
⇒ (a, c) ∈ R, ∀ a, b, c ∈ A
(viii) Equivalence Relation A relation R is said to be an equivalence relation, if it is simultaneously reflexive, symmetric and transitive on A.
(ix) Partial Order Relation A relation R is said to be a partial order relation, if it is simultaneously reflexive, symmetric and anti-symmetric on A.
(x) Total Order Relation A relation R on a set A is said to be a total order relation on A, if R is a partial order relation on A.
Inverse Relation
If A and B are two non-empty sets and R be a relation from A to B, such that R = {(a, b) : a ∈ A, b ∈ B}, then the inverse of R, denoted by R-1 , i a relation from B to A and is defined by
R-1 = {(b, a) : (a, b) ∈ R}
Equivalence Classes of an Equivalence Relation
Let R be equivalence relation in A (≠ Φ). Let a ∈ A.
Then, the equivalence class of a denoted by [a] or {a} is defined as the set of all those points of A which are related to a under the relation R.
Composition of Relation
Let R and S be two relations from sets A to B and B to C respectively, then we can define relation SoR from A to C such that (a, c) ∈ So R ⇔ ∃ b ∈ B such that (a, b) ∈ R and (b, c) ∈ S.
This relation SoR is called the composition of R and S.
(i) RoS ≠ SoR
(ii) (SoR)-1 = R-1oS-1
known as reversal rule.
Congruence Modulo m
Let m be an arbitrary but fixed integer. Two integers a and b are said to be congruence modulo m, if a – b is divisible by m and we write a ≡ b (mod m).
i.e., a ≡ b (mod m) ⇔ a – b is divisible by m.
Important Results on Relation
• If R and S are two equivalence relations on a set A, then R ∩ S is also on ‘equivalence relation on A.
• The union of two equivalence relations on a set is not necessarily an equivalence relation on the set.
• If R is an equivalence relation on a set A, then R-1 is also an equivalence relation on A.
• If a set A has n elements, then number of reflexive relations from A to A is 2n2 – 2
• Let A and B be two non-empty finite sets consisting of m and n elements, respectively. Then, A x B consists of mn ordered pairs. So, total number of relations from A to B is 2nm.
Binary Operations
Closure Property
An operation * on a non-empty set S is said to satisfy the closure ‘ property, if
a ∈ S, b ∈ S ⇒ a * b ∈ S, ∀ a, b ∈ S
Also, in this case we say that S is closed for *.
An operation * on a non-empty set S, satisfying the closure property is known as a binary operation.
or
Let S be a non-empty set. A function f from S x S to S is called a binary operation on S i.e., f : S x S → S is a binary operation on set S.
Properties
• Generally binary operations are represented by the symbols * , +, … etc., instead of letters figure etc.
• Addition is a binary operation on each one of the sets N, Z, Q, R and C of natural numbers, integers, rationals, real and complex numbers, respectively. While addition on the set S of all irrationals is not a binary operation.
• Multiplication is a binary operation on each one of the sets N, Z, Q, R and C of natural numbers, integers, rationals, real and complex numbers, respectively. While multiplication on the set S of all irrationals is not a binary operation.
• Subtraction is a binary operation on each one of the sets Z, Q, R and C of integers, rationals, real and complex numbers, respectively. While subtraction on the set of natural numbers is not a binary operation.
• Let S be a non-empty set and P(S) be its power set. Then, the union and intersection on P(S) is a binary operation.
• Division is not a binary operation on any of the sets N, Z, Q, R and C. However, it is not a binary operation on the sets of all non-zero rational (real or complex) numbers.
• Exponential operation (a, b) → ab is a binary operation on set N of natural numbers while it is not a binary operation on set Z of integers.
Types of Binary Operations
(i) Associative Law A binary operation * on a non-empty set S is said to be associative, if (a * b) * c = a * (b * c), ∀ a, b, c ∈ S.
Let R be the set of real numbers, then addition and multiplication on R satisfies the associative law.
(ii) Commutative Law A binary operation * on a non-empty set S is said to be commutative, if
a * b = b * a, ∀ a, b ∈ S.
Addition and multiplication are commutative binary operations on Z but subtraction not a commutative binary operation, since
2 — 3 ≠ 3— 2 .
Union and intersection are commutative binary operations on the power P(S) of all subsets of set S. But difference of sets is not a commutative binary operation on P(S).
(iii) Distributive Law Let * and o be two binary operations on a non-empty sets. We say that * is distributed over o., if
a * (b o c)= (a * b) o (a * c), ∀ a, b, c ∈ S also called (left distribution) and (b o c) * a = (b * a) o (c * a), ∀ a, b, c ∈ S also called (right distribution).
Let R be the set of all real numbers, then multiplication distributes addition on R.
Since, a.(b + c) = a.b + a.c,∀ a, b, c ∈ R.
(iv) Identity Element Let * be a binary operation on a non-empty set S. An element e a S, if it exist such that
a * e = e * a = a, ∀ a ∈ S.
is called an identity elements of S, with respect to *.
For addition on R, zero is the identity elements in R.
Since, a + 0 = 0 + a = a, ∀ a ∈ R
For multiplication on R, 1 is the identity element in R.
Since, a x 1 =1 x a = a,∀ a ∈ R
Let P (S) be the power set of a non-empty set S. Then, Φ is the identity element for union on P (S) as
A ∪ Φ =Φ ∪ A = A, ∀ A ∈ P(S)
Also, S is the identity element for intersection on P(S).
Since, A ∩ S=A ∩ S=A, ∀ A ∈ P(S).
For addition on N the identity element does not exist. But for multiplication on N the idenitity element is 1.
(v) Inverse of an Element Let * be a binary operation on a non-empty set ‘S’ and let ‘e’ be the identity element.
Let a ∈ S. we say that a-1 is invertible, if there exists an element b ∈ S such that a * b = b * a = e
Also, in this case, b is called the inverse of a and we write, a-1 = b
Addition on N has no identity element and accordingly N has no invertible element.
Multiplication on N has 1 as the identity element and no element other than 1 is invertible.
Let S be a finite set containing n elements. Then, the total  number of binary operations on S in nn2
Let S be a finite set containing n elements. Then, the total number of commutative binary operation on S is n [n(n+1)/2].
CBSE Class 12 Maths Notes : Matrices
May 28, 2014 by Anuj William Leave a Comment
A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as

matrix is enclosed by [ ] or ( ) or | | | |
Compact form the above matrix is represented by [aij]m x n or A = [aij].
1. Element of a Matrix The numbers a11, a12 … etc., in the above matrix are known as the element of the matrix, generally represented as aij , which denotes element in ith row and jth column.
2. Order of a Matrix In above matrix has m rows and n columns, then A is of order m x n.
Types of Matrices
1. Row Matrix A matrix having only one row and any number of columns is called a row matrix.
2. Column Matrix A matrix having only one column and any number of rows is called column matrix.
3. Rectangular Matrix A matrix of order m x n, such that m ≠ n, is called rectangular matrix.
4. Horizontal Matrix A matrix in which the number of rows is less than the number of columns, is called a horizontal matrix.
5. Vertical Matrix A matrix in which the number of rows is greater than the number of columns, is called a vertical matrix.
6. Null/Zero Matrix A matrix of any order, having all its elements are zero, is called a null/zero matrix. i.e., aij = 0, ∀ i, j
7. Square Matrix A matrix of order m x n, such that m = n, is called square matrix.
8. Diagonal Matrix A square matrix A = [aij]m x n, is called a diagonal matrix, if all the elements except those in the leading diagonals are zero, i.e., aij = 0 for i ≠ j. It can be represented as
A = diag[a11 a22… ann]
9. Scalar Matrix A square matrix in which every non-diagonal element is zero and all diagonal elements are equal, is called scalar matrix. i.e., in scalar matrix
aij = 0, for i ≠ j and aij = k, for i = j
10. Unit/Identity Matrix A square matrix, in which every non-diagonal element is zero and every diagonal element is 1, is called, unit matrix or an identity matrix.

11. Upper Triangular Matrix A square matrix A = a[ij]n x n is called a upper triangular matrix, if a[ij], = 0, ∀ i > j.
12. Lower Triangular Matrix A square matrix A = a[ij]n x n is called a lower triangular matrix, if a[ij], = 0, ∀ i < j.
13. Submatrix A matrix which is obtained from a given matrix by deleting any number of rows or columns or both is called a submatrix of the given matrix.
14. Equal Matrices Two matrices A and B are said to be equal, if both having same order and corresponding elements of the matrices are equal.
15. Principal Diagonal of a Matrix In a square matrix, the diagonal from the first element of the first row to the last element of the last row is called the principal diagonal of a matrix.

16. Singular Matrix A square matrix A is said to be singular matrix, if determinant of A denoted by det (A) or |A| is zero, i.e., |A|= 0, otherwise it is a non-singular matrix.
Algebra of Matrices
1. Addition of Matrices
Let A and B be two matrices each of order m x n. Then, the sum of matrices A + B is defined only if matrices A and B are of same order.
If A = [aij]m x n , A = [aij]m x n
Then, A + B = [aij + bij]m x n
Properties of Addition of Matrices If A, B and C are three matrices of order m x n, then
1. Commutative Law A + B = B + A
2. Associative Law (A + B) + C = A + (B + C)
3. Existence of Additive Identity A zero matrix (0) of order m x n (same as of A), is additive identity, if
A + 0 = A = 0 + A
4. Existence of Additive Inverse If A is a square matrix, then the matrix (- A) is called additive inverse, if
A + ( – A) = 0 = (- A) + A
5. Cancellation Law
A + B = A + C ⇒ B = C (left cancellation law)
B + A = C + A ⇒ B = C (right cancellation law)
2. Subtraction of Matrices
Let A and B be two matrices of the same order, then subtraction of matrices, A – B, is defined as
A – B = [aij – bij]n x n,
where A = [aij]m x n, B = [bij]m x n
3. Multiplication of a Matrix by a Scalar
Let A = [aij]m x n be a matrix and k be any scalar. Then, the matrix obtained by multiplying each element of A by k is called the scalar multiple of A by k and is denoted by kA, given as
kA= [kaij]m x n
Properties of Scalar Multiplication If A and B are matrices of order m x n, then
1. k(A + B) = kA + kB
2. (k1 + k2)A = k1A + k2A
3. k1k2A = k1(k2A) = k2(k1A)
4. (- k)A = – (kA) = k( – A)
4. Multiplication of Matrices
Let A = [aij]m x n and B = [bij]n x p are two matrices such that the number of columns of A is equal to the number of rows of B, then multiplication of A and B is denoted by AB, is given by

where cij is the element of matrix C and C = AB
Properties of Multiplication of Matrices
1. Commutative Law Generally AB ≠ BA
2. Associative Law (AB)C = A(BC)
3. Existence of multiplicative Identity A.I = A = I.A,
I is called multiplicative Identity.
4. Distributive Law A(B + C) = AB + AC
5. Cancellation Law If A is non-singular matrix, then
AB = AC ⇒ B = C (left cancellation law)
BA = CA ⇒B = C (right cancellation law)
6. AB = 0, does not necessarily imply that A = 0 or B = 0 or both A and B = 0
Important Points to be Remembered
(i) If A and B are square matrices of the same order, say n, then both the product AB and BA are defined and each is a square matrix of order n.
(ii) In the matrix product AB, the matrix A is called premultiplier (prefactor) and B is called postmultiplier (postfactor).
(iii) The rule of multiplication of matrices is row column wise (or → ↓ wise) the first row of AB is obtained by multiplying the first row of A with first, second, third,… columns of B respectively; similarly second row of A with first, second, third, … columns of B, respectively and so on.
Positive Integral Powers of a Square Matrix
Let A be a square matrix. Then, we can define
1. An + 1 = An. A, where n ∈ N.
2. Am. An = Am + n
3. (Am)n = Amn, ∀ m, n ∈ N
Matrix Polynomial
Let f(x)= a0xn + a1xn – 1 -1 + a2xn – 2 + … + an. Then
f(A)= a0An + a1An – 2 + … + anIn
is called the matrix polynomial.
Transpose of a Matrix
Let A = [aij]m x n, be a matrix of order m x n. Then, the n x m matrix obtained by interchanging the rows and columns of A is called the transpose of A and is denoted by A’ or AT.
A’ = AT = [aij]n x m
Properties of Transpose
1. (A’)’ = A
2. (A + B)’ = A’ + B’
3. (AB)’ = B’A’
4. (KA)’ = kA’
5. (AN)’ = (A’)N
6. (ABC)’ = C’ B’ A’
Symmetric and Skew-Symmetric Matrices
1. A square matrix A = [aij]n x n, is said to be symmetric, if A’ = A.
i.e., aij = aji , ∀i and j.
2. A square matrix A is said to be skew-symmetric matrices, if i.e., aij = — aji, di and j
Properties of Symmetric and Skew-Symmetric Matrices
1. Elements of principal diagonals of a skew-symmetric matrix are all zero. i.e., aii = — aii 2ii = 0 or aii = 0, for all values of i.
2. If A is a square matrix, then
(a) A + A’ is symmetric.
(b) A — A’ is skew-symmetric matrix.
3. If A and B are two symmetric (or skew-symmetric) matrices of same order, then A + B is also symmetric (or skew-symmetric).
4. If A is symmetric (or skew-symmetric), then kA (k is a scalar) is also symmetric for skew-symmetric matrix.
5. If A and B are symmetric matrices of the same order, then the product AB is symmetric, iff BA = AB.
6. Every square matrix can be expressed uniquely as the sum of a symmetric and a skew-symmetric matrix.
7. The matrix B’ AB is symmetric or skew-symmetric according as A is symmetric or skew-symmetric matrix.
8. All positive integral powers of a symmetric matrix are symmetric.
9. All positive odd integral powers of a skew-symmetric matrix are skew-symmetric and positive even integral powers of a skew-symmetric are symmetric matrix.
10. If A and B are symmetric matrices of the same order, then
(a) AB – BA is a skew-symmetric and
(b) AB + BA is symmetric.
11. For a square matrix A, AA’ and A’ A are symmetric matrix.
Trace of a Matrix
The sum of the diagonal elements of a square matrix A is called the trace of A, denoted by trace (A) or tr (A).
Properties of Trace of a Matrix
1. Trace (A ± B)= Trace (A) ± Trace (B)
2. Trace (kA)= k Trace (A)
3. Trace (A’ ) = Trace (A)
4. Trace (In)= n
5. Trace (0) = 0
6. Trace (AB) ≠ Trace (A) x Trace (B)
7. Trace (AA’) ≥ 0
Conjugate of a Matrix
The matrix obtained from a matrix A containing complex number as its elements, on replacing its elements by the corresponding conjugate complex number is called conjugate of A and is denoted by A.
Properties of Conjugate of a Matrix
If A is a matrix of order m x n, then

Transpose Conjugate of a Matrix
The transpose of the conjugate of a matrix A is called transpose conjugate of A and is denoted by A0 or A*.
i.e., (A’) = A‘ = A0 or A*
Properties of Transpose Conjugate of a Matrix
(i) (A*)* = A
(ii) (A + B)* = A* + B*
(iii) (kA)* = kA*
(iv) (AB)* = B*A*
(V) (An)* = (A*)n
Some Special Types of Matrices
1. Orthogonal Matrix
A square matrix of order n is said to be orthogonal, if AA’ = In = A’A Properties of Orthogonal Matrix
(i) If A is orthogonal matrix, then A’ is also orthogonal matrix.
(ii) For any two orthogonal matrices A and B, AB and BA is also an orthogonal matrix.
(iii) If A is an orthogonal matrix, A-1 is also orthogonal matrix.
2. ldempotent Matrix
A square matrix A is said to be idempotent, if A2 = A.
Properties of Idempotent Matrix
(i) If A and B are two idempotent matrices, then
• AB is idempotent, if AB = BA.
• A + B is an idempotent matrix, iff
AB = BA = 0
• AB = A and BA = B, then A2 = A, B2 = B
(ii)
• If A is an idempotent matrix and A + B = I, then B is an idempotent and AB = BA= 0.
• Diagonal (1, 1, 1, …,1) is an idempotent matrix.
• If I1, I2 and I3 are direction cosines, then

is an idempotent as |Δ|2 = 1.
A square matrix A is said to be involutory, if A2 = I
4. Nilpotent Matrix
A square matrix A is said to be nilpotent matrix, if there exists a positive integer m such that A2 = 0. If m is the least positive integer such that Am = 0, then m is called the index of the nilpotent matrix A.
5. Unitary Matrix
A square matrix A is said to be unitary, if A‘A = I
Hermitian Matrix
A square matrix A is said to be hermitian matrix, if A = A* or
= aij, for aji only.
Properties of Hermitian Matrix
1. If A is hermitian matrix, then kA is also hermitian matrix for any non-zero real number k.
2. If A and B are hermitian matrices of same order, then λ1A + λB, also hermitian for any non-zero real number λ1, and λ.
3. If A is any square matrix, then AA* and A* A are also hermitian.
4. If A and B are hermitian, then AB is also hermitian, iff AB = BA
5. If A is a hermitian matrix, then A is also hermitian.
6. If A and B are hermitian matrix of same order, then AB + BA is also hermitian.
7. If A is a square matrix, then A + A* is also hermitian,
8. Any square matrix can be uniquely expressed as A + iB, where A and B are hermitian matrices.
Skew-Hermitian Matrix
A square matrix A is said to be skew-hermitian if A* = – A or aji for every i and j.
Properties of Skew-Hermitian Matrix
1. If A is skew-hermitian matrix, then kA is skew-hermitian matrix, where k is any non-zero real number.
2. If A and B are skew-hermitian matrix of same order, then λ1A + λ2B is also skew-hermitian for any real number λ1 and λ2.
3. If A and B are hermitian matrices of same order, then AB — BA is skew-hermitian.
4. If A is any square matrix, then A — A* is a skew-hermitian matrix.
5. Every square matrix can be uniquely expressed as the sum of a hermitian and a skew-hermitian matrices.
6. If A is a skew-hermitian matrix, then A is a hermitian matrix.
7. If A is a skew-hermitian matrix, then A is also skew-hermitian matrix.
Adjoint of a Square Matrix
Let A[aij]m x n be a square matrix of order n and let Cij be the cofactor of aij in the determinant |A| , then the adjoint of A, denoted by adj (A), is defined as the transpose of the matrix, formed by the cofactors of the matrix.
Properties of Adjoint of a Square Matrix
If A and B are square matrices of order n, then
1. A (adj A) = (adj A) A = |A|I
2. adj (A’) = (adj A)’
3. adj (AB) = (adj B) (adj A)
4. adj (kA) = kn – 1(adj A), k ∈ R
5. adj (Am) = (adj A)m
6. adj (adj A) = |A|n – 2 A, A is a non-singular matrix.
7. |adj A| =|A|n – 1 ,A is a non-singular matrix.
8. |adj (adj A)| =|A|(n – 1)2 A is a non-singular matrix.
9. Adjoint of a diagonal matrix is a diagonal matrix.
Inverse of a Square Matrix
Let A be a square matrix of order n, then a square matrix B, such that AB = BA = I, is called inverse of A, denoted by A-1.
i.e., 
or AA-1 = A-1A = 1
Properties of Inverse of a Square Matrix
1. Square matrix A is invertible if and only if |A| ≠ 0
2. (A-1)-1 = A
3. (A’)-1 = (A-1)’
4. (AB)-1 = B-1A-1
In general (A1A1A1 … An)-1 = An-1An – 1-1 … A3-1A2-1A1-1
5. If a non-singular square matrix A is symmetric, then A-1 is also symmetric.
6. |A-1| = |A|-1
7. AA-1 = A-1A = I
8. (Ak)-1 = (A-1)Ak k ∈ N

Elementary Transformation
Any one of the following operations on a matrix is called an elementary transformation.
1. Interchanging any two rows (or columns), denoted by Ri←→Rj or Ci←→Cj
2. Multiplication of the element of any row (or column) by a non-zero quantity and denoted by
Ri → kRi or Ci → kCj
3. Addition of constant multiple of the elements of any row to the corresponding element of any other row, denoted by
Ri → Ri + kRj or Ci → Ci + kCj
Equivalent Matrix
• Two matrices A and B are said to be equivalent, if one can be obtained from the other by a sequence of elementary transformation.
• The symbol≈ is used for equivalence.
Rank of a Matrix
A positive integer r is said to be the rank of a non-zero matrix A, if
1. there exists at least one minor in A of order r which is not zero.
2. every minor in A of order greater than r is zero, rank of a matrix A is denoted by ρ(A) = r.
Properties of Rank of a Matrix
1. The rank of a null matrix is zero ie, ρ(0) = 0
2. If In is an identity matrix of order n, then ρ(In) = n.
3. (a) If a matrix A does’t possess any minor of order r, then ρ(A) ≥ r.
(b) If at least one minor of order r of the matrix is not equal to zero, then ρ(A) ≤ r.
4. If every (r + 1)th order minor of A is zero, then any higher order – minor will also be zero.
5. If A is of order n, then for a non-singular matrix A, ρ(A) = n
6.  ρ(A’)= ρ(A)
7. ρ(A*) = ρ(A)
8. ρ(A + B) &LE; ρ(A) + ρ(B)
9. If A and B are two matrices such that the product AB is defined, then rank (AB) cannot exceed the rank of the either matrix.
10. If A and B are square matrix of same order and ρ(A) = ρ(B) = n, then p(AB)= n
11. Every skew-symmetric matrix,of odd order has rank less than its order.
12. Elementary operations do not change the rank of a matrix.
Echelon Form of a Matrix
A non-zero matrix A is said to be in Echelon form, if A satisfies the following conditions
1. All the non-zero rows of A, if any precede the zero rows.
2. The number of zeros preceding the first non-zero element in a row is less than the number of such zeros in the successive row.
3. The first non-zero element in a row is unity.
4. The number of non-zero rows of a matrix given in the Echelon form is its rank.
Homogeneous and Non-Homogeneous System of Linear Equations
A system of equations AX = B, is called a homogeneous system if B = 0 and if B ≠ 0, then it is called a non-homogeneous system of equations.
Solution of System of Linear Equations
The values of the variables satisfying all the linear equations in the system, is called solution of system of linear equations.
1 . Solution of System of Equations by Matrix Method
(i) Non-Homogeneous System of Equations Let AX = B be a system of n linear equations in n variables.
• If |A| ≠ 0, then the system of equations is consistent and has a unique solution given by X = A-1B.
• If |A| = 0 and (adj A)B = 0, then the system of equations is consistent and has infinitely many solutions.
• If |A| = 0 and (adj A) B ≠ 0, then the system of equations is inconsistent i.e., having no solution
(ii) Homogeneous System of Equations Let AX = 0 is a system of n linear equations in n variables.
• If I |A| ≠ 0, then it has only solution X = 0, is called the trivial solution.
• If I |A| = 0, then the system has infinitely many solutions, called non-trivial solution.
2. Solution of System of Equations by Rank Method
(i) Non-Homogeneous System of Equations Let AX = B, be a system of n linear equations in n variables, then
• Step I Write the augmented matrix [A:B]
• Step II Reduce the augmented matrix to Echelon form using elementary row-transformation.
• Step III Determine the rank of coefficient matrix A and augmented matrix [A:B] by counting the number of non-zero rows in A and [A:B].
Important Results
1. If ρ(A) ≠ ρ(AB), then the system of equations is inconsistent.
2. If ρ(A) =ρ(AB) = the number of unknowns, then the system of equations is consistent and has a unique solution.
3. If ρ(A) = ρ(AB) < the number of unknowns, then the system of equations is consistent and has infinitely many solutions.
(ii) Homogeneous System of Equations
• If AX = 0, be a homogeneous system of linear equations then, If ρ(A) = number of unknown, then AX = 0, have a non-trivial solution, i.e., X = 0.
• If ρ(A) < number of unknowns, then AX = 0, have a non-trivial solution, with infinitely many solutions.
CBSE Class 12 Maths Notes : Determinant
May 28, 2014 by Neepur Garg Leave a Comment
Determinant
Every square matrix A is associated with a number, called its determinant and it is denoted by det (A) or |A| .
Only square matrices have determinants. The matrices which are not square do not have determinants
(i) First Order Determinant
If A = [a], then det (A) = |A| = a
(ii) Second Order Determinant

|A| = a11a22 – a21a12
(iii) Third Order Determinant

Evaluation of Determinant of Square Matrix of Order 3 by Sarrus Rule
 then determinant can be formed by enlarging the matrix by adjoining the first two columns on the right and draw lines as show below parallel and perpendicular to the diagonal.

The value of the determinant, thus will be the sum of the product of element. in line parallel to the diagonal minus the sum of the product of elements in line perpendicular to the line segment. Thus,
Δ = a11a22a33 + a12a23a31 + a13a21a32 – a13a22a31 – a11a23a32 – a12a21a33.
Note This method doesn’t work for determinants of order greater than 3.
Properties of Determinants
(i) The value of the determinant remains unchanged, if rows are changed into columns and columns are changed into rows e.g.,
|A’| = |A|
(ii) If A = [aij]n x n , n > 1 and B be the matrix obtained from A by interchanging two of its rows or columns, then
det (B) = – det (A)
(iii) If two rows (or columns) of a square matrix A are proportional, then |A| = O.
(iv) |B| = k |A| ,where B is the matrix obtained from A, by multiplying one row (or column) of A by k.
(v) |kA| = kn|A|, where A is a matrix of order n x n.
(vi) If each element of a row (or column) of a determinant is the sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants, e.g.,

(vii) If the same multiple of the elements of any row (or column) of a determinant are added to the corresponding elements of any other row (or column), then the value of the new determinant remains unchanged, e.g.,

(viii) If each element of a row (or column) of a determinant is zero, then its value is zero.
(ix) If any two rows (columns) of a determinant are identical, then its value is zero.
(x) If each element of row (column) of a determinant is expressed as a sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants.
Important Results on Determinants
(i) |AB| = |A||B| , where A and B are square matrices of the same order.
(ii) |An| = |A|n
(iii) If A, B and C are square matrices of the same order such that ith column (or row) of A is the sum of i th columns (or rows) of B and C and all other columns (or rows) of A, Band C are identical, then |A| =|B| + |C|
(iv) |In| = 1,where In is identity matrix of order n
(v) |On| = 0, where On is a zero matrix of order n
(vi) If Δ(x) be a 3rd order determinant having polynomials as its elements.
(a) If Δ(a) has 2 rows (or columns) proportional, then (x – a) is a factor of Δ(x).
(b) If Δ(a) has 3 rows (or columns) proportional, then (x – a)2 is a factor of Δ(x). ,
(vii) A square matrix A is non-singular, if |A| ≠ 0 and singular, if |A| =0.
(viii) Determinant of a skew-symmetric matrix of odd order is zero and of even order is a non-zero perfect square.
(ix) In general, |B + C| ≠ |B| + |C|
(x) Determinant of a diagonal matrix = Product of its diagonal elements
(xi) Determinant of a triangular matrix = Product of its diagonal elements
(xii) A square matrix of order n, is non-singular, if its rank r = n i.e., if |A| ≠ 0, then rank (A) = n



(xiv) If A is a non-singular matrix, then |A-1| = 1 / |A| = |A|-1
(xv) Determinant of a orthogonal matrix = 1 or – 1.
(xvi) Determinant of a hermitian matrix is purely real .
(xvii) If A and B are non-zero matrices and AB = 0, then it implies |A| = 0 and |B| = 0.
Minors and Cofactors
 then the minor Mij of the element aij is the determinant obtained by deleting the i row and jth column.

The cofactor of the element aij is Cij = (- 1)i + j Mij
Adjoint of a Matrix – Adjoint of a matrix is the transpose of the matrix of cofactors of the give matrix, i.e.,

Properties of Minors and Cofactors
(i) The sum of the products of elements of .any row (or column) of a determinant with the cofactors of the corresponding elements of any other row (or column) is zero, i.e., if

then a11C31 + a12C32 + a13C33 = 0 ans so on.
(ii) The sum of the product of elements of any row (or column) of a determinant with the cofactors of the corresponding elements of the same row (or column) is Δ


Differentiation of Determinant

Integration of Determinant

If the elements of more than one column or rows are functions of x, then the integration can be done only after evaluation/expansion of the determinant.
Solution of Linear equations by Determinant/Cramer’s Rule
Case 1. The solution of the system of simultaneous linear equations
a1x + b1y = C1 …(i)
a2x + b2y = C2 …(ii)
is given by x = D1 / D, Y = D2 / D

(i) If D ≠ 0, then the given system of equations is consistent and has a unique solution given by x = D1 / D, y = D2 / D
(ii) If D = 0 and Dl = D2 = 0, then the system is consistent and has infinitely many solutions.
(iii) If D = 0 and one of Dl and D2 is non-zero, then the system is inconsistent.
Case II. Let the system of equations be
a1x + b1y + C1z = d1
a2x + b2y + C2z = d2
a3x + b3y + C3z = d3
Then, the solution of the system of equation is
x = D1 / D, Y = D2 / D, Z = D3 / D, it is called Cramer’s rule.

(i) If D ≠ 0, then the system of equations is consistent with unique solution.
(ii) If D = 0 and atleast one of the determinant D1, D2, D3 is non-zero, then the given system is inconsistent, i.e., having no solution.
(iii) If D = 0 and D1 = D2 = D3 = 0, then the system is consistent, with infinitely many solutions.
(iv) If D ≠ 0 and D1 = D2 = D3 = 0, then system has only trivial solution, (x = y = z = 0).
Cayley-Hamilton Theorem
Every matrix satisfies its characteristic equation, i.e., if A be a square matrix, then |A – xl| = 0 is the characteristics equation of A. The values of x are called eigenvalues of A.
i.e., if x3 – 4x2 – 5x – 7 = 0 is characteristic equation for A, then
A3 – 4A2 + 5A – 7I = 0
Properties of Characteristic Equation
(i) The sum of the eigenvalues of A is equal to its trace.
(ii) The product of the eigenvalues of A is equal to its determinant.
(iii) The eigenvalues of an orthogonal matrix are of unit modulus.
(iv) The feigen values of a unitary matrix are of unit modulus.
(v) A and A’ have same eigenvalues.
(vi) The eigenvalues of a skew-hermitian matrix are either purely imaginary or zero.
(vii) If x is an eigenvalue of A, then x is the eigenvalue of A* .
(viii) The eigenvalues of a triangular matrix are its diagonal elements.
(ix) If x is the eigenvalue of A and |A| ≠ 0, then (1 / x) is the eigenvalue of A-1.
(x) If x is the eigenvalue of A and |A| ≠ 0, then |A| / x is the eigenvalue of adj (A).
(xi) If x1, x2,x3, … ,xn are eigenvalues of A, then the eigenvalues of A2 are x22, x22,…, xn2.
Cyclic Determinants


Applications of Determinant in Geometry
Let three points in a plane be A(x1, y1), B(x2, y2) and C(x3, y3), then

= 1 / 2 [x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)]

Maximum and Minimum Value of Determinants

where ais ∈ [α1, α2,…, αn]
Then, |A|max when diagonal elements are
{ min (α1, α2,…, αn)}
and non-diagonal elements are
{ max (α1, α2,…, αn)}
Also, |A|min = – |A|max
CBSE Class 12 Maths Notes : Probability
May 1, 2014 by Neepur Garg Leave a Comment
Some Basic Terms
Coin
A coin has two sides, Head and Tail. If an event consists of more than one coins, then coins are considered as. distinct, if not otherwise stated.
Die
A die has six face marked 11, 2, 3, 4, 5 and 6. If we have more than one dice, then all dice are considered as distinct, if not otherwise stated.
Playing Cards
A pack of playing cards has 52 cards. There are 4 suits (spade, heart, diamond and club) each having 13 cards. There are two colours, red (heart and diamond) and black (spade and club) each having 26 cards.
In 13 cards of each suit, there are 3 face cards namely king, queen and jack so there are in all ’12 face cards. Also, there are 16 honour cards,
4 of each suit namely ace, king, queen and jack.
Types of Experiments
1. Deterministic Experiment
Those experiments, which when repeated under identical conditions produce the same result or outcome are known as deterministic experiment,
2. Probabilistic/Random Experiment
Those experiments, which when repeated under identical conditions, do not produce the same outcome every time but the outcome in a trial is one of the several possible outcomes, called random experiment.
Important Definitions
(i) Trial Let a random experiment, be repeated under identical conditions, then the experiment is called a Trial.
(ii) Sample Space The set of all possible outcomes of an experiment is called the sample space of the experiment and it is denoted by S.
(iii) Event A subset of the sample space associated with a random experiment is called event or case.
(iv) Sample Points The outcomes of an experiment is called the sample point.
(v) Certain Event An event which must occur, whatever be the outcomes, is called a certain or sure event.
(vi) Impossible Event An event which cannot occur in a particular random experiment, is called an impossible event.
(vii) Elementary Event An event certaining only one sample point is called elementary event or indecomposable events.
(viii) Favourable Event Let S be the sample space associated with a random experiment and let E ⊂ S. Then, the elementary events belonging to E are known as the favourable event to E .
(ix)  Compound Events An event certaining more than one sample points is called compound events or decomposable events.
Probability
If there are n elementary events associated with a random experiment and m of them are favourable to an event A, then the probability of happening or occurrence of A, denoted by P(A), is given by
P(A) = m / n = Number of favourable cases / Total number of possible cases
Types of Events
(i) Equally Likely Events The given events are said to be equally likely, if none of them is expected to occur in preference to the other.
(ii) Mutually Exclusive Events A set of events is said to be mutually exclusive, if the happening of one excludes the happening of the other.
If A and B are mutually exclusive, then P(A ∩ B) = 0
(iii) Exhaustive Events A set of events is said to be exhaustive, if the performance of the experiment always results in the occurrence of atleast one of them.
If E1, E2, … , En are exhaustive events, then El ∪ E2 ∪ … ∪ En = S i.e., P(E1 ∪ E2 ∪ E3 ∪ … ∪ En) = 1
(iv) Independent Events Two events A and B associated to a random experiment are independent, if the probability of occurrence or non-occurrence of A is not affected by the occurrence or non-occurrence of B.
i.e., P(A ∩ B) = P(A) P(B)
Complement of an Event
Let A be an event in a sample space S~the complement of A is the set of all sample points of the space other than the sample point in A and it is denoted by,
A’ or A = {n : n ∈ S, n ∉ A}
(i) P(A ∪ A’) = S
(ii) P(A ∩ A’) = φ
(iii) P(A’)’ = A
Partition of a Sample Space
The events A1, A2,…., An represent a partition of the sample space S, if they are pairwise disjoint, exhaustive and have non-zero probabilities. i.e.,
(i) Ai ∩ Aj = φ; i ≠ j; i,j= 1,2, …. ,n
(ii) A1 ∪ A2 ∪ … ∪ An = S
(iii) P(Ai) > 0, ∀ i = 1,2, …. ,n
Important Results on Probability
(i) If a set of events A1, A2,…., An are mutually exclusive, then
A1 ∩ A2 ∩ A3 ∩ …∩ An = φ
P(A1 ∪ A2 ∪ A3 ∪… ∪ An) = P(A1) + (A2) + … + P(An)
and A1 ∩ A2 ∩ A3 ∩ …∩ An = 0
(ii) If a set of events A1, A2,…., An are exhaustive, then
P(A1 ∪ A2 ∪ … ∪ An) = 1
(iii) Probability of an impossible event is O. i.e., P(A) = 0, if A is impossible event. ,
(iv) Probability of any event in a sample space is 1. i.e., P(A) = 1
(v) Odds in favour of A = P(A) / P(A)
(v) Odds in Against of A = P(A) / P(A)
(vii) Addition Theorem of Probability
(a) For two events A and B
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
(b) For three events A, Band C
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) -P(A ∩ B) – P(B ∩ C) – P(A ∩ C) + P(A ∩ B ∩ C)
(c) For n events A1, A2,…., An

(viii) Booley’s Inequality
If A1, A2,…., An are n events associated with a random experiment, then

(ix) If A and B are two events, then
P(A ∩ B) ≤ P(A) ≤ P(A ∪ B) ≤ P(A) + P(B)
(x) If A and B are two events associated with a random experiment, then
(a) P(A ∩ B) = P(B) – P(A ∩ B)
(b) P(A ∩ B) = P(A) – P(A ∩ B)
(c)P [(A ∩ B) ∪ (A ∩ B)] = P(A) + P(B) – 2P(A ∩ B)
(d) P(A ∩ B) = 1- P(A ∪ B)
(e) P(A ∪ B) = 1- P(A ∩ B)
(f) P(A) = P(A ∩ B) + P(A ∩ B).
(g) P(B) = P(A ∩ B) + P(B ∩ A)
(xi) (a) P (exactly one of A, B occurs)
= P(A) + P(B) – 2P(A ∩ B) = P(A ∪ B) – P(A ∩ B)
(b) P(neither A nor B) = P(A’ ∩ B’) = 1 – P(A ∪ B)
(xii) If A, Band C are three events, then
(a) P(exactly one of A, B, C occurs)
= P(A) + P(B) + P(C) – 2P(A ∩ B) – 2P(B ∩ C) – 2P(A ∩ C) + 3P(A ∩ B ∩ C)
(b) P (atleast two of A, B, C occurs)
= P(A ∩ B) + P(B ∩ C) + P(C ∩ A) – 2P(A ∩ B ∩ C)
(c) P (exactly two of A, B, C occurs) .
= P(A ∩ B) + P(B ∩ C) + P(A ∩ C) – 3P(A ∩ B ∩ C)
(xiii) (a) P(A ∪ B) = P(A) + P(B), if A and B are mutually exclusive events.
(b) P(A ∪ B ∪ C) = P(A) + P(B) + P(C), if A, Band C are mutually exclusive events.
(xiv) P(A) = 1- P(A)
(xv) P(A ∪ B) = P(S) = 1, P(φ) = 0
(xvi) P(A ∩ B) = P(A) x P(B) , if A and B are independent events.
(xvii) If A and B are independent events associated with a random experiment, then
(a) A and B are independent events.
(b) A and B are independent events.
(c) A and B are independent events.
(xviii) If A1, A2,…., An are independent events associated with a random experiment, then probability of occurrence of atleast one
= P(A1 ∪ A2 ∪…. ∪ An) = 1 – P(A1 ∪ A2 ∪…. ∪ An)
= 1 – P(A1)P(A2)…P(An)
(xix) If B ⊆ A, then P(A ∩ B) = P(A) – P(B)
Conditional Probability
Let A and B be two events associated with a random experiment, Then, the probability of occurrence of event A under the condition that B has already occurred and P(B) ≠ 0, is called the conditional probability.
i.e., P(A/B) = P(A ∩ B) / P(B)
If A has already occurred and P (A) ≠ 0, then
P(B/A) = P(A ∩ B) / P(A)
Also, P(A / B) + P (A / B) = 1
Multiplication Theorem on Probability
(i) If A and B are two events associated with a random experiment, then
P(A ∩ B) = P(A)P(B /A), IF P(A) ≠ 0
OR
P(A ∩ B) = P(B)P(A /B), IF P(B) ≠ 0
(ii) If A1, A2,…., An are n events associated with a random experiment, then
P(A1 ∩ A2 ∩…. ∩ An) = P(A1) P(A2 / A1) P(A3 / (A1 ∩ A2)) …P(An / (A1 ∩ A2 ∩ A3 ∩…∩A n – 1))
Total Probability
Let S be the sample space and let E1, E2,…., En be n mutually exclusive and exhaustive events associated with a random experiment. If A is any event which occurs with E1 or E2 or … or En then
P(A) = P(E1)P(A / E1) + P(E2)P(A / E2) + … + P(En) P(A / En)

Baye’s Theorem
Let S be the sample space and let E1, E2,…,En, be n mutually exclusive and exhaustive events associated With a random experiment. If A is any event which occurs with E1 or E2 or … or En then probability of occurrence of Ei, when A occurred,

where, P (Ei), i = 1,2, , n are known as the priori probabilities
P (A / Ei), i = 1,2, , n are called the likelyhood probabilities
P (Ei / A), i = 1, 2, … ,n are called the posterior probabilities
Random Variable
Let U or S be a sample space associated with a given random experiment. A real valued function X defined on U or S, i:e.,
X : U → R is called a random variable.
There are two types of random variable.
(i) Discrete Random Variable – If the range of the real function X: U → R is a finite set or an infinite set of real numbers, it is called a discrete random variable.
(ii) Continuous Random Variable – If the range of X is an interval (a, b) of R, then X is called a continuous random variable. e.g., In tossing of two coins S = {HH, HT, TH , TT}, let X denotes number of heads in tossing of two coins, then
X (HH) = 2,X (TH) = 1, X (TT) = 0
Probability Distribution
If a random variable X takes values X1, X2,…., Xn with respective probabilities P1, P2,…., Pn then

is known as the probability distribution of X.
or
Probability distribution gives the values of the random variable along with the corresponding probabilities.
Mathematical Expectation/Mean
If X is a discrete random variable which assume values X1, X2,…., Xn with respective probabilities P1, P2,…., Pn then the mean x of X is defined as
E(X) = X = P1X1 + P2X2 + … + PnXn = Σni = 1 PiXi
Important Results
(i) Variance V(X) = σ2x = E(X2) – (E(X))2
where, E(X2) = Σni = 1 x2iP(xi)
(ii) Standard Deviation
√V(X) = σx = √E(X2) – (E(X))2
(iii) If Y = a X + b, then
(a) E(Y) = E(aX + b) = aE(X) + b
(b) σ2y = a2V(Y) = a2σ2x
(c) σy = √V(Y) = |a|σx
(iv) If Z = aX2 + bX + c, then
E(Z) = E(aX2 + bX + c)
= aE(X2) + bE(X) + c
Binomial Distribution
Bernaulli Trial
In a random experiment, if there are any two events, “Success and Failure” and the sum of the probabilities of these two events is 1, then any outcome of such experiment is- known as a Bernaulli Trial.
Binomial Distribution
The probability of r successes in n independent Bernaulli Trials is denoted by P(X = r) and is given by
P(X = r) = nCrprqn – r,
where p = probability of success,
q = probability of failure
and p+q=l
Important Results
(i) If P = q, then probability of r successes in n trials is nCrpn
(ii) If the total number of trials is n in any attempt and if there are N such attempts, then the total number of r successes is N(nCrprqn – r)
(iii) Mean = E(X) = x= np
(iv) Variance = σ2x = npq
(v) Standard Deviation = σ2x = √npq
(vi) Mean is always greater than variance
Poisson’s Distribution
It is the limiting case of binomial distribution under the following conditions
(i) Number of trials are very large, i.e., n → ∞
(ii) p → 0
(iii) np → λ, a finite quantity (λ A is called parameter)
The probability of r success for Poisson’s distribution is given by
P(X = r) = e – λλ’ / r!, r = 0, 1, 2,…
For Poisson’s distribution
Mean = Variance = λ = np
Geometrical Probability
If the total number’ of outcomes of a trial in a random experiment is infinite, in such cases, the definitioin, of probability is modified and the
general expression for the probability P of occurrence of an event is given by
p = Measure of the specifie part of the region / Measure of the whole region
where, measure means length or al’~a or volume of the region, if we are dealing with one, two or three dimensional space respectively.
Application Based Result
(i) When two dice are thrown, the number of ways of getting a total r is
(a) (r – 1), if 2 ≤ r ≤ 7 and (b) (13 – r), if 8 ≤ r ≤ 12
(ii) Experiment with insertion of n letters in n addressed envelopes.
(a) Probability of inserting all the n letters in right envelopes
= 1 / n!
(b) Probability that all letters does not in right envelopes
1 – 1 / n!
(c) Probability of keeping al1 the letters in wrong envelope
1 / 2! – 1 / 3!+…+ (-1)n / n!
(d) Probability that exactly letters are in right envelopes
= 1 / r! [1 / 2! – 1 / 3!+ 1 / 4 -…+ (-1)n – r / (n – r)!]
(iii) (a) Selection of shoes from a Cupboard Out of n pair of shoes, if k shoes are selected at random, the probability that there is no pair is
p = nCk2k / 2nCk
(b) The probability that there is atleast one pair is (1- p).
(iv) Selection of Squares from the Chessboard – If r squares are selected from a chessboard, then probability that they lie on a diagonal is
4[7Cr + 6Cr +… + 1Cr] + 2(8Cr) / 64Cr
(v) If A and B are two finite sets and if a mapping is selected at random from the set of all mapping from A into B, then the probability that the mapping is
(a) a one-one function = n(B)Pn(A) / n(B)n(A)
(b) a many-one function = 1 – n(B)Pn(A) / n(B)n(A)
(c) a constant function = n(B) / n(B)n(A)
(d) a one-one onto function = n(A)! / n(B)n(A)
CBSE Class 11 and 12 Maths Notes : Inverse T – Functions and T – Equations
May 1, 2014 by Neepur Garg Leave a Comment
Inverse Function
If y = f(x) and x = g(y) are two functions such that f (g(y)) = y and g (f(y)) = x, then f and y are said to be inverse of each other
i.e., g = f-1
IF y = f(x), then x = f-1(y)
Inverse Trigonometric Functions
If y = sin X-1, then x = sin-1 y, similarly for other trigonometric functions.
This is called inverse trigonometric function .
Now, y = sin-1(x), y ∈ [π / 2 , π / 2] and x ∈ [-1,1].
(i) Thus, sin-1x has infinitely many values for given x ∈ [-1, 1].
(ii) There is only one value among these values which lies in the interval [π / 2 , π / 2]. This value is called the principal value.
Domain and Range of Inverse Trigonometric FunctionsGraphs of Inverse Trigonometric Functions
 

Properties of Inverse Trigonometric Functions
Property I


Property II

Property III

Property IV

Property V

Property VI

Property VII

Property VIII


Property IX

Property X


Property XI

Property XII


Important Results

where Sk denotes the sum of the product of x1,x2,…xn takes k at a time.
Inverse Hyperbolic Functions
If sinh y = x, then y is called the inverse hyperbolic sine of x and it is written as y = sinh-1 x.
Similarly, cosh-1 x, tan h-1 x etc., can be defined,
Domain and Range of Inverse Hyperbolic Functions

Relation between Inverse Circular Functions and Inverse Hyperbolic Functions
(i) sinh-1x = – i sin-1(ix)
(ii) cosh-1x = – i cos-1 x .
(iii) tanh--1x = – i tan-1(ix)
Important Results

Trigonometric Equation
An equation involving one or more trigonometrical ratios of unknown angle is called a trigonometric equation .
Solution/Roots of a Trigonometric Equation
A value of the unknown angle which satisfies the given equation, is called a solution or root of the equation.
The trigonometric equation may have infinite number of solutions.
(i) Principal Solution – The least value of unknown angle which satisfies the given equation, is called a principal solution of trigonometric equation.
(ii) General Solution – We know that, trigonometric function are periodic and solution of trigonometric equations can be generalised with the help of the periodicity of the trigonometric functions. The solution consisting of all possible solutions of a trigonometric equation is called its general solution.
Important Results


Important Points to be Remembered
(i) While solving an equation, we have to square it, sometimes the resulting roots does not satisfy the original equation.
(ii) Do not cancel common factors involving the unknown angle on LHS and RHS.Because it may be the solution of given equation.
(iii) (a) Equation involving sec θ or tan θ can never be a solution of the form (2n + 1) π / 2.
(b) Equation involving coseca or cote can never be a solution of the form θ = nπ.
CBSE Class 12 Maths Notes : Functions
May 28, 2014 by Neepur Garg Leave a Comment
Let A and B be .two non-empty sets, then a function f from set A to set B is a rule which associates each element of A to a unique element of B.
It is represented as f: A → B and function is also called mapping.
f : A → B is called a real function, if A and B are subsets of R.
Domain and Codomain of a Real Function
Domain and codomain of a function f is a set of all real numbers x for which f(x) is a real number. Here, set A is domain and set B is codomain.
Range of a Real Function
Range of a real function, f is a set of values f(x) which it attains on the points of its domain.
Classification of Real Functions
Real functions are generally classified under two categories algebraic functions and transcendental functions.
1. Algebraic Functions
Some algebraic functions are given below
(i) Polynomial Functions If a function y = f(x) is given by

where, a0, a1, a2,…, an are real numbers and n is any non -negative integer, then f (x) is called a polynomial function in x.
If a0 ≠ 0, then the degree of the polynomial f(x) is n. The domain of a polynomial function is the set of real number R.
e.g., y = f(x) = 3x5 – 4x2 – 2x + 1
is a polynomial of degree 5.
(ii) Rational Functions If a function y = f(x) is given by
f(x) = φ(x) / Ψ(x)
where, φ(x) and Ψ(x) are polynomial functions, then f(x) is called rational function in x.
(iii) Irrational Functions The algebraic functions containing one or more terms having non-integral rational power x are called irrational functions.
e.g., y = f(x) = 2√x – 3√x + 6
2. Transcendental Function
A. function, which is not algebraic, is called a transcendental function. Trigonometric, Inverse trigonometric, Exponential, Logarithmic, etc are transcendental functions.
Explicit and Implicit Functions
(i) Explicit Functions A function is said to be an explicit function, if it is expressed in the form y = f(x).
(ii) Implicit Functions A function is said to be an implicit function, if it is expressed in the form f(x, y) = C, where C is constant.
e.g., sin (x + y) – cos (x + y) = 2
Intervals of a Function
(i) The set of real numbers x, such that a ≤ x ≤ b is called a closed interval and denoted by [a, b] i.e., {x: x ∈ R, a ≤ x ≤ b}.
(ii) Set of real number x, such that a < x < b is called open interval and is denoted by (a, b)
i.e., {x: x ∈ R, a < x < b}
(iii) Intervals [a,b) = {x: x ∈ R, a ≤ x ≤ b} and (a, b] = {x: x ≠ R, a < x ≤ b} are called semi-open and semi-closed intervals.
Graph of Real Functions
1. Constant Function
Let c be a fixed real number. The function that associates to each real number x, this fixed number c is called a constant function i.e., y = f{x) = c for all x ∈ R.
Domain of f{x) = R
Range of f{x) = {c}

2. Identity Function
The function that associates to each real number x for the same number x, is called the identity function.
i.e., y = f(x) = x, ∀ x ∈ R.
Domain of f(x) = R
Range f(x) = R

3. Linear Function
If a and b be fixed real numbers, then the linear function is defmed as y = f(x) = ax + b, where a and b are constants.
Domain of f(x) = R
Range of f(x) = R
The graph of a linear function is given in the following diagram, which is a straight line with slope a.

4. Quadratic Function
If a, b and c are fixed real numbers, then the quadratic function is expressed as
y = f(x) = ax2 + bx + c, a ≠ 0
⇒ y = a (x + b / 2a)2 + 4ac – b2 / 4a
which is equation of a parabola in downward, if a < 0 and upward, if a > 0 and vertex at ( – b / 2a, 4ac – b2 / 4a).
Domain of f(x) = R
Range of f(x) is [ – ∞, 4ac – b2 / 4a], if a < 0 and [4ac – b2 / 4a, ∞], if a > 0

5. Square Root Function
Square root function is defined by y = F(x) = √x, x ≥ 0.
Domain of f(x) = [0, ∞)
Range of f(x) = [0, ∞)

6. Exponential Function
Exponential function is given by y = f(x) = ax, where a > 0, a ≠ 1.

The graph of the function is as shown, which is increasing, if a > 1 and decreasing, if 0 < a < 1.
Domain of f(x) = R
Range of f(x) = (0, ∞)

7. Logarithmic Function
A logarithmic function may be given by y = f(x) = loga x, where a > 0, a ≠ 1 and x > 0.
The graph of the function is as shown below. which is increasing, if a > 1 and decreasing, if 0 < a < 1.

Domain of f(x) = (0, ∞)
Range of f(x) = R
8. Power Function
The power function is given by y = f(x) = xn ,n ∈ I, n ≠ 1, 0.
The domain and range of the graph y = f(x), is depend on n.
(a) If n is positive even integer.

i.e., f(x) = x2, x4 ,….
Domain of f(x) = R
Range of f(x) = [0, ∞)
(b) If n is positive odd integer.

i.e., f(x) = x3, x5 ,….
Domain of f(x) = R
Range of f(x) = R
(c) If n is negative even integer.
i.e., f(x) = x– 2, x – 4 ,….

Domain of f(x) = R – {0}
Range of f(x) = (0, ∞)
(d) If n is negative odd integer.

i.e., f(x) = x– 1, x – 3 ,….
Domain of f(x) = R – {0}
Range of f(x) = R – {0}
9. Modulus Function (Absolute Value Function)
Modulus function is given by y = f(x) = |x| , where |x| denotes the absolute value of x, that is
|x| = {x, if x ≥ 0, – x, if x < 0

Domain of f(x) = R
Range of f(x) = [0, &infi;)

Domain of f(x) = R
Range of f(x) = {-1, 0, 1}
11. Greatest Integer Function

The greatest integer function is defined as
y = f(x) = [x]
where, [x] represents the greatest integer less than or equal to x. i.e., for any integer n, [x] = n, if n ≤ x < n + 1 Domain of f(x) = R Range of f(x) = I
Properties of Greatest Integer Function
(i) [x + n] = n + [x], n ∈ I
(ii) x = [x] + {x}, {x} denotes the fractional part of x.
(iii) [- x] = – [x], -x ∈ I
(iv) [- x] = – [x] – 1, x ∈ I
(v) [x] ≥ n ⇒ x ≥ n,n ∈ I
(vi) [x] > n ⇒ x ⇒ n+1, n ∈ I
(vii) [x] ≤ n ⇒ x < n + 1, n ∈ I
(viii) [x] < n ⇒ x < n, n ∈ I
(ix) [x + y] = [x] + [y + x – [x}] for all x, y ∈ R
(x) [x + y] ≥ [x] + [y]
(xi) [x] + [x + 1 / n] + [x + 2 / n] +…+ [x + n – 1 / n] = [nx], n ∈ N
12. Least Integer Function
The least integer function which is greater than or equal to x and it is denoted by (x).
Thus, (3.578) = 4, (0.87) = 1, (4) = 4, (- 8.239) = – 8, (- 0.7) = 0

In general, if n is an integer and x is any real number between n and (n + 1).
i.e., n < x ≤ n + 1, then (x) = n + 1
∴ f(x) = (x)
Domain of f = R
Range of f= [x] + 1
13. Fractional Part Function
It is denoted as f(x) = {x} and defined as
(i) {x} = f, if x = n + f, where n ∈ I and 0 ≤ f < 1
(ii) {x} = x – [x]

i.e., {O.7} = 0.7, {3} = 0, { – 3.6} = 0.4
(iii) {x} = x, if 0 ≤ x ≤ 1
(iv) {x} = 0, if x ∈ I
(v) { – x} = 1 – {x}, if x ≠ I
Graph of Trigonometric Functions
1. Graph of sin x

(i) Domain = R
(ii) Range = [-1,1]
(iii) Period = 2π
2. Graph of cos x

(i) Domain = R
(ii) Range = [-1,1]
(iii) Period = 2π
3. Graph of tan x

(i) Domain = R ~ (2n + 1) π / 2, n ∈ I
(ii) Range = [- &infi;, &infi;]
(iii) Period = π
4. Graph of cot x

(i) Domain = R ~ nπ, n ∈ I
(ii) Range = [- &infi;, &infi;]
(iii) Period = π
5. Graph of sec x

(i) Domain = R ~ (2n + 1) π / 2, n ∈ I
(ii) Range = [- &infi;, 1] ∪ [1, &infi;)
(iii) Period = 2π
6. Graph of cosec x

(i) Domain = R ~ nπ, n ∈ I
(ii) Range = [- &infi;, – 1] ∪ [1, &infi;)
(iii) Period = 2π
Operations on Real Functions
Let f: x → R and g : X → R be two real functions, then
(i) Sum The sum of the functions f and g is defined as
f + g : X → R such that (f + g) (x) = f(x) + g(x).
(ii) Product The product of the functions f and g is defined as
fg : X → R, such that (fg) (x) = f(x) g(x)
Clearly, f + g and fg are defined only, if f and g have the same domain. In case, the domain of f and g are different. Then, Domain of f + g or fg = Domain of f ∩ Domain of g.
(iii) Multiplication by a Number Let f : X → R be a function and let e be a real number .
Then, we define cf: X → R, such that (cf) (x) = cf (x), ∀ x ∈ X.
(iv) Composition (Function of Function) Let f : A → B and g : B → C be two functions.
We define gof : A → C, such that
got (c) = g(f(x)), ∀ x ∈ A
Alternate There exists Y ∈ B, such that
if f(x) = y and g(y) = z, then got (x) = z
Periodic Functions
A function f(x) is said to be a periodic function of x, provided there exists a real number T > 0, such that
F(T + x) = f(x), ∀ x ∈ R
The smallest positive real number T, satisfying the above condition is known as the period or the fundamental period of f(x) ..
Testing the Periodicity of a Function
(i) Put f(T + x) = f(x) and solve this equation to find the positive values of T independent of x.
(ii) If no positive value of T independent of x is obtained, then f(x) is a non-periodic function.
(iii) If positive val~es ofT independent of x are obtained, then f(x) is a periodic function and the least positive value of T is the period of the function f(x).
Important Points to be Remembered
(i) Constant function is periodic with no fundamental period.
(ii) If f(x) is periodic with period T, then 1 / f(x) and. √f(x) are also periodic with f(x) same period T.
{iii} If f(x) is periodic with period T1 and g(x) is periodic with period T2, then f(x) + g(x) is periodic with period equal to LCM of T1 and T2, provided there is no positive k, such that f(k + x) = g(x) and g(k + x) = f(x).
(iv) If f(x) is periodic with period T, then kf (ax + b) is periodic with period T / |a|’ where a, b ,k ∈ R and a, k ≠ 0.
(v) sin x, cos x, sec x and cosec x are periodic functions with period 2π.
(vi) tan x and cot x are periodic functions with period π.
(vii) |sin x|, |cos x|, |tan x|, |cot x|, |sec x| and |cosec x| are periodic functions with period π.
(viii) sinn x, cosn x, secn x and cosecnx are periodic functions with period 2π when n is odd, or π when n is even .
(ix) tann x and cotn x are periodic functions with period π.
(x) |sin x| + |cos x|, |tan x| + |cot x| and |sec x| + |cosec x| are periodic with period π / 2.
(xi) If f(x) is a periodic function with period T and g(x) is any function, such that domain of f ⊂ domain of g, then gof is also periodic with period T.
Even and Odd Functions
Even Functions A real function f(x) is an even function, if f( -x) = f(x).
Odd Functions A real function f(x) is an odd function, if f( -x) = – f(x).
Properties of Even and Odd Functions
(i) Even function ± Even function = Even function.
(ii) Odd function ± Odd function = Odd function.
(iii) Even function * Odd function = Odd function.
(iv) Even function * Even function = Even function.
(v) Odd function * Odd function = Even function.
(vi) gof or fog is even, if anyone of f and g or both are even.
(vii) gof or fog is odd, if both of f and g are odd.
(viii) If f(x) is an even function, then d / dx f(x) or ∫ f(x) dx is odd and if dx .. f(x) is an odd function, then d / dx f(x) or ∫ f(x) dx is even.
(ix) The graph of an even function is symmetrical about Y-axis.
(x) The graph of an odd function is symmetrical about origin or symmetrical in opposite quadrants.
(xi) An even function can never be one-one, however an odd function mayor may not be one-one.
Different Types of Functions (Mappings)
1. One-One and Many-One Function
The mapping f: A → B is a called one-one function, if different elements in A have different images in B. Such a mapping is known as injective function or an injection.
Methods to Test One-One
(i) Analytically If x1, x2 ∈ A,
then f(x1) = f(x2) => x1 = x2
or equivalently x1 ≠ x2 => f(x1) ≠ f(x2)
(ii) Graphically If any .line parallel to x-axis cuts the graph of the function atmost at one point, then the function is one-one.
(iii) Monotonically Any function, which is entirely increasing or decreasing in whole domain, then f(x) is one-one.
Number of One-One Functions Let f : A → B be a function, such that A and B are finite sets having m and n elements respectively, (where, n > m).
The number of one-one functions
n(n – 1)(n – 2) …(n – m + 1) = { nPm, n ≥ m, 0, n < m
The function f : A → B is called many – one function, if two or more than two different elements in A have the same image in B.
2. Onto (Surjective) and Into Function
If the function f: A → B is such that each element in B (codomain) is the image of atleast one element of A, then we say that f is a function of A ‘onto’ B. Thus, f: A → B, such that f(A) = B.
i.e., Range = Codomain
Note Every polynomial function f: R → R of degree odd is onto.
Number of Onto (surjective) Functions Let A and B are finite sets having m and n elements respectively, such that 1 ≤ n ≤ m, then number of onto (surjective) functions from A to B is
nΣr = 1 (- 1)n – r nCr rm
= Coefficient of xn in n! (ex – 1)r
If f : A → B is such that there exists atleast one element in codomain which is not the image of any element in domain, then f(x) is into.
Thus, f : A → B, such that f(A) ⊂ B
i.e., Range ⊂ Codomain
Important Points to be Remembered
(i) If f and g are injective, then fog and gof are injective.
(ii) If f and g are surjective, then fog is surjective.
(iii) Iff and g are bijective, then fog is bijective.
Inverse of a Function
Let f : A → B is a bijective function, i.e., it is one-one and onto function.

We define g : B → A, such that f(x) = y => g(y) = x, g is called inverse of f and vice-versa. Symbolically, we write g = f-1
Thus, f(x) = y => f-1(y) = x
CBSE Class 12 Maths Notes : Limits, Continuity and Differentiablity
May 28, 2014 by Neepur Garg Leave a Comment
Limit
Let y = f(x) be a function of x. If at x = a, f(x) takes indeterminate form, then we consider the values of the function which is very near to a. If these values tend to a definite unique number as x tends to a, then the unique number, so obtained is called the limit of f(x) at x = a and we write it as  .
Left Hand and Right Hand Limits
If values of the function at the points which are very near to a on the left tends to a definite unique number as x tends to a, then the unique number, so obtained is called the left hand limit of f(x) at x = a. We write it as

Uniqueness of Limit If  exists, then it is unique. There cannot be two distinct numbers l1 and l2 such that when x tends to a, the function f(x) tends to both l1 and l2.
Fundamental Theorems on Limits

Important Results on Limit
1. Trigonometric Limits

2. Exponential Limits

3. Logarithmic Limits

4. Based on the Form 1∞

Methods of Evaluating Limits
1. Determinate Forms (Limits by Direct Substitution)
To find   we substitute x = a in the function. If the value x –> a comesout to be a definite value, it is the limit. That is  = f(a) provided it exists.
2. Indeterminate Forms
If direct substitution of x = a while evaluating   leads to one of the following form

Then, it is called interminate form these limits can be counted by  using L’ Hospitals’s rule or some other method given below.
(i) Limits by Factorisation If  attains 0/0  form, the x-a  must be a factor of numerator and denominator which ca be cancelled out.

(ii) Limits by Substitution In order to evaluate   a may substitute x = a + h or a – h, so that as x – a, h → 0. Thus

This method is applied to bring the limit at zero as the most formulae are given as  .
(iii) Limits of Functions as x → ∞
If   is of the form ∞/∞ and f(x) and g(x) are both polynomial of x. Then, we divide numerator and denominator by the highest power of x and put 0 for -1/x.
If m and n are positive integers and a0, b0 # 0 are non-zero real numbers, then

L’Hospital’s Rule

Limit Using Expansions
Many limits can be ecaluated very easily by applying expansion series, some of the standard expansions are

Important Result

Use of newton-Leibnitz’s formula in Evaluating the Limits

Sandwich theorem


Continuity
If the graph of a function has no break or gap, then it is continuous, otherwise it is discontinuous. A function which is not continuous is called a discontinuous function.
e.g., Graph of sin x is continuous

While f(x) = 1/x is discontinuous at x=0.

Continuity at a Point

Cauchy’s Definition of Continuity
A function f is said to be continuous at a point a of its domain differentiate for every ε > 0 there exists ε > 0 (dependent on ε) such that |x-a| < δ ⇒ |f(x) – f(a)| < ε.
Heine’s Definition of Continuity
A function f is said to be continuous at a point ‘a’ of its domain D, if for every sequence < an > of the points in D converging to a, then the sequence < f(an) > converges to f(a) i.e., lim an = a = lim f (an) )= 1(a).
Discontinuity of a Function
The function f(x) can be discontinuous at a point x = a in any one of the following ways.

Important Points to be Remembered
(i) If f (x) is continuous and g(x) is discontinuous at x = a, then the product function φ(x) = f(x).g(x) is not necessarily be discontinuous at x = a.
(ii) If f (x) and g (x) both are discontinuous at x = a, then the product function φ(x) =f(x) g(x) is not necessarily be discontinuous at x = a.
Continuity of a Function in an Interval
(i) A function f(x) is said to be continuous in an open interval (a, b), if f(x) is continuous at every point of the interval.
(ii) A function f(x) is said to be continuous in a closed interval [a, b], if (x) is continuous in (a, b). In addition, f(x) is continuous at x = a from right limit and f(x) is continuous at x = b, from left limit.

Fundamental Theorems of Continuity
(i) If f and g are continuous functions, then
•  f ± g and fg are continuous.
• cf is continuous, where c is a constant.
• f/g  is continuous at those points, where g(x) ≠ 0.
(ii) If g is continuous at a point a and f is continuous at g(a), then fog is continuous at a.
(iii) If f is continuous in [a, b] , then it is bounded in [a, b] i.e. , there exist m and M such that
m ≤ f(x) ≤ M, ∀ x ∈ [a, b]
where m and M are called minimum and maximum values f(x) respectively in the interval [a, b].
(iv) If f is continuous in [a, b], then f assumes atleast once eve value between minimum and maximum values of f(x).
Thus, a ≤ x ≤ b ⇒ m ≤ f(x) ≤ M or range of f(x) = [m, M], x ε [a, b].
(v) If f is continuous in its domain, then |f| is also continuous in it domain.
(vi) If f is continuous at a and f (a) ≠ 0, then there exists an ope interval (a — δ, a + δ) such that for all x ε (a — δ, a + δ), f (x) has the same sing as f(a).
(vii) If f is a continuous function defined on [a, b] such that f (a) an f (b) are of opposite sign, then there exists atleast one solution the equation f(x)= 0 in the open interval (a, b).
(viii) If f is continuous on [a, b] and maps [a, b] into [a, b], then for some x ε [a , b], we have f (x)= x.
(ix) If f is continuous in domain D, then 1/f  is also continuous in D – {x : f(x)= 0}.
(x) A function f(x) is said to be everywhere continuous, if it is continuous on the entire real line (-∞,∞)
Differentiability of a Function at a Point
The function f(x) is differentiable at a point P iff there exists a unique tangent at point P.
In other words, f(x) is differentiable at a point P iff the curve does not have P as a corner point i.e., the function is not differentiable at those points on which function has holes or sharp edges.
Let us consider the function f(x) = |x – 1|.

Differentiability in an Interval
A function f(x) is said to be differentiable in an interval (a, b), if f(x) is differentiable at every point of this interval (a, b).
A function f(x) is said to be differentiable in a closed interval [a, b], if f(x) is differentiable in (a, b), in addition f(x) is differentiable at x = a from right hand limit and differentiable at x = b from left hand limit.
Relation between Continuity and Differentiability
(i) If a function f(x) is differentiable at x = a, then f(x) is necessarily continuous at x = a but the converse is not necessary true.
(ii) The sum, difference, product and quotient of two differentiable function is differentiable. The composition of differentiable function is a differentiable funciton converse of (i) is not necessarily true i.e., if a function f(x) is continuous at x = a, then it is not necessarily differentiable at x = a e.g., f(x) =I xl is continuous at x = 0 but not differentiable at x = 0.
Continuity and Differentiability of Different Functions


CBSE Class 12 Maths Notes : Differentiation
January 23, 2015 by Neepur Garg Leave a Comment
Derivative
The rate of change of a quantity y with respect to another quantity x is called the derivative or differential coefficient of y with respect to x .
Differentiation of a Function
Let f(x) is a function differentiable in an interval [a, b]. That is, at every point of the interval, the derivative of the function exists finitely and is unique. Hence, we may define a new function g: [a, b] → R, such that, ∀ x ∈ [a, b], g(x) = f'(x).
This new function is said to be differentiation (differential coefficient) of the function f(x) with respect to x and it is denoted by df(x) / d(x) or Df(x) or f'(x).

Differentiation ‘from First Principle
Let f(x) is a function finitely differentiable at every point on the real number line. Then, its derivative is given by

Standard Differentiations
1. d / d(x) (xn) = nxn – 1, x ∈ R, n ∈ R
2. d / d(x) (k) = 0, where k is constant.
3. d / d(x) (ex) = ex
4. d / d(x) (ax) = ax loge a > 0, a ≠ 1


Fundamental Rules for Differentiation

(v) if d / d(x) f(x) = φ(x), then d / d(x) f(ax + b) = a φ(ax + b)
(vi) Differentiation of a constant function is zero i.e., d / d(x) (c) = 0.
Geometrically Meaning of Derivative at a Point
Geometrically derivative of a function at a point x = c is the slope of the tangent to the curve y = f(x) at the point {c, f(c)}.
Slope of tangent at P = lim x → c f(x) – f(c) / x – c = {df(x) / d(x)} x = c or f’ (c).
Different Types of Differentiable Function
1. Differentiation of Composite Function (Chain Rule)
If f and g are differentiable functions in their domain, then fog is also differentiable and
(fog)’ (x) = f’ {g(x)} g’ (x)
More easily, if y = f(u) and u = g(x), then dy / dx = dy / du * du / dx.
If y is a function of u, u is a function of v and v is a function of x. Then,
dy / dx = dy / du * du / dv * dv / dx.
2. Differentiation Using Substitution
In order to find differential coefficients of complicated expression involving inverse trigonometric functions some substitutions are very helpful, which are listed below .

3. Differentiation of Implicit Functions
If f(x, y) = 0, differentiate with respect to x and collect the terms containing dy / dx at one side and find dy / dx.
Shortcut for Implicit Functions For Implicit function, put d /dx {f(x, y)} = – ∂f / ∂x / ∂f / ∂y, where ∂f / ∂x is a partial differential of given function with respect to x and ∂f / ∂y means Partial differential of given function with respect to y.
4. Differentiation of Parametric Functions
If x = f(t), y = g(t), where t is parameter, then
dy / dx = (dy / dt) / (dx / dt) = d / dt g(t) / d / dt f(t) = g’ (t) / f’ (t)
5. Differential Coefficient Using Inverse Trigonometrical Substitutions
Sometimes the given function can be deducted with the help of inverse Trigonometrical substitution and then to find the differential coefficient is very easy.


Logarithmic Differentiation Function
(i) If a function is the product and quotient of functions such as y = f1(x) f2(x) f3(x)… / g1(x) g2(x) g3(x)… , we first take algorithm and then differentiate.
(ii) If a function is in the form of exponent of a function over another function such as [f(x)]g(x) , we first take logarithm and then differentiate.
Differentiation of a Function with Respect to Another Function
Let y = f(x) and z = g(x), then the differentiation of y with respect to z is
dy / dz = dy / dx / dz / dx = f’ (x) / g’ (x)
Successive Differentiations
If the function y = f(x) be differentiated with respect to x, then the result dy / dx or f’ (x), so obtained is a function of x (may be a constant).
Hence, dy / dx can again be differentiated with respect of x.
The differential coefficient of dy / dx with respect to x is written as d /dx (dy / dx) = d2y / dx2 or f’ (x). Again, the differential coefficient of d2y / dx2 with respect to x is written as
d / dx (d2y / dx2) = d3y / dx3 or f”'(x)……
Here, dy / dx, d2y / dx2, d3y / dx3,… are respectively known as first, second, third, … order differential coefficients of y with respect to x. These alternatively denoted by f’ (x), f” (x), f”’ (x), … or y1, y2, y3…., respectively.
Note dy / dx = (dy / dθ) / (dx / dθ) but d2y / dx2 ≠ (d2y / dθ2) / (d2x / dθ2)
Leibnitz Theorem
If u and v are functions of x such that their nth derivative exist, then

nth Derivative of Some Functions


Derivatives of Special Types of Functions

(vii) Differentiation of a Determinant

(viii) Differentiation of Integrable Functions If g1 (x) and g2 (x) are defined in [a, b], Differentiable at x ∈ [a, b] and f(t) is continuous for g1(a) ≤ f(t) ≤ g2(b), then

Partial Differentiation
The partial differential coefficient of f(x, y) with respect to x is the ordinary differential coefficient of f(x, y) when y is regarded as a constant. It is a written as ∂f / ∂x or Dxf or fx.

e.g., If z = f(x, y) = x4 + y4 + 3xy2 + x4y + x + 2y
Then, ∂z / ∂x or ∂f / ∂x or fx = 4x3 + 3y2 + 2xy + 1 (here, y is consider as constant)
∂z / ∂y or ∂f / ∂y or fy = 4y3 + 6xy + x2 + 2 (here, x is consider as constant)
Higher Partial Derivatives
Let f(x, y) be a function of two variables such that ∂f / ∂x , ∂f / ∂y both exist.
(i) The partial derivative of ∂f / ∂y w.r.t. ‘x’ is denoted by ∂2f / ∂x2 / or fxx.
(ii) The partial derivative of ∂f / ∂y w.r.t. ‘y’ is denoted by ∂2f / ∂y2 / or fyy.
(iii) The partial derivative of ∂f / ∂x w.r.t. ‘y’ is denoted by ∂2f / ∂y ∂x / or fxy.
(iv) The partial derivative of ∂f / ∂x w.r.t. ‘x’ is denoted by ∂2f / ∂y ∂x / or fyx.
Note ∂2f / ∂x ∂y = ∂2f / ∂y ∂x
These four are second order partial derivatives.
Euler’s Theorem on Homogeneous Function
If f(x, y) be a homogeneous function in x, y of degree n, then
x (&partf / ∂x) + y (&partf / ∂y) = nf
Deduction Form of Euler’s Theorem
If f(x, y) is a homogeneous function in x, y of degree n, then
(i) x (∂2f / ∂x2) + y (∂2f / ∂x ∂y) = (n – 1) &partf / ∂x
(ii) x (∂2f / ∂y ∂x) + y (∂2f / ∂y2) = (n – 1) &partf / ∂y
(iii) x2 (∂2f / ∂x2) + 2xy (∂2f / ∂x ∂y) + y2 (∂2f / ∂y2) = n(n – 1) f(x, y)
Important Points to be Remembered
If α is m times repeated root of the equation f(x) = 0, then f(x) can be written as
f(x) =(x – α)m g(x), where g(α) ≠ 0.
From the above equation, we can see that
f(α) = 0, f’ (α) = 0, f” (α) = 0, … , f(m – l) ,(α) = 0.
Hence, we have the following proposition
f(α) = 0, f’ (α) = 0, f” (α) = 0, … , f(m – l) ,(α) = 0.
Therefore, α is m times repeated root of the equation f(x) = 0.
CBSE Class 12 Maths Notes : Application of Derivatives
May 28, 2014 by Anuj William Leave a Comment
Tangents and Normals
The derivative of the curve y = f(x) is f ‘(x) which represents the slope of tangent and equation of the tangent to the curve at P is

where (x, y) is an arbitrary point on the tangent.

The equation of normal at (x, y) to the curve is

1. If    then the equations of the tangent and normal at (x, y) are (Y – y) = 0 and (X – x) = 0, respectively.
2. If    then the equation of the tangent and normal at (x, y) are (X – x) = 0 and (Y – y) = 0, respectively.
Slope of Tangent
(i) If the tangent at P is perpendicular to x-axis or parallel to y-axis,

(ii) If the tangent at P is perpendicular to y-axis or parallel to x-axis,

Slope of Normal

(ii) If   , then normal at (x, y) is parallel to y-axis and perpendicular to x-axis.
(iii) If   then normal at (x, y) is parallel to x-axis and perpendicular to y-axis.
Length of Tangent and Normal
(i) Length of tangent, PA = y cosec θ =

(ii) Length of normal,

(iii) Length of subtangent,

(iv) Length of subnormal,


Angle of Intersection of Two Curves
Let y = f1(x) and y = f2(x) be the two curves, meeting at some point P (x1, y1), then the angle between the two curves at P (x1, y1) = The angle between the tangents to the curves at P (x1, y1)

The other angle between the tangents is (180 — θ). Generally, the smaller of these two angles is taken to be the angle of intersection.
∴ The angle of intersection of two curves θ is given by

Derivatives as the Rate of Change
If a variable quantity y is some function of time t i.e., y = f(t), then small change in Δt time At have a corresponding change Δy in y.
Thus, the average rate of change = (Δy/Δt)
When limit At Δt→ 0 is applied, the rate of change becomes instantaneous and we get the rate of change with respect to at the instant x.

So, the differential coefficient of y with respect to x i.e., (dy/dx) is nothing but the rate of increase of y relative to x.
Rolle’s Theorem
Let f be a real-valued function defined in the closed interval [a, b], such that
1. f is continuous in the closed interval [a, b].
2. f(x) is differentiable in the open interval (a, b).
3. f(a)= f(b)
Then, there is some point c in the open interval (a, b), such that f’ (c) = 0.
Geometrically Under the assumptions of Rolle’s theorem, the graph of f(x) starts at point (a, 0) and ends at point (b, 0) as shown in figures.

The conclusion is that there is at least one point c between a and b, such that the tangent to the graph at (c, f(c)) is parallel to the x-axis.
Algebraic Interpretation of Rolle’s Theorem
Between any two roots of a polynomial f(x), there is always a root of its derivative f’ (x).
Lagrange’s Mean Value Theorem
Let f be a real function, continuous on the closed interval [a, b] and differentiable in the open interval (a, b). Then, there is at least one point c in the open interval (a, b), such that

Geometrically Any chord of the curve y = f(x), there is a point on the graph, where the tangent is parallel to this chord.
Remarks In the particular case, where f(a) = f(b).
The expression [f(b) – f(a)/(b – a)] becomes zero. Thus, when
f(a) = f (b), f ‘ (c) = 0 for some c in (a, b).
Thus, Rolle’s theorem becomes a particular case of the mean value theorem.
Approximations and Errors
1. Let y = f(x) be a given function. Let Ax denotes a small increment in Δx, corresponding which y increases by Δy. Then, for small increments, we assume that

2. Let Δx be the error in the measurement of independent variable x and Δy is corresponding error in the measurement of dependent variable y. Then,

• Δy = Absolute error in measurement of y
• (Δy/y) = Relative error in measurement of y
• (Δy/y) * 100 = Percentage error in measurement of y
Monotonicity of Functions
1. Monotonic Function
A function f(x) is said to be monotonic on an interval (a, b), if it is either increasing or decreasing on (a, b).
2. Strictly Increasing Function
f(x) is said to be increasing in D1, if for every x1, x2 ∈ D1, x1 > x2 ⇒ f(x1) > f(x2). It means that there is a certain increase in the value of f(x) with an increase in the value of x.
3. Classification of Strictly Increasing Function

4. Non-Decreasing Function
f(x) is said to be non-decreasing in D1, if for every x1, x2 ∈ D1, x1 > x2 ⇒ f(x1) ≥ f(x2). It means that the value of f(x) would new decrease with an increase in the value of x.
5. Strictly Decreasing Function
f(x) is said to be decreasing in D1, if for every x1, x2 ∈ D1, x1 > x2 ⇒ f(x1) < f(x2). It means that there is a certain decrease in the value c f(x) with an increase in the value of x.
Classification of Strictly Decreasing Function

6. Non-increasing Function
f(x) is said to be non-increasing in D1, if for every x1, x2 ∈ D1, x1 > x2 ⇒ f(x1) ≤ f(x2). It means that the value of f(x) would never increase with an increase in the value of x.
If a function is either strictly increasing or strictly decreasing, then it is also a monotonic function.
Important Points to be Remembered
(i) A function f (x) is said to be increasing (decreasing) at point x0, if there is an interval (x0 — h, x0 + h) containing x0, such that f(x) is increasing (decreasing) on (x0 — h, x0 + h).
(ii) A function f (x) is said to be increasing on [a , b], if it is increasing (decreasing) on (a ,b) and it is also increasing at x = a and x = b.
(iii) If (x) is increasing function on (a , b), then tangent at every point on the curve y = f(x) makes an acute angle θ with the positive direction of x-axis.
∴ 
(iv) Let f be a differentiable real function defined on an open interval (a, b).
• If  f ‘ (x) > 0 for all x ∈ (a, b), then f (x) is increasing on (a, b).
• If  f ‘ (x) < 0 for all x ∈ (a , b), then f (x) is decreasing on (a, b).
(v) Let f be a function defined on (a, b).
• If  f ‘(x) > 0 for all x ∈ (a, b) except for a finite number of points, where f ‘ (x) = 0, then f(x) is increasing on (a, b).
• If  f ‘(x) < 0 for all x ∈ (a , b) except for a finite number of points, where f ‘(x) = 0, then f(x) is decreasing on (a , b).
Properties of Monotonic Functions
1. If f(x)is strictly increasing function on an interval [a, b], then f-1 exist and also a strictly increasing function.
2. If f(x) is strictly increasing function on [a, b], such that it is continuous, then f-1 is continuous on [f(a), f(b)].
3. If f(x) and g(x) are strictly increasing (or decreasing) function on [a, b], then gof(x) is strictly increasing (or decreasing) function on [a, b].
4. If one of the two functions f(x) and g(x) is strictly increasing and other a strictly decreasing, then gof(x) is strictly decreasing on [a, b].
5. If f(x) is continuous on [a, b], such that f’ (c) ≥ 0 (f ‘ (c) > 0) for each c ∈ (a, b) is strictly increasing function on [a, b].
6. If f(x) is continuous on [a, b] such that f ‘(c) ≤ (f ‘ (c) < 0) for each c ∈ (a, b), then f(x) is strictly decreasing function on [a, b].

Maxima and Minima of Functions
1. A function y = f(x) is said to have a local maximum at a point x = a. If f(x) ≤ f(a) for all x ∈ (a – h, a + h), where h is somewhat small but positive quantity.

The point x = a is called a point of maximum of the function f(x) and f(a) is known as the maximum value or the greatest value or the absolute maximum value of f(x).
2. The function y = f(x) is said to have a local minimum at a point x = a, if f(x) ≥ f(a) for all x ∈ (a – h, a + h), where h is somewhat small but positive quantity.

The point x = a is called a point of minimum of the function f(x) and f(a) is known as the minimum value or the least value or the absolute minimum value of f(x).
Properties of Maxima and Minima
1. If f(x) is continuous function in its domain, then at least one maxima and one minima must lie between two equal values of x.
2. Maxima and minima occur alternately, i.e., between two maxima there is one minima and vice-versa.
3. If f(x) → ∞ as x → a or b and f ‘ (x) = 0 only for one value of x (sayc) between a and b, then f(c) is necessarily the minimum and the least value.
4. If f(x) → p -∞ as x → a or b and f(c) is necessarily the maximum and the greatest value.
Important Points to be Remembered
1. If f(x) be a differentiable functions, then f ‘(x) vanishes at every local maximum and at every local minimum.
2. The converse of above is not true, i.e., every point at which f’ (x) vanishes need not be a local maximum or minimum. e.g., if f(x) = x3 then f ‘(0) = 0, but at x =0. The function has neither minimum nor maximum. In general these points are point of inflection.
3. A function may attain an extreme value at a point without being derivable there at, e.g., f(x) = |x| has a minima at x = 0 but f'(0) does not exist.
4. A function f(x) can has several local maximum and local minimum values in an interval. Thus, the maximum and minimum values of f(x) defined above are not necessarily the greatest and the least values of f(x) in a given interval.
5. A minimum value at some point may even be greater than a maximum values at some other point.
Maximum and Minimum Values in a Closed Interval
Let y = f(x) be a function defined on [a, b]. By a local maximum (or local minimum) value of a function at a point c ∈ [a, b] we mean the greatest (or the least) value in the immediate neighbourhood of x = c. It does not mean the greatest or absolute maximum (or the least or absolute minimum) of f(x) in the interval [a, b]. A function may have a number of local maxima or local minima in a given interval and even a local minimum may be greater than a relative maximum.
Local Maximum
A function f(x) is said to attain a local maximum at x = a, if there exists a neighbourhood (a – δ, a + δ), of c such that,
f(x) < f(a), ∀ x ∈ (a – δ, α + δ), x ≠ a
or f(x) – f(a) < 0, ∀ x ∈ (a – δ, α + δ), x ≠ a
In such a case f(a) is called to attain a local maximum value of f(x) at x = a.
Local Minimum
f (x) > f(a), ∀ x ∈ (a – δ, α + δ), x ≠ a
or f(x) – f(a) > 0, ∀ x ∈ (a – δ, α + δ), x ≠ a
In such a case f(a) is called the local minimum value of f(x) at x = a.
Methods to Find Local Extremum
1. First Derivative Test
Let f(x) be a differentiable function on an interval I and a ∈ I. Then,
1. (i) Point a is a local maximum of f(x), if
(a) f ‘(a) = 0
(b) f ‘(x) > 0, if x ∈ (a – h, a) and f’ (x) < 0, if x ∈ (a, a + h), where h is a small but positive quantity.
2. (ii) Point a is a local minimum of f(x), if
(a) f ‘(a) = 0
(b) f ‘(a) < 0, if x ∈ (a – h, a) and f ‘(x) > 0, if x ∈ (a, a + h), where h is a small but positive quantity.
3. (iii) If f ‘(a) = 0 but f ‘(x) does not changes sign in (a – h, a + h), for any positive quantity h, then x = a is neither a point of minimum nor a point of maximum.
2. Second Derivative Test
Let f(x) be a differentiable function on an interval I. Let a ∈ I is such that f “(x) is continuous at x = a. Then,
1. x = a is a point of local maximum, if f ‘(a) = 0 and f “(a) < 0.
2. x = a is a point of local minimum, if f ‘(a) = 0 and f”(a) > 0.
3. If f ‘(a) = f “(a) = 0, but f” (a) ≠ 0, if exists, then x = a is neither a point of local maximum nor a point of local minimum and is called point of inflection.
4. If f ‘(a) = f “(a) = f ‘”(a) = 0 and f iv(a) < 0, then it is a local maximum. And if f iv > 0, then it is a local minimum.
nth Derivative Test
Let f be a differentiable function on an interval / and let a be an interior point of / such that
(i) f ‘(a) = f “(a) = f ‘”(a) = … f n – 1(a) = 0 and
(ii) fn (a) exists and is non-zero, then
• If n is even and f n (a) < 0 ⇒ x = a is a point of local maximum.
• If n is even and f n (a) > 0 ⇒ x = a is a point of local minimum.
• If n is odd ⇒ x = a is a point of local maximum nor a point of local minimum.
Important Points to be Remembered
1. To Find Range of a Continuous Function Let f(x) be a continuous function on [a, b], such that its least value in [a, b1 is m and the greatest value in [a, b] is M. Then, range of value of f(x) for x ∈ [a, b] is [m, M].
2. To Check for the injectivity of a Function A strictly monotonic function is always one-one (injective). Hence, a function f (x) is one-one in the interval [a, b], if f ‘(x) > 0 , ∀ x ∈ [a, b] or f’ (x) < 0 , ∀ x ∈ [a, b].
3. The points at which a function attains either the local maximum value or local minimum values are known as the extreme points or turning points and both local maximum and local minimum values are called the extreme values of f(x). Thus, a function attains an extreme value at x = a, if f(a) is either a local maximum value or a local minimum value. Consequently at an extreme point ‘a’, f (x) — f (a) keeps the same sign for all values of x in a deleted nbd of a.
4. A necessary condition for (a) to be an extreme value of a function (x) is that f ‘(a) = 0 in case it exists.
5. This condition is only a necessary condition for the point x = a to be an extreme point. It is not sufficient. i.e., f ‘(a) = 0 does not necessarily imply that x = a is an extreme point. There are functions for which the derivatives vanish at a point but do not have an extreme value. e.g., the function f(x) = x3 , f ‘(0) = 0 but at x = 0 the function does not attain an extreme value.
6. Geometrically the above condition means that the tangent to the curve y = f(x) at a point where the ordinate is maximum or minimum is parallel to the x-axis.
7. All x,for which f ‘(x) = 0, do not give us the extreme values. The values of x for which f ‘(x) = 0 are called stationary values or critical values of x and the corresponding values of f(x) are called stationary or turning values of f(x).
Critical Points of a Function
Points where a function f(x) is not differentiable and points where its derivative (differentiable coefficient) is z ?,ro are called the critical points of the function f(x).
Maximum and minimum values of a function f(x) can occur only at critical points. However, this does not mean that the function will have maximum or minimum values at all critical points. Thus, the points where maximum or minimum value occurs are necessarily critical Points but a function may or may not have maximum or minimum value at a critical point.
Point of Inflection
Consider function f(x) = x3. At x = 0, f ‘(x)= 0. Also, f “(x) = 0 at x = 0. Such point is called point of inflection, where 2nd derivative is zero. Consider another function f(x) = sin x, f “(x)= – sin x. Now, f “(x)= 0 when x = nπ, then this points are called point of inflection.
At point of inflection
1. It is not necessary that 1st derivative is zero.
2. 2nd derivative must be zero or 2nd derivative changes sign in the neighbourhood of point of inflection.
Concept of Global Maximum/Minimum
• Let y = f(x) be a given function with domain D.
• Let [a, b] ⊆ D, then global maximum/minimum of f(x) in [a, b] is basically the greatest/least value of f(x) in [a, b].
• Global maxima/minima in [a, b] would always occur at critical points of f(x) with in [a, b] or at end points of the interval.
Global Maximum/Minimum in [a, b]
In order to find the global maximum and minimum of f(x) in [a, b], find out all critical points of f(x) in [a, b] (i.e., all points at which f ‘(x)= 0) and let f(c1), f(c2) ,…, f(n) be the values of the function at these points.
Then, M1  → Global maxima or greatest value.
and M1 → Global minima or least value.
where M1 = max { f(a), f(c1), f(c1) ,…, f(cn), f(b)}
and M1 = min { f(a), f(c1), f(c2) ,…, f(cn), f(b)}
Then, M1 is the greatest value or global maxima in [a, b] and M1 is the least value or global minima in [a, b].
CBSE Class 12 Maths Notes : Indefinite Integrals
May 28, 2014 by Neepur Garg Leave a Comment
Let f(x) be a function. Then, the collection of all its primitives is called the indefinite integral of f(x) and is denoted by ∫f(x)dx. Integration as inverse operation of differentiation. If d/dx {φ(x)) = f(x), ∫f(x)dx = φ(x) + C, where C is called the constant of integration or arbitrary constant.
Symbols f(x) → Integrand
f(x)dx → Element of integration
∫→ Sign of integral
φ(x) → Anti-derivative or primitive or integral of function f(x)
The process of finding functions whose derivative is given, is called anti-differentiation or integration.
Elementary Standard Integrals

Geometrical Interpretation of Indefinite Integral
If d/dx {φ(x)} = f (x), then ∫f(x)dx = φ(x) + C. For different values of C, we get different functions, differing only by a constant. The graphs of these functions give us an infinite family of curves such that at the points on these curves with the same x-coordinate, the tangents are parallel as they have the same slope φ'(x) = f(x).

Consider the integral of 1/2√x
i.e., ∫1/2√xdx = √x + C, C ∈ R
Above figure shows some members of the family of curves given by y = + C for different C ∈ R.
Comparison between Differentiation and Integration
(i) Both differentiation and integration are linear operator on functions as
d/dx {af(x) ± bg(x)} = a d/dx{f(x) ± d/dx{g(x)}
and ∫[a.f(x) ± b.g(x)dx = a ∫f(x)dx ± b ∫g(x)dx
(ii) All functions are not differentiable, similarly there are some function which are not integrable.
(iii) Integral of a function is always discussed in an interval but derivative of a function can be discussed in a interval as well as on a point.
(iv) Geometrically derivative of a function represents slope of the tangent to the graph of function at the point. On the other hand, integral of a function represents an infinite family of curves placed parallel to each other having parallel tangents at points of intersection of the curves with a line parallel to Y-axis.
Rules of Integration

Method of Substitution

Basic Formulae Using Method of Substitution

If degree of the numerator of the integrand is equal to or greater than that of denominator divide the numerator by the denominator until the degree of the remainder is less than that of denominator i.e.,
(Numerator / Denominator) = Quotient + (Remainder / Denominator)
Trigonometric Identities Used for Conversion of Integrals into the Integrable Forms

Standard Substitution

Special Integrals

Important Forms to be converted into Special Integrals
(i) Form I

(ii) Form II

Put px + q = λd / dx (ax2 + bx + c) + mu;
Now, find values of λ and mu; and integrate.
(iii) Form III

when P(x) is a polynomial of degree 2 or more carry out the dimension and express in the form
P(x) / (ax2 + bx + c)  = Q(x) + R(x) / (ax2 + bx + c), where R(x) is a linear expression or constant, then integral reduces to the form discussed earlier.
(iv) Form IV

After dividing both numerator and denominator by x2, put x – a2 / x = t or x + (a2 / x) = t.
(v) Form V

To evaluate the above type of integrals, we proceed as follow
(a) Divide numerator and denominator by cos2x.
(b) Rreplace sec2x, if any in denominator by 1 + tan2 x.
(c) Put tan x = t, then sec2xdx = dt
(vi) Form VI

(vii) Form VII

(viii) Form VIII

(ix) Form IX

To evaluate the above type of integrals, we proceed as follows
• Divide numerator and denominator by x2
• Express the denominator of integrands in the form of (x + 1/x)2 ± k2
• Introduce (x + 1/x) or d (x – 1/x) or both in numerator.
• Put x + 1/x = t or x – 1/x = t as the case may be.
• Integral reduced to the form of ∫ 1 / x2 + a2dx or ∫ 1 / x2 + a2dx
(x) Form X

Integration by Parts
This method is used to integrate the product of two functions. If f(x) and g(x) be two integrable functions, then

(i) We use the following preferential order for taking the first function. Inverse→ Logarithm→ Algebraic → Trigonometric→ Exponential. In short we write it HATE.
(ii) If one of the function is not directly integrable, then we take it a the first function.
(iii) If only one function is there, i.e., ∫log x dx, then 1 (unity) is taken as second function.
(iv) If both the functions are directly integrable, then the first function is chosen in such a way that its derivative vanishes easily or the function obtained in integral sign is easilY integrable.
Integral of the Form

Integration Using Partial Fractions
(i) If f(x) and g(x) are two polynomials, then f(x) / g(x) defines a rational algebraic function of x. If degree of f(x) < degree of g(x), then f(x) / g(x) is called a proper rational function.
(ii) If degree of f(x) ≥ degree of g(x), then f(x) /g(x) is called an improper g(x) rational function.
(iii) If f(x) / g(x) isan improper rational function, then we divide f(x) by g(x) g(x) and convert it into a proper rational function as f(x) / g(x) = φ(x) + h(x) / g(x).
(iv) Any proper rational function f(x) / g(x) can be expressed as the sum of rational functions each having a simple factor of g(x). Each such fraction is called a partial fraction and the process of obtaining them, is called the resolution or decomposition of f(x) /g(x) partial fraction.

Shortcut for Finding Values of A, B and C etc.
Case I. When g(x) is expressible as the product of non-repeated line factors.

Trick To find Ap put x = a in numerator and denominator after P deleting the factor (x — ap).
Case II. When g(x) is expressible as product of repeated linear factors.

Here, all the constant cannot be calculated by using the method in Case I. However, Bl, B2, B3, … , Bn can be found using the same method i.e., shortcut can be applied only in the case of non-repeated linear factor.
Integration of Irrational Algebraic Function
Irrational function of the form of (ax + b)1/n and x can be evaluated by substitution (ax + b) = tn, thus

Integrals of the Type (bxm + bxn)P
Case I. If P ∈ N (natural number) we expand the binomial theorem and integrate.
Case II. If P ∈ Z (integers), put x = pk, where k denominator of m and n.
Case III. If (m+1)/n is an integer, we put (a + bxn) = rk, where k is th denominator of the fraction.
Integration of Hyperbolic Functions
• ∫sinh x dx = cosh x + C
• ∫cosh x dx = sinh x + C
• ∫sech2x dx = tanh x + C
• ∫cosech2x dx = – coth x + C
• ∫sech x tanh x dx = – sech x + C
• ∫cosech x coth x dx = – cosech x + C
Case IV If {(m+1)/n} + P is an integer, we put (a + bxn) = rkxn is the denominator of the fraction p.

CBSE Class 12 Maths Notes : Definite Integrals and its Application
May 28, 2014 by Anuj William Leave a Comment
Let f(x) be a function defined on the interval [a, b] and F(x) be its anti-derivative. Then,

The above is called the second fundamental theorem of calculus.
 is defined as the definite integral of f(x) from x = a to x = b. The numbers and b are called limits of integration. We write

Evaluation of Definite Integrals by Substitution
Consider a definite integral of the following form

Step 1 Substitute g(x) = t
⇒ g ‘(x) dx = dt
Step 2 Find the limits of integration in new system of variable i.e.. the lower limit is g(a) and the upper limit is g(b) and the g(b) integral is now 
Step 3 Evaluate the integral, so obtained by usual method.
properties of Definite Integral

13. Leibnitz Rule for Differentiation Under Integral Sign
(a) If Φ(x) and ψ(x) are defined on [a, b] and differentiable for every x and f(t) is continuous, then

(b) If Φ(x) and ψ(x) are defined on [a, b] and differentiable for every x and f(t) is continuous, then

14. If f(x) ≥ 0 on the interval [a, b], then 
15. If (x) ≤ Φ(x) for x ∈ [a, b], then 
16. If at every point x of an interval [a, b] the inequalities
g(x) ≤ f(x) ≤ h(x)
are fulfilled, then

18. If m is the least value and M is the greatest value of the function f(x) on the interval [a, bl. (estimation of an integral), then

19. If f is continuous on [a, b], then there exists a number c in [a, b] at which

is called the mean value of the function f(x) on the interval [a, b].
20. If f22 (x) and g2 (x) are integrable on [a, b], then

21. Let a function f(x, α) be continuous for a ≤ x ≤ b and c ≤ α ≤ d.
Then, for any α ∈ [c, d], if

22. If f(t) is an odd function, then  is an even function.
23. If f(t) is an even function, then  is an odd function.
24. If f(t) is an even function, then for non-zero a,  is not necessarily an odd function. It will be an odd function, if

25. If f(x) is continuous on [a, α], then  is called an improper integral and is defined as 

27. Geometrically, for f(x) > 0, the improper integral  gives area of the figure bounded by the curve y = f(x), the axis and the straight line x = a.
Integral Function
Let f(x) be a continuous function defined on [a, b], then a function φ(x) defined by   is called the integral function of the function f.
Properties of Integral Function
1. The integral function of an integrable function is continuous.
2. If φ(x) is the integral function of continuous function, then φ(x) is derivable and of φ ‘ = f(x) for all x ∈ [a, b].
Gamma Function
If n is a positive rational number, then the improper integral  is defined as a gamma function and it is denoted by Γn

Properties of Gamma Function


Summation of Series by Definite Integral

The method to evaluate the integral, as limit of the sum of an infinite series is known as integration by first principle.

Area of Bounded Region
The space occupied by the curve along with the axis, under the given condition is called area of bounded region.
(i) The area bounded by the curve y = F(x) above the X-axis and between the lines x = a, x = b is given by 

(ii) If the curve between the lines x = a, x = b lies below the X-axis, then the required area is given by

(iii) The area bounded by the curve x = F(y) right to the Y-axis and the lines y = c, y = d is given by

(iv) If the curve between the lines y = c, y = d left to the Y-axis, then the area is given by

(v) Area bounded by two curves y = F (x) and y = G (x) between x = a and x = b is given by

(vi) Area bounded by two curves x = f(y) and x = g(y) between y=c and y=d is given by 

(vii) If F (x) ≥. G (x) in [a, c] and F (x) ≤ G (x) in [c,d], where a < c < b, then area of the region bounded by the curves is given as

Area of Curves Given by Polar Equations
Let  f(θ) be a continuous function,  θ ∈ (a, α), then the are t bounded by the curve r = f(θ) and radius α, β(α < β) is

Area of Parametric Curves
Let x = φ(t) and y = ψ(t) be two parametric curves, then area bounded by the curve, X-axis and ordinates x = φ(t1), x = ψ(t2) is

Volume and Surface Area
If We revolve any plane curve along any line, then solid so generated is called solid of revolution.
1. Volume of Solid Revolution
1. The volume of the solid generated by revolution of the area bounded by the curve y = f(x), the axis of x and the ordinates  it being given that f(x) is a continuous a function in the interval (a, b).
2. The volume of the solid generated by revolution of the area bounded by the curve x = g(y), the axis of y and two abscissas y = c and y = d is  it being given that g(y) is a continuous function in the interval (c, d).
Surface of Solid Revolution
(i) The surface of the solid generated by revolution of the area bounded by the curve y = f(x), the axis of x and the ordinates

is a continuous function in the interval (a, b).
(ii) The surface of the solid generated by revolution of the area bounded by the curve x = f (y), the axis of y and y = c, y = d is  continuous function in the interval (c, d).
Curve Sketching
1. symmetry
1. If powers of y in a equation of curve are all even, then curve is symmetrical about X-axis.
2. If powers of x in a equation of curve are all even, then curve is symmetrical about Y-axis.
3. When x is replaced by -x and y is replaced by -y, then curve is symmetrical in opposite quadrant.
4. If x and y are interchanged and equation of curve remains unchanged curve is symmetrical about line y = x.
2. Nature of Origin
1. If point (0, 0) satisfies the equation, then curve passes through origin.
2. If curve passes through origin, then equate low st degree term to zero and get equation of tangent. If there are two tangents, then origin is a double point.
3. Point of Intersection with Axes
1. Put y = 0 and get intersection with X-axis, put x = 0 and get intersection with Y-axis.
2. Now, find equation of tangent at this point i. e. , shift origin to the point of intersection and equate the lowest degree term to zero.
3. Find regions where curve does not exists. i. e., curve will not exit for those values of variable when makes the other imaginary or not defined.
4. Asymptotes
1. Equate coefficient of highest power of x and get asymptote parallel to X-axis.
2. Similarly equate coefficient of highest power of y and get asymptote parallel to Y-axis.
5. The Sign of (dy/dx)
Find points at which (dy/dx) vanishes or becomes infinite. It gives us the points where tangent is parallel or perpendicular to the X-axis.
6. Points of Inflexion
  and solve the resulting equation.If some point of inflexion is there, then locate it exactly.
Taking in consideration of all above information, we draw an approximate shape of the curve.
Shape of Some Curves is Given Below

CBSE Class 12 Maths Notes : Differential Equations
May 28, 2014 by Neepur Garg Leave a Comment
An equation that involves an independent variable, dependent variable and differential coefficients of dependent variable with respect to the independent variable is called a differential equation.
e.g., (i) x2(d2y / dx2) + x3 (dy / dx)3 7x2y2
(ii) (x2 + y2) dx = (x2 – y2) dy
Order and Degree of a Differential Equation
The order of a differential equation is the order of the highest derivative occurring in the equation. The order of a differential equation is always a positive integer.
The degree of a differential equation is the degree (exponent) of the derivative of the highest order in the equation, after the equation is free from negative and fractional powers of the derivatives.
Linear and Non-Linear Differential Equations
A differential equation is said to be linear, if the dependent variable and all of its derivatives occurring in the first power and there are no product of these. A linear equation of nth order can be written in the form

where, P0, P1, P2,…, Pn – 1 and Q must be either constants or functions of x only.
A linear differential equation is always of the first degree but every differential equation of the first degree need not be linear.
e.g., The equations d2y / dx2 + (dy / dx)2 + xy = 0 and
x(d2y / dx2) + y (dy / dx) + y = x3, (dy / dx) d2y / dx2 + y = 0
are not linear.
Solution of Differential Equations
A solution of a differential equation is a relation between the variables, not involving the differential coefficients, such that this relation and the derivative obtained from it satisfy the given differential equation.
e.g., Let d2y / dx2 + y = 0
Integrating above equation twicely, we get y = A cos x + B sin x.
General Solution
If the solution of the differential equation contains as many independent arbitrary constants as the order of the differential equation, then it is called the general solution or the complete integral of the differential equation.
e.g., The general solution of d2y / dx2 + y = 0 is y = A cos x + B sin x because it contains two arbitrary constants A and B, which is equal to the order of the equation.
Particular Solution
Solution obtained by giving particular values to the arbitrary constants in the general solution is called a particular solution. e.g., In the
previous example, if A = B = 1, then y = cos x + sin x is a particular solution of the differential equation d2y / dx2 + y = 0.
Solution of a differential equation is also called its primitive.
Formation of Differential Equation
Suppose, we have a given equation with n arbitrary constants f(x, y, c1, c2,…, cn) = 0.
Differentiate the equation successively n times to get n equations.
Eliminating the arbitrary constants from these n + 1 equations leads to the required differential equations.
Solutions of Differential Equations of the First Order and First Degree
A differential equation of first degree and first order can be solved by following method.
1. Inspection Method
If the differential equation’ can be written as f [f1(x, y) d {f1(x, y)}] + φ [f2(x, y) d {f2(x, y)}] +… = 0] then each term can be integrated separately.
For this, remember the following results

2. Variable Separable Method
If the equation can be reduced into the form f(x) dx + g(y) dy = 0, we say that the variable have been separated. On integrating this reduced, form, we get ∫ f(x) dx + ∫ g(y) dy = C, = C, where C is any arbitrary constant.
3. Differential Equation Reducible to Variables Separable Method
A differential equation of the form dy / dx = f(ax + by + c) can be reduced to variables separable form by substituting
ax + by + c = z => a + b dy / dx = dz / dx
The given equation becomes
1 / b (dz / dx – a) f(z) => dz / dx = a + b f(z)
=> dz / a+ bf(z) = dx
Hence, the variables are separated in terms of z and x.
4. Homogeneous Differential Equation
A function f(x, y) is said to be homogeneous of degree n, if
f(λx, λy) = λn f(x, y)
Suppose a differential equation can be expressed in the form
dy / dx = f(x, y) / g(x, y) = F (y / x)
where, f(x, y) and g(x, y) are homogeneous function of same degree. To solve such types of equations, we put y = vx
=> dy / dx = v + x dv / dx.
The given equation, reduces to
v + x dv / dx = F(v)
=> x dv / dx = F(v) – v
∴ dv / F(v) – v = dx / x
Hence, the variables are separated in terms of v and x.
5. Differential Equations Reducible to Homogeneous Equation
The differential equation of the form
dy / dx = a1x + b1y + c1 / a2x + b2y + c2 ……(i)
put X = X + h and y = Y + k
∴ dY / dX = a1 X + b1 Y + (a1h + b1k + c1) / a2X + b2 Y + (a2h + b2k + c2) ……(ii)
We choose h and k, so as to satisfy a1h + b1k + c1 = 0 and a2h + b2k + c2 = 0.
On solving, we get
h / b1c2 – b2c1 = k / c1a2 – c2a1 = 1 / a1b2 – a2b1
∴ h = b1c2 – b2c1 / a1b2 – a2b1 and k = c1a2 – c2a1 / a1b2 – a2b1
Provided a1b2 – a2b1 ≠ 0 , a1 / a2 ≠ ba / b2
Then, Eq, (ii) reduces to dY / dX = (a1 X + b1 Y) / (a2X + b2 Y), which is a homogeneous form and will be solved easily.
6. Exact Differential Equation
Differential equation M(x,y) dy + N(x,y) dy = 0 is called an exact differential equation.
If a function u (x, y) exist such that,
du = Mdx + Ndy.
Necessary and Sufficient Condition for an Equation to be an Exact Differential Equation
Differential equation Mdx + Ndy = 0 where, M and N are the functions •of x and y, will be an exact differential equation, if
∂N / ∂y = ∂N / ∂x
Solution of Exact Differential Equation

7. Linear Differential Equation
A linear differential equation of the first order can be either of the following forms
(i) dy / dx + Py = Q, where P and Q are functions of x or constants.
(ii) dx / dy + Rx = S, where Rand S are functions of y or constants.
Consider the differential Eq. (i) i.e., dy / dx + Py = Q

Similarly, for the second differential equation dx / dy + Rx = S, the integrating factor, IF = e ∫R dy and the general solution is
x (IF) = ∫ S (IF) dy + C
8. Differential Equation Reducible to Linear Form
Bernoulli’s Equation An equation of the form dy / dx + Py = Qyn, where P and Q are functions of x along or constants, is called
Bernoulli’s equation.
Divide both the sides by yn, we get
y-n dy / dx + Py-n + 1 = Q
Put y-n + 1 = z
=> (-n + 1)y-n dy / dx = dz / dx
The equation reduces to
1 / 1 – n dz / dx + Pz = Q => dz / dx + (1 – n) Pz = Q (1 – n)
which is linear in z and can be solved in the usual manner.
9. Clairaut Form for Differential Equation
Differential equation y = Px + f(p), where P= dy / dx … (i)
is called clairaut form of differential equation. In which, get its general solution by replacing P from C.
Now, differential on both sides of Eq, (i) with respect to x and put dy / dx = P.
P = P + x dp / dx + f’ (P) dp / dx = 0
=> [x + f’ (p)] dp / dx = 0
=> dp / dx = 0 => p = C
10. Orthogonal Trajectory
Any curve, which cuts every member of a given family of curves at right angle, is called an orthogonal trajectory of the family.
Procedure for finding the Orthogonal Trajectory
(i) Let f(x,y,c)= 0 be the equation of the given family of curves, where ‘c’ is an arbitrary parameter.
(ii) Differentiate f = 0, with respect to ‘x’ and eliminate 0, i.e., from a differential equation.
(iii) Substitute (- dx / dy) for (dy / dx) in the above differential equation.
This will give the differential equation of the orthogonal trajectories.
(iv) By solving this differential equation, we get the required orthogonal trajectories.
CBSE Class 12 Maths Notes : Vectors
May 28, 2014 by Neepur Garg Leave a Comment
A vector has direction and magnitude both but scalar has only magnitude.
Magnitude of a vector a is denoted by |a| or a. It is non-negative scalar.
Equality of Vectors
Two vectors a and b are said to be equal written as a = b, if they have (i) same length (ii) the same or parallel support and (iii) the same sense.
Types of Vectors
(i) Zero or Null Vector A vector whose initial and terminal points are coincident is called zero or null vector. It is denoted by 0.
(ii) Unit Vector A vector whose magnitude is unity is called a unit vector which is denoted by nˆ
(iii) Free Vectors If the initial point of a vector is not specified, then it is said to be a free vector.
(iv) Negative of a Vector A vector having the same magnitude as that of a given vector a and the direction opposite to that of a is called the negative of a and it is denoted by —a.
(v) Like and Unlike Vectors Vectors are said to be like when they have the same direction and unlike when they have opposite direction.
(vi) Collinear or Parallel Vectors Vectors having the same or parallel supports are called collinear vectors.
(vii) Coinitial Vectors Vectors having same initial point are called coinitial vectors.
(viii) Coterminous Vectors Vectors having the same terminal point are called coterminous vectors.
(ix) Localized Vectors A vector which is drawn parallel to a given vector through a specified point in space is called localized vector.
(x) Coplanar Vectors A system of vectors is said to be coplanar, if their supports are parallel to the same plane. Otherwise they are called non-coplanar vectors.
(xi) Reciprocal of a Vector A vector having the same direction as that of a given vector but magnitude equal to the reciprocal of the given vector is known as the reciprocal of a.
i.e., if |a| = a, then |a-1| = 1 / a.
Addition of Vectors
Let a and b be any two vectors. From the terminal point of a, vector b is drawn. Then, the vector from the initial point O of a to the terminal point B of b is called the sum of vectors a and b and is denoted by a + b. This is called the triangle law of addition of vectors.

Parallelogram Law
Let a and b be any two vectors. From the initial point of a, vector b is drawn and parallelogram OACB is completed with OA and OB as adjacent sides. The vector OC is defined as the sum of a and b. This is called the parallelogram law of addition of vectors.
The sum of two vectors is also called their resultant and the process of addition as composition.

Properties of Vector Addition
(i) a + b = b + a (commutativity)
(ii) a + (b + c)= (a + b)+ c (associativity)
(iii) a+ O = a (additive identity)
(iv) a + (— a) = 0 (additive inverse)
(v) (k1 + k2) a = k1 a + k2a (multiplication by scalars)
(vi) k(a + b) = k a + k b (multiplication by scalars)
(vii) |a+ b| ≤ |a| + |b| and |a – b| ≥ |a| – |b|
Difference (Subtraction) of Vectors
If a and b be any two vectors, then their difference a – b is defined as a + (- b).

Multiplication of a Vector by a Scalar
Let a be a given vector and λ be a scalar. Then, the product of the vector a by the scalar λ is λ a and is called the multiplication of vector by the scalar.
Important Properties
(i) |λ a| = |λ| |a|
(ii) λ O = O
(iii) m (-a) = – ma = – (m a)
(iv) (-m) (-a) = m a
(v) m (n a) = mn a = n(m a)
(vi) (m + n)a = m a+ n a
(vii) m (a+b) = m a + m b
Vector Equation of Joining by Two Points
Let P1 (x1, y1, z1) and P2 (x2, y2, z2) are any two points, then the vector joining P1 and P2 is the vector P1 P2.

The component vectors of P and Q are
OP = x1i + y1j + z1k
and OQ = x2i + y2j + z2k
i.e., P1 P2 = (x2i + y2j + z2k) – (x1i + y1j + z1k)
= (x2 – x1) i + (y2 – y1) j + (z2 – z1) k
Its magnitude is
P1 P2 = √(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2
Position Vector of a Point
The position vector of a point P with respect to a fixed point, say O, is the vector OP. The fixed point is called the origin.
Let PQ be any vector. We have PQ = PO + OQ = — OP + OQ = OQ — OP = Position vector of Q — Position vector of P.

i.e., PQ = PV of Q — PV of P
Collinear Vectors
Vectors a and b are collinear, if a = λb, for some non-zero scalar λ.
Collinear Points
Let A, B, C be any three points.
Points A, B, C are collinear <=> AB, BC are collinear vectors.
<=> AB = λBC for some non-zero scalar λ.
Section Formula
Let A and B be two points with position vectors a and b, respectively and OP= r.
(i) Let P be a point dividing AB internally in the ratio m : n. Then,
r = m b + n a / m + n

Also, (m + n) OP = m OB + n OA
(ii) The position vector of the mid-point of a and b is a + b / 2.
(iii) Let P be a point dividing AB externally in the ratio m : n. Then,
r = m b + n a / m + n
Position Vector of Different Centre of a Triangle
(i) If a, b, c be PV’s of the vertices A, B, C of a ΔABC respectively, then the PV of the centroid G of the triangle is a + b + c / 3.
(ii) The PV of incentre of ΔABC is (BC)a + (CA)b + (AB)c / BC + CA + AB
(iii) The PV of orthocentre of ΔABC is
a(tan A) + b(tan B) + c(tan C) / tan A + tan B + tan C
Scalar Product of Two Vectors
If a and b are two non-zero vectors, then the scalar or dot product of a and b is denoted by a * b and is defined as a * b = |a| |b| cos θ, where θ is the angle between the two vectors and 0 < θ < π .
(i) The angle between two vectors a and b is defined as the smaller angle θ between them, when they are drawn with the same initial point.
Usually, we take 0 < θ < π.Angle between two like vectors is O and angle between two unlike vectors is π .
(ii) If either a or b is the null vector, then scalar product of the vector is zero.
(iii) If a and b are two unit vectors, then a * b = cos θ.
(iv) The scalar product is commutative
i.e., a * b= b * a
(v) If i , j and k are mutually perpendicular unit vectors i , j and k, then
i * i = j * j = k * k =1
and i * j = j * k = k * i = 0
(vi) The scalar product of vectors is distributive over vector addition.
(a) a * (b + c) = a * b + a * c (left distributive)
(b) (b + c) * a = b * a + c * a (right distributive)
Note Length of a vector as a scalar product
If a be any vector, then the scalar product
a * a = |a| |a| cosθ ⇒ |a|2 = a2 ⇒ a = |a|
Condition of perpendicularity a * b = 0 <=> a ⊥ b, a and b being non-zero vectors.
Important Points to be Remembered
(i) (a + b) * (a – b) = |a|22 – |b|2
(ii) |a + b|2 = |a|22 + |b|2 + 2 (a * b)
(iii) |a – b|2 = |a|22 + |b|2 – 2 (a * b)
(iv) |a + b|2 + |a – b|2 = (|a|22 + |b|2) and |a + b|2 – |a – b|2 = 4 (a * b)
or a * b = 1 / 4 [ |a + b|2 – |a – b|2 ]
(v) If |a + b| = |a| + |b|, then a is parallel to b.
(vi) If |a + b| = |a| – |b|, then a is parallel to b.
(vii) (a * b)2 ≤ |a|22 |b|2
(viii) If a = a1i + a2j + a3k, then |a|2 = a * a = a12 + a22 + a32
Or
|a| = √a12 + a22 + a32
(ix) Angle between Two Vectors If θ is angle between two non-zero vectors, a, b, then we have
a * b = |a| |b| cos θ
cos θ = a * b / |a| |b|
If a = a1i + a2j + a3k and b = b1i + b2j + b3k
Then, the angle θ between a and b is given by
cos θ = a * b / |a| |b| = a1b1 + a2b2 + a3b3 / √a12 + a22 + a32 √b12 + b22 + b32
(x) Projection and Component of a Vector
Projection of a on b = a * b / |a|
Projection of b on a = a * b / |a|
Vector component of a vector a on b

Similarly, the vector component of b on a = ((a * b) / |a2|) * a
(xi) Work done by a Force
The work done by a force is a scalar quantity equal to the product of the magnitude of the force and the resolved part of the displacement.
∴ F * S = dot products of force and displacement.
Suppose F1, F1,…, Fn are n forces acted on a particle, then during the displacement S of the particle, the separate forces do quantities of work F1 * S, F2 * S, Fn * S.

Here, system of forces were replaced by its resultant R.
Vector or Cross Product of Two Vectors
The vector product of the vectors a and b is denoted by a * b and it is defined as
a * b = (|a| |b| sin θ) n = ab sin θ n …..(i)
where, a = |a|, b= |b|, θ is the angle between the vectors a and b and n is a unit vector which is perpendicular to both a and b, such that a, b and n form a right-handed triad of vectors.
Important Points to be Remembered
(i) Let a = a1i + a2j + a3k and b = b1i + b2j + b3k

(ii) If a = b or if a is parallel to b, then sin θ = 0 and so a * b = 0.
(iii) The direction of a * b is regarded positive, if the rotation from a to b appears to be anti-clockwise.
(iv) a * b is perpendicular to the plane, which contains both a and b. Thus, the unit vector perpendicular to both a and b or to the plane containing is given by n = a * b / |a * b| = a * b / ab sin θ
(v) Vector product of two parallel or collinear vectors is zero.
(vi) If a * b = 0, then a = O or b = 0 or a and b are parallel on collinear.
(vii) Vector Product of Two Perpendicular Vectors
If θ = 900, then sin θ = 1, i.e. , a * b = (ab)n or |a * b| = |ab n| = ab
(viii) Vector Product of Two Unit Vectors If a and b are unit vectors, then
a = |a| = 1, b = |b| = 1
∴ a * b = ab sin θ n = (sin theta;).n
(ix) Vector Product is not Commutative The two vector products a * b and b * a are equal in magnitude but opposite in direction.
i.e., b * a =- a * b ……..(i)
(x) The vector product of a vector a with itself is null vector, i. e., a * a= 0.
(xi) Distributive Law For any three vectors a, b, c
a * (b + c) = (a * b) + (a * c)
(xii) Area of a Triangle and Parallelogram
(a) The vector area of a ΔABC is equal to 1 / 2 |AB * AC| or 1 / 2 |BC * BA| or 1 / 2 |CB * CA|.
(b) The area of a ΔABC with vertices having PV’s a, b, c respectively, is 1 / 2 |a * b + b * c + c * a|.
(c) The points whose PV’s are a, b, c are collinear, if and only if a * b + b * c + c * a
(d) The area of a parallelogram with adjacent sides a and b is |a * b|.
(e) The area of a Parallelogram with diagonals a and b is 1 / 2 |a * b|.
(f) The area of a quadrilateral ABCD is equal to 1 / 2 |AC * BD|.
(xiii) Vector Moment of a Force about a Point
The vector moment of torque M of a force F about the point O is the vector whose magnitude is equal to the product of |F| and the perpendicular distance of the point O from the line of action of F.

∴ M = r * F
where, r is the position vector of A referred to O.
(a) The moment of force F about O is independent of the choice of point A on the line of action of F.
(b) If several forces are acting through the same point A, then the vector sum of the moments of the separate forces about a point O is equal to the moment of their resultant force about O.
(xiv) The Moment of a Force about a Line

Let F be a force acting at a point A, O be any point on the given line L and a be the unit vector along the line, then moment of F about the line L is a scalar given by (OA x F) * a
(xv) Moment of a Couple
(a) Two equal and unlike parallel forces whose lines of action are different are said to constitute a couple.
(b) Let P and Q be any two points on the lines of action of the forces – F and F, respectively.

The moment of the couple = PQ x F
Scalar Triple Product
If a, b, c are three vectors, then (a * b) * c is called scalar triple product and is denoted by [a b c].
∴ [a b c] = (a * b) * c
Geometrical Interpretation of Scalar Triple Product
The scalar triple product (a * b) * c represents the volume of a parallelepiped whose coterminous edges are represented by a, b and c which form a right handed system of vectors.
Expression of the scalar triple product (a * b) * c in terms of components
a = a1i + a1j + a1k, b = a2i + a2j + a2k, c = a3i + a3j + a3k is

Properties of Scalar Triple Products
1. The scalar triple product is independent of the positions of dot and cross i.e., (a * b) * c = a * (b * c).
2. The scalar triple product of three vectors is unaltered so long as the cyclic order of the vectors remains unchanged.
i.e., (a * b) * c = (b * c) * a= (c * a) * b
or
[a b c] = [b c a] = [c a b].
3. The scalar triple product changes in sign but not in magnitude, when the cyclic order is changed.
i.e., [a b c] = – [a c b] etc.
4. The scalar triple product vanishes, if any two of its vectors are equal.
i.e., [a a b] = 0, [a b a] = 0 and [b a a] = 0.
5. The scalar triple product vanishes, if any two of its vectors are parallel or collinear.
6. For any scalar x, [x a b c] = x [a b c]. Also, [x a yb zc] = xyz [a b c].
7. For any vectors a, b, c, d, [a + b c d] = [a c d] + [b c d]
8. [i j k] = 1

11. Three non-zero vectors a, b and c are coplanar, if and only if [a b c] = 0.
12. Four points A, B, C, D with position vectors a, b, c, d respectively are coplanar, if and only if [AB AC AD] = 0.
i.e., if and only if [b — a c— a d— a] = 0.
13. Volume of parallelepiped with three coterminous edges a, b,c is | [a b c] |.
14. Volume of prism on a triangular base with three coterminous edges a, b,c is 1 / 2 | [a b c] |.
15. Volume of a tetrahedron with three coterminous edges a, b,c is 1 / 6 | [a b c] |.
16. If a, b, c and d are position vectors of vertices of a tetrahedron, then
Volume = 1 / 6 [b — a c — a d — a].
Vector Triple Product
If a, b, c be any three vectors, then (a * b) * c and a * (b * c) are known as vector triple product.
∴ a * (b * c)= (a * c)b — (a * b) c
and (a * b) * c = (a * c)b — (b * c) a
Important Properties
(i) The vector r = a * (b * c) is perpendicular to a and lies in the plane b and c.
(ii) a * (b * c) ≠ (a * b) * c, the cross product of vectors is not associative.
(iii) a * (b * c)= (a * b) * c, if and only if and only if (a * c)b — (a * b) c = (a * c)b — (b * c) a, if and only if c = (b * c) / (a * b) * a
Or if and only if vectors a and c are collinear.
Reciprocal System of Vectors
Let a, b and c be three non-coplanar vectors and let
a’ = b * c / [a b c], b’ = c * a / [a b c], c’ = a * b / [a b c]
Then, a’, b’ and c’ are said to form a reciprocal system of a, b and c.
Properties of Reciprocal System
(i) a * a’ = b * b’= c * c’ = 1
(ii) a * b’= a * c’ = 0, b * a’ = b * c’ = 0, c * a’ = c * b’= 0
(iii) [a’, b’, c’] [a b c] = 1 ⇒ [a’ b’ c’] = 1 / [a b c]
(iv) a = b’ * c’ / [a’, b’, c’], b = c’ * a’ / [a’, b’, c’], c = a’ * b’ / [a’, b’, c’]
Thus, a, b, c is reciprocal to the system a’, b’ ,c’.
(v) The orthonormal vector triad i, j, k form self reciprocal system.
(vi) If a, b, c be a system of non-coplanar vectors and a’, b’, c’ be the reciprocal system of vectors, then any vector r can be expressed as r = (r * a’ )a + (r * b’)b + (r * c’) c.
Linear Combination of Vectors
Let a, b, c,… be vectors and x, y, z, … be scalars, then the expression x a yb + z c + … is called a linear combination of vectors a, b, c,….
Collinearity of Three Points
The necessary and sufficient condition that three points with PV’s b, c are collinear is that there exist three scalars x, y, z not all zero such that xa + yb + zc ⇒ x + y + z = 0.
Coplanarity of Four Points
The necessary and sufficient condition that four points with PV’s a, b, c, d are coplanar, if there exist scalar x, y, z, t not all zero, such that xa + yb + zc + td = 0 rArr; x + y + z + t = 0.
If r = xa + yb + zc…
Then, the vector r is said to be a linear combination of vectors a, b, c,….
Linearly Independent and Dependent System of Vectors
(i) The system of vectors a, b, c,… is said to be linearly dependent, if there exists a scalars x, y, z, … not all zero, such that xa + yb + zc + … = 0.
(ii) The system of vectors a, b, c, … is said to be linearly independent, if xa + yb + zc + td = 0 rArr; x + y + z + t… = 0.
Important Points to be Remembered
(i) Two non-collinear vectors a and b are linearly independent.
(ii) Three non-coplanar vectors a, b and c are linearly independent.
(iii) More than three vectors are always linearly dependent.
Resolution of Components of a Vector in a Plane
Let a and b be any two non-collinear vectors, then any vector r coplanar with a and b, can be uniquely expressed as r = x a + y b, where x, y are scalars and x a, y b are called components of vectors in the directions of a and b, respectively.

∴ Position vector of P(x, y) = x i + y j.
OP2 = OA2 + AP2 = |x|2 + |y|2 = x2 + y2
OP = √x2 + y2. This is the magnitude of OP.
where, x i and y j are also called resolved parts of OP in the directions of i and j, respectively.
Vector Equation of Line and Plane
(i) Vector equation of the straight line passing through origin and parallel to b is given by r = t b, where t is scalar.
(ii) Vector equation of the straight line passing through a and parallel to b is given by r = a + t b, where t is scalar.
(iii) Vector equation of the straight line passing through a and b is given by r = a + t(b – a), where t is scalar.
(iv) Vector equation of the plane through origin and parallel to b and c is given by r = s b + t c, where s and t are scalars.
(v) Vector equation of the plane passing through a and parallel to b and c is given by r = a + sb + t c, where s and t are scalars.
(vi) Vector equation of the plane passing through a, b and c is r = (1 – s – t)a + sb + tc, where s and t are scalars.
Bisectors of the Angle between Two Lines
(i) The bisectors of the angle between the lines r = λa and r = μb are given by r = &lamba; (a / |a| &plumsn; b / |b|)
(ii) The bisectors of the angle between the lines r = a + λb and r = a + μc are given by r = a + &lamba; (b / |b| &plumsn; c / |c|).
CBSE Class 12 Maths Notes : Three Dimensional Geometry
May 28, 2014 by Neepur Garg Leave a Comment
Coordinate System
The three mutually perpendicular lines in a space which divides the space into eight parts and if these perpendicular lines are the coordinate axes, then it is said to be a coordinate system.

Sign Convention

Distance between Two Points
Let P(x1, y1, z1) and Q(x2, y2, z2) be two given points. The distance between these points is given by
PQ √(x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2
The distance of a point P(x, y, z) from origin O is
OP = √x2 + y2 + z2
Section Formulae
(i) The coordinates of any point, which divides the join of points P(x1, y1, z1) and Q(x2, y2, z2) in the ratio m : n internally are
(mx2 + nx1 / m + n, my2 + ny1 / m + n, mz2 + nz1 / m + n)
(ii) The coordinates of any point, which divides the join of points P(x1, y1, z1) and Q(x2, y2, z2) in the ratio m : n externally are
(mx2 – nx1 / m – n, my2 – ny1 / m – n, mz2 – nz1 / m – n)
(iii) The coordinates of mid-point of P and Q are
(x1 + x2 / 2 , y1 + y2 / 2, z1 + z2 / 2)
(iv) Coordinates of the centroid of a triangle formed with vertices P(x1, y1, z1) and Q(x2, y2, z2) and R(x3, y3, z3) are
(x1 + x2 + x3 / 3 , y1 + y2 + y3 / 3, z1 + z2 + z3 / 3)
(v) Centroid of a Tetrahedron
If (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) and (x4, y4, z4) are the vertices of a tetrahedron, then its centroid G is given by
(x1 + x2 + x3 + x4  / 4 , y1 + y2 + y3 + y4 / 4, z1 + z2 + z3 + z4 / 4)
Direction Cosines
If a directed line segment OP makes angle α, β and γ with OX , OY and OZ respectively, then Cos α, cos β and cos γ are called direction cosines of up and it is represented by l, m, n.
i.e.,
l = cos α
m = cos β
and n = cos γ

If OP = r, then coordinates of OP are (lr, mr , nr)
(i) If 1, m, n are direction cosines of a vector r, then
(a) r = |r| (li + mj + nk) ⇒ r = li + mj + nk
(b) l2 + m2 + n2 = 1
(c) Projections of r on the coordinate axes are
(d) |r| = l|r|, m|r|, n|r| / √sum of the squares of projections of r on the coordinate axes
(ii) If P(x1, y1, z1) and Q(x2, y2, z2) are two points, such that the direction cosines of PQ are l, m, n. Then,
x2 – x1 = l|PQ|, y2 – y1 = m|PQ|, z2 – z1 = n|PQ|
These are projections of PQ on X , Y and Z axes, respectively.
(iii) If 1, m, n are direction cosines of a vector r and a b, c are three numbers, such that l / a = m / b = n / c.
Then, we say that the direction ratio of r are proportional to a, b, c.
Also, we have
l = a / √a2 + b2 + c2, m = b / √a2 + b2 + c2, n = c / √a2 + b2 + c2
(iv) If θ is the angle between two lines having direction cosines l1, m1, n1 and 12, m2, n2, then
cos θ = l112 + m1m2 + n1n2
(a) Lines are parallel, if l1 / 12 = m1 / m2 = n1 / n2
(b) Lines are perpendicular, if l112 + m1m2 + n1n2
(v) If θ is the angle between two lines whose direction ratios are proportional to a1, b1, c1 and a2, b2, c2 respectively, then the angle θ between them is given by
cos θ = a1a2 + b1b2 + c1c2 / √a21 + b21 + c21 √a22 + b22 + c22
Lines are parallel, if a1 / a2 = b1 / b2 = c1 / c2
Lines are perpendicular, if a1a2 + b1b2 + c1c2 = 0.
(vi) The projection of the line segment joining points P(x1, y1, z1) and Q(x2, y2, z2) to the line having direction cosines 1, m, n is
|(x2 – x1)l + (y2 – y1)m + (z2 – z1)n|.
(vii) The direction ratio of the line passing through points P(x1, y1, z1) and Q(x2, y2, z2) are proportional to x2 – x1, y2 – y1 – z2 – z1 Then, direction cosines of PQ are
x2 – x1 / |PQ|, y2 – y1 / |PQ|, z2 – z1 / |PQ|
Area of Triangle
If the vertices of a triangle be A(x1, y1, z1) and B(x2, y2, z2) and C(x3, y3, z3), then


Angle Between Two Intersecting Lines
If l(x1, m1, n1) and l(x2, m2, n2) be the direction cosines of two given lines, then the angle θ between them is given by
cos θ = l112 + m1m2 + n1n2
(i) The angle between any two diagonals of a cube is cos-1 (1 / 3).
(ii) The angle between a diagonal of a cube and the diagonal of a face (of the cube is cos-1 (√2 / 3)
Straight Line in Space
The two equations of the line ax + by + cz + d = 0 and a’ x + b’ y + c’ z + d’ = 0 together represents a straight line.
1. Equation of a straight line passing through a fixed point A(x1, y1, z1) and having direction ratios a, b, c is given by
x – x1 / a = y – y1 / b = z – z1 / c, it is also called the symmetrically form of a line.
Any point P on this line may be taken as (x1 + λa, y1 + λb, z1 + λc), where λ ∈ R is parameter. If a, b, c are replaced by direction cosines 1, m, n, then λ, represents distance of the point P from the fixed point A.
2. Equation of a straight line joining two fixed points A(x1, y1, z1) and B(x2, y2, z2) is given by
x – x1 / x2 – x1 = y – y1 / y2 – y1 = z – z1 / z2 – z1
3. Vector equation of a line passing through a point with position vector a and parallel to vector b is r = a + λ b, where A, is a parameter.
4. Vector equation of a line passing through two given points having position vectors a and b is r = a + λ (b – a) , where λ is a parameter.
5. (a) The length of the perpendicular from a point  on the line r – a + λ b is given by

(b) The length of the perpendicular from a point P(x1, y1, z1) on the line

where, 1, m, n are direction cosines of the line.
6. Skew Lines Two straight lines in space are said to be skew lines, if they are neither parallel nor intersecting.
7. Shortest Distance If l1 and l2 are two skew lines, then a line perpendicular to each of lines 4 and 12 is known as the line of shortest distance.
If the line of shortest distance intersects the lines l1 and l2 at P and Q respectively, then the distance PQ between points P and Q is known as the shortest distance between l1 and l2.
8. The shortest distance between the lines

9. The shortest distance between lines r = a1 + λb1 and r = a2 + μb2 is given by

10. The shortest distance parallel lines r = a1 + λb1 and r = a2 + μb2 is given by

11. Lines r = a1 + λb1 and r = a2 + μb2 are intersecting lines, if (b1 * b2) * (a2 – a1) = 0.
12. The image or reflection (x, y, z) of a point (x1, y1, z1) in a plane ax + by + cz + d = 0 is given by
x – x1 / a = y – y1 / b = z – z1 / c = – 2 (ax1 + by1 + cz1 + d) / a2 + b2 + c2
13. The foot (x, y, z) of a point (x1, y1, z1) in a plane ax + by + cz + d = 0 is given by
x – x1 / a = y – y1 / b = z – z1 / c = – (ax1 + by1 + cz1 + d) / a2 + b2 + c2
14. Since, x, y and z-axes pass through the origin and have direction cosines (1, 0, 0), (0, 1, 0) and (0, 0, 1), respectively. Therefore, their equations are
x – axis : x – 0 / 1 = y – 0 / 0 = z – 0 / 0
y – axis : x – 0 / 0 = y – 0 / 1 = z – 0 / 0
z – axis : x – 0 / 0 = y – 0 / 0 = z – 0 / 1
Plane
A plane is a surface such that, if two points are taken on it, a straight line joining them lies wholly in the surface.
General Equation of the Plane
The general equation of the first degree in x, y, z always represents a plane. Hence, the general equation of the plane is ax + by + cz + d = 0. The coefficient of x, y and z in the cartesian equation of a plane are the direction ratios of normal to the plane.
Equation of the Plane Passing Through a Fixed Point
The equation of a plane passing through a given point (x1, y1, z1) is given by a(x – x1) + b (y — y1) + c (z — z1) = 0.
Normal Form of the Equation of Plane
(i) The equation of a plane, which is at a distance p from origin and the direction cosines of the normal from the origin to the plane are l, m, n is given by lx + my + nz = p.
(ii) The coordinates of foot of perpendicular N from the origin on the plane are (1p, mp, np).

Intercept Form
The intercept form of equation of plane represented in the form of
x / a + y / b + z / c = 1
where, a, b and c are intercepts on X, Y and Z-axes, respectively.
For x intercept Put y = 0, z = 0 in the equation of the plane and obtain the value of x. Similarly, we can determine for other intercepts.
Equation of Planes with Given Conditions
(i) Equation of a plane passing through the point A(x1, y1, z1) and parallel to two given lines with direction ratios

(ii) Equation of a plane through two points A(x1, y1, z1) and B(x2, y2, z2) and parallel to a line with direction ratios a, b, c is

(iii) The Equation of a plane passing through three points A(x1, y1, z1), B(x2, y2, z2) and C(x3, y3, z3) is

(iv) Four points A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3) and D(x4, y4, z4) are coplanar if and only if

(v) Equation of the plane containing two coplanar lines

Angle between Two Planes
The angle between two planes is defined as the angle between the normal to them from any point.
Thus, the angle between the two planes
a1x + b1y + c1z + d1 = 0
and a2x + b2y + c2z + d2 = 0

is equal to the angle between the normals with direction cosines
± a1 / √Σ a21, ± b1 / √Σ a21, ± c1 / √Σ a21
and ± a2 / √Σ a22, ± b2 / √Σ a22, ± c2 / √Σ a22
If θ is the angle between the normals, then
cos θ = ± a1a2 + b1b2 + c1c2 / √a21 + b21 + c21 √a22 + b22 + c22
Parallelism and Perpendicularity of Two Planes
Two planes are parallel or perpendicular according as the normals to them are parallel or perpendicular.
Hence, the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0
are parallel, if a1 / a2 = b1 / b2 = c1 / c2 and perpendicular, if a1a2 + b1b2 + c1c2 = 0.
Note The equation of plane parallel to a given plane ax + by + cz + d = 0 is given by ax + by + cz + k = 0, where k may be determined from given conditions.
Angle between a Line and a Plane
In Vector Form The angle between a line r = a + λ b and plane r *• n = d, is defined as the complement of the angle between the line and normal to the plane:
sin θ = n * b / |n||b|
In Cartesian Form The angle between a line x – x1 / a1 = y – y1 / b1 = z – z1 / c1
and plane a2x + b2y + c2z + d2 = 0 is sin θ = a1a2 + b1b2 + c1c2 / √a21 + b21 + c21 √a22 + b22 + c22
Distance of a Point from a Plane
Let the plane in the general form be ax + by + cz + d = 0. The distance of the point P(x1, y1, z1) from the plane is equal to


If the plane is given in, normal form lx + my + nz = p. Then, the distance of the point P(x1, y1, z1) from the plane is |lx1 + my1 + nz1 – p|.
Distance between Two Parallel Planes
If ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 be equation of two parallel planes. Then, the distance between them is

Bisectors of Angles between Two Planes
The bisector planes of the angles between the planes
a1x + b1y + c1z + d1 = 0, a2x + b2y + c2z + d2 = 0 is
a1x + b1y + c1z + d1 / √Σa21 = ± a2x + b2y + c2z + d2 / √Σa22
One of these planes will bisect the acute angle and the other obtuse angle between the given plane.
Sphere
A sphere is the locus of a point which moves in a space in such a way that its distance from a fixed point always remains constant.
General Equation of the Sphere
In Cartesian Form The equation of the sphere with centre (a, b, c) and radius r is
(x – a)2 + (y – b)2 + (z – c)2 = r2 …….(i)
In generally, we can write
x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0
Here, its centre is (-u, v, w) and radius = √u2 + v2 + w2 – d
In Vector Form The vector equation of a sphere of radius a and Centre having position vector c is |r – c| = a
Important Points to be Remembered
(i) The general equation of second degree in x, y, z is ax2 + by2 + cz2 + 2hxy + 2kyz + 2lzx + 2ux + 2vy + 2wz + d = 0
represents a sphere, if
(a) a = b = c (≠ 0)
(b) h = k = 1 = 0
The equation becomes
ax2 + ay2 + az2 + 2ux + 2vy + 2wz + d – 0 …(A)
To find its centre and radius first we make the coefficients of x2, y2 and z2 each unity by dividing throughout by a.
Thus, we have
x2+y2+z2 + (2u / a) x + (2v / a) y + (2w / a) z + d / a = 0 …..(B)
∴ Centre is (- u / a, – v / a, – w / a)
and radius = √u2 / a2 + v2 / a2 + w2 / a2 – d / a
= √u2 + v2 + w2 – ad / |a| .
(ii) Any sphere concentric with the sphere
x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0
is x2 + y2 + z2 + 2ux + 2vy + 2wz + k = 0
(iii) Since, r2 = u2 + v2 + w2 — d, therefore, the Eq. (B) represents a real sphere, if u2 +v2 + w2 — d > 0
(iv) The equation of a sphere on the line joining two points (x1, y1, z1) and (x2, y2, z2) as a diameter is
(x – x1) (x – x1) + (y – y1) (y – y2) + (z – z1) (z – z2) = 0.
(v) The equation of a sphere passing through four non-coplanar points (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) and (x4, y4, z4) is

Tangency of a Plane to a Sphere
The plane lx + my + nz = p will touch the sphere x2 + y2 + z2 + 2ux + 2vy + 2 wz + d = 0, if length of the perpendicular from the centre ( – u, – v,— w)= radius,
i.e., |lu – mv – nw – p| / √l2 + m2 + n2 = √u2 + v2 + w2 – d
(lu – mv – nw – p)2 = (u2 + v2 + w2 – d) (l2 + m2 + n2)
Plane Section of a Sphere
Consider a sphere intersected by a plane. The set of points common to both sphere and plane is called a plane section of a sphere.
In ΔCNP, NP2 = CP2 – CN2 = r2 – p2
∴ NP = √r2 – p2

Hence, the locus of P is a circle whose centre is at the point N, the foot of the perpendicular from the centre of the sphere to the plane.
The section of sphere by a plane through its centre is called a great circle. The centre and radius of a great circle are the same as those of the sphere.
CBSE Class 12 Maths Notes : Linear Programming Problem
May 28, 2014 by Neepur Garg Leave a Comment
Linear Programming
It is an important optimization (maximization or minimization) technique used in decision making is business and everyday life for obtaining the maximum or minimum values as required of a linear expression to satisfying certain number of given linear restrictions.
Linear Programming Problem (LPP)
The linear programming problem in general calls for optimizing a linear function of variables called the objective function subject to a set of linear equations and/or linear inequations called the constraints or restrictions.
Objective Function
The function which is to be optimized (maximized/minimized) is called an objective function.
Constraints
The system of linear inequations (or equations) under which the objective function is to be optimized is called constraints.
Non-negative Restrictions
All the variables considered for making decisions assume non-negative values.
Mathematical Description of a General Linear Programming Problem
A general LPP can be stated as (Max/Min) z = clxl + c2x2 + … + cnxn (Objective function) subject to constraints

and the non-negative restrictions
xl, x2,….., xn ≥ 0 where all al1, al2,…., amn; bl, b2,…., bm; cl, c2,…., cn are constants and xl, x2,…., xn are variables.
Slack and Surplus Variables
The positive variables which are added to left hand sides of the constraints to convert them into equalities are called the slack variables. The positive variables which are subtracted from the left hand sides of the constraints to convert them into equalities are called the surplus variables.
Important Definitions and Results
(i) Solution of a LPP A set of values of the variables xl, x2,…., xn satisfying the constraints of a LPP is called a solution of the LPP.
(ii) Feasible Solution of a LPP A set of values of the variables xl, x2,…., xn satisfying the constraints and non-negative restrictions of a LPP is called a feasible solution of the LPP.
(iii) Optimal Solution of a LPP A feasible solution of a LPP is said to, be optimal (or optimum), if it also optimizes the objective function of the
problem.
(iv) Graphical Solution of a LPP The solution of a LPP obtained by graphical method i.e., by drawing the graphs corresponding to the constraints and the non-negative restrictions is called the graphical solution of a LPP.
(v) Unbounded Solution If the value of the objective function can be increased or decreased indefinitely, such solutions are called unbounded solutions.
(vi) Fundamental Extreme Point Theorem An optimum solution of a LPP, if it exists, occurs at one of the extreme points (i.e., corner points) of the convex
Polygon of the set of all feasible solutions.
Solution of Simultaneous Linear Inequations
The graph or the solution set of a system of simultaneous linear inequations is the region containing the points (x, y) which satisfy all the inequations of the given system simultaneously.
To draw the graph of the simultaneous linear inequations, we find the region of the xy-plane, common to all the portions comprWng the solution sets of the given inequations. If there is no region common to all the solutions of the given inequations, we say that the solution set of the system of inequations is empty.
Note The solution set of simultaneous linear inequations may be an empty set or it may be the region bounded by the straight lines corresponding to given linear inequations or it may be an unbounded region with straight line boundaries.
Graphical Method to Solve a Linear Programming Problem
There are two techniques of solving a LPP by graphical method
1. Corner point method and
2. Iso-profit or Iso-cost method
1. Corner Point Method
This method of solving a LPP graphically is based on the principle of extreme point theorem.
Procedure to Solve a LPP Graphically by Corner Point Method
(i) Consider each constraint as an equation.
(ii) Plot each equation on graph, as each one will geometrically represent a straight line.
(iii) The common region, thus obtained satisfying all the constraints and the non-negative restrictions is called the feasible region. It is a convex polygon.
(iv) Determine the vertices (corner points) of the convex polygon. These vertices are known as the extreme points of corners of the feasible region.
(v) Find the values of the objective function at each of the extreme points. The point at which the value of the objective function is optimum (maximum or minimum) is the optimal solution of the given LPP.
2. Isom-profit or Iso-cost Method
Procedure to Solve a LPP Graphically by Iso-profit or Iso-cost Method
(i) Consider each constraint as an equation.
(ii) Plot each equation on graph as each one will geometrically represent a straight line.
(iii) The polygonal region so obtained, satisfying all the constraints and the non-negative restrictions is the convex set of all feasible solutions of the given LPP, which is also known as feasible region.
(iv) Determine the extreme points of the feasible region.
(v) Give some convenient value k to the objective function Z and draw the corresponding straight line in the xy-plane.
(vi) If the problem is of maximization, then draw lines parallel to the line Z = k and obtain a line which is farthest from the origin and has atleast one point common to the feasible region. If the problem is of minimization, then draw lines parallel to the line Z = k and obtain a line, which is nearest to the origin and has atleast one point common to the feasible region.
(vii) The common point so obtained is the optimal solution of the given LPP.
Working Rule for Marking Feasible Region
Consider the constraint ax + by ≤ c, where c > 0.
First draw the straight line ax + by = c by joining any two points on it. For this find two convenient points satisfying this equation.
This straight line divides the xy-plane in two parts. The inequation ax + by c will represent that part of the xy-plane which lies to that side of the line ax + by = c in which the origin lies.
Again, consider the constraint ax + by ≥ c, where c > 0.
Draw the straight line ax + by = c by joining any two points on it.
This straight line divides the xy-plane in two parts. The inequation ax + by ≥ c will represent that part of the xy-plane, which lies to that side of the line ax + by = c in which the origin does not lie.
Important Points to be Remembered
(i) Basic Feasible Solution A BFS is a basic solution which also satisfies the non-negativity restrictions.
(ii) Optimum Basic Feasible Solution A BFS is said to be optimum, if it also optimizes (Max or min) the objective function.
Important Definitions
1. Point Sets Point sets are sets whose elements are points or vectors in En or Rn (n-dimensional euclidean space).
2. Hypersphere A hypersphere in En with centre at ‘a’ and radius ∈ > 0 is defined to be the set of points
X = -{x:|x — a| = ∈}
3. An ∈ neighbourhood An & neighbourhood about the point ‘a is defined as the set of points lying inside the hypersphere with centre at ‘a’ and radius ∈ > 0.
4. An Interior Point A point ‘a’ is an interior point of the set S, if there exists an ∈ neighbourhood about ‘a’ which contains only points of the set S.
5. Boundary Point A point ‘a’ is a boundary point of the set S if every ∈ neighbourhood about ‘a’ contains points which are in the set and the points which are not in the set.
6. An Open. Set A set S is said to be an open set, if it contain only the interior points.
7. A Closed Set A set S is said to be a closed set, if it contains a its boundary points.
8. Lines In En the line through the two points x1 and x2, x1 ≠ x2 is defined to be the set of points.
X = {x: x = λ x1 + (1 — λ) x2, for all real λ}
9. Line Segments In En, the line segment joining two point x1 and x2 is defined to be the set of points.
X = {x:x = λ x1 + (1 — λ)x2, 0 ≤ λ ≤ 1}
10. Hyperplane A hyperplane is defined as the set of points satisfying
c1x1+ c2x2 + …+ cnxn = z (not all ci = 0)
or cx = z
for prescribed values of c1, c2,…, cn and z.
11. Open and Closed Half Spaces
A hyperplane divides the whole space En into three mutually disjoint sets given by
X1 = {x : cx >z}
X2 = {x : cx = z}
X3 = {x : cx < z}
The sets x1 and x2 are called ‘open half spaces’. The sets {x : cx ≤ z} and { x : cx ≥ z} are called ‘closed half spaces’.
12. Parallel Hyperplanes Two hyperplanes c1x = z1 and c2x = z2 are said to be parallel, if they have the same unit normals i.e., if c1 = Xc2 for λ, λ being non-zero.
13. Convex Combination A convex combination of a finite number of points x1, x2,…., xn is defined as a point x = λ1 x1 + λ2x2 + …. + λnxn, where λi is real and ≥ 0, ∀ and

14. Convex Set A set of points is said to be convex, if for any two points in the set, the line segment joining these two points is also in the set.
or
A set is convex, if the convex combination of any two points in the set, is also in the set.

15 Extreme Point of a Convex Set A point x in a convex set c is called an ‘extreme point’, if x cannot be expressed as a convex combination of any two distinct points x1 and x2 in c.
16. Convex Hull The convex hull c(X) of any given set of points X is the set of all convex combinations of sets of points from X.
17. Convex Function A function f(x) is said to be strictly convex at x, if for any two other distinct points x1 and x2.
f{ λx1 + (1 — λ)x2} < λf(x1) + (1— λ)f(x2), where 0 < λ < 1.
And a function f(x) is strictly concave, if — f(x) is strictly convex.
18. Convex Polyhedron The set of all convex combinations of finite number of points is called the convex polyhedron generated by these points.
Important Points to be Remembered
(i) A hyperplane is a convex set.
(ii) The closed half spaces H1 = {x : cx ≥ z} and H2 = {x : cx ≤ z} are convex sets.
(iii) The open half spaces : {x : cx > z} and {x : cx < z} are convex sets.
(iv) Intersection of two convex sets is also a convex sets.
(v) Intersection of any finite number of convex sets is also a convex set.
(vi) Arbitrary intersection of convex sets is also a convex set.
(vii) The set of all convex combinations of a finite number of points X1, X2,…., Xn is convex set.
(viii) A set C is convex, if and only if every convex linear combination of points in C, also belongs to C.
(ix) The set of all feasible solutions (if not empty) of a LPP is a convex set.
(x) Every basic feasible solution of the system Ax = b,x ≥ 0 is an extreme point of the convex set of feasible solutions and conversely.
(xi) If the convex set of the feasible solutions of Ax = b,x ≥ 0 is a convex polyhedron, then atleast one of the extreme points gives an optimal solution.
(xii) If the objective function of a LPP assumes its optimal value at more than one extreme point, then every convex combination of these extreme points gives the optimal value of the objective function.