XII - 11.24 Solution of a homogeneous system of linear equations - Video Lectures

Solving a Homogeneous System
NightingaleMath
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XII - 11.23 Rank method - Video Lectures

9 May


A number r is said to be the rank of a an "m x n" marix if
i) every square sub matrix of it of order (r+1) or more is singular, and
ii) there exists at least on square matrix of order r which is non-singular.

In other words, the rank of a m x n matrix is the order of the highest order non-singular square submatrix of it.


Solving 4x4 Linear Equations by Rank of Matrix Method_Detailed Step by Step Explanation
Sujoy Krishna Das
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XII - 11.22 Solution of a non-homogeneous system of linear equations - Video Lectures

9 May


Non-Homogeneous system of equation with infinite solution
Rahul Abhang
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Nonhomogeneous System Solutions
TheTrevTutor

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XII - 11.21 System of simultaneous linear equations - Video Lectures

Consistency of a System of Linear Equations
MathDoctorBob
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Consistent And Inconsistent System of Equations Example - 1 / Matrices / Maths Algebra
We Teach Academy Maths
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XI - 2.4 Some results on relations - Video Lectures

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.



Proof of Set operations in Relations -1 / NCERT Std XI Mathematics
MathsMynd
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XI - 2.5 Composition of relations - Video Lectures

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^-


Composition of relations
Math 290, GMU 
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XII - 11.20 Echelon form of a matrix - Video Lectures

Elementary Linear Algebra: Echelon Form of a Matrix, Part 1
James Hamblin
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XII - 11.19 Equivalent matrices - Video Lectures

Operations that Produce Row Equivalent Matrices
psccmath
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XI - 3.6 Function as a relation - Video Lectures

Maths Relation and Functions Part 1 (Relation function concept) Mathematics CBSE Class X1
ExamFearVideos
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XI - 3.5 Number of functions - Video Lectures

Number of functions
Gate Lectures by Ravindrababu Ravula
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XI - 3.4 Equal function - Video Lectures

Example On Equal Functions / Maths Algebra
We Teach Academy Maths
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XI - 3.2 Domain, Co-Domain and range of a function - Video Lectures

Domain, Codomain, and Range
Worldwide Center of Mathematics
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XI - 3.1 Function - Video Lectures

Functions, Lecture 4 , Maths IIT JEE ( Definition of functions)
Collegepedia.in
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XI - 3.8 Composition of functions - Video Lectures

Composition of Functions
ProfRobBob
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XI - 3.10 Inverse of an element - Video Lectures

Inverse elements for Binary operations : ExamSolutions Maths Revision
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Examsolutions

XI - 3.11 Inverse of a function - Video Lectures

Inverse Functions
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ProfRobBob

XI - 3.12 Properties of inverse of a function - Video Lectures

15 May


Properties of the Inverse Image of a Function on Sets: Practice With Proof
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XI - 2.3 Types of relations - Video Lectures

XI -

2.3 Types of relations - Video Lectures

2.3 Types of relations


Void relation
Universal relation
Identity relation
Reflexive relation
Symmetric relation
Transitive relation
Antisymmetric relation
Equivalence relation


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________________
We teach academy mathematics

XI - 2.2 Relations - Video Lectures

XI - 2.2 Relations - Video Lectures

2.2 Relation

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.

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We teach academy mathematics


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Kadas Learning

XI - 2.1 Cartesian product of sets - Video Lectures

XI - 2.1 Cartesian product of sets - Video Lectures

2.1 Cartesian product of sets

Cartesian product is an operation on sets.


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

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

Theorems

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)
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IMA

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IMA

XII - 11.14 Elementary transformations of Elementary Operations of a matrix - Video Lectures

XII -

11.14 Elementary transformations of Elementary Operations of a matrix - Video Lectures


1. Interchange of two rows or columns.
2. Multiplication of all elements of a row or column of a matrix by a non-zero scalar,
3. Addition to the elements of a row or column of the corresponding elements of any other row (to a row) or any other column (to a column) multiplied by a scalar k.


Elementary matrix: A matrix obtained from an identity matrix by a single elementary operation (transformation) is called an elementary matrix.

Elementary Operation of matrix - all three operations - Video

https://www.youtube.com/watch?v=1k7-qh3mj4k

Finding Inverse of a Matrix Using Elementary Transformations

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maths1122

XII - 11.13 Inverse of a matrix - Video Lectures

Let A be a square matrix of order n

If AB = I_n = BA

The B is inverse of A and is written as
A^-1 = B



Theorems related to Inverses of matrices

1. Every invertible matrix possesses a unique inverse

2. A square matrix is invertible iff it is nonsingular.

3. A^-1 = (1/|A|)adj A

4. Cancellation laws: Let A, B, and C be square matrices of the same order n. If A is a non-singular matrix, then

(i) AB = AC => B = C … (left cancellation law)
(ii) BA = CA => B = C … (right cancellation law)

This law is true only when |A| ≠ 0. Otherwise, there may be matrices such that AB = AC but B≠C.

5. Reversal law: If A and B are invertible matrices of the same order, then AB is invertible and

(AB) ^-1 = B^-1A^-1

6.If A,B,C are invertible matrices then
(ABC) ^-1 = C^-1B^-1A^-1

7.If A is an invertible square matrix, then A^T is also invertible and
(A^T)^-1 = (A^-1)^T

8. Let A be a non-singular square matrix of order n. Then

|adj A| = |A|^n-1

9. If A and B are non-singular square matrices of the same order, then

adj AB = (adj B) (adj A)

10. If A is an invertible square matrix, then

adj A^T = (adj A) ^T


11. If A is a non-singular square matrix, then

adj(adj A) = |A|^n-2A

Inverse of 2x2 matrix

Math Meeting


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Inverse of 3x3 matrix
Math Meeting
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Matrix Inverse Properties
__________________


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slcmath@pc

XII - 11.12 Adjoint of a matrix - Video Lectures

XII - 11.12 Adjoint of a matrix - Video Lectures



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Exam Fear Videos


Adjoint of matrix order 2X2
FreeTutorialsWorld
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Adjoint of a 3x3 matrix
Astryl

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XII - 11.11 Singular matrix - Video Lectures

XII - 11.11 Singular matrix - Video Lectures

A square matrix is a singular matrix if its determinant is zero

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KhanAcademy

XII - 11.9 Symmetric and skew symmetric matrices - Video Lectures

XII -

11.9 Symmetric and skew symmetric matrices - Video Lectures

Symmetric matrix

A square matrix is called a symmetric matrix iff a_ij = a_ji for all I,j.
It means (A)_ij = (A^T) _ij

skew symmetric matrix
A square matrix is called a skew-symmetric matrix iff a_ij = -a_ji for all I,j.
It means (A)_ij = -(A^T) _ij
It means A^T = -A


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Techtud



Problem on 11.9 Symmetric and skew symmetric matrices - Video Lectures
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Khanacademy

XI - 1.11 Some important results on number of elements in sets - Video Lectures

XI -
1.11 Some important results on number of elements in sets - Video Lectures

Finding the Number of Elements in a Set

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MATH 110

Important Results on Number of Elements on Sets

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AppuSeriesAcademy

XII - 11.8 Transpose of a matrix - Video Lectures

XII -
11.8 Transpose of a matrix - Video Lectures


Tranpose of a matrix A^T is obtained from A by changing its rows into columns and its columns into rows.
The first row of A is the first column of A^T.

Properties of Transpose

1. (A^T)^T = A
2. (A+B) ^T = A^T+B^T ( A and B must have the same order)
3. (kA) ^T = kA^T., (k is any scalar)
4. (AB) ^T = B^TA^T

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https://www.youtube.com/watch?v=uZYIZ5M2DaU
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Example Problem
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Ram Polepeddi

XII - 11.6 Subtraction of Matrices - Video Lectures

Class XII - Chapter Matrices

11.6 Subtraction of Matrices - Video Lectures


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_______________________
numericalmethodsguy

XII - 11.7 Multiplication of matrices - Video Lectures

Class XII - Chapter Matrices

11.7 Multiplication of matrices - Video Lectures
__________________


__________________
ProfRobBob

XII - 11.5 Multiplication of a matrix by a scalar - Video Lectures

Class XII - Chapter Matrices
11.5 Multiplication of a matrix by a scalar - Video Lectures

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ProfRobBob

1.9 Laws of algebra of sets - Video Lectures

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eduarrow.com

IIT JEE Mathematics Study Plan 1. Sets

R.D. Sharma, Objective Mathematics, Chapter 1
Video Lectures


1.1 Sets
1.2 Description of a set
1.3 Types of sets
1.4 Subsets
1.5 Universal set
1.6 Power set
1.7 Venn diagrams
1.8 Operations on sets
1.9 Laws of algebra of sets
1.10 More results on operations on sets
1.11 Some important results on number of elements in sets

Sets Chapter - Study Plan

(1 May to 7 May)

Day 1 ( 1 May)

1.1 Sets
1.2 Description of a set
1.3 Types of sets
1.4 Subsets
1.5 Universal set
1.6 Power set
1.7 Venn diagrams

       Video Lectures - Sets 

Day 2  (2 May)

1.8 Operations on sets

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      More Video Lectures on 1.8 Operations on Sets

1.9 Laws of algebra of sets

Day 3  (3 May)

1.10 More results on operations on sets

Day 4  (4 May)

1.11 Some important results on number of elements in sets

Day 5

Obj. Exercises 1 to 27

Day 6
Fill in the blanks 1 to 5
True/False 1 to 13

Day 7
Practice Exercises 1 to 21


For reviewing the concepts, formulas, and theorems of the chapters visit
Ch. 1. Sets - Concept Review


Updated   1 May 2016,  10 Apr 2016,  7 May 2015

IIT JEE Mathematics Study Plan 11. Matrices

11.1 Matrix
11.2 Types of matrices
11.3 Equality of matrices
11.4 Algebra of matrices
11.5 Multiplication of a matrix by a scalar (scalar multiplication)
11.6 Subtraction of matrices (definition)
11.7 Multiplication of matrices
11.8 Transpose of a matrix
11.9 Symmetric and skew symmetric matrices
11.10 Determinants
11.11 Singular matrix
11.12 Adjoint of a matrix
11.13 Inverse of a matrix
11.14 elementary transformations of elementary operations of a matrix
11.15 Orthogonal matrix
11.16 Submatrix
11.17 Rank of a matrix
11.18 Some theorems on rank of a matrix
11.19 Equivalent matrices
11.20 Echelon form of a matrix
11.21 System of simultaneous linear equations
11.22 Solution of a non-homogeneous system of linear equations
11.23 Rank method
11.24 Solution of a homogeneous system of linear equations


Study Plan


Day 1

11.1 Matrix
11.2 Types of matrices
11.3 Equality of matrices
11.4 Algebra of matrices

Day 2

11.5 Multiplication of a matrix by a scalar (scalar multiplication)
11.6 Subtraction of matrices (definition)
11.7 Multiplication of matrices

Day 3
11.8 Transpose of a matrix
Objective Types questins 1 to 6,
Practice Exercises 1 to 10

Day 4

11.9 Symmetric and skew symmetric matrices
Ex 1 to 8


Day 5

11.10 Determinants
11.11 Singular matrix
11.12 Adjoint of a matrix

Day 6

11.13 Inverse of a matrix
11.14 elementary transformations of elementary operations of a matrix

Day 7
11.15 Orthogonal matrix
11.16 Submatrix
11.17 Rank of a matrix
11.18 Some theorems on rank of a matrix

Day 8
11.19 Equivalent matrices
11.20 Echelon form of a matrix
Objective Type Exercises 8 to 20

Day 9
11.21 System of simultaneous linear equations
11.22 Solution of a non-homogeneous system of linear equations
11.23 Rank method


Day 10

11.24 Solution of a homogeneous system of linear equations
Revision of concepts in the chapter



Day 11
OTE 21 to 40

Day 12
OTE 41 to 60

Day 13
OTE 61 to 80

Day 14
OTE 81 to 91
Fill in the blanks 1 to 17

Day 15

True/false questions 1 to 30



Day 16
Practice Exercises 11 to 20

Day 17
Practice Exercises 21 to 33


Day 18
Revision - Theory, Formulas and Difficult Problems

Day 19
Revision - Theory, Formulas and Difficult Problems


Day 20
Revision - Theory, Formulas and Difficult Problems


Updated 1 May 2016,  7 Nov 2008

1.8 Operations on Sets - Video Lectures

Union of Sets
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Intersection of Sets
_________________

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Difference of Sets and Complement of a Set
_________________

_________________

Video Lectures - Sets - Chapter 1

Playlist of 15 videos on Sets Chapter 1
_______________

_______________
Examfear Videos


IIT JEE Mathematics Study Plan 1. Sets
http://iit-jee-maths.blogspot.com/2008/11/1-sets.html

Video Lectures - Sets - Chapter 1 Hindi

44 minutes video lecture
________________

________________
studyezee


IIT JEE Mathematics Study Plan 1. Sets
http://iit-jee-maths.blogspot.com/2008/11/1-sets.html

Please inform other interesting videos on the topic

Real Functions - Chapter Revision Points

Sections in the Chapter

1.Introduction

2 Description of real functions

3 Intervals (Closed and open)

4 Domains and ranges of real functions

5 Real functions - Examples

6 Operations on real functions

7 Even and odd functions

8 Extension of a function

9 Periodic function


1 Description of real functions

2 Intervals (Closed and open)

3 Domains and ranges of real functions

4 Real functions - Examples

5 Operations on real functions

6 Even and odd functions

7 Extension of a function

8 Periodic function




If the domain and co-domain of a function are subsets of R (set of all real numbers), it is called a real valued function or in short a real function.


Updated 17 Jan 2016, 2 Dec 2008

27. Definite Integrals - Revision Facilitator

Revision Facilitator


Try to recollect relevant points on the topic.

If required right click on the topic and click on open in a new window to read the relevant material.

Close the window and come back.




1 The definite integral

2 Evaluation of definite integrals

3 Geometric interpretation of definite integral

4 Evaluation of integrals by substitution

5 Properties of definite integrals

6 Integral function

7 Summation of series using definite integral as the limit of a sum

8 Gamma function

26. Indefinite Integrals - Revision Facilitator

1. Indefinite Integral - Antiderivative – Primitive

2. Integrals of some standard functions

3. Integration – Some standard formulas

a. ∫kf(x)dx =

b. ∫[f(x)± g(x)]dx =

c. d/dx [∫f(x)dx] =

4. Integration by substitution

5. Integrals of the form [f '(x)/f(x)]dx

6. Integrals of the form sin ^m x cos ^n x dx

7. Integrals of the functional form 1/(x²±a²)

8. Integrals of the form [1/(ax²+bx+c)]dx

9. Integrals of the form [1/√(ax²+bx+c)]dx

10. Integrals of the form [(px+q)/(ax²+bx+c)]dx

11. Integrals of the functional form [P(x)/(ax²+bx+c)]dx

12. Integrals of the form [(px+q)/√(ax²+bx+c)]dx

13. Integrals of the functional form [1/(a sin²x + b cos²x +c)]dx

14. Integrals of the functional form [1/(a sin x + b cos x +c)]dx

15. Integrals of the functional form [(a sin x + b cos x)/(c sin x + d cos x)]dx

16. Integrals of [(a sin x+b cos x +c)/(p sin x + q cos x +r)] dx


17. Integration by parts

18. Integral of e^x [f(x)+f'(x)]dx


19. Integrals of e^ax sinbx dx,e^ax cos bx dx

20. Integrals of √(a²±x²) and √(x²-a²)

21.Integrals of the functions of the form √(ax²+bx+c)dx

22. Integrals of the functions of the form (px+q)[√(ax²+bx+c)]dx

23. Integration of Rational Algebraic Functions by Using Partial Fractions

24. Integration of [(x²+1)/(x^4+λx²+1)]dx

25. Integration of Function [G(x)/(P√Q)]dx

24. Increasing and Decreasing Functions - Study Plan

Sections in the Chapter

1 Increasing and Decreasing Functions - Definitions

2 Necessary and sufficient conditions for monotonicity of functions

3 Properties of monotonic functions

Study Plan

Day 1

1 Increasing and Decreasing Functions - Definitions

2 Necessary and sufficient conditions for monotonicity of functions

3 Properties of monotonic functions

Day 2

Illustrative Objective Type Examples: 1 to 15

Day 3

Objective Type Exercise: 1 to 20

Day 4
O.T.E.: 21 to 38

Day 5 Fill in the blanks type exercise 1 to 8
Practice Exercise 1 to 20

Revision period

Day 6
Concept review

Day 7
Formula review

Day 8
Difficult problem review

Day 9 and 10
Test paper problems



Revision Facilitator


Try to recollect relevant points on the topic.

If required right click on the topic and click on open in a new window to read the relevant material.

Close the window and come back.


1 Increasing and Decreasing Functions - Definitions

2 Necessary and sufficient conditions for monotonicity of functions

3 Properties of monotonic functions


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Ch. 12 Determinants - 1

12.1 Definition

Every square matrix can be associated to an expression or a number which is known as determinant.

If the matrix has only one element a₁₁ then a₁₁ is the determinant.

If the matrix is of order 2 that 2 by 2 matrix

|A| =

|a11 a12|
|a21 a22| =

a11*a22 – a12*a21

Determinant of a square matrix of order 3

Determinant of a square matrix of order 3 is the sum of the product of elements a1j in the first row with (-1) ^1+j times the determinant of a 2×2 sub-matrix obtained by leaving the first row and column passing through the element.



(i) Only square matrices have determinants.
(ii) The determinant of a square matrix of order three can be expanded along any row or column.

Determinant of a square matrix of order 4 or more

(iii) Determinant of a square matrix of order 4 or more can be determined following the procedure of finding the determinant of a square matrix of order 3. But in this case, especially in the case of 4×4 matrix, when we omit the rows and columns containing the elements of a row, we get 3×3 sub-matrices and we have to find determinants for them.

12.2 Singular matrix

A square matrix is a singular matrix if its determinant is zero.
Otherwise it is a non-singular matrix.


12.3 Minors and cofactors

Minor: For a square matrix [a_ij] or order n, the minor M_ij of a_ij, in A is the determinant of the square sub-matrix of order (n-1), obtained by leaving (or striking off) i^th row and j^th column of A.

Cofactor: Cofactor of an element a_ij in a square matrix [a_ij] is termed C_ij.

C_ij = (-1) ^i+j M_ij

M_ij is the minor of element a_ij in a square matrix [a_ij].

Minors and cofactors are defined for elements of a square matrix only. They are not defined for determinants.

12.4 Properties of determinants

1. For a square matrix, the sum of the product of elements of any row (or column) with their cofactors is always equal to determinants of the matrix.

2. For a square matrix, the sum of the product of elements of any row (or column) with the cofactors of corresponding elements of some other row (or column) is zero.

3. The value of a determinant remains unchanged if its rows and columns are interchanged.

4. If any two rows (or columns) of a determinant are interchanged, then the value of the determinant changes by minus sign only.

5. If any two rows or columns of a determinant are identical then its value is zero.

6. If each element of a row (or a column) of a determinant is multiplied by a constant k, then the value of the new determinant is ‘k’ times the value of the original determinant.

7. If each element of a row (or a 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 of the same order.

8. If each element of a row (or a column) of a determinant is multiplied by the same constant and then added to the corresponding elements of some other row (column) then the value of the determinant remains same.

9. If each element of a row (or column) in a determinant is zero, then its value is zero.

10. If the matrix is a diagonal square matrix then its determinant is the product of all the diagonal elements.

11. If A and B are square matrices of the same order, then

|AB| = |A| |B|

12. If a matrix is a triangular matrix of order n, then its determinant is the product of all the diagonal elements.

12.5 Evaluation of Determinants

To evaluate determinants or large matrices, we use the properties of determinants given in section 12.4 above, to create many zeroes in the elements of a row or column and then expand the determinant using elements and cofactors of that row or column.

12.6 Evaluation of Determinants by using Factor Theorem

If f(x) is a polynomial and f(α) = 0 the, (x- α) is a factor of f(x).

If a determinant is a polynomial in x, then (x- α) is factor of the determinant if its value is zero when we put x = α.

Using this rule we can find determinant as a product of its factors.

12.7 Product of Determinants

A definition of product of determinants is similar to the rule of multiplication of matrices.




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Updated 17 Jan 2016, 7 June 2008

23. Tangents, Normals and other Applications of Derivatives - Revision Facilitator

1 Slopes of the tangent and the norm

2 Equations of tangent and normal

3 Angle of intersection of two curves

4 Lengths of tangent, normal

5.Subtangent, and subnormal

6.Rolle’s theorem

7. Lagrange’s mean value theorem

Limits - Chapter Revision Points

Sections in the Chapter

20.1 Informal approach to limit
20.2 Formal approach to limit
20.3 Evaluation of left hand and right hand limits
20.4 Difference between the value of a function at a point and the limit at a point
20.5 The algebra of limits
20.6 Evaluation of limits




We can approach a given number ‘a’ on the real line from its left hand side by increasing numbers which are less than ‘a’. It means starting from a- δ and increasing to reach a.

We can also approach a given number ‘a’ on the real line from its right hand side by decreasing numbers which are greater than ‘a’. It means starting from a+δ and decreasing to reach a.

Hence there are two types of limits – left hand limit and right hand limit.

For some functions both these limits are equal at a point and for some functions they are not equal.

If both are equal we say lim (x→a) f(x) exists. Otherwise it does not exist.


Updated 17 Jan 2016, 2 Dec 2008

Circle - Chapter Revision Points

Sections in the chapter

1. Definition
2. Standard equation of a circle
3. Some particular cases of the central form of the equation of a circle
4. General equation of a circle
5. Equation of a circle when the co-ordinates of end points of a diameter are given
6. Intercepts on the axis
7. Position of a point with respect to a circle
8. Equation of a circle in parametric form
9. Intersection of a straight line and a circle
10. The length of the intercept cut off from a line by a circle
11. Tangent to a circle at a given point.
12. Normal to a circle at a given point
13. Length of the tangent from a point to a circle
14. Pair of tangents drawn from a point to a given circle
15. Combined equation of pair of tangents
16. Director circle and its equation
17. Chord of contact of tangents
18. Pole and polar
19. Equation of the chord bisected at a given point
20. Diameter of a circle
21. Common tangents to two circles
22. Common chord of two circles
23. Angle of intersection of two curves and the condition of orthogonality of two circles.
24. Radical axis
25. Equation of a circle through the intersection of a circle and line
26. Circle through the intersection of two circles
27. Coaxial system of circles

Revision Points

Equation of circle in various forms

a. Centre (h,k) and radius a

b. Centre (h,k) and passing through origin

c. Centre (h,k) and circle touches the axis of x

d. Centre (h,k) and circle touches the axis of y

e. Centre (h,k) and and circle touches both axes.

f. general equation

g. circle whose diameter is the line joining two points

h. Circle through three given points

i. Parametric equation of the circle

h. equation of a circle that touches given circle at a given point

j. Equation of a circle passing through the intersection of given circles S1 = 0 and S2 = 0






The standard equation of a circle with center C(h,k) and radius r is as follows:

(x - h)² + (y - k)² = r²





Parametric Equations
The equation of a circle, centred at the origin, is: x2 + y2 = a2, where a is the radius.

Suppose we have a curve which is described by the following two equations:

x = acosθ (1)
y = asinθ (2)

We can eliminate q by squaring and adding the two equations:

x² + y² = a²cos²θ + a²sin²θ = a² .

Hence equations (1) and (2) together also represent a circle centred at the origin with radius a and are known as the parametric equations of the circle. θ is known as the parameter. As θ varies between 0 and 2π, x and y vary.

http://www.mathsrevision.net/alevel/pages.php?page=97




Updated  17 Jan 2016, 28 April 2008

Ch. 18 Hyperbola - Chapter Revision Points

Hyperbola - Sections in the R D Sharma Chapter

1. Hyperbola - Definitions

2. Equation of hyperbola in its standard form

3. Second focus and second directrix of the hyperbola

4. Vertices, major and minor axes, foci, directrices and centre of the hyperbola

5. Eccentricity

6. Length of latus rectum

7. Focal distances of a point

8. Conjugate hyperbola

9 .Parametric equations and parametric coordinates

10. Equation of the chord joining any two points on a hyperbola

11. Intersection of a line and a hyperbola

12. Condition of a line to be a tangent to a hyperbola

13. Equation of tangent in different forms

14. Number of tangents drawn from a point to a hyperbola

15. Equation of the pair of tangents from a point to a hyperbola

16. Equation of the chord of contacts of tangents


17. Equation of normal in different forms

18 Number of normals

19. Equation of the chord of a hyperbola bisected at a given point

20 Asymptotes of a hyperbola

21. Rectangular hyperbola

Revision Points

1. Introduction




2. Equation of hyperbola in its standard form

x² /a² - y² /b² = 1

The hyperbola intersects x axis at (a,0) and (-a,0).
The hyperbola does not intersect teh y axis.

x² /a² = 1 +y²/b²
Hence x² /a²≥1
x² ≥a²
x≥a or x≤-a
Hence hyperbola lies on the rightof the line x=a and on the left of the lie x = -a.
the hyperbola consists of two separate branches

Hyperbola is symmetric about both axes.

b² = a²(e²-1)
Where e = eccentricity

Focus is (ae,0)
Directrix is the line x = a/e

Length of latus rectum = 2b²/a







3. Second focus and second directrix of the hyperbola




4. Vertices, major and minor axes, foci, directrices and centre of the hyperbola

At vertices, the curve meets the line joining foci. Vertices for the hyperbola
x² /a² - y² /b² = 1 are (a,0) and (-a,0).

the straight line joining the vertices is called the transverse axis of the hyperbola. Its length is 2a.




5. Eccentricity




6. Length of latus rectum




7. Focal distances of a point




8. Conjugate hyperbola

For the hyperbola x²/a² - y²/b² = 1, the congugate hyperbola is
-x²/a² + y²/b² = 1




9 .Parametric equations and parametric coordinates

x - a sec θ and y = b tan θ are the parametric coordinates

x = a cosh θ and y = b sinh θ are also parametric coordinates.

cosh θ = [e^θ +e^θ]/2
sinh θ = [e^θ - ^θ]/2




10. Equation of the chord joining any two points on a hyperbola

The points are taken as P(a sec θ1,b tan θ1), Q(a sec θ2,b tan θ2)

The equation of the chord joining P and Q is

y -b tan θ1 = [(b tan θ2 - b tan θ1)/(a sec θ2 - a sec θ1)]*(x-a sec θ1)






11. Intersection of a line and a hyperbola



12. Condition of a line to be a tangent to a hyperbola




13. Equation of tangent in different forms





14. Number of tangents drawn from a point to a hyperbola




15. Equation of the pair of tangents from a point to a hyperbola





16. Equation of the chord of contacts of tangents





17. Equation of normal in different forms





18 Number of normals





19. Equation of the chord of a hyperbola bisected at a given point





20 Asymptotes of a hyperbola






21. Rectangular hyperbola


Updated 17 Jan 2016, 5 June 2008

JEE Mathematics 13. Cartesian System of Coordinates and Straight Lines - Revision Points and Facilitator

Topics of the chapter

1. Introduction

2. Catesian coordinate system

3. Distance between two points

4. Area of a triangle

5. Section Formulae

6. coordinates of the centroid, in-centre, and ex centre of a triangle

7. Locus and equation to a locus

8. Shifting of origin

9. Rotation of axes

10. Definition of a straight line

11. Slope (Gradient) of a straight line

12. Angle between two straight lines

13. Intercepts of a line on the axes

14. Equations of lines parallel to the coordinate axes

15. Different forms of the equation of a straight line

16. Transformation of general equation in different standard forms

17. point of intersection of two lines

18. Coordination of concurrency of three lines

19. Lines parallel and perpendicular to a given line

20. Angle between two straight lines when their equations are given.

21. Distance of a point from a line

22. Positions of points relative to a line

23. Equations of straight lines passing through a given point and making a given angle with a given line

24. Equations of the bisectors of the angles between two straight lines

25. Some important points of a triangle

26. Family of lines through the intersection of two given lines


Revision Points


Euclidean Geometry and Analytic Geometry - Difference

Distance between two points

Area of a triangle

Section Formulae

coordinates of the centres related to triangle

Definition of a straight line

Slope (Gradient) of a straight line

Angle between two straight lines

Intercepts of a line on the axes

Joint equation of a pair of straight lines

Ch. 8 Permutations and Combinations Revision Points - 1

1. Factorial

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)!


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

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

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.


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 ⁿ C_r.

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


Theorem 2

P(n,r) = ⁿ C_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


8.5 Permutations under certain conditions
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

Revision Points (Text R.D. Sharma)

8.6 Permutations of Objects not all Distinct

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!


8.7 Permutations when Objects can Repeat
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 .

8.8 Circular Permutations
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)!..



_{Updated  10 Jan 2016, 7 June 2008}

Ch. 7. Quadratic Equations and Expressions - Revision Points 1

Some definitions

Real polynomial

Complex polynomial

Degree of a polynomial

Polynomial equation

Some Results on roots of an equation


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

3. Position of roots of a polynomial equation
4 Descartes rule of signs
5. Relations between roots and coefficients
6. Formation of a polynomial equation form given roots
7. Transformation of equations




8. Roots of a quadratic equation with real coefficients

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 .="" br="">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.




9. Quadratic expression and its graph

Graph of a quadratic expression is a parabola.

10. Sign of a quadratic expression for real values of the variable
11. Solution of inequations
12. Position of roots of a quadratic equation
13. Common roots
14. Values of a rational expression P(x)/Q(x) for real values of x, where P(x) and Q(x) are quadratic expressions
15. Condition for resolution into linear factors of a quadratic function
16. Algebraic interpretation of Rolle’s theorem




Updated 9 Jan 2016, 7 June 2008