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
Sunday, January 17, 2016
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
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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
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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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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
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
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 = [eθ +eθ]/2
sinh θ = [eθ - 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
Saturday, January 16, 2016
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
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
Sunday, January 10, 2016
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 Ep(n) denote the exponent of the prime p in the positive integer n. Then,
Ep(n!) = Ep(1.2.3…(n-1).n)
This will be equal to Ep(p.2p.3p…[n/p]p)
= [n/p]+ Ep(1.2.3...[n/p])
This process continues and we get the answer
Ep(n!) = [n/p] + [n/p²]+…+[n/ps]
Where s is the largest positive integer such that ps≤n≤ps+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 Cr.
Then P(n,r) = n Cr = n(n-1)(n-2)…(n-(r-1))
Theorem 2
P(n,r) = n Cr = 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-1Cr-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-1Cr-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-2Cr-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
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 Ep(n) denote the exponent of the prime p in the positive integer n. Then,
Ep(n!) = Ep(1.2.3…(n-1).n)
This will be equal to Ep(p.2p.3p…[n/p]p)
= [n/p]+ Ep(1.2.3...[n/p])
This process continues and we get the answer
Ep(n!) = [n/p] + [n/p²]+…+[n/ps]
Where s is the largest positive integer such that ps≤n≤ps+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 Cr.
Then P(n,r) = n Cr = n(n-1)(n-2)…(n-(r-1))
Theorem 2
P(n,r) = n Cr = 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-1Cr-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-1Cr-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-2Cr-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
Saturday, January 9, 2016
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
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
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