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jnvLudhiana
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SIMPLIFIED
MATERIAL
FORM
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Relations and Functions
Basic Concepts and Formulae:
Relations:
A relation in set A is a subset of A X A. We also write it as R = {(a, b) ∈ AXA / aRb}
• Type of Relation :
I.
Reflexive Relation: A relation R in a set X is called reflexive if
( x, x ) ∈ R ,
II.
∀ x∈ X.
Symmetric Relation: A relation R in a set X is called symmetric if
( x, y ) ∈ R ⇒ ( y , x ) ∈ R ,
III.
∀ x, y ∈ X .
Transitive Relation: A relation R in a set X is called transitive if
(x, y )∈ R and ( y, z )∈ R ⇒ (x, z )∈ R,
∀ x, y , z ∈ X .
Equivalence relation: A relation which reflexive, symmetric and transitive is
called equivalence relation.
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• Function:
Let X and Y are two non empty sets. The function
is a rule or formula which associate each element
x ∈ X
f
from X into Y
with unique element y ∈ Y .
Mathematically f : X → Y f (x ) = y.
Here X is called domain of
f
and Y is called co domain of f . The set of images of
of all the elements of X is called Range of f .
• Types of Functions
1. One – One (Injective) Function: A function f : X → Y is said to be
One-One if distinct elements of X has distinct images of Y.
Mathematically, for ∀ x1 , x 2 ∈ X , f ( x1 ) = f (x 2 ) ⇒ x1 = x 2 .
2. Onto (Surjective) Function: A function f : X → Y is said to be Onto if
every element of Y is the image of at least one element of X.
Mathematically, for y ∈ Y ∃ x ∈ X such that f (x ) = y. Note that for onto
function co domain is equal to the range set.
3. One to one (bijective) function: A function which is both one-one and
onto is called one to one function.
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• Composite function: Let f : X → Y and g : Y → Z are two functions. The
composite of f and g , written as fog is defined as the function
that
fog : X → Z
such
fog ( y ) = f (g ( y )), ∀ y ∈ Y .
Note that fog and gof are two different functions.
• Invertible function: A f : X → Y function is said to be invertible if it satisfy the
following equivalent conditions:
1. f must be one to one function.
2.
∃ g :Y → X
such that gof = I x and fog = I y .
The function g is called the inverse of
f
and is denoted by f −1 .
• Binary Operations: Let A be a non empty set. Then binary operation * on A is
a function from A × A to A. Mathematically * ( x . y ) = x * y .
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• Properties of binary operations: Let
1. * is associative if
x, y, z ∈ X .
Then
x * ( y * z ) = (x * y )* z .
2. * is commutative if
x * y = y * x.
3.
e ∈ X
is called identity element of * , if
4.
i∈ X
is called inverse element of x w. r. t. * , if
x * e = x = e * x.
x * i = e = i * x.
INVERSE TRIGONOMETRIC FUNCTIONS
Basic Concepts and Formulae:
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• Domain and Range of trigonometric functions:
FUNCTION
sin x
cos x
tan x
cot x
sec x
cos ecx
DOMAIN
R
R
RANGE
π


R −  x : x = (2n + 1) , n ∈ Z 
2


R − {x : x = n π , n ∈ Z }
π


R −  x : x = (2n + 1) , n ∈ Z 
2


R − {x : x = n π , n ∈ Z }
R
[− 1, 1]
[− 1, 1]
R
R − (− 1 , 1 )
R − (− 1 , 1 )
• Domain, Range and Principal value branch for inverse trigonometric
functions:
FUNCTION
DOMAIN
x
[− 1, 1]
−1
x
[− 1, 1]
tan
−1
x
R
cot
−1
x
R
sec
−1
x
R − (− 1 , 1 )
sin
−1
cos
Range(PRINCIPAL VALUE BRANCH)
 π π
− 2 , 2 


[0 , π ]
 π π
− , 
 2 2
(0 , π )
[0, π ] − π
2
cos ec
−1
x
R − (− 1 , 1 )
 π π
− 2 , 2 


Properties And Formulas of inverse trigonometrical
function
Set ‘A’
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Sin −1 (sin x ) = x , x ∈ ( −
cos
−1
(cos x ) = x
−1
(tan x ) = x
cot
−1
(cot x ) = x
sec
−1
(sec x ) = x
tan
cos ce
−1
Π Π
, )
2 2
.
(cos ecx ) = x
Set ‘B’
sin
−1
1
  = cos ec x , x ≥ 1orx ≤ 1
x
−1
1
cos −1   = sec −1 x , x ≥ 1orx ≤ 1
x
1
tan −1   = cot −1 x x>0
x
1
cos ec −1   = sin −1 x
x
,
sec
1
−1
  = cos x
x
 
−1
1
cot −1   = tan −1 x
x
Set ‘C’
sin
cos
−1
(− x ) =
−1
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− sin
(− x ) = π
−1
x
− cos
−1
x
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tan
cos
−1
(− x ) =
ec
−1
− tan
(− x ) =
−1
x
− cos ec
−1
sec
−1
(− x ) = π
− sec
−1
x
cot
−1
(− x ) = π
− cot
−1
x
x
Set ‘D’
π
sin −1 x + cos −1 x =
sec −1 x + cos ec −1 x =
tan −1 x + cot −1 x =
2
π
2
π
2
Set ‘E’
tan −1 x + tan −1 y = tan −1 x + y
1 − xy
tan
−1
x − tan
−1
y = tan
−1
x− y
1 + xy
if xy <1
if xy>-1
2
2x
−1  2 x 
−1  1 − x 
−1


sin
=
cos
=
tan
= 2 tan −1x
 2
2
2


1− x
1+ x 
1 + x 
Set ‘F’
o
sin
−1
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x + sin
−1
y = sin
−1
(x
1− y2 + y 1− x2
)
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(
o
sin −1 x − sin −1 y = sin −1 x 1 − y 2 − y 1 − x 2
o
cos −1 x + cos −1 y = cos −1 xy − 1 − y 2 1 − x 2
o
cos −1 x − cos −1 y = cos −1 xy + 1 − y 2 1 − x 2
)
(
)
(
)
While writing inverse trigonometrical functions in simplest form, we can use
the following substitution
for a 2 − x 2 ; we substitute x=a sinө or x=a cosө
for a 2 + x 2 ; we substitute x=a tanө or x=a cotө
for x 2 − a 2 ; we substitute x=a secө or x=a cosecө
Matrices
Basic Concepts and Formulae:
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1. Matrix : A set of mn numbers arranged in the form of a rectangular array of m rows and
n columns is called an m × n matrix.
2. Elements of a Matrix : Each of the mn entries of m × n matrix is called an element or an
entry of the matrix.
3. To define a matrix ,we must define its order and its elements.
4. Row matrix : A matrix having only one row is called a row matrix.
Hence, Row matrix is of order 1 × n .
5.
Column Matrix: A Matrix having only one column is called a column matrix. Hence ,
Column matrix is of order m × 1.
6. Square Matrix : A matrix in which the number of rows is equal to the number of
columns is called a square matrix.
[ ]
7. Diagonal Matrix : A square matrix A = aij
m×n
is called a diagonal matrix if all the
elements except those in the leading diagonal , are zero i.e. aij = 0 , i
[ ]
8. Scalar Matrix : Square matrix A = aij
(ii) aij = C , C
≠ 0
m×n
≠ j
.
is called a scalar matrix if (i) aij = 0 , i
≠ j
.and
.
9. Null Matrix : A matrix whose all elements are zero is called a null matrix or Zero matrix.
[ ]
10. Upper Triangular Matrix : A square matrix A= aij is called an upper triangular matrix if
aij=0 for all i>j.
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[ ]
11. Lower Triangular Matrix : A square matrix A= aij is called an lower triangular matrix
if aij=0 for all i<j.
[ ]
12. Equality of Matrices : Two matrices A = aij
m×n
[ ]
and B = bij
r×s
are equal if
(i) m=r , the number of rows in A equatls to number of rows in B.
(ii) n=s , the number of columns in A equals to number of columns in B
(iii)
aij = bij , for all i = 1,2,3,….. and
1,2,3,…..
j=
13. Addition of Matrices: Let A , B be two matrices , each of order m × n . Then, their sum A
+ B is a matrix of order m × n and is obtained by adding the corresponding elements of A
and B.
14. Commutative : If A and B are two m × n matrices , then A+B=B+A i.e. matrix addition
is commutative.
15. Associative : If A ,B,C are three matrices of the same order , then (A+B)+C=A+(B+C)
16. Existence of Identity : The null matrix is the identity element for matrix addition i.e.
A+O = A = O+A
[ ]
17. Existence of Inverse : For every matrix A = aij
m×n
[
,there exist a matrix A = − aij
]
m× n
,
denoted by –A such that A+(-A) =O =(-A) +A
[ ]
18. Multiplication of a Matrix by a Scalar : Let A = aij
m×n
be an m × n matrix and k be any
number called a scalar .Then the matrix obtained by multiplying every element of A by k
is called the scalar multiple of A by k and is denoted by kA.
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[ ]
19. Multiplication of Matrices : Let A= aij
m×n
[ ]
and B= a jk
n× p
be two matrices , given in such
way that the number of columns in A is equal to the number of rows in B . Then their
product AB is an m × p matrix , given by
AB = [cik ]m× p where cik = ai1b1k + ai 2 b2 k + ..... + ain bnk
20. If A ,B, C be three matrices of order m × n , n × p, p × q respectively , then (AB)C=A(BC).
21. If A ,B, C be three matrices of order m × n , n × p, n × p respectively , then AB+AC
=A(B+C)
22. Integral power of Matrices exists only for Square matrices.
23. If A is any square matrix , we write
A2 for A × A.
24. Transpose of a Matrix : Let A be an m × n . Then the n × m matrix, obtained by
interchanging the rows and columns of A , is called the transpose of A and is denoted by
A’. Thus
25.
(i)
if the order of A is m × n , then order of A’ os n × m.
(ii)
(i, j) th element of A = ( j, i )th element of A’.
(A')'= A
26. (A+B)’=A’+B’
27. (kB)’ = k B’
28. (AB)’=B’A’
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29. A square matrix A = [ aij ] is said to be a symmetric matrix iff A’ = A i.e. aij = aji
∀ i,j.
30. A square matrix A = [ aij ] is said to be a skew symmetric matrix iff A’ = -A i.e. aij =- aji
∀ i,j.
31. Invertible Matrices : If A is a square matrix of order n and if there exists another square
matrix of the same order n such that AB= BA = I ,then B is called the inverse of matrix A
and denoted by
A−1
32. Inverse of Matrix , if it exists , can be found by using either row or column operations
but not both simultaneous.
33. If after doing one or more than one elementary operation, we obtain all 0’s in one or
more rows of the given matrix, then
A−1 does not exist.
Commit to Memory
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1. A square matrix A = [ aij ] is said to be a symmetric matrix iff A’ = A i.e. aij = aji
∀ i,j.
2. A square matrix A = [ aij ] is said to be a skew symmetric matrix iff A’ = -A i.e. aij =- aji
∀ i,j.
3. (kB)’ = k B’
4. (AB)’=B’A’
5. ( AB ) −1 = B −1 A −1
6. ( A −1 )' = ( A' ) −1
7.
( A n ) −1 = ( A −1 ) n
8. If A and B are invertible matrices of same order , then ( AB ) −1 = B −1 A −1
9. If A and B are invertible matrices of same order , then adj(AB)=(adjA)(adjB)
10. ( A T ) −1 = ( A −1 ) T
11. Inverse of Matrix , if it exists , can be found by using either row or column operations but
not both simultaneous .
12. If after doing one or more than one elementary operation, we obtain all 0’s in one or more
rows of the given matrix, then
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A−1 does not exist.
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Determinant
Basic Concepts and Formulae
a b 
1. Determinant of a square matrix of order two : Let 
 be a square matrix of order 2 ,
c d 
a b
then det(A) = A =
= ad-bc
c d
a
2. Determinant of a square matrix of order three : Let  d

 g
a b c
e f
d f
d
3 , then det(A) = A = d e f = a
-b
+c
h i
g i
g
g h i
b
e
h
c
f  be a square matrix of order
i 
e
h
3. Area of triangle : Area of Triangle ABC whose vertices are A ( x 1 , y 1 ) , B ( x 2 , y 2 ) and
x1 y1 1
1
C ( x3 , y3 ) is given by Area of triangle = x 2 y 2 1
2
x3 y 3 1
4. A Matrix A is said to be a singular matrix if A = 0
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5. Adjoint of a Matrix : The adjoint of a matrix A is the transpose of a matrix of cofactors of
elements of matrix A.
[ ]
6. Inverse of a square matrix : Let A = aij be asqaure matrix of oreder n . Then inverse of
amatrix A is defined as
A
−1
and is given by
A
−1
=
1
adj( A)
A
Commit to Memory
1. Expanding a determinant along any row or column gives the same value.
2. A Matrix A is said to be a singular matrix if A = 0
3. A(adjA)=(adjA)A= A I
4. For matrices /a and /b of same order AB = A B
5. If A is a square matrix of order n , then adjA = A
n −1
6. If A is a square matrix of order n , then kA = k n A
7. If A is an invertible matrix then (adjA)’ = adj(A’)
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8. adj(adjA)= A
9.
A−1 =
n−2
A
1
A
10. Since area is a positive quantity , so always take the absolute value of determinant.
11. A System of equations is said to be consistent if its solution exists.
CONTINUOUTY AND
DIFFERENTIABILITY
Basic Concepts and Formulae:
1. Let f be a real valued function on the subset of real numbers and let a be any point in the
domain of f. Then f is said to be a continous at x = a if
Lt x → a f ( x ) = f ( a )
Or Lt x → a + f ( x ) = f ( a ) = Lt x → a − f ( x )
2. A function is discontinous at x = a if
(i) f (a ) does not exist
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(ii) Lt x→ a f (x ) does not exist
(iii) Both exist but are not equal.
3. A function is said to be continuous if it is continous at every point in its domain.
4. If f and g are two continuous functions at a point a then
(i)f + g is continuous at a
(ii) f - g is continuous at a
(iii) f . g is continuous at a
(iv)
f
is continuous at a
g
(v) c. f is continuous at a, where c is a constant.
5. For a function y = f ( x ) ,
dy
represent the instantaneous rate of change of y with respect to x.
dx
dy
f ( x + h) − f ( x)
= Lt h → 0
dx
h
where h is a very small increment in h.
6. List of some useful formulae
FARMULAE:
Derivatives
(1)
d  x n+1 

 = x n
dx  n + 1 
(2)
d
(sin x ) = cos x
dx
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(3)
d
(cos x ) = − sin x
dx
(4)
d
(tan x ) = sec 2 x
dx
(5)
d
(cot x ) = − cos ec 2 x
dx
(6)
d
(sec x ) = sec x + tan x
dx
(7)
d
(cos ecx ) = − cos ecx cot x
dx
(8)
d
1
(sin −1 x) =
dx
1− x2
(9)
d
1
(cos −1 x) = −
dx
1− x2
(10)
d
1
(tan −1 x ) =
dx
1+ x2
(11)
d
1
(cot −1 ) = −
dx
1+ x2
(12)
d
1
(sec −1 x) =
dx
x x2 −1
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(13)
d
1
(cos ec −1 x) = −
dx
x x2 −1
(14)
d x
e = ex
dx
(15)
d
1
(log x ) =
dx
x
7. Product Rule – Let u and v be two functions of x, then
d (u.v )
dv
du
=u
+v
dx
dx
dx
8. Quotient Rule – If u and v be two functions of x , then
d u
 =
dx  v 
v
du
dv
−u
dx
dx
v2
9. Every differential function is continuous but converse may not be true.
10.Logarithmic Differentiation
Logarithmic differentiation are used for differentiation of functions which consists of the
product or quotients of a number of functions or the given function is of type [ f ( x)]
g ( x)
. In this
method we take the logarithm on both the sides of the function and then differentiate it with
respect to x
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11.Parametric form : Sometimes we come across the function when both s and y are expressed in
dy
terms of another variable say t, i.e. x = f (t ) and y = g (t ) .This form is parametric and
is found
dx
dy dy dx
by applying the formula
=
÷
dx dt dt
12. Rolle’s Theorem : If f(x) be a real valued function defined in a closed interval [a , b ] such
that
i) It is continuous in closed interval [a , b ]
ii) It is differentiable in open interval ( a , b )
iii) f ( a ) = f ( b )
then there exist at least one real value c ∈ ( a , b ) such that f ' ( c ) = 0
13. Lagrange’s Mean Value Theorem : If f(x) be a real valued function defined in a closed
interval [a , b ] such that
i) It is continuous in closed interval [a , b ]
ii) It is differentiable in open interval ( a , b )
then there exist at least one real value c ∈ ( a , b ) such that f ' ( c ) =
f (b ) − f ( a )
b−a
APPLICATION OF DERIVATIVES
Basic Concepts and Formulae
1. For the curve y = f ( x ) , dy represent the slope of the tangent to the curve y = f ( x ) .At any
dx
dy
point ( x 1 , y 1 ) it is represented by  
.
 dx  ( x1 , y1 )
2. If dy = 0 ,then tangent is parallel to x axis and vice versa.
dx
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3. If dy is not defined then tangent is parallel to y axis.
dx
4. The equation of tangent to the curve y = f ( x ) at
( x1 , y1 )
is given by
( x1 , y1 )
is given by
 dy 
y − y1 =  
( x − x1 )
 dx  ( x1 , y1 )
5. The equation of normal to the curve y = f ( x ) at
y − y1 = −
1
( x − x1 )
 dy 
 
 dx  ( x1 , y1 )
6. A function is said to be increasing in an interval ( a,b ) if f ' ( x ) > 0
7. A function is said to be decreasing in an interval ( a,b ) if f ' ( x ) < 0
8. If ‘f’ be a function defined in the closed interval I and there exist a point ‘a’ in the interval I
such that f ( a ) ≥ f ( x ) for all x ∈ I .Then a function is said to attain absolute maximum at x
= a and f (a ) is absolute maximum value.
9. If ‘f’ be a function defined in the closed interval I and there exist a point ‘a’ in the interval I
such that f ( a ) ≤ f ( x ) for all x ∈ I .Then a function is said to attain absolute minimum at x =
a and f (a ) is absolute maximum value.
10.To find absolute maximum value and absolute minimum value for function f(x)
Defined in [a , b]
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i)Find solution of f ' ( x ) = 0 .Let these are x 1 , x2 , x3,…….
ii)Find f ( a ), f ( x 1 ), f ( x 2 )......... ....., f ( b )
iii)Maximum and minimum values among f ( a ), f ( x 1 ), f ( x 2 )......... ....., f ( b ) are
the absolute maximum and absolute minimum values and corresponding points are the
points of absolute maxima and absolute minima.
11) For any function f ( x ) , all such points where f ' ( x ) = 0 are called the turning points.
12) To get a point of maxima for any function f ( x ) we must have f ' ( x ) = 0 and f ' ' ( x ) < 0 .In
the same way for getting a point of minima for any function f ( x ) we must have
f ' ( x ) = 0 and f ' ' ( x ) > 0 .
Integrals and its applications
Basic Concepts and Formulae:
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1.Integration:- Integration is the process of finding the function
whose derivatives is given.
2.Indefinite Integral:- Indefinite means not unique. Actually
there exists infinitely many integrals of a function, which can be
obtained by choosing arbitrary the value of C from the real
numbers.
3.Methods of Integrations:-
(a)Integration by substitution. (Used to integrate the function by suitable substitution)
(b) Integration by Parts.(Used to integrate the product of two functions. The first function is
taken as per order of ILATE )
( c ) Integration by Partial Fraction. (Used to integrate the rational algebraic functions)
4.Important Therems:-
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(a) The Indefinite Integral of an algebraic sum of two or more functions is equal to the algebraic
sum of their integrals
i.e.
dx =
dx +
dx
(b) A Constant term may be taken outside from the integral sign. i.e
dx =
dx where k is a constant.
( c ) If the numerator in an integral is the exact derivative of denominator then its integral is
logarithmic of denominator. i.e
dx =
(d)
+c
dx =
+c
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( e) Integral of the functions whose numerator is unity and denominator is a homogeneous
function of First degree can be found easily by substitution of
sin 2x =
, cos 2x =
and tan x = t
(f) Integral of the functions whose numerator is unity and denominator is a homogeneous function
of Second degree in sin x , cos x or both can be found easily by Dividing both numerator and
dnominator by
and then substituting tan x = t.
DIFFERENTIAL EQUATION
Basic Concepts and Formulae:
Definition:
A differential equation is an equation which involves unknown function and their
derivatives w.r.t one or more independent variables. For a given function g, find
a function f such that
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dy
= g (x)
dx
where y= f(x)
……………..i
An equation of the form (1) is known as a differential equation.
001111
d2y
dy
+ 9y = 0
−6
2
dx
dx
……..(4)
Note :We use following notation
2
3
dy
= y ′ d y = y ′′ d y = y ′′′
dx
dx 2
dx 3
etc.
Order of differential equation
The order of the differential equation is the order of the highest order derivatives
occurring in the differential equation
d2y
dy
−6
+ 9y = 0
2
dx
dx
d ′ ′′y
d ′′y
−7
+ 11 y = 0
d x ′′′
d x ′′
Here The order of the differential equation first is 2 and The order of the differential
equation second is 3.
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Degree of differential equation
The Degree of the differential equation is the index of the highest order derivatives which
appears in the differential equation after making it free from negative and fractional
power.
d2y
dy
−6
+ 9y = 0
2
dx
dx
d ′ ′′y
d ′′y
−7
+ 11 y = 0
d x ′′′
d x ′′
Here The degree of the differential equation first is 1 and The order of the differential
equation second is 1.
General and particular solution of the differential equation
Formulate of differential equation
1.
Order of the differential equation is equal to the
number of independent arbitrary constants in the equation.
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2.
Satisfied by the given equation.
3.
Free from arbitrary constants.
Equation with Variable Separable
An equation whose variable are separable can be put into the form
f
1
( x ) dx
+
f
2
( x ) dy
=
0
Integrating the general soluation
∫
f 1 ( x ) dx +
∫
f 2 ( x ) dy = C
where C is arbitrary constant.
Homogeneous Differential Equation
Definition :
A differential equation is an equation which involves unknown faction and their of
the form dy = f ( x , y )
dx
g ( x , y )
Where f(x,y) and g(x,y) are homogeneous fuction
Like
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dy x 3 − 3x 2 y
=
dx 3xy 2 − y 3
(x
3
)
(
)
− 3 xy 2 dx = y 3 − 3 x 2 y dy
dy
3xy
=
dx 3xy − y 2
To solve homogeneous function
1. Put y= vx
2. Find derivatives dy = v + x dv
dx
dx
3. Substituting value of y and
dy
dx
in the given equation
4. After that it will be reducible to variable separable and we know how
to solve it in the previous method
Linear Differential Equation (of first
order)
Definition:
A Linear differential equation is an equation which is form
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dy
dx
+
Py
=
Q
Where P& Q are fuction of x only
Like
dy 2
+ y = 3x
dx x
dy cos x
+
y = 3 x cos x
dx sin x
cos x 2
dy
+ y cos x = sin x
dx
To solve homogeneous equation
1. Make the equation in the form
dy
dx
+
Py
=
Q
2. Find the value of P & Q
3. Find integral factor i.e
I .F . = e ∫
Pdx
4. FIND THE SOLUTION
pdx
Pdx
∫
∫
ye = ∫Qe dx+c
VECTORS
Basic Concepts and Formulae:
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•
Section Formulae for vectors: The position vector of a
point C dividing a ρline segment joining the points A and B whose position
vectors are aρ and b respectively
in the ratio m:n
ρ
ρ
na + mb
.
1} internally is given by
m+n
ρ ρ
mb − na
2} externally is given by
.
m−n
Dot Product between vectors:
•
ρ
1} If aρ and b are any two vectors and θ ( 0≤θ≤π ) is the angle between
ρ ρ ρρ
them, a ⋅ b = a b cos θ .
ρ
)
ρ ρ
ρ
)
)
2} Let a = a1i + a 2 j + a 3 k and b = b1i + b2 j + b3 k then a ⋅ b = a1b1 + a 2 b2 + a 3 b3
3} If
ρ
a
2
ρ
and b are any two vectors and θ is the angle between them then
ρ ρ
a ⋅b
cos θ = ρ ρ .
ab
4} Two vectors
ρ
a
ρ
ρ ρ
and b are perpendicular then a ⋅ b = 0 .
• Projectionρof a vector:
Let aρ and b are any vector. Then Projection of
ρ aρ ⋅ bρ
b is ρ .
b
•
ρ
a
in the direction of
Cross Product between vectors:ρ
1} If θ is the angle between two vectors aρ and b then their cross product
is
ρ ρ ρρ
)
a × b = a b sin θ n , Where
Containing
ρ
a
ρ
)
n
is a unit vector perpendicular to the plane
ρ
and b such that aρ , b and
)
n
form right handed system
Of coordinate axes.
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i
j
b1
a2
b2
ρ
)
)
)
ρ ρ
ρ
2} Let a = a1i + a 2 j + a 3 k and b = b1i + b2 j + b3 k , then a × b = a1
k
a3 .
b3
3} Note that the cross product of two vectors is a vector perpendicular to
both
of them.
4} Relation between
ρ
a
ρ
ρ
a ⋅ b
ρ2
ρ2
ρ ρ
and a × b : we have aρ × b = a 2 b 2 − aρ ⋅ b .
5}. Area of a parallelogram whose adjacent sides are represented by
ρ
and b is
ρ ρ
axb .
ρ
6} Area of a parallelogram whose diagonals are represented by aρ and b is
1 ρ ρ
a xb .
2
ρ ρ
a
xb
ρ
7} Area of a triangle whose two sides are represented by aρ and b is
.
2
ρρ
ρ
axb
8} ρ ρ is a unit vector which is perpendicular to both aρ and b .
axb
9} Two vectors
ρ
a
ρ
ρ ρ
and b are parallel then axb = 0 .
ρ
• Scalar triple product: If aρ , b and
ρ
c
are any three vectors then scalar
a1 a2
ρ ρ ρ
ρ ρ ρ
triple product denoted as a ⋅ (b × c ) and a • (b × c ) = b1 b2
c1
c2
a3
b3 .
c3
THREE DIMENSIONAL GEOMETRY
Basic Concepts and Formulae:
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The direction cosines of line with direction ratios a, b, c are
•


a
b
c

.
,
,
 2
2
2
2
2
2
2
2
2 
a +b +c
a +b +c 
 a +b +c
•
Also if the line makes angles α , β , and γ respectively with the
coordinate axis then the direction cosines are
cos α , cos β and cos γ respectively.
•
Angle between two lines whose direction ratios are a1 , b1 , c1 and
a 2 , b2 , c 2 is
a1a2 + b1b2 + c1c2
cosθ =
2
2
2
2
2
(a 21 + b1 + c1 )(a2 + b2 + c2 )
1} If the lines are perpendicular then a1 a 2 + b1b2 + c1c 2 = 0
2} If the lines are parallel then
a1
b
c
= 1 = 1 .
a 2 b2 c 2
TABLE OF EQUATION AND FORMULAE:
Topic
Equation of
straight line
passing through
a point and
parallel to a
given vector
Equation of
straight line
passing through
two given point
Shortest distance
between two
skew lines whose
equations
are
ρ
ρ ρ
r = a1 + λ b1 and
ρ
ρ ϖ
r = a 2 + µb2
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Vector form
ρ
ρ ρ
r = a + λb
ρ ρ
ρ ρ
r = a + λ (b − a )
ρ ρ
ρ ρ
( a 2 − a1 ) ⋅ (b1 × b2 )
ρ ρ
b1 × b2
Cartesian form
x − x1
y − y1
z − z1
=
=
a
b
c
=λ
x − x1
y − y1
z − z2
=
=
x 2 − x1 y 2 − y1 z 2 − z1
x 2 − x1
a1
a2
y 2 − y1
b1
b2
z 2 − z1
c1
c2
(b1c 2 − b2 c1 ) 2 + (c1 a 2 − c 2 a1 ) 2 + ( a1b2 − a 2 b1 ) 2
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ρ
ρ
ρ
Shortest distance
( a 2 − a1 ) × b
ρ
between two
b
parallel planes
whose equations
ρ
ρ ρ
are r = a1 + λb
ρ
ρ
and r = a 2 + µb
ρ
r . nˆ = d
Equation of
plane in normal
form
ρ
Equation of a
(rρ − aρ).N = 0
plane passing
through a given
point with
position vector
ρ
a and
perpendicular toρ
a given vector N
ρ
Equation of
(rρ − aρ).[(b − aρ)× (cρ − aρ)]
plane passing
=0
through three
non collinear
points with
position
vectors
ρ
------------------
lx + my + nz = d
A( x − x1 ) + B ( y − y1 ) + C ( z − z1 ) = 0
x − x1
y − y1
z − z1
x2 − x1
x3 − x1
y 2 − y1
y3 − y1
z 2 − z1 = 0
z3 − z1
ρ
ρ
a , b and c
Intercept form of
plane with a, b
and c as x, y and
z intercepts
Equation of a
plane through
the line of
intersection of
two planes
whose vector
equations are
ρ ρ
r ⋅ n1 = d1 and
---------
ρ ρ
(r ⋅ n1 − d1 ) +
ρ ρ
λ ( r ⋅ n2 − d 2 ) = 0
x y z
+ + =1
a b c
A1 x + B1 y + C1 z + D1 +
λ ( A2 x + B2 y + C2 z + D2 ) = 0
ρ ρ
r ⋅ n2 = d 2
Distance of a
point
(x1 , y1 , z1 ) from
plane
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-----------
Ax1 + By1 + Cz1 − d
A2 + B 2 + C 2
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Ax + By + Cz = d
Distance of a
point with
position vector
from plane
ϖ
a
ρ ρ
a⋅n −d
ρ
n
-----------
ρ ρ
r ⋅n = d.
SOME OTHER CONCEPTS RELATED TO LINE AND PLANE:
•
•
Angle between two straight lines whose vector equations are
ρ ρ
ρ
ρ
ρ ρ
ρ ϖ
b1 ⋅ b2
r = a1 + λ b1 and r = a 2 + µb2 is cos θ = ρ ρ .
b1 b2
Equation of a plane which is perpendicular to any vector
and which
is at a distance p from the origin is rρ ⋅ nρ = d where
ρ
n
ρ
d = n p.
•
•
Equation of a plane in Cartesian form is Ax + By + Cz =
Angle between two planes whose vector equations are
ρ ρ
r ⋅ n1 = d1 and
d.
ρ ρ
n1 ⋅ n2
ρρ
r .n2 = d 2 is cosθ = ρ ρ .
n1 n2
•
ρ
ρ ρ
r = a + λb
and
Angle between a line and a plane whose vector equations are
ρ ρ
r ⋅n = d
ρ ρ
b ⋅n
is sin θ = ρ ρ . .
b n
ρ ρ
ρ ρ
Distance between two parallel planes r ⋅ n = d1 and r ⋅ n = d 2 is
•
d1 − d 2
ρ .
n
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•
Equation of a plane passing through two given points
(x1 , y1 , z1 ) and (x 2 , y 2 , z 2 ) and perpendicular to a plane Ax + By + Cz
x − x1
y − y1
z − z1
x 2 − x1
A
y 2 − y1
B
z 2 − z1 = 0.
C
is
Equation of a plane passing through two given points
•
(x1 , y1 , z1 ) and (x 2 , y 2 , z 2 ) and parallel to a line
•
= d
x − x1
y − y1
z − z1
x 2 − x1
a
y 2 − y1
b
z 2 − z1 = 0.
c
x − x1 y − y1 z − z1
=
=
is
a
b
c
Equation of a plane passing through a given point (x1 , y1 , z1 ) and
perpendicular to two planes A1 x + B1 y + C1 z = d1 and A2 x + B 2 y + C 2 z = d 2 is
x − x1
y − y1
A1
A2
B1
B2
z − z1
C1 = 0.
C2
Equation of a plane passing through a given point (α , β , γ ) and
•
parallel to two lines x − x1 = y − y1 = z − z1 and x − x 2 = y − y 2 = z − z 2 is
a1
x −α
y−β
z −γ
a1
a2
b1
b2
c1
c2
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b1
c1
a2
b2
c2
= 0.
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LINEAR PROGRAMMING
Basic Concepts and Formulae:
In simple language Linear Programming Problem ( LPP ), stand for
optimization ( to maximize or minimize ) a certain thing ( which may be profit, cost,
use of resources, use of workers, usability of machines etc.) with respect to certain
restrictions. The thing to optimize is known as objective function and the restrictions
are known as constraints. In this chapter we shall learn to form an LPP and solve it
graphically and also the types of LPP.
It is very useful in formation of certain policies in corporate world. It is a very
useful Mathematics tool in hands of Businessmen, by which they can maximize the
profit or minimize the losses/ costs/ expenditure where the affecting variables are
two in counting.
Basic Concepts -
Meaning of Linear Programming Problem (LPP) –
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A linear Programming Problem is one that is concerned with finding the optimal
value (i.e. maximum or minimum value) of a linear function (called objective
function ) of several variables (in XII class syllabus we are restricted to only two
variables say x and y ), subject to the conditions that the variables are non-negative
and satisfy a set of linear inequalities ( called linear constraints). The term linear
implies that all the mathematical relations used in the problem are linear relations
while the programming refers to the method of determining a particular programme
or plan of action.
The general LPP is given by
Max. Z (Min. Z) = ax + by ,………………(1)
Subject to the constraints:
a1 x + b1 y ≤ (≥)c1
a x + b y ≤ (≥)c
2
2
 2
a3 x + b3 y ≤ (≥)c3
……………………….(2)

..........

..........

a n x + bn y ≤ (≥)cn
x ≥ 0 , andy ≥ 0 . ……………………………(3)
Here (1) is called objective function, (2) contains the constraints and (3) is called non
negativity condition.
Types of Linear Programming Problem (LPP) –
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Manufacturing problems-
In these problems, we determine the
number of units of different products which should be produced and sold by a firm
when each product requires a fixed manpower, machine hours, labor hour per unit
of product, warehouse space per unit of the output etc., in order to make maximum
profit.
Diet problems -In these problems, we determine the amount of different
kinds of constituents/nutrients which should be included in a diet so as to minimize
the cost of the desired diet such that it contains a certain minimum amount of each
constituent/nutrients.
Transportation problems
-In these problems, we determine a
transportation schedule in order to find the cheapest way of transporting a product
from plants/factories situated at different locations to different markets.
Steps to solve a LPP using Graphical Method -
1. Construct the LPP using given word problem.
2. Write the auxiliary equations, by merely changing the sign of inequality to
equality in all the constraints.
3. Write the graph table of all the linear equations formed in the step two.
4. Using graph table draw the graphs of all auxiliary equation, and find the
solution region.
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5. (Bounded Region) If solution region is bounded, then find its corner points say
A, B, C,…etc.
6. Find the value of Z at A, B, C,……. Then the max.(min.) value of Z will represent
the solution. It is called the optimal value and the corresponding pont will
gives the value of x and y.
7. (Un-Bounded Region) If in step 5, the solution is not bounded i.e. not a closed
region, then again find the corner point and find the value of Z on these values,
say m is the largest/ maximum value of Z and n in the minimum value of Z,
then check whether the region given by
Z ≥ m
or
Z ≤ n
have any common point
with solution region or not. In case it has any common point with solution
region then the corresponding max. or min. solution does not exists. In
general in this case the maximum solution does not exist, and minimum
solution may or may not exist.
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Probability
Basic Concepts and Formulae:
Till now you are well familiar with the topic “Probability”. All of us know about
the basic concepts of probability i.e. a trial, random experiment, outcomes of an
experiments, events, complimentary events, exhaustive events and mutually events
etc.
In this chapter we shall discuss the concept of conditional probability of an
event, given that another event has occurred, which will be helpful in understanding
the Multiplication theorem of probability, Independence of events, Total probability
theorem and hence Baye’s theorem. We shall also learn the concept of a Random
Variable ( rv ) and Distribution of a random variable. At last we shall study the
concept of discrete probability distribution called Binomial Distribution.
Conditional Probability-
The probability of occurrence an event E, given that event F has occurred, is called
Conditional Probability of event E w.r.t. F, written as P(E/F) and is given by
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P(E/F)=
P (E ∩ F )
, provided P(F) ≠ 0.
P( F )
The probability of occurrence an event F, given that event E has occurred, is called
Conditional Probability of event F w.r.t. E, written as P(F/E) and is given by
P(F/E)=
P (E ∩ F )
, provided P(E) ≠ 0.
P( E )
Properties Conditional Probability-
•
The Conditional Probability of an event E, given that event F has already
occurred, is always greater than or equal to 0 and less than or equal to 1, i.e.
0 ≤ P ( E / F ) ≤ 1.
•
P(S/F) = 1, where S represent the sample space.
•
If A, B and F are the events if the sample space S, then
P ( A ∪ B / F ) = P ( A / F ) + P ( B / F ) − P ( A ∩ B / F ). and
P ( A ∪ B / F ) = P ( A / F ) + P ( B / F ). if A and B are mutually disjoint, i.e. A ∩ B = φ .
Multiplication Theorem of Probability-
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If E and F are two events given experiment, whose sample space is S, then the
probability of simultaneous happening of E and F is given by
P ( E ∩ F ) = P ( E ) P ( F / E ), P ( E ) ≠ 0 . OR P ( E ∩ F ) = P ( F ) P ( E / F ), P ( F ) ≠ 0 .
Multiplication rule of probability for more than two
Events-
If E, F and G are three events of given experiment, whose sample space is S, then the
probability of simultaneous happening of E, F and G is given by
P ( E ∩ F ∩ G ) = P ( E ) P ( F / E ) P ( G / E ∩ F ).
Independent Events-
If E and F are two events of given experiment(s), then these are said to be
independent if happening of one does not affect the probability of other.
i.e. either P ( E / F ) = P ( E ), P ( F ) ≠ 0 . Or P ( F / E ) = P ( F ), P ( E ) ≠ 0 .
If E and F are two events of same experiment, then these are said to be independent if
P ( E ∩ F ) = P ( E ). P ( F ).
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Multiplication Theorem of Probability For
Independent Events -
If E and F are two independent events of given experiment, whose sample space is S,
then the probability of simultaneous happening of E and F is given by
P ( E ∩ F ) = P ( E ) P ( F ).
Partition of a Sample Space –
The collection of subsets E1 , E 2 , E3 ,.......E n , of a sample space S of an experiment, is
said to be a partition of the sample space S, if
1.
P ( E i ) φ 0.∀i = 1,2,3,......n.
2. Ei ∩ E j = φ.∀i ≠ j, andi, j = 1,2,3,......n.
3. E1 ∪ E 2 ∪ E 3 ....... ∪ E n = S .
In other words the collection of mutually disjoint and exhaustive events is called a partition of the
given sample space.
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Theorem of Total Probability –
Let the events E1 , E 2 , E3 ,.......E n , form a partition of the sample space S, of an
experiment. If A is any event of S, then
n
P ( A ) = P ( E 1 ) P ( A / E 1 ) + P ( E 2 ) P ( A / E 2 ) + .......... ...... P ( E n ) P ( A / E n ) = Σ P ( E i ) P ( A / E i )
i =1
Baye’s Theorem-
Let E1, E2, E3, ……….., En be a partition of sample space S. If A is any random event of
S, with P(A) > 0, then
P ( E i / A) =
P ( E i ).P ( A / E i )
n
;1 ≤ i ≤ n.
∑ P ( E i ).P ( A / E i )
i =1
Random Variable (rv) –
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A random variable is a variable which takes real values depending upon the
outcomes of an experiment. In other words we say that a rv is a function on the
sample space of an experiment to the set of real numbers.
Here in this class we deal with only discrete rv.
A discrete random variable is a variable which takes integral values depending upon
the outcomes of the experiment.
Probability Distribution of rv –
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The probability distribution of a rv X is given by
X
:
P(X) :
x1
x2
P(x1)
P(x2)
………………xn
…………….P(xn)
Where the rv X takes possible values x1 ,x2, x3, ………, xn, with probabilities p(x1),
n
p(x2), p(x3), …………….p(xn) respectively, and p ( x i ) > 0 , ∑ p ( x i ) = 1, i = 1, 2 ,3,........, n.
i =1
Mean, Variance and Standard Deviation of a rv –
Let X be a rv whose possible values are x1 ,x2, x3, ………, xn occur with probabilities
p(x1), p(x2), p(x3), …………….p(xn) respectively. Then Mean of rv written by E(X),
variance of X written by Var(X), are given by
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µ = E ( X ) = ∑ x i p ( x i ).
n
Var ( X ) = σ = ∑ ( x i − µ ) 2 p ( x i ) = E ( X − µ ) 2
2
x
i =1
n
S tan dardDeviat ion = Var ( X ) =
∑ (x
i
− µ ) 2 p ( xi ) .
i =1
Bernoulli Trials –
Trials of a random experiment are called Bernoulli Trials, if they satisfy the following
conditions:
1. There should be a finite number of trials.
2. The trials should be independent.
3. Each trial has exactly two outcomes: called success or failure.
4. The probability of success (failure) remains same in each trial.
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Binomial Distribution –
Let we have n Bernoulli trials, with p as probability of success and q as probability of
failure. Let X be a rv denotes the number of successes, then the probability of x
n!
successes is given by P ( X = x ) =
p x q n − x = n C x p x q n − x ; x = 1, 2,3,........ n.( q = 1 − p )
x! ( n − x )!
This type of rv distribution is called, binomial distribution.
It is written as X~B(n, p), X is called binomial variate.
Here in this case E(X) = np, and Var(X) = npq. (in syllabus without proof).
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