Saturday, April 9, 2016
Video Lectures - Sets - Chapter 1
Playlist of 15 videos on Sets Chapter 1
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IIT JEE Mathematics Study Plan 1. Sets
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Video Lectures - Sets - Chapter 1 Hindi
44 minutes video lecture
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Sunday, January 17, 2016
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
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
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If required right click on the topic and click on open in a new window to read the relevant material.
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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
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If required right click on the topic and click on open in a new window to read the relevant material.
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1 Increasing and Decreasing Functions - Definitions
2 Necessary and sufficient conditions for monotonicity of functions
3 Properties of monotonic functions
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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 [aij] or order n, the minor Mij of aij, in A is the determinant of the square sub-matrix of order (n-1), obtained by leaving (or striking off) ith row and jth column of A.
Cofactor: Cofactor of an element aij in a square matrix [aij] is termed Cij.
Cij = (-1) i+j Mij
Mij is the minor of element aij in a square matrix [aij].
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
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 [aij] or order n, the minor Mij of aij, in A is the determinant of the square sub-matrix of order (n-1), obtained by leaving (or striking off) ith row and jth column of A.
Cofactor: Cofactor of an element aij in a square matrix [aij] is termed Cij.
Cij = (-1) i+j Mij
Mij is the minor of element aij in a square matrix [aij].
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
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