Learning Mathematics for IIT JEE

Home Page - IIT JEE Mathematics - Study Plans and Revision Notes

2012-03-31T00:56:00.000-07:00

Study Plans and Revision Notes are created on the basis of the chapters in R.D. Sharma's Book

1. Sets
Study guide and notes
2. Cartesian product of sets and relations
Study guide and notes
3. Functions
Study guide and notes
4. Binary operations
Study guide and notes
5. Complex numbers
Study guide and notes
6. Sequences and series
Study guide and notes
7. Quadratic equations and expressions
Study guide and notes
8. Permutations and Combinations
Study guide and notes
9. Binomial theorem
Study guide and notes
10. Exponential and logarithmic series
Study guide and notes
11.Matrices
Study guide and notes
12. Determinants
Study guide and notes
13 Cartesian System of Rectangular Coordinates and straight lines
Study guide and notes
14. Family of lines
Study guide and notes
Ch.15 Circle
Study guide and notes
Ch. 16. Parabola
Study guide and notes
Ch. 17. Ellipse
Study guide and notes
Ch. 18. Hyperbola
Study guide and notes
19. Real Functions
Study guide and notes
20. Limits
Study guide and notes
21. Continuity and Differentiability
Study guide and notes
22. Differentiation
Study guide and notes
23. Tangents , Normals and other applications of derivatives
Study guide and notes
24. Increasing and decreasing functions
Study guide and notes
25 Maximum and minimum values
Study guide and notes
26. Indefinite integrals
Study guide and notes
27. Definite Integration
Study guide and notes
28. Areas of Bounded regions
Study guide and notes
29. Differential equations
Study guide and notes
30. VECTORS
Study guide and notes
31. THREE DIMENSIONAL GEOMETRY
Study guide and notes
32. Probability
Study guide and notes
33. Trigonometric ratios, Identities and Maximum & Minimum Values of Trigonometrical Expressions
Study guide and notes
Ch.34 properties of Triangles and circles connected with them
Study guide and notes
Ch. 35. Trigonometrical equations
Study guide and notes
36. Inverse Trigonometrical functions
Study guide and notes
Ch. 37 Solution of Triangles
Study guide and notes
Ch. 38 Heights and distances
Study guide and notes

Vector Calculus - Online Book by Michael Corral

2012-01-16T19:58:00.001-08:00

Visit

http://www.mecmath.net/

Trigonometry - Introduction - Online Book by Michael Corral

2012-01-16T19:54:00.000-08:00

Download pdf copy from

http://www.mecmath.net/trig/trigbook.pdf

Chapters in R.D Sharma Mathematics Book

2011-12-18T00:47:00.000-08:00

1. Sets
2. Cartesian product of sets and relations
3. Functions
4. Binary operations
5. Complex numbers
6. Sequences and series
7. Quadratic equations and expressions
8. Permutations and Combinations
9. Binomial theorem
10. Exponential and logarithmic series
11.Matrices
12. Determinants
13 Cartesian System of Rectangular Coordinates and straight lines
14. Family of lines
Ch.15 Circle
Ch. 16. Parabola
Ch. 17. Ellipse
Ch. 18. Hyperbola
19. Real Functions
20. Limits
21. Continuity and Differentiability
22. Differentiation
23. Tangents , Normals and other applications of derivatives
24. Increasing and decreasing functions
25 Maximum and minimum values
26. Indefinite integrals
27. Definite Integration
28. Areas of Bounded regions
29. Differential equations
30. VECTORS
31. THREE DIMENSIONAL GEOMETRY
32. Probability
33. Trigonometric ratios, Identities and Maximum & Minimum Values of Trigonometrical Expressions
Ch.34 properties of Triangles and circles connected with them
Ch. 35. Trigonometrical equations
36. Inverse Trigonometrical functions
Ch. 37 Solution of Triangles
Ch. 38 Heights and distances

Mathematics Videos for IIT JEE

2011-05-31T08:25:00.000-07:00

There are many videos now available on internet especially Youtube to help us in learning mathematics.

Some of them are being collected for various chapters of IIT JEE Syllabus in a Knol Book of Videos

Analytical Geometry - Online Book by Donald Vossler

2011-04-02T09:10:00.000-07:00

http://www.descarta2d.com/BookHTML/Table_of_Contents.html

Calculus - Online Book - Tutorhomework Book

2011-04-02T09:06:00.000-07:00

http://www.tutor-homework.com/Math_Help/Calculus.html

Calculus - Online Book by Faraz Hussain

2011-04-02T08:48:00.000-07:00

1. Why Study Calculus?

4. The Derivative

http://www.understandingcalculus.com/

New articles on mathematics on google knol

2009-09-23T06:17:00.000-07:00

On google knol publishing platform some more articles on algebra are posted.

http://knol.google.com/k/narayana-rao-kvss/knol-sub-directory-algebra-new-knols/2utb2lsm2k7a/1658#

Algebra - New Posts on Google Knol

2009-08-22T20:50:00.000-07:00

Knol Sub-Directory - Algebra - New Knols

http://knol.google.com/k/narayana-rao-kvss/-/2utb2lsm2k7a/1658#

More articles or posts are appearing on knol in the area of mathematics.

Yesterday I made a subdirectory of knols on geometry. Today I saw an author writing on combinatorics and number theory. They may not be focussed on JEE but for having a look at a different treatment of the topic they are good. Additional advantage is that you can ask a doubt and the author is likely to respond to your question. That interactive learning is possible when you read knols.

Mathematics - Interesting Essays

2009-07-20T00:40:00.000-07:00

http://knol.google.com/k/narayana-rao-kvss/knol-sub-directory-mathematics/2utb2lsm2k7a/1455#

List of interesting articles on Mathematics.
You can search for more using knol search engine.

IIT JEE Learning Mathematics - Recall Facilitation

2009-05-08T01:06:00.000-07:00

Memorizing vs. Rote Learning & Drilling

Ideas of Michael Paul Goldenberg
Ann Arbor, MI, United States

I know of no one who opposes memorization (which by the way is NOT the same as rote learning.)

MINDLESS rote learning of things that can be learned effectively, possibly MUCH more effectively, in other ways needs to be stopped.

Mneomonic methods are, however, extremely easy to understand and put into practical use for a wide range of applications. These methods pay off in inverse proportion to the arbitrariness of the material being memorized. It means is that when faced with , say, a random or arbitrary list of items, dates, facts, etc., the more random the list and therefore the less conceptual links or "common knowledge" might be involved, the more a person using mnemonics would gain from using these techniques. Because otherwise the main option would be some variation on pure rote.

However, less time was needed for memorizing information with more structure, because the "inherent logic" or interconnectedness of the information helped one memorize.

Mathematics already is based on logical and conceptual links. Hence, it is often the case that what needs to be "memorized" in the sense mentioned above is minimal.

What sorts of things would need to be memorized in mathematics? Well, things like Order of Operations, which consists of conventions, not something that simply HAS to be. Terminology. Notation. Axioms. Things that do not follow from first precepts.

Even going beyond that, it is undoubtedly true that we need to "memorize" certain fundamental relationships and identities in specific areas of mathematics in order to not have to tediously look them up for every single instance in which they arise. In trigonometry, for example, understanding the definition of sine, cosine, and tangent in right triangle trigonometry is a key "fact" that one does much better to have at one's mental fingertips than not.

The amount that "must" be memorized is often far smaller than one originally believes, because of underlying relationships and concepts that create natural connections among a smaller set of facts. Anyone who is led to believe that entire chapters of a mathematics book and solutions of all problems need to be memorized by drill or rote is being mistaught.

That which is understood conceptually has a better chance of lasting, and can be more readily recreated through the concepts even if the "at one's fingertips" recall has been weakened or extinguished. Most people are well aware of this through personal experience.

http://rationalmathed.blogspot.com/2009/04/memorizing-vs-rote-learning-drilling.html

IIT JEE Mathematics Blog Status

2009-04-30T08:48:00.000-07:00

I am presently preparing study plans for each chapter based on the text book by R D Sharma. I completed up to chapter 30.

You can see all chapters by clicking on labels revision facilitator or study plan.

Feel free to give your comments on the plans.

Interesting Arithmetic Relations

2009-04-10T22:01:00.000-07:00

1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321

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

1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 +10= 1111111111

-------------------------------
9 x 9 + 7 = 88
98 x 9 + 6 = 888
987 x 9 + 5 = 8888
9876 x 9 + 4 = 88888
98765 x 9 + 3 = 888888
987654 x 9 + 2 = 8888888
9876543 x 9 + 1 = 88888888
98765432 x 9 + 0 = 888888888

-------------------------------
1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321

Blog Status - IIT JEE Mathematics

2009-03-11T09:03:00.000-07:00

Presently developing chapterwise study plans for JEE 2010 and JEE 2011

Blog Status - 26 January

2009-01-26T00:26:00.001-08:00

Presently working on trigonometry chapters

Concept of Congruency of Triangles

2009-01-23T08:10:00.000-08:00

Concept of Congruency
(From 9th Class Text)
(Used in Trigonometry Chapters of XI )

When two triangles have the same size, then they are said to be congruent triangles.

Two congruent triangles are equal in all respects and when one is placed on the other, both exactly coincide. This means each part of one triangle is equal to the corresponding part of the other.

If ABC and DEF are congruent triangles, when DEF is placed over ABC, both will coincide and this proves that they are congruent. This process of proof is known as proof by superposition.

But one need not check for all the six parameters (three sides and three angles) for proving congruency.

Conditions for Congruency

1. If two sides and the included angle of a triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

2. If two angles and a side of a triangle are equal to the two angles and the corresponding side of another triangle, then the triangles are congruent.

3. If the three sides of the first triangle are equal to the corresponding three sides of the second triangle, then the triangles are congruent.

4. In case of right angled triangles, if the hypotenuse and a side of a triangle are equal to the corresponding side and hypotenuse of another triangle, then the triangles are congruent.

More briefly the rules are
(i) Two sides and the included angle (S.A.S.)
(ii) Two angles and corresponding side (A.A.S.)
(iii) Three sides (S.S.S.)
(iv) Right angle, hypotenuse, and one side (R.H.S.)

Ask questions and answer questions about IIT JEE Subjects

2009-01-01T01:07:00.000-08:00

KNOWLEDGE QUESTION AND ANSWER BOARD
http://knol.google.com/k/narayana-rao-kvss/-/2utb2lsm2k7a/654#

Set: Explanation

2008-12-20T04:38:00.000-08:00

Set is synonymous with the words, ‘collection’, aggregate’, ‘class’, and is comprised of elements.

The words ‘element’, ‘object’, and ‘member’ are synonymous.

Sets designated by specific letters.

N: natural numbers
Z : integers
Z⁺: positive integers
Q: rational numbers
Q⁺: positive rational numbers
R: real numbers
R⁺: positive real numbers
C: complex numbers

Description of a set

2008-12-20T04:37:00.001-08:00

Sets can be described by roster method or set-builder method.

Roster method:
In this method, the set is described by listing all the elements within braces { }, separated by commas.

Example: {2,4,6,8,10}
It is a set having 5 elements.

Set-builder method:
In this method, a set is described by a property of x where x represents the elements. If the property of x is represented by P(x), the set description is given by
{x : P(x) is satisfied} or {x| P(x) is satisfied}

Example: {x| x is an even number less than or equal to 10}
This description will give {2,4,6,8,10} in roster form.

Types of sets

2008-12-20T04:35:00.000-08:00

Empty set (ф)
A set is said t be empty or null or void set if it has no element and it is denoted by ф.

Singleton set
A set consisting of single element.

Finite set
A set is called a finite set if it is either void set or its elements can be listed (counted or labeled) by natural numbers 1,2,3 … and the counting of number of elements stops at a certain natural number of say (n).

The number of elements in a finite set (n) is called the cardinal number or order of a finite set A and is denoted by n(A).

Infinite set
A set who elements cannot be listed by the natural numbers however large the number may be is called an infinite set.

Equivalent set
Two finite sets are equivalent if their cardinal numbers or number of elements are same.

Equal set
Two sets A and B are equal if every element in A is a member of B and every element of B is a member of A.

Subset
When A and B are two sets, if every element of A is an element of B, then A is called a subset of B.

Universal set (U)
In discussions of sets, the superset that contains all other sets in discussion is called the universal set.

Power set
When A is a set, the collection or family of all subsets of A is called the power set of A and is denoted by P(A).

Power set is a set of subsets or elements of a power set are subsets of a set.
P(A) = {S: S is a subset of A}

If A is a finite set having n elements, the P(A) has 2ⁿ elements.

Complement of a set
If U is a universal set, the complement of a set A with respect to U is denoted as A’ or A^c or U – A . It is a set of those elements of U which are not in A.

A’ = {x| x є U, and x is does not belong to A}

Theorems on subsets

2008-12-20T04:33:00.000-08:00

1. Every set is a subset of itself.
2. The empty set is a subset of every set.
3. The total number of subsets of a finite set containing n elements is 2ⁿ

Operations on sets

2008-12-20T04:32:00.000-08:00

Union of sets
The union of sets A and B is th set of all those elements which belong either to A or to B or to both A and B.

The symbol used to denote union of sets and A and B is A U B.

x Є (A U B) implies x Є A or x Є B
x does not belong to A U B implies x does not belong to A and also x does not belong to B.

Intersection of sets
The intersection of sets A and B is the set of all those elements that belong to both A and B.

The intersection of sets A and B is denoted by A ∩ B.

x Є (A ∩ B) implies x Є A and also x Є B.

Difference of sets
The difference of sets A and B, written as A-B is the set of all those elements of A which do not belong B.

It means x Є (A - B) implies x Є A or x does not belong to B.

Symmetric difference of sets
The symmetric difference of sets A and B is the set (A-B)U((B-A) and is denoted by AΔB.

AΔB = (A-B)U((B-A) = {x: x does not belong to A ∩ B).

Types of sets based on operations

2008-12-20T04:31:00.002-08:00

Disjoint sets

Complement of a set
When U is the universal set and A is a subset of U, the complement of A with respect to U is denoted by A’ or A₀ or U-A and it is defined as the set of all those elements of U which are not in A.

A’ = {x: x does not belong to A but x ЄU}
x ЄA’ implies x does not belong to A.

Laws of algebra of sets

2008-12-20T04:31:00.001-08:00

1. Idempotent laws

(i) A U A = A
(ii) A ∩ A = A

2. Identity laws

(i) A U ф = A
(ii) A ∩ U = A

3. Commutative law
(i) A U B = B U A
(ii) A ∩ B = B ∩ A

4. Associative laws

(i) (A U B) U C = A U (B U C)
(ii) (A ∩ B) ∩ C = A ∩ (B ∩ C)

5. Distributive laws

(i) A U (B ∩ C) = (A U B) ∩ (A U C)
(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

6. De-morgan’s laws

(i) (A U B)’ = A’ ∩ B’
(ii) (A ∩ B)’ = A’ U B’

Some more deductions/theorems/ related to operations on sets

2008-12-20T04:30:00.000-08:00

If A and B are two sets

(i) A – B = A ∩ B’
(ii) B – A = B ∩ A’
(iii) A – B = A <=> A ∩ B = ф
(iv) (A – B) U B = A U B
(v) (A-B) ∩ B = ф
(vi) A is a sub set of B <=> B’ is a subset of A’
(vii) (A-B) U (B-A) = (A U B) – (A ∩ B)

If A, B and C are three sets, then

(i) A – (B ∩ C) = (A-B) U (A-C)
(ii) A – (B U C) = (A-B) ∩ (A-C)
(iii) A ∩ (B-C) = (A ∩ B) - (A ∩ C)
(iv) A ∩ (B Δ C) = (A∩B) Δ (A∩C)

Number of elements in sets n(A) and Some Results on Them

2008-12-20T04:27:00.000-08:00

Note union operation and universal set have the same symbol in these pages. Hence identify appropriately.

n(A) denotes the number of elements in the set A. Similarly n(B) and n(C).

If A,B and C are finite sets. U is the finite universal set, then

(i) n(A U B) = n(A) +n(B) – n(A∩B)

(ii) n(A U B) = n(A) +n(B) <=> A, B are disjoint non-void sets.

(iii) n(A-B) = n(A) –n(A∩B)

(iv) n(A ΔB) = Number of elements which belong to exactly one of A or B
= n((A-B) U (B-A))

(v) n(A U B U C) = n(A) + n(B) + n(C) – n(A∩B) – n(B∩C) – n(A∩C)+n(A∩B∩C)

(vi) No. Of elements in exactly two of the sets A,B,C
= n(A∩B) + n(B∩C)+n(C∩A)-3n(A∩B∩C)

(vii) No. 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)

(viii) n(A’ U B’) = n((A∩B)’) = n(U) – n(A∩B)

(ix) n(A’∩B’) = n((AUB)’) = n(U)-n(A∩B)

Ordered Pair

2008-12-20T00:10:00.000-08:00

An Ordered pair consists of two objects or elements in a given fixed order.

For example when A and B are any two sets, a pair (a,b) where a ЄA and b ЄB is an ordered pair. The fixed order comes from the two sets A and B and the first element is from A and the second element is from B.

Equality of ordered pairs: Two ordered pairs (a1,b1) and (a2,b2) are equal to if a1 = a2 and b1 = b2.

Cartesian product of sets

2008-12-20T00:09:00.000-08:00

Cartesian product is an operation on sets.

Let A and B be any two non empty sets. The set of all ordered pairs (a,b) such that aЄA and bЄB is called the Cartesian product of the sets A and B and is denoted by A×B

A×B represents Cartesian product of sets and it is a set of ordered pairs.

Theorems on Cartesian product of sets

2008-12-20T00:08:00.000-08:00

Theorem 1; For any three sets

(i) A×(B U C) = (A×B) U (A×C)
(ii) A×(B∩C) = (A×B) ∩(A×C)

Theorem 2: For any three sets

A×(B – C) = (A×B) – (A×C)

Theorem 3: If and A and B are any two non-empty sets, then

A×B = B×A  A = B

Theorem 4: If A is a subset of B, A×A is a sub set of (A×B) ∩(B×A)

Theorem 5
If A is a subset of B, (A×C) is a subset of (B×C) for any set C.

Theorem 6
If A is a subset of B and C is a subset of D, (A×C) is a subset of (B×D)

Theorem 7
For any sets A,B,C , D,

(A×B) ∩(C×D) = (A∩C) ×(B∩D)

Theorem 8

For any three sets A,B,C

i. (A×(B’ U C’) = (A×B) ∩(A×C)
ii. (A×(B’ ∩ C’) = (A×B) U(A×C)

Theorem 9
When A and B are two non-empty sets having n elements in common, (A×B) and (B×A) will have n² elements in common.

Relation - Definition

2008-12-20T00:07:00.002-08:00

Let A and B be two sets. Then a relation R from A to B is a subset of A×B.

R is a relation from A to B <=> R is a subset of A×B.

Total number of relations: If A and B are two non empty sets with m and n elements respectively, A×B consists of mn ordered pairs.
Since each subset defines a relation from A to B, so total number of relations from A to B is 2^mn.

Domain and Range of Relation

2008-12-20T00:07:00.001-08:00

R is a relation means that it is a set of ordered paris.

Domain of a relation

When R is a relation from a set A to set B, the set of all first components of the ordered pairs belonging to R is called the domain of R.

Range of a relation

When R is a relation from a set A to set B, the set of all second components of the ordered pairs belonging to R is called the range of R.

In a relation from set A to set B, domain of the relation will be a subset of A and range of the relation will be a subset of B.

Relation on Set A and Inverse Relation

2008-12-20T00:06:00.000-08:00

Relation on Set A
A relation from A to A i.e., a subset of A×A, is called a relation on set A.

Inverse relation
When a relation R is from set A to set B, a relation from set B to set A denoted by R^-1 is the inverse of R. That is if R is a set of (a,b), then R^-1 is a set of (b,a).

In the case Domain of relation R = Range of relation R^-1

Range of relation R = Domain of relation R^-1

Types of relations

2008-12-20T00:05:00.001-08:00

Void or empty relation
When A is a set, ф is a subset of A×A and so it is a relation on A. This relation is called the void or empty relation on A.

Universal relation
When A is a set, A×A is a subset of A×A and so it is a relation on A. This relation is called the universal relation.

Identity relation
When A is a set, the relation I_A = {(a,a):a ЄA} on A is called the identity relation on A.

Reflexive relation
A relation on a set A is said to be reflexive if every element of A is related to itself.

Think. What is the difference between identity relation and reflexive relation?

Symmetric relation
A relation R on a set A is said to be a symmetric relation iff

(a,b) Є R implies (b,a) ЄR for all a,b Є A.

i.e. aRb implies bRa for a,b Є.

Transitive relation
A relation R on a set A is said to be a transitive relation iff
(a,b) ЄR and (b,c) ЄR implies (a,c) ЄR for a,b,c ЄA.

Antisymmetric relation
A relation R on set A is said t be an antisymmetric relation iff
(a,b) Є R and(b,a) Є R implies a =b for all a,b ЄA.

Equivalence relation
A relation R on set A is said to be an equivalence relation on A iff
i. it is reflexive.
ii. it is symmetric
iii. it is transitive.

Some more properties and results on relations

2008-12-20T00:04:00.000-08:00

1. If R and S are two equivalence relations on a set A, then R∩S is also an equivalence relation on A.
2. The union of two equivalence relations on a set is not necessarily an equivalence relation on the set.
3. If R is an equivalence relation on a set A, the R^-1 is also an equivalence relation on A.

Composition of Relations

2008-12-20T00:00:00.000-08:00

When r and S are two relations from set A to B and B to C respectively, we can define a relation SoR from A to C such that

(a.c) Є SoR imples for all b Є B subject to the relations (a,b) ЄR and (b.c) ЄS.

SoR is called the composition of R and S.

Properties of SoR

In general RoS is not equal to SoR.

(SoR) ^- = R^-oS^-

Function – definition

2008-12-19T07:30:00.002-08:00

Let A and B be two non-empty sets. Then a function ‘f’ from set A to set B is a rule or method or correspondence which associates elements of set A to elements of set B such that

(i) all elements of set A are associated to elements in set B.
(ii) an element of set A is associated to a unique element in set B.

Domain, Co-Domain and Range of a Function

2008-12-19T07:30:00.001-08:00

Let f: A → B. Then, the set A is known as the domain of f and the set B is known as the co-domain of f.

The set of f-images of elements of A (elements in set B associated with elements in set A) is known as the range of f or image set of A under f and is denoted by f(A)

Thus range of f = f(A) = {f(x): x Є A}

F(A) is a subset of B.

Range of f is a subset of co-domain of f.

Description of a Function

2008-12-19T07:29:00.000-08:00

A function f:A→B can be described by giving A and f(a) for every element in A if it is a finite set.

If A is infinite, the functions are described by a formula.
For example A function f:Z→Z is given by the formula f(x) = x²+1

Equal functions

2008-12-19T07:28:00.002-08:00

Two functions f and g are equal or f = g iff
The domain of f = domain of g,
The co-domain of f = the codomain of g and
f(x) = g(x) for every x belonging to their common domain.

Number of functions

2008-12-19T07:28:00.001-08:00

Let A and B be two finite sets having m and n elements.
The total number of functions from A to B is n^m.

Function as a relation

2008-12-19T07:27:00.000-08:00

A relation f from A to B (both being non empty sets), that is a sub set of A×B is called a function from A to B if
For each a ЄA there exists b ЄB such that (a,b) Єf and if (a,b) Єf and (a,c) Єf means b = c.

Kinds of functions

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Kinds of Functions

One-One function

A function f:A→B is said to be a one-one function if different elements of A have different images in B.

Many-One function

A function f:A→B is said to be many-one function if two or more elements of set A have the same image in B.

Onto function

A function f:A→B is said to be an onto function if every element of B is the f image of some element of A.

Range of is the codomain of f. Codomain of f is B.

Into function

A function f:A→B is said to be an into function if there exists an element in B having no pre-image in A.

One-one onto function

A function which is one-one as well as onto function.

Composition of functions

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If f:A→B and g:B→C are two functions, then a function gof: A→C is defined by

(gog)(x) = g(f(x), for all x ЄA and is called the composition of f and g.

Properties of composition of functions

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The composition of functions is not commutative fog ≠ gof.

The composition of functions is associative. If f,g, and h are functions such that (fog)oh and fo(goh) exist, then
(fog)oh = fo(goh)

The composition of two bijections is also a bijection.

Inverse of an element

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In case of f:A→B, if a ЄA is associated with b ЄB, then b is called the f image of ‘a’ and is written as b = f(a). a is said to be the pre-image or inverse element of ‘b’ under f and we write a = f^-1(b).

Inverse of a function

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If If f:A→B is a bijection meaning x ЄA and y ЄB then we associate y ЄB with x ЄA and such a function is known as the inverse function and is denoted by f^-1.

Properties of inverse of a function

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The inverse of a bijection is unique

The inverse of a bijection is also a bijection

Binary Operations and Types of Binary Operations

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When S is a nonempty set, a function f:S×S→S is called a binary operation.

Each ordered pair (a,b)Є(S×S) is associated to a unique element f(a,b) in S.

Examples: addition of a and b (both natural numbers). a+b is also a natural number.

a belongs to N, b belongs to N, and a+B also belongs to N.

Types

Commutative binary operation

Associative binary operation

Distributive binary operation

Logarithm - Definition

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If a^x = y then log_ay = x

log is the abbreviation of the word logarithm

Fundamental Laws of Logarithms

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log MN = log M + log N

log M/N = log M - log N

log Mⁿ = n log M

log 1 = 0

log_aa = 1 (log of any positive quantity of the same base is always one.

Systems of Logarithms

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Common logarithms to the base 10

Natural logarithms to the base e

Standard Form of Decimal

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If k is a positive number we can express it as

k = m*10^p

Where m is a decimal number such that 1≤m≤10 and p is an integer.

This form m*10^p is called standard form of decimal.

Finding Log N from tables

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Find the characteristic of N.
Find the mantisssa
Log N = Charateristic + Mantissa

Antilogarithms

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If log n = m

n = antilog m

Definition of sets of numbers

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N = set of natural numbers

I = set of integers

Q = set of all rational numbers

R = set of real numbers rational and irrational

C = set of complex numbers

Imaginary unity - Iota

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Sqrt(-1) = i
“i” is called imaginary unity

Integral powers of IOTA (i)

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i³ = i*i² = i*(-1) = -i

Imaginary quantities

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Square roots of numbers -3, -5 etc are called imaginary quantities

Complex numbers - definition

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Number of the form a+ib (ex: 4+i3) is called a complex number.

a is called real part Re(z) and b is called imaginary part Im(z).

Equality of complex numbers

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Two complex numbers z1 = a1+ib1 and z2 = a2+ib2 are equal if a1 = a2 and b1 = b2.

It means z1 = z2 if Re(z1) = Re(z2) and Im(z1) = Im(z2).

Addition of complex numbers

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When z1 = a1+ib1 and z2 = a2+ib2 the addition of the two z1 + z2 is defined as the complex number (a1+a2) + i(b1+b2)

Re(z1+z2) = Re(z1) + Re(z2)

Im(z1+z2) = Im(z1)+Im(z2)

Subtraction of complex numbers

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When z1 = a1+ib1 and z2 = a2+ib2 the subtraction of the two z1 - z2 is defined as z1+(-z2)

Multiplication of complex numbers

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(a1+ib1) (a2+ib2) by multiplying and simplifying we get

(a1a2 – b1b2) + i(a1b2+a2b1)

Multiplicative inverse of a+ib = a/(a² + b²) - ib/(a² + b²))

Division of complex numbers

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z1/z2 = z1* Multiplicative inverse of z2

Conjugate of a complex number

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conjugate of z (= a+ib) = a-ib (is termed as z bar)

Modulus of a complex number

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|z| = |a+ib| = SQRT(a² +b²)

Properties of Modulus

If z is a complex number, then

(i) |z| = 0 <=> z = 0
(ii) |z| = |conjugate of z| = |-z| = |-conjugate of z|
(iii) -|z| ≤Re(z) ≤|z|
(iv) -|z| ≤Im(z) ≤|z|
(v) z*congulage of z = |z|²

Reciprocal of a complex number

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Multiplicative inverse and reciprocal are same

Complex number as a rotating arrow in the argand plane

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Roots of a complex number

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Write the give number in polar form

Add 2mπ to the argument

Apply the De Moivre's theorem

Put m = 0,1,2,...,(n-1) i.e. one less than the number in the denominator of the given index in the lowest form.

Sequences and Series - Definitions

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Sequence

A sequence is a function whose domain is the set N of natural numbers.

Sequence is denoted by ‘a’ and the nth term in the sequence a(n) is denoted by a_{n/sub>.

A sequence whose range is a subset of R is called a real sequence.

Representation of a sequence

One way is to list its first few terms till the rule for writing down other terms becomes clear.

Another way is to represent a real sequence is to give a rule of writing the nth term of the sequence.

Series

If a1, a2, a3, … is a sequence, then the expression a1+a2+a3+… is a series.

Progressions

It is not necessary that the terms of a sequence always follow a certain pattern or they are described by some explicit formula for the nth term. Those sequences whose terms follow certain patterns are called progressions.}

General term of an A.P.

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nth term = an = a+(n-1)d

Selection of terms in an A.P.

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a-d, a, a+d

Sum of n terms of an A.P.

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Sn = n(a+1)/2

Sum of n terms of a G.P.

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S_n = a[(rⁿ-1)/(r-1)]

Sum of an Infinite G.P

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S = a/(1-r)
r is less than one

Sum to n terms - some special sequences

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1. Natural numbers

n(n+1)/2

2. Squares of natural numbers

n(n+1)(2n+1)/6

3. Cubes of natural numbers

[n(n+1)/2]²

Harmonic progression

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A sequence a1, a2, a2,…,an of non-zero numbers is called a Harmonic progression if the sequence 1/a1,1,a2,..,1/an,.. is an A.P.

Example: The sequence 1,1/4,1/7,1/10,… is a H.P. because the sequence 1,4,7,10,… is in A.P.

d of the corresponding AP = 1/a2 -1/a1

an of H.P. 1/[a+(n-1)d] where a = 1/a1

Insertion of n harmonic means between two give numbers a and b

a,H1,H2,…,Hn,b are in H.P.
=> 1/a, 1/H1,1/H2,…,1/Hn,1/b are in A.P.

Let d be common difference of this A.P.
The last term in AP 1/b is the (n+2)th term.

So 1/b = 1/a +(n+1)d
=> d = (1/b -1/a)/(n+1) = (a-b)/ab(n+1)
=> 1/H1 = (1/a) +d
1/H2 = (1/a)+2d

1/Hn = (1/a)+nd

Harmonic mean of n numers

If a1, a2, ..., an are n non-zero numbers, then the harmonic mean H of these numbers is given by

1/H = [1/a1 +1/a2+…+1/an]/n

Properties of arithmetic, geometric, and harmonic means between two given numbers (a and b)

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A.M. = A = (a+b)/2
G.M. = G = √ab
H.M. = H = 2ab/(a+b)

1. A≥G≥H
2. A,G,H form a GP, i.e., G² = AH
3. The equation x²-2Ax+G² has as its roots a and b.
4. The equation x³-3A x²+3G³x/H-G³ = 0 has as its roots a,b,c when A,G,and H are arithmetic mean, geometric mean, and harmonic mean of three numbers a,b,and c.

Quadratic Equations and Expressions - Definitions

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Real Polynomials: Coefficients are real numbers and variables take real values.

Complex Polynomials: Coefficients are complex numbers and variable is varying complex number.

Polynomial equation:
f(x) = 0

Roots of an equation: The values of the variable satisfying the given equation are called its roots.

Some conclusions on Roots of a Polynomial Equation

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1. An equation of degree n has n roots, real or imaginary.
2. Surd and imaginary roots always occur in pairs, i.e. if 5-3i is a root of an equation, then 5 +3i is also its root. Similarly, if 3+SQRT(5) is a root of a given equation, then 3-SQRT(5) is also its root.
3. An odd degree equation has at least one real root, whose sign is opposite to that of its last term, provided that the coefficient of highest degree term is positive.
4. Every equation of an even degree whose constant term is negative and the coefficient of highest degree term is positive, has at least two real roots, one positive and one negative

Descartes Rule of Signs

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The maximum number of positive real roots of a polynomial equation f(x) = 0 is the number of changes of signs from positive to negative and negative to positive in f(x).

Relations between roots and coefficients

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S1 = α1+α2+...+αn = -a1/a0

S2 = α1α2 + α1α3+... = Σαiαj; i≠j = (-1)² (a2/a0)

Sn = α1α2...αn = (-1)ⁿ(const. term/a0) {constant term = an)

Formation of a polynomial equation from given roots

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If α1, α2, α3,...,αk are the roots of an nth degree equation

xⁿ-S1x^n-1+S2x^n-2-S3x^n-3++...+(-1)ⁿSn = 0

where Sk denotes the sum of the products of roots taken k at a time.

Roots of a quadratic equation with real coefficients

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ax²+bx+c where a≠0, a,b,c Є R is a quadratic equation with real coefficients.

The quantity D = b²-4ac is the called the discriminant of the quadratic equation.

1. The roots are real and distinct if and only if D>0.
2. The roots are real and equal if and only D = 0
3. The roots are complex with non-zero imaginary part if and only if D<0.
4. The roots are rational iff a,b,c are rational and D is a proper square.
5. The roots are of the form p+√q (p,q Є Q), iff a,b,c are rational and D is not a perfrect square.
6. If a =1, b,c ЄI and the roots are rational numbers, then these roots must be integers.
7. If a quadratic equation in x has more than two roots, then it is an identity in x that is a=b=c=o.

Graph of Quadratic Expression

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Graph of a quadratic expression is a parabola.

Sign of a quadratic expression for real values of the variable

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For real values of x, the sign of the quadratic expression f(x) = ax² +bx+c is the same as that of 'a' except when the roots of the equation ax²+bx+c are real and distinct and x lies between them.

ax²+bx+c is greater than 0 for all x Є R iff a is greater than 0 and D is less than zero. (D is discriminant b²-ac), and

ax²+bx+c is lesser than 0 for all x Є R iff a is less than 0 and D is less than zero.

Values of a rational expression P(x)/Q(x) for real values of x, where P(x) and Q(x) are quadratic expressions

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Factorial

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Factorial: the continued product of first n natural numbers is called the “n factorial” and is denoted by n! or

i.e. n! = 1*2*3…*(n-1)*n

4! = 1*2*2*4

n! is defined for positive integers only.

0! is defined as 1.

n! = n*(n-1)!

Exponent of prime number p in factorial of n (n!)

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The exponent of prime number of 3 in 100! is 48.
It means 100! is divisible by 3⁴⁸

How do you find it?
Let p be a prime number and n be a positive integer. Then find the last integer in the sequence 1,2,…,n which is divisible by p.
Express this integer as [n/p]p.
[n/p] denotes the greatest integer less than or equal to n/p

In case of 3 (p) and 100 (n); [n/p] is 33 and n/p is 33 and 1/3.

Let E_p(n) denote the exponent of the prime p in the positive integer n. Then,

E_p(n!) = E_p(1.2.3…(n-1).n)

This will be equal to E_p(p.2p.3p…[n/p]p)
= [n/p]+ E_p(1.2.3...[n/p])

This process continues and we get the answer

E_p(n!) = [n/p] + [n/p²]+…+[n/p^s]
Where s is the largest positive integer such that p^s≤n≤p^s+1

Hence applying the formula to find exponent of prime 3 in 100!

E₃(100!) = [100/3] + [100/3²] + [100/3³] + [100/3⁴]
= 33+11+3+1 = 48

Note: remember the meaning of notation [100/3] or [n/p]

Fundamental principle of multiplication

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If there are two jobs such that one of them can be completed in m ways, and when it has been completed in any one of these m ways, second job can be completed in n ways, then the two jobs in succession can be completed in m*n ways.

Fundamental principle of addition

If there are two jobs such that they can be performed independently in m and n ways respectively, then either of the two jobs can be performed in (m+n) ways.

Permutations Definition and Theorems

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Each of the arrangement which can be made by taking some or all of a number of things is called a permutation.

Theorem 1

Let r and n be positive integers such that 1≤r≤n. then the number of all permutations of n distinct things taken r at a time is given by

n(n-1)(n-2)…(n-(r-1))

Notation: Let r and n be positive integers such that 1≤r≤n. then the number of all permutations of n distinct things taken r at a time is denoted by the symbol P(n,r) or ⁿC_r.

Then P(n,r) = ⁿC_r = n(n-1)(n-2)…(n-(r-1))

Theorem 2

P(n,r) = ⁿC_r = n!/(n-r)!

Theorem 3

The number of all permutations of n distinct things taken all at a time is n!.

Theorem 4

0! = 1

Permutations under certain conditions

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Three theorems

Theorem 1
The number of all permutations of n different objects taken r at a time, when a particular object is to be always included in each arrangement is r.^n-1C_r-1

Theorem 2

The number of all permutations of n different objects taken r at a time, when a particular object is never taken in each arrangement is, ^n-1C_r-1

Theorem 3

The number of all permutations of n different objects taken r at a time, when two specified objects always occur together is 2!(r-1) ^n-2C_r-2

Permutations of Objects not all Distinct

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Theorems and Formulas

Theorem
The number of mutually distinguishable permutations of n things, taken all at a time, of which p are alike of one kind, q alike of second such that p+q = n, is

n!/p!q!

Formulas based on the above theorem

1. The number of mutually distinguishable permutations of n things, taken all at a time, of which p1 are alike of one kind, p2 alike of second,…, pr alike of of rth kind such that p1+p2+…pr = n, is

n!/p1!p2!…pr!

2. The number of mutually distinguishable permutations of n tings, of which p are alike of one kind, q alike of second and remaining all are distinct is
n!/p!q!

3. suppose there are r things to be arranged, allowing repetitions. Let further p1,p2,…,pr be the integers such that the first object occurs exactly p1 times, the second occurs exactly p2 times, etc. Then the total number of permutations of these r objects to the above condition is

(p1+p2+…+pr)!/p1!p2!…pr!

Permutations when Objects can Repeat

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Theorem
The number of permutations of n different things, taken r at a time, when each may be repeated any number of times in each arrangement is n².

Circular Permutations

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If we arrange objects along a closed curve for example a circle, the permutations are known as circular permutations. In a circular permutation, we have to consider one object as fixed and the remaining are arranged as in case of linear arrangement.

Linear arrangement is arrangement in a row.

Theorem
The number of circular permutations of n distinct objects is (n-1)!.

Anti-clock wise and clockwise order of arrangements are considered as distinct permutations in the above theorem.

If the anticlockwise and clockwise order is not distinct as in case of a garland which can be turned over easily, the number of distinct permutations will be ½ (n-1)!

Combinations and Difference between Combinations and Permutations

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Each of the different selections made by taking some or all of a number of objects, irrespective of their arrangement is called a combination.

Difference between Combinations and Permutations

In a combination, the ordering of the selected objects is immaterial whereas in a permutation, the ordering is essential. For example AB and BA are same as combinations, but different as permutations.

Associate the word selection for combinations and arrangement for permuations.

Notation and Theorem for Combinations

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Notation

The number of all combinations of n objects, taken r at a time is denoted by C(n,r) or ⁿC_r.

ⁿC_r is defined when n and r are non-negative numbers.

Theorem

The number of all combinations of n distinct objects, taken r at a time is given by

ⁿC_r = n!/(n-r)!r!

Results from the theorem

ⁿC_r = [n(n-1)(n-2)...(n-r+1)]/(1.2.3...r)

ⁿC_n =1

ⁿC₀ = 1

ⁿC_r = ⁿP_r/r!

Properties of C(n,r)

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Properties of ⁿC_r or C(n,r)

1. ⁿC_r = ⁿC_n-r

Note: If x=y = n

ⁿC_x = ⁿC_y

2. Let n and r be non-negative integers such that r≤n. Then

ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r

3. Let n and r be non-negative integers such that 1≤ r≤n. Then
ⁿC_r = (n/r) ^n-1C_r-1

4. If 1≤ r≤n, then

n.^n-1C_r-1 = (n-r+1)ⁿC_r-1

5. ⁿC_x = ⁿC_y implies x+y = n

6. If n is even, then the greatest value of ⁿC_r [0≤ r≤n] is ⁿC_n/2.

7. If n is odd, then the greatest value of ⁿC_r [0≤ r≤n] is ⁿC_(n+1)/2 or ⁿC_(n-1)/2.

Selection of one or more items

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Selection from different items

The number of ways of selecting one or more items from a group of n distinct items is 2ⁿ - 1.