D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 1
HAND BOOK OF MATHEMATICS (Definitions and Formulae)
CLASS β 12
SUBJECT: MATHEMATICS
D.SREENIVASULU PGT(Mathematics) KENDRIYA VIDYALAYA
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 2
CLASS 12 : CBSE MATHEMATICS
RELATIONS AND FUNCTIONS
TYPES OF RELATIONS:
EMPTY RELATION: A relation π in a set π΄ is called empty relation, if no element
of π΄ is related to any element of π΄, i.e., π = β β π΄ Γ π΄.
UNIVERSAL RELATION: A relation π in a set π΄ is called universal relation, if each
element of π΄ is related to every element of π΄, i.e. , π = π΄ Γ π΄.
TRIVIAL RELATIONS: Both the empty relation and the universal relation are sometimes
called trivial relations.
A relation R in a set A is called a) Reflexive, if (π₯, π₯) β π πππ ππ£πππ¦ π₯ β π΄
b) Symmetric, ππ (π₯, π¦) β π πππππππ π‘βππ‘ (π¦, π₯) β π πππ πππ π₯, π¦ β π΄
c) Transitive, ππ (π₯, π¦) β π πππ (π¦, π§) β π πππππππ π‘βππ‘ (π₯, π§) β π πππ πππ π₯, π¦, π§ β π΄
EQUIVALENCE RELATION: π΄ relation π in a set π΄ is said to be an equivalence relation
if π is reflexive, symmetric and transitive.
EQUIVALENCE CLASS: Let π be an equivalence relation on a non-empty set π΄ and
π β π΄. Then the set of all those elements of π΄ which are related to π, is
called the equivalence class determined by π and is denoted by [π].
i.e [π] = {π₯ β π΄ βΆ (π₯, π) β π }
TYPES OF FUNCTIONS:
ONE-ONE (INJECTIVE) FUNCTION: A function π βΆ π β π is defined to be one-one
(or injective), if the images of distinct elements of X under f are distinct, i.e., for
every π₯1 , π₯2 β π, π(π₯1) = π( π₯2 ) πππππππ π₯1 = π₯2. Otherwise, f is called many-one.
ONTO (SURJECTIVE) FUNCTION : A function π βΆ π β π is said to be onto (or surjective),
if every element of π is the image of some element of π under f. i.e., for every
π¦ β π, there exists an element π₯ ππ π such that π(π₯) = π¦.
NOTE: f : X β Y is onto if and only if Range of f = Codomain. BIJECTIVE FUNCTION: A function π βΆ π β π is said to be bijective, if π is both one-
one and onto.
COMPOSITION OF FUNCTIONS: Let π βΆ π΄ β π΅ and π βΆ π΅ β πΆ be two functions. Then
the composition of π πππ π, denoted by πππ, is defined as the function
πππ βΆ π΄ β πΆ πππ£ππ ππ¦ πππ (π₯) = π(π (π₯)), β π₯ β π΄.
INVERTIBLE FUNCTION : A function π βΆ π β π is defined to be invertible, if there
exists a function π βΆ π β π such that πππ = πΌπ₯ πππ πππ = πΌπ¦ (i.e πππ(π₯) = π₯
and πππ(π¦) = π¦) The function π is called the inverse of π and is denoted by πβ1
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 3
NOTE: If π is invertible, then π must be one-one and onto and conversely, if
π is one-one and onto, then π must be invertible.
BINARY OPERATIONS:
BINARY OPERATION : A binary operation β on a set π΄ is a function ββΆ π΄ Γ π΄ β π΄.
We denote β (π, π) ππ¦ π β π Note: In general, β is said to be binary operation on π΄ if π, π β π΄ β π β π β π΄
COMMUTATIVE: A binary operation β on the set π is called commutative, if
π β π = π β π, for every π, π β π.
ASSOCIATIVE: A binary operation ββΆ π΄ Γ π΄ β π΄ is said to be associative if
(π β π) β π = π β (π β π), β a, b, c, β A.
IDENTITY: Given a binary operation ββΆ π΄ Γ π΄ β π΄, an element π β π΄, if it exists, is
called identity for the operation β, if π β π = π = π β π β π β π΄.
INVERSE: Given a binary operation ββΆ π΄ Γ π΄ β π΄ with the identity element e in
A, an element a β A is said to be invertible with respect to the operation β, if
there exists an element b in A such that π β π = π = π β π and b is called the
inverse of a and is denoted by πβ1.
INVERSE TRIGONOMETRIC FUNCTIONS
PRINCIPAL VALUE BRANCHES:
FUNCTION DOMAIN RANGE (Principal Value
Branch)
sinβ1 π₯ [β1, 1] [βπ
2,π
2]
cosβ1 π₯ [β1, 1] [0, π]
tanβ1 π₯ R (βπ
2,π
2)
cosecβ1 π₯ π β (β1, 1) [βπ
2,π
2] β {0}
secβ1 π₯ π β (β1, 1) [0, π] β {π
2}
cotβ1 π₯ R (0, π)
PROPERTIES OF INVERSE TRIGONOMETRIC FUNCTIONS:
sin(sinβ1 π₯) = π₯ , π₯ β [β1, 1]
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 4
sinβ1(sin π₯) = π₯ , π₯ β [βπ
2,π
2]
sinβ1 1
π₯ = cosocβ1 π₯ , π₯ β₯ 1 ππ π₯ β€ β1
cosβ1 1
π₯ = secβ1 π₯ , π₯ β₯ 1 ππ π₯ β€ β1
tanβ1 1
π₯= cotβ1 π₯ , π₯ > 0
sinβ1(βπ₯) = β sinβ1 π₯ , π₯ β [β1, 1]
cosecβ1(βπ₯) = β cosecβ1 π₯, |π₯| β₯ 1
tanβ1(βπ₯) = β tanβ1 π₯ , π₯ β π
cosβ1(βπ₯) = π β cosβ1 π₯, π₯ β [β1, 1],
secβ1(βπ₯) = π β secβ1 π₯, |π₯| β₯ 1
cotβ1(βπ₯) = π β cotβ1 π₯, π₯ β π
sinβ1 π₯ + cosβ1 π₯ =π
2 , π₯ β [β1, 1]
tanβ1 π₯ + cotβ1 π₯ =π
2, π₯ β π
cosecβ1 π₯ + secβ1 π₯ =π
2 , |π₯| β₯ 1
tanβ1 π₯ + tanβ1 π¦ = tanβ1 (π₯+π¦
1βπ₯π¦) , π₯π¦ < 1
tanβ1 π₯ + tanβ1 π¦ = π + tanβ1 (π₯+π¦
1βπ₯π¦) , π₯π¦ > 1
tanβ1 π₯ β tanβ1 π¦ = tanβ1 (π₯βπ¦
1+π₯π¦) , π₯π¦ > β1
2 tanβ1 π₯ = tanβ1 (2π₯
1βπ₯2) , |π₯| β€ 1
2 tanβ1 π₯ = cosβ1 (1βπ₯2
1+π₯2) , π₯ β₯ 0
2 tanβ1 π₯ = sinβ1 (2π₯
1+π₯2) , π₯ β (β1, 1)
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 5
MATRICES
ORDER OF A MATRIX : A general matrix of order π Γ π can be written as
= [πππ]πΓπ , π€βπππ π = 1,2, β¦π πππ π = 1,2, β¦ π
ππ’ππππ ππ πππ€π = π πππ ππ’ππππ ππ ππππ’πππ = π
TYPES OF MATRICES:
COLUMN MATRIX: A matrix is said to be a column matrix if it has only one column.
Examples: π΄ = [2
β9] πππππ ππ πππ‘πππ₯ π΄ ππ 2 Γ 1
πππ π΅ = [ββ50
β12
] πππππ ππ πππ‘πππ₯ π΅ ππ 3 Γ 1
ROW MATRIX: A matrix is said to be a row matrix if it has only one row
Examples: π΄ = [14 26] πππππ ππ πππ‘πππ₯ π΄ ππ 1 Γ 2
π© = [0 β7 12] πππππ ππ πππ‘πππ₯ π΅ ππ 1 Γ 3
SQUARE MATRIX: A matrix in which the number of rows is equal to the number of
columns, is said to be a square matrix.
Examples: A = [2 46 β8
] πππππ ππ πππ‘πππ₯ π΄ ππ 2 Γ 2
X = [
5 0 β81
2β2 14
7 β8 4
] order of matrix X is 3 Γ 3
DIAGONAL MATRIX: A square matrix π΄ = [πππ]πΓπ is said to be a diagonal matrix
if all its non-diagonal elements are zero
Example: π΄ = [6 0 0
0 β6 00 0 9
]
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 6
SCALAR MATRIX : A diagonal matrix is said to be a scalar matrix if its diagonal
elements are equal
Example : π΄ = [5 0 00 5 00 0 5
]
IDENTITY MATRIX: A square matrix in which elements in the diagonal are all 1 and
rest are all zero is called an identity matrix.
Example : π΄ = [1 0 00 1 00 0 1
], generally it is denoted by I.
ZERO MATRIX: A matrix is said to be zero matrix or null matrix if all its elements
are zero.
Examples : [0] , [0 0 00 0 0
] ,
[0 0 00 0 00 0 0
] πππ πππ ππππ πππ‘πππππ , πππππππππ¦ πππππ‘ππ ππ¦ πΆ.
EQUALITY OF MATRICES: Two matrices π΄ = [πππ]πΓπ πππ π΅ = [πππ]πΓπ
are
said to be equal if (i) they are of the same order (ii) each element of π΄ is equal to
the corresponding element of π΅, that is πππ = πππ πππ πππ π πππ π.
Example: Let π΄ = [2 18 64 β5
] πππ π΅ = [2 18 64 β5
] , we say that π΄ = π΅
OPERATION OF MATRICES:
ADDITION OF MATRICES: Let π΄ = [πππ]πΓπ πππ π΅ = [πππ]πΓπ
be two matrices of
the same order. Then π΄ + π΅ is defined to be the matrix of order of π Γ π obtained
by adding corresponding elements of π΄ πππ π΅
i.e π΄ + π΅ = [πππ + πππ]πΓπ
DIFFERENCE OF MATRICES: Let π΄ = [πππ]πΓπ πππ π΅ = [πππ]πΓπ
be two matrices of
the same order. Then π΄ β π΅ is defined to be the matrix of order of π Γ π obtained
by subtracting corresponding elements of π΄ πππ π΅
i.e π΄ β π΅ = [πππ β πππ]πΓπ
MULTIPLICATION OF MATRICES: The product of two matrices π΄ and π΅ is defined if the
number of columns of π΄ is equal to the number of rows of π΅.
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 7
Let π΄ = [πππ]πΓπ πππ π΅ = [πππ]πΓπ
. Then the product of the matrices π΄
and π΅ is the matrix πΆ of order m Γ p. To get the (π, π)π‘β πππππππ‘ πππ matrix πΆ, we take
the i th row of π΄ and π th column of π΅, multiply them elementwise and take the sum of
all these products. i.e πΆππ = β πππ.ππ=1 πππ
Example: Let π¨ = [π π ππ π π
] πππ π© = [π ππ ππ π
]
π¨π© = [π Γ π + π Γ π + π Γ π π Γ π + π Γ π + π Γ ππ Γ π + π Γ π + π Γ π π Γ π + π Γ π + π Γ π
]
= [π + ππ + ππ π + ππ + πππ + ππ + ππ π + ππ + ππ
] = [ππ ππππ πππ
]
MULTIPLICATION OF A MATRIX BY A SCALAR: Let π΄ = [πππ]πΓπand π is a scalar, then
ππ΄ = π[πππ]πΓπ= [π. πππ]πΓπ
Example: π¨ = [π π βππ π π
] βΉ ππ¨ = [π(π) π(π) π(βπ)ππ ππ ππ
] = [π ππ βππππ ππ ππ
]
TRANSPOSE OF A MATRIX: If π΄ = [πππ]πΓπ be an m Γ n matrix, then the matrix
obtained by interchanging the rows and columns of π΄ is called the transpose of π΄.
Transpose of the matrix π΄ is denoted by π΄β² or π΄T .
If π΄ = [πππ]πΓπ, then π΄β² = [πππ]πΓπ
Example: π¨ = [π π ππ π ππ π π
] βΉ π¨π» = [π π ππ π ππ π π
]
SYMMETRIC MATRIX: A square matrix If π΄ = [πππ] is said to be symmetric if
π¨β² = π¨, that is, [πππ] =[πππ] for all possible values of π πππ π
Example: π¨ = [π π πππ π πππ π π
] , πππππππ π¨β² = π¨.
SKEW-SYMMETRIC MATRIX: A square matrix π΄ = [πππ] is said to be skew
symmetric matrix if Aβ² = β A, that is πππ = βπππ for all possible values of i and j and
πππ = 0 for all i.(all the diagonal elements are zero).
Example: π¨ = [π π βππ
βπ π βπππ π π
] , πππππππ π¨β² = βπ¨
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 8
TRANSFORMATION OF A MATRIX:
The interchange of any two rows or two columns. The interchange of π th and π th rows is
denoted by π π β π π and interchange of π th and π th column is denoted by πΆπ β πΆπ .
The multiplication of the elements of any row or column by a non-zero number. The
multiplication of each element of the π th row by π, where π β 0 is denoted by
Ri β k Ri. The corresponding column operation is denoted by Ci β kCi
The addition to the elements of any row or column, the corresponding
elements of any other row or column multiplied by any non-zero number. The
addition to the elements of i th row, the corresponding elements of j th row
multiplied by k is denoted by Ri β Ri + kRj . The corresponding column operation is
denoted by Ci β Ci + kCj .
INVERTIBLE MATRICES: If A is a square matrix of order m, and if there exists
another square matrix B of the same order m, such that AB = BA = I, then B is
called the inverse matrix of A and it is denoted by Aβ 1. In that case A is said to be
invertible.
PROPERTIES OF MATRICES:
π΄ + π΅ = π΅ + π΄
π΄ β π΅ β π΅ β π΄
π΄π΅ β π΅π΄
(π΄π΅)πΆ = π΄(π΅πΆ)
(π΄β²)β² = π΄
π΄πΌ = πΌπ΄ = π΄
π΄π΅ = π΅π΄ = πΌ, π‘βππ π΄β1 = π΅ πππ π΅β1 = π΄
π΄π΅ = 0 βΉ ππ‘ ππ πππ‘ πππππ π πππ¦ π‘βππ‘ πππ π‘βπ πππ‘πππ₯ ππ π§πππ.
π΄(π΅ + πΆ) = π΄π΅ + π΄πΆ
Every square matrix can possible to express as the sum of symmetric and
skew-symmetric matrices.
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 9
π¨ =π
π(π¨ + π¨β²) +
π
π(π¨ β π¨β²), where (π΄ + π΄β²) is symmetric matrix and (π΄ β π΄β²) is
skew-symmetric matrices.
Apply a sequence of row operation on A = IA till we get, I = BA. The matrix
B will be the inverse of A. Similarly, if we wish to find Aβ1 using column
operations, then, write A = AI and apply a sequence of column operations on
A = AI till we get, I = AB. The matrix B will be the inverse of A.
After applying one or more elementary row (column) operations on
A = IA (A = AI), if we obtain all zeros in one or more rows of the matrix A
on L.H.S., then Aβ1 does not exist.
_________________________________________________________
DETERMINANTS
DETERMINANT:
πΏππ‘ π΄ = [π ππ π
] , π‘βππ πππ‘(π΄) = |π΄| = ππ β ππ
πΏππ‘ π΄ = [
π π ππ π ππ β π
] , π‘βππ |π΄| = π |π πβ π
| β π |π ππ π
| + π |π ππ β
|
PROPERTIES OF DETERMINANTS:
If rows and columns are interchanged, then the value of the determinant
remains same.
If any two rows (or columns) of a determinant are interchanged, then sign of
determinant changes
If any two rows (or columns) of a determinant are identical (all
corresponding elements are same), then value of determinant is zero.
If each element of a row (or a column) of a determinant is multiplied by a
constant k, then its value gets multiplied by k.
If, to each element of any row or column of a determinant, the equimultiples
of corresponding elements of other row (or column) are added, then value of
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 10
determinant remains the same, i.e., the value of determinant remain same if
we apply the operation Ri β Ri + kRj or Ci β Ci + k Cj .
If some or all elements of a row or column of a determinant are expressed
as sum of two (or more) terms, then the determinant can be expressed as
sum of two (or more) determinants
MINORS: Minor of an element πππ of a determinant is the determinant obtained by
deleting its ith row and jth column in which element πππ lies. Minor of an element πππ
is denoted by πππ.
CO-FACTORS: Cofactor of an element πππ , denoted by π΄ππ is defined by π΄ππ =
(β1)π+π. πππ , where πππ is minor of πππ
ADJOINT OF A MATRIX: The adjoint of a square matrix π΄ = [πππ] is defined as the
transpose of the matrix [π΄ππ] , where π΄ππ is the cofactor of the element πππ . Adjoint
of the matrix A is denoted by adj A.
INVERSE OF A MATRIX: Let A be a square matrix.
π΄β1 =1
|π΄|ππππ΄
SOLUTION OF SYSTEM OF LINEAR EQUATIONS BY USING MATRIX METHOD:
Let the system of linear equations be
π1π₯ + π1π¦ + π1π§ = π1
π2π₯ + π2π¦ + π2π§ = π2
π3π₯ + π3π¦ + π3π§ = π3
These equations can be written as
[
π1 π1 π1π2 π2 π2
π3 π3 π3
] [π₯π¦π§] = [
π1
π2
π3
]
π΄π = π΅
π = π΄β1π΅
π΄β1ππ₯ππ π‘π , ππ |π΄| β 0 π. π π‘βπ π πππ’π‘πππ ππ₯ππ π‘π πππ ππ‘ ππ π’ππππ’π.
πβπ π π¦π π‘ππ ππ πππ’ππ‘ππππ ππ π πππ π‘π ππ ππππ ππ π‘πππ‘ ππ π‘βπ π πππ’π‘πππ ππ₯ππ π‘π .
ππ |π΄| = 0 , π‘βππ π€π πππππ’πππ‘π (ππππ΄)π΅.
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 11
πΌπ |π΄| = 0 πππ (ππππ΄)π΅ β π , (O being zero matrix), then solution does not
exist and the system of equations is called inconsistent.
πΌπ |π΄| = 0 πππ (ππππ΄)π΅ = π, then system may be either consistent or
inconsistent according as the system have either infinitely many solutions or
no solution.
IMPORTANT NOTES:
The matrix A is singular if |π΄| = 0
|ππ΄| = ππ|π΄|,π€βπππ π = πππππ ππ πππ‘πππ₯ π΄
π΄(ππππ΄) = (ππππ΄)π΄ = |π΄|πΌ
|ππππ΄| = |π΄|πβ1, π€βπππ π = πππππ ππ πππ‘πππ₯ π΄
|π΄(ππππ΄)| = |π΄|π, π€βπππ π = πππππ ππ πππ‘πππ₯ π΄
|π΄π΅| = |π΄||π΅|
(π΄π΅)β1 = π΅β1π΄β1
|π΄β1| = |π΄|β1
|π΄π| = |π΄|
CONTINUITY AND DIFFERENTIABLITY
CONTINUITY: Suppose f is a real function on a subset of the real numbers and let a
be a point in the domain of f. Then f is continuous at a limπ₯βΆπ
π(π₯) = π(π)
i.e πΏπ»πΏ = π π»πΏ = π(π)
limπ₯βπβ
π(π₯) = limπ₯βπ+
π(π₯) = π(π)
DIFFERENTIATION:
FIRST PRINCIPLE:
Let π¦ = π(π₯), π‘βππ ππ¦
ππ₯= lim
ββ0
π(π₯+β)βπ(π₯)
β
π¦ = ππππ π‘πππ‘ βππ¦
ππ₯= 0
π¦ = π₯π βππ¦
ππ₯= ππ₯πβ1
π¦ = π πππ₯ βππ¦
ππ₯= πππ π₯
π¦ = πππ π₯ βππ¦
ππ₯= βπ πππ₯
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 12
π¦ = π‘πππ₯ βππ¦
ππ₯= π ππ2π₯
π¦ = πππ πππ₯ βππ¦
ππ₯= βπππ πππ₯. πππ‘π₯
π¦ = π πππ₯ βππ¦
ππ₯= π πππ₯. π‘πππ₯
π¦ = πππ‘π₯ βππ¦
ππ₯= βπππ ππ2π₯
π¦ = sinβ1 π₯ βππ¦
ππ₯=
1
β1βπ₯2
π¦ = cosβ1 π₯ βππ¦
ππ₯= β
1
β1βπ₯2
π¦ = tanβ1 π₯ βππ¦
ππ₯=
1
1+π₯2
π¦ = cosecβ1 π₯ βππ¦
ππ₯= β
1
π₯βπ₯2β1
π¦ = secβ1 π₯ βππ¦
ππ₯=
1
π₯βπ₯2β1
π¦ = cotβ1 π₯ βππ¦
ππ₯= β
1
1+π₯2
π¦ = ππ₯ βππ¦
ππ₯= ππ₯
π¦ = ππ₯ βππ¦
ππ₯= ππ₯. ππππ
π¦ = ππππ₯ βππ¦
ππ₯=
1
π₯
Product Rule: π¦ = π’. π£ βππ¦
ππ₯= π’. π£β² + π£. π’β², π€βπππ π’β² =
ππ’
ππ₯ , π£β² =
ππ£
ππ₯
Quotient Rule: π¦ =π’
π£β
ππ¦
ππ₯=
π£.π’β²βπ’.π£β²
π£2, π€βπππ π’β² =
ππ’
ππ₯ , π£β² =
ππ£
ππ₯
Chain Rule: Let π¦ = π(π‘)πππ π₯ = π(π‘) π‘βππ ππ¦
ππ₯=
ππ¦
ππ‘.ππ‘
ππ₯
π¦ = π(ππ₯ + π) βΉππ¦
ππ₯= π. πβ²(ππ₯ + π),
πΈπ₯: π¦ = sin(4π₯ + 9) βΉππ¦
ππ₯= 4. cos (4π₯ + 9)
π¦ = [π(π₯)]π βππ¦
ππ₯= π. [π(π₯)]πβ1. πβ²(π₯)
Logarithmic Differentiation: Let π¦ = [π’(π₯)]π£(π₯)
ππππ¦ = π£. ππππ’ βΉ1
π¦.ππ¦
ππ₯=
π£
π’. π’β² + π£β². ππππ’
ππ¦
ππ₯= π¦ [
π£
π’. . π’β² + π£β². ππππ’]
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 13
Mean Value Theorem: Let f : [a, b] β R be a continuous function
on [a, b] and differentiable on (a, b). Then there exists some c in (a, b)
such that some π β (π, π) π π’πβππ‘ π‘βππ‘ πβ²(π) =π(π)βπ(π)
πβπ
Rolleβs Theorem: Let f : [a, b] β R be continuous on [a, b] and
differentiable on (a, b), such that f(a) = f(b), where a and b are some
real numbers. Then there exists some c in (a, b) such that fβ²(c) = 0.
Note: In above all formulas, ππππ = πππππ
APPLICATION OF DERIVATIVES
RATE OF CHANGE OF QUANTITIES
1) Area of circle (A) = ππ2
Rate of change of area = ππ΄
ππ‘= 2ππ
ππ
ππ‘
2) Circumference of circle (C) = 2ππ
Rate of change of Circumference = ππΆ
ππ‘= 2π
ππ
ππ‘
3) Perimeter of a rectangle (P) = 2(π₯ + π¦), where x = length , y = width
Rate of change of Perimeter = ππ
ππ‘= 2(
ππ₯
π₯π‘+
ππ¦
ππ‘)
4) Area of rectangle (A) = π₯. π¦ , where x = length , y = width
Rate of change of area = ππ΄
ππ‘= π₯.
ππ¦
ππ‘+ π¦.
ππ₯
ππ‘
5) Volume of cube (V) =π₯3, where x = edge of cube
Rate of change of Volume = ππ
ππ‘= 3π₯2 ππ₯
ππ‘
6) Surface area of cube (S) = 6π₯2
Rate of change of Surface area = ππ
ππ‘= 6π₯
ππ₯
ππ‘
7) Volume of sphere (V) = 4
3ππ3
Rate of change of Volume = ππ
ππ‘=
4
3(3ππ2)
ππ
ππ‘
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 14
8) Surface area of Sphere (S) = 4ππ2
Rate of change of Surface area = ππ
ππ‘= 8ππ
ππ
ππ‘
9) Total cost = C(x), where C(x) isi Rupees of the production of x units
Marginal cost = ππΆ
ππ₯
10) Total Revenue = R(x)
Marginal Revenue = ππ
ππ₯
INCREASING AND DECREASING FUNCTION
Let I be an interval contained in the domain of a real valued function f. Then
f is said to be
(i) increasing on I ππ π₯ < π¦ ππ πΌ π‘βππ π(π₯) β€ π(π¦), πππ πππ π₯, π¦ β πΌ.
(ii) strictly increasing on I ππ π₯ < π¦ ππ πΌ π‘βππ π(π₯) < π(π¦), πππ πππ π₯, π¦ β πΌ
(iii) decreasing on I ππ π₯ < π¦ ππ πΌ π‘βππ π(π₯) β₯ π(π¦), πππ πππ π₯, π¦ β πΌ.
(iv) strictly decreasing on I ππ π₯ < π¦ ππ πΌ π‘βππ π(π₯) > π(π¦), πππ πππ π₯, π¦ β πΌ
(a) f is strictly increasing in (π, π) ππ πβ(π₯) > 0 πππ πππβ π₯ β (π, π)
(b) f is strictly decreasing in (π, π) ππ πβ(π₯) < 0 πππ πππβ π₯ β (π, π)
A function will be increasing or decreasing in R if it is so in every interval of R
f is a constant function in [π, π] ππ πβ(π₯) = 0 πππ πππβ π₯ β (π, π)
TANGENTS AND NORMALS
Let given curve be y=f(x)
Slope of tangent to the curve at (π₯1 , π¦1) ππ π = [ππ¦
ππ₯]π₯=π₯1
Slope of normal to the curve at (π₯1 , π¦1) = β1
π
Equation of tangent at (ππ , ππ) ππ π β ππ = π(π β ππ)
Equation of normal at (ππ , ππ) ππ π β ππ = βπ
π(π β ππ)
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 15
APPROXIMATION
Let y = f(x)
βπ¦ =ππ¦
ππ₯. βπ₯
π¦ + βπ¦ = π¦ +ππ¦
ππ₯. βπ₯
i.e π(π₯ + βπ₯) = π(π₯) + πβ²(π₯). βπ₯
MAXIMA AND MINIMA
First Derivative Test:
Let f be a function defined on an open interval I. Let f be continuous at a critical
point c in I. Then
(i) If f β²(x) changes sign from positive to negative as x increases through c, then
c is a point of local maxima and maximum value of π(π₯) = π( π) .
(ii) If f β²(x) changes sign from negative to positive as x increases through c, then
c is a point of local minima and minimum value of π(π₯) = π( π) .
(iii) If f β²(x) does not change sign as x increases through c, then c is neither a
point of local maxima nor a point of local minima. Infact, such a point is called
point of inflexion.
Second Derivative Test
Let f be a function defined on an interval I and c β I. Let f be twice differentiable
at c. Then
(i) π₯ = π is a point of local maxima if π β²(π) = 0 and π β³(π) < 0 The values f (c) is
local maximum value of f .
(ii) (ii) x = c is a point of local minima if π β²(π) = 0 and π β³(π) > 0 In this case, f (c)
is local minimum value of f .
(iii) The test fails if π β²(π) = 0 and π β³(π) = 0. In this case, we go back to the
first derivative test and find whether c is a point of maxima, minima or a point
of inflexion. Absolute maxima and absolute minima (maxima and minima in a closed interval)
Given π(π₯) πππ πππ‘πππ£ππ [π, π]
Find πβ(π₯)
Let πβ(π₯) = 0
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 16
Find critical values. (i.e find the values of x if πβ(π₯) = 0 ). say
π₯ = π₯1 πππ π₯ = π₯2
Calculate π(π), π(π₯1), π(π₯2) πππ π(π).
Identify maxima and minima values of f(x).
INTEGRALS
INDEFINITE INTEGRALS
1) β«1ππ₯ = π₯ + π
2) β«π₯ ππ₯ =π₯2
2+ π
3) β«π₯π ππ₯ =π₯π+1
π+1+ π, π β β1
4) β« π πππ₯ ππ₯ = βπππ π₯ + π
5) β« πππ π₯ ππ₯ = π πππ₯ + π
6) β« π‘πππ₯ ππ₯ = πππ|π πππ₯| + π
7) β« πππ πππ₯ ππ₯ = πππ|πππ πππ₯ β πππ‘π₯| + π
8) β« π πππ₯ ππ₯ = πππ|π πππ₯ + π‘πππ₯| + π
9) β« πππ‘π₯ ππ₯ = πππ|π πππ₯| + π
10) β« π ππ2π₯ ππ₯ = π‘πππ₯ + π
11) β« πππ ππ2π₯ ππ₯ = βπππ‘π₯ + π
12) β« π πππ₯. π‘πππ₯ ππ₯ = π πππ₯ + π
13) β« πππ πππ₯. πππ‘π₯ ππ₯ = βπππ πππ₯ + π
14) β«1
β1βπ₯2ππ₯ = sinβ1 π₯ + π ππ β cosβ1 π₯ + π
15) β«1
1+π₯2 ππ₯ = tanβ1 π₯ + π ππ β cotβ1 π₯ + π
16) β«1
π₯βπ₯2β1ππ₯ = secβ1 π₯ + π ππ β cosecβ1 π₯ + π
17) β« ππ₯ ππ₯ = ππ₯ + π
18) β«1
π₯ππ₯ = πππ|π₯| + π
19) β«ππ₯ ππ₯ = ππ₯ππππ + π
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 17
20) β«1
π₯2βπ2ππ₯ =
1
2ππππ |
π₯βπ
π₯+π| + π
21) β«1
π2βπ₯2 ππ₯ =1
2ππππ |
π+π₯
πβπ₯| + π
22) β«1
π₯2+π2ππ₯ =
1
πtanβ1 π₯
π+ π
23) β«1
βπ₯2βπ2ππ₯ = πππ|π₯ + βπ₯2 β π2| + π
24) β«1
βπ₯2+π2ππ₯ = πππ|π₯ + βπ₯2 + π2| ππ₯ = +π
25) β«1
βπ2βπ₯2ππ₯ = sinβ1 π₯
π+ π
26) β«βπ₯2 β π2 ππ₯ =π₯
2βπ₯2 β π2 β
π2
2πππ|π₯ + βπ₯2 β π2| + π
27) β«βπ₯2 + π2 ππ₯ =π₯
2βπ₯2 + π2 +
π2
2πππ|π₯ + βπ₯2 + π2| + π
28) β«βπ2 β π₯2 ππ₯ =π₯
2βπ2 β π₯2 +
π2
2sinβ1 π₯
π+π
29) β« ππ₯[π(π₯) + πβ²(π₯)] ππ₯ = ππ₯π(π₯) + π
30) β«π’. π£ ππ₯ = π’ β« π£ππ₯ β β«[π’β² β« π£ππ₯] ππ₯ + π,π€βπππ π’ = π’(π₯) πππ π£ = π£(π₯)
31) β«[π(π₯) Β± π(π₯)]ππ₯ = β«π(π₯)ππ₯ Β± β«π(π₯)ππ₯
32) β«π. π(π₯)ππ₯ = π β«π(π₯)ππ₯,π€βπππ π ππ ππππ π‘πππ‘.
Note: Let antiderivative of π(π₯) = πΉ(π₯)
i.e. β«π(π₯)ππ₯ = πΉ(π₯) + π, then
β«π(ππ₯ + π)ππ₯ =1
ππΉ(ππ₯ + π) + π
Partial fractions
The rational function π(π₯)
π(π₯) is said to be proper if the degree of π(π₯) is
less than the degree of π(π₯)
Partial fractions can be used only if the integrand is proper rational
function
S.No Form of rational function Form of Partial fraction
1 1
(π₯ β π)(π₯ β π)
π΄
π₯ β π+
π
π₯ β π
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 18
2 ππ₯ + π
(π₯ β π)(π₯ β π)
π΄
π₯ β π+
π
π₯ β π
3 1
(π₯ β π)(π₯ β π)(π₯ β π)
π΄
π₯ β π+
π
π₯ β π+
1
π₯ β π
4 ππ₯ + π
(π₯ β π)(π₯ β π)(π₯ β π)
π΄
π₯ β π+
π
π₯ β π+
1
π₯ β π
5 ππ₯2 + ππ₯ + π
(π₯ β π)(π₯ β π)(π₯ β π)
π΄
π₯ β π+
π
π₯ β π+
1
π₯ β π
6 1
(π₯ β π)2(π₯ β π)
π΄
π₯ β π+
π΅
(π₯ β π)2+
πΆ
π₯ β π
7 ππ₯ + π
(π₯ β π)2(π₯ β π)
π΄
π₯ β π+
π΅
(π₯ β π)2+
πΆ
π₯ β π
8 ππ₯2 + ππ₯ + π
(π₯ β π)2(π₯ β π)
π΄
π₯ β π+
π΅
(π₯ β π)2+
πΆ
π₯ β π
9
ππ₯2 + ππ₯ + π
(π₯ β π)(π₯2 + ππ₯ + π)
where π₯2 + ππ₯ + π cannot
be factorized further
π΄
π₯ β π+
π΅π₯ + πΆ
π₯2 + ππ₯ + π
Integral of the type
πΏπππππ
ππ’πππππ‘ππ ,
πΏπππππ
βππ’πππππ‘ππ ,πΏπππππβππ’πππππ‘ππ
πΏπππππ = π΄π
ππ₯(ππ’πππππ‘ππ) + π΅
DEFINITE INTEGRALS
Definite integralas the limit of a sum
β«π(π₯)
π
π
ππ₯ = limββ0
β. [π(π) + π(π + β) + π(π + 2β) + β―+ π(π + (π β 1)β]
π€βπππ β =π β π
π β πβ = π β π
Properties Of Definite Integrals
1) β« π(π₯)ππ₯π
π= 0
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 19
2) β« π(π₯)π
πππ₯ = β« π(π‘)
π
πππ‘
3) β« π(π₯)π
πππ₯ = ββ« π(π₯)ππ₯
π
π
4) β« π(π₯)π
πππ₯ = β« π(π₯)ππ₯
π
π+ β« π(π₯)ππ₯,π€βπππ π < π < π
π
π
5) β« π(π₯)π
πππ₯ = β« π(π + π β π₯)
π
πππ₯
6) β« π(π₯)ππ₯π
0= β« π(π β π₯)ππ₯
π
0
7) β« π(π₯)2π
0ππ₯ = β« π(π₯)ππ₯
π
0+ β« π(2π β π₯)ππ₯
π
0
8) β« π(π₯)ππ₯2π
0= {
β« π(π₯)ππ₯ , ππ π(2π β π₯) = π(π₯)π
0
0 , ππ π(2π β π₯) = βπ(π₯)
9) β« π(π₯)π
βπππ₯ = {
2β« π(π₯)π
0ππ₯ , ππ π(π₯)ππ ππ£ππ. π. π π(βπ₯) = π(π₯)
0 , ππ π(π₯)ππ πππ. π. π π(βπ₯) = βπ(π₯)
APPLICATION OF INTEGRALS
CURVE β LINE
β’ CIRCLE β LINE
β’ PARABOLA β LINE
β’ ELLIPSE - LINE
CURVE β CURVE
β’ PARABOLA β PARABOLA
β’ PARABOLA β CIRCLE
β’ CIRCLE -CIRCLE
AREA OF TRIANGLE
β’ CO-ORDINATES OF VERTICES ARE GIVEN
β’ EQUATIONS OF SIDES ARE GIVEN
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 20
STEPS
DRAW THE DIAGRAM
MAKE A SHADED REGION
FIND INTERSECTION POINTS
IDENTIFY THE LIMITS
WRITE THE INTEGRAL(S) FOR THE REGION
EVALUATE THE INTEGRAL
THE VALUE SHOULD BE POSITIVE
1) π΄πππ ππ π βππππ ππππππ = β« π(π₯)ππ₯π
π
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 21
DIFFERENTIAL EQUATIONS
Methods of solving First Order and First Degree Differential Equations
Differential Equations with Variables seperables
Homogeneous differential equations
Linear differential equations.
Differential Equations with Variables separables
β’ Let the differential equation be ππ¦
ππ₯=
π(π₯)
π(π¦)
then π(π¦)ππ¦ = π(π₯)ππ₯
then integrate on both sides β«π(π¦)ππ¦ = β«π(π₯)ππ₯
β’ Let the differential equation be ππ¦
ππ₯=
π(π¦)
π(π₯)
then ππ¦
π(π¦)=
ππ₯
π(π₯)
then integrate on both sides β«ππ¦
π(π¦)= β«
ππ₯
π(π₯)
β’ Let the differential equation be ππ¦
ππ₯= π(π₯). π(π¦)
then ππ¦
π(π¦)= π(π₯)ππ₯
then integrate on both sides β«ππ¦
π(π¦)= β«π(π₯)ππ₯
Homogeneous differential equations
A function πΉ(π₯, π¦) is said to be homogeneous function of degree n if
πΉ(ππ₯, ππ¦) = πππΉ(π₯, π¦)
A differential equation of the form ππ¦
ππ₯= πΉ(π₯, π¦) is saidto be homogeneous if F(x,y)
is a homogeneous function of degree zero
i.e. if πΉ(ππ₯, ππ¦) = π0πΉ(π₯, π¦)
Steps to solve the homogeneous differential equation of the type: π π
π π= π(
π
π)
Let π¦ = π£π₯
ππ¦
ππ₯= π£ + π₯
ππ£
ππ₯
Substitute π¦ = π£π₯ and ππ¦
ππ₯= π£ + π₯
ππ£
ππ₯ in
ππ¦
ππ₯= π(
π¦
π₯)
Then use variables and separables in terms of π¦ and π£ only
Steps to solve the homogeneous differential equation of the type: π π
π π= π(
π
π)
Let π₯ = π£π¦
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 22
ππ₯
ππ¦= π£ + π¦
ππ£
ππ¦
Substitute π₯ = π£π¦ and ππ₯
ππ¦= π£ + π¦
ππ£
ππ¦ in
ππ₯
ππ¦= π(
π₯
π¦)
Then use variables and separables in terms of π₯ and π£ only
Linear differential equation
Steps to solve the Linear differential equation of the type: π π
π π+ π·(π)π = πΈ(π)
ππ¦
ππ₯+ π(π₯)π¦ = π(π₯)
πΌππ‘πππππ‘ππ πΉπππ‘ππ (πΌπΉ) = πβ«π(π₯)ππ₯
ππππ’π‘πππ ππ π¦. (πΌπΉ) = β«(πΌπΉ). π(π₯)ππ₯
Steps to solve the Linear differential equation of the type: π π
π π+ π·(π)π = πΈ(π)
ππ₯
ππ¦+ π(π¦)π₯ = π(π¦)
πΌππ‘πππππ‘ππ πΉπππ‘ππ (πΌπΉ) = πβ«π(π¦)ππ¦
ππππ’π‘πππ ππ π₯. (πΌπΉ) = β«(πΌπΉ). π(π¦)ππ¦
VECTORS
Position vector of the point π΄(π , π , π ) is ππ΄ = π₯π + π¦π + π§οΏ½οΏ½
π΄π΅ = ππ΅ β ππ΄
Let π = π₯π + π¦π + π§οΏ½οΏ½ then |π | = βπ₯2 + π¦2 + π§2
Unit vector of π =π
|π | , is denoted by οΏ½οΏ½
Let π = ππ + ππ + ποΏ½οΏ½ is said to be a unit vector if |π | = 1
Projection of π ππ οΏ½οΏ½ =π . οΏ½οΏ½
|οΏ½οΏ½ |
THREE DIMENSIONAL GEOMETRY
Direction cosines of a line are the cosines of the angles made by the line with the
positive directions of the coordinate axes.
Let a line making the angles with π₯, π¦, π§ axis are πΌ, π½, πΎ repectively.
Direction cosines are π = πππ πΌ,π = πππ π½, π = πππ πΎ
If π, π, π are the direction cosines of a line, then l2 + m2 +n2 = 1.
Direction ratios of a line joining two points π(π₯1, π¦1, π§1) πππ π(π₯2, π¦2, π§2) are
π = π₯2 β π₯1, π = π¦2 β π¦1, π = π¦2 β π¦1
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 23
If π, π, π are the direction cosines and π, π, π are the direction ratios of a line then
π = Β±π
βπ2+π2+π2, π = Β±
π
βπ2+π2+π2, π = Β±
π
βπ2+π2+π2
Direction cosines of a line joining two points
π(π₯1, π¦1, π§1) πππ π(π₯2, π¦2, π§2) πππ π₯2βπ₯1
ππ,π¦2βπ¦1
ππ,π§2βπ§1
ππ
where PQ= β(π₯2 β π₯1)2 + (π¦2 β π¦1)2 + (π§2 β π§1)2
Direction ratios of a line are the numbers which are proportional to the direction
cosines of a line.
Skew lines are lines in space which are neither parallel nor intersecting. They lie in
different planes.
Angle between skew lines is the angle between two intersecting lines drawn from any
point (preferably through the origin) parallel to each of the skew lines.
If π1, π1, π1 πππ π2, π2, π2 are the direction cosines of two lines; and π is the acute angle
between the two lines; then πππ π|π1π2 + π1π2 + π1π2|
If π1, π1, π1πππ π2, π2, π2 are the direction ratios of two lines and π is the acute angle
between the two lines; then πππ π = |π1π2+π1π2+π1π2
βπ12+π1
2+π12βπ2
2+π22+π2
2|
Vector equation of a line that passes through the given point whose position vector is π
and parallel to a given vectorοΏ½οΏ½ ππ π = π + ποΏ½οΏ½ .
Equation of a line through a point (π₯1, π¦1, π§1) and having direction cosines π, π, π is π₯βπ₯1
π=
π¦βπ¦1
π=
π§βπ§1
π
The vector equation of a line which passes through two points whose position vectors
are π πππ οΏ½οΏ½ ππ π = π + π(οΏ½οΏ½ β π )
Cartesian equation of a line that passes through two points
(π₯1, π¦1, π§1) πππ (π₯2, π¦2, π§2) ππ π₯βπ₯1
π₯2βπ₯1=
π¦βπ¦1
π¦2βπ¦1=
π§βπ§1
π§2βπ§1 .
If π is the acute angle between π = π1 + ππ1 πππ π = π2 + ππ2
, π‘βππ πππ π = |π1 .π2
|π1 ||π2 ||
If π₯βπ₯1
π1=
π¦βπ¦1
π1=
π§βπ§1
π1πππ
π₯βπ₯2
π2=
π¦βπ¦2
π2=
π§βπ§2
π2 are the equations of two lines, then the
acute angle between the two lines is given by πππ π = |π1π2 + π1π2 + π1π2|.
Shortest distance between two skew lines is the line segment perpendicular to both the
lines.
Shortest distance between π = π1 + ππ1 πππ π = π2 + ππ2
ππ |(π1 Γπ2 ).(π2 βπ1 )
|π1 Γπ2 ||
Shortest distance between the lines: π₯βπ₯1
π1=
π¦βπ¦1
π1=
π§βπ§1
π1 πππ
π₯βπ₯2
π2=
π¦βπ¦2
π2=
π§βπ§2
π2 ππ
|
π₯2βπ₯1 π¦2βπ¦1 π§2βπ§1π1 π1 π1π2 π2 π2
|
β(π1π2βπ2π1)2+(π1π2βπ2π1)2+(π1π2βπ2π1)2
Distance between parallel lines π = π1 + ποΏ½οΏ½ πππ π = π2 + ποΏ½οΏ½ ππ |οΏ½οΏ½ Γ(π2 βπ1 )
|οΏ½οΏ½ ||
D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA. Page 24
In the vector form, equation of a plane which is at a distance π from the origin, and οΏ½οΏ½ is
the unit vector normal to the plane through the origin is π . οΏ½οΏ½ = π.
Equation of a plane which is at a distance of π from the origin and the direction cosines
of the normal to the plane as π, π, π ππ ππ₯ + ππ¦ + ππ§ = π.
The equation of a plane through a point whose position vector is π and perpendicular to
the vector οΏ½οΏ½ ππ (π β π ). οΏ½οΏ½ = 0.
Equation of a plane perpendicular to a given line with direction ratios A,B,C and passing
through a given point (π₯1,π¦1, π§1) ππ π΄(π₯ β π₯1) + π΅(π¦ β π¦1) + πΆ(π§ β π§1) = 0
Equation of a plane passing through three non collinear points
(π₯1, π¦1, π§1), (π₯2, π¦2, π§2)πππ (π₯3, π¦3, π§3) ππ |
π₯ β π₯1 π¦ β π¦1 π§ β π§1
π₯2 β π₯1 π¦2 β π¦1 π§2 β π§1
π₯3 β π₯1 π¦3 β π¦1 π§3 β π§1
| = 0
Vector equation of a plane that contains three non collinear points having position
vectors π , οΏ½οΏ½ πππ π ππ (π β π ). [(οΏ½οΏ½ β π ) Γ (π β π )] = 0
Equation of a plane that cuts the coordinates axes at
(π, 0,0), (0, π, 0) πππ (0,0, π) ππ π₯
π+
π¦
π+
π§
π= 1
Vector equation of a plane that passes through the intersection of planes
π . π1 = π1 πππ π . π2 = π2 ππ π . (π1 + ππ2 ) = π1 + ππ2, π€βπππ πis any nonzero
constant.
Vector equation of a plane that passes through the intersection of two given planes
π΄1π₯ + π΅1π¦ + πΆ1π§ + π·1 = 0 πππ π΄2π₯ + π΅2π¦ + πΆ2π§ + π·2 = 0 ππ
(π΄1π₯ + π΅1π¦ + πΆ1π§ + π·1) + π(π΄2π₯ + π΅2π¦ + πΆ2π§ + π·2) = 0
Two planes π = π1 + ππ1 πππ π = π2 + ππ2
are coplanar if (π2 β π1 ). (π1 Γ π2
) = 0
Two planes π1π₯ + π1π¦ + π1π§ + π1 = 0 πππ π2π₯ + π2π¦ + π2π§ + π2 = 0 are coplanar if
|
π₯2 β π₯1 π¦2 β π¦1 π§2 β π§1
π1 π1 π1
π2 π2 π2
| = 0.
In the vector form, if π is the angle between the two planes,
π . π1 = π1 πππ π . π2 = π2, π‘βππ π = πππ β1 |οΏ½οΏ½ 1.π2 |
|π1 ||π2 |.
The angle π between the line π = π + ποΏ½οΏ½ and the plane π . οΏ½οΏ½ = π ππ sin π = |οΏ½οΏ½ .οΏ½οΏ½
|οΏ½οΏ½ ||οΏ½οΏ½||
The angle π between the planes π΄1π₯ + π΅1π¦ + πΆ1π§ + π·1 = 0 πππ π΄2π₯ + π΅2π¦ + πΆ2π§ +
π·2 = 0 is given by cos π = ||π΄1π΄2+π΅1π΅2+πΆ1πΆ2
βπ΄12+π΅1
2+πΆ12βπ΄2
2+π΅22+πΆ2
2
||
The distance of a point whose position vector is π from the plane π . οΏ½οΏ½ = π ππ |π β π . οΏ½οΏ½|
The distance from a point (π₯1, π¦1, π§1) to the plane π΄π₯ + π΅π¦ + πΆπ§ + π· =
0 ππ |π΄π₯1+π΅π¦1+πΆπ§1+π·
βπ΄2+π΅2+πΆ2|