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Relations and Functions Relation

7. Equivalence Classes Given, an arbitrary equivalence relation R on an arbitrary set X, R divides X into mutually disjoint subsets Ai called partitions satisfying Ai ∪ Aj = X and Ai ∩ Aj = φ, i ≠ j . The subset Ai are called equivalence class and it is denoted by [a].

Let A be a non-empty set and R ⊆ A × A. Then, R is called a relation on A. If (a, b) ∈ R, then we say that a is related to b and we write aRb. If (a, b) ∉ R, then we write b R/ a.

Types of Relation

Function

1. Empty Relation

Let A and B be two non-empty sets. Then, a relation f from A to B which associates to each element x ∈ A, a unique element of f(x) ∈ B is called a function from A to B and we write f : A → B. Here, A is called the domain of f, i.e., dom(f ) = A, B is called the codomain of f. Also, {f(x) : x ∈ A} ⊆ B is called the range of f.

A relation R on a set A is called empty relation, if no element of A is related to any element of A i.e., R = φ ⊂ A × A is the empty relation.

2. Universal Relation A relation R on a set A is called universal relation, if each element of A is related to every element of A i.e., R = A × A.

Every function is a relation but every relation is not a function.

Types of Function

Both the empty relation and the universal relation are sometimes called trivial relations.

1. One-one (Injective) Function

3. Reflexive Relation

A function f : A → B is said to be one-one, if f ( x1 ) = f ( x2 ) ⇒ x1 = x2 or x1 ≠ x2 ⇒ f ( x1 ) ≠ f ( x2 )

A relation R defined on set A is said to be reflexive, if ( x, x ) ∈ R, ∀ x ∈ A.

where, x1, x2 ∈ A

4. Symmetric Relation

2. Many-one Function

A relation R defined on set A is said to be symmetric, if ( x, y ) ∈ R ⇒ ( y, x ) ∈ R, ∀ x, y ∈ A

A function f : A → B is said to be many-one, if two or more than two elements in A have the same image in B, i . e ., if x1 ≠ x2 , then f ( x1 ) = f ( x2 ).

5. Transitive Relation A relation R defined on set A is said to be transitive, if ( x, y ) ∈ R and ( y, z ) ∈ R ⇒ ( x, z ) ∈ R, ∀ x, y, z ∈ A

3. Onto (Surjective) Function A function f : A → B is said to be onto, if every element in B has its pre-image in A, i.e., if for each y ∈ B, there exists an element x ∈ A, such that f ( x ) = y.

6. Equivalence Relation A relation R on a set A is called an equivalence relation, if it is reflexive, symmetric and transitive.

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Fast Track Revision Notes

Mathematics-XII

4. Into Function

Binary Operation

A function f : A → B is said to be into, if atleast one element of B do not have a pre-image.

Let S be a non-empty set and ∗ be an operation on S such that a ∈ S, b ∈ S ⇒ a ∗ b ∈ S , ∀ a, b ∈ S Then, ∗ is called a binary operation on S.

5. One-one and Onto (Bijective) Function A function f : X → Y is said to be one-one and onto, if f is both one-one and onto.

6. Composite Function

Group

Let f : A → B and g : B → C, then

An algebraic structure (S ∗ ) consisting of a non-void set S and a binary operation ∗ defined on S is called a group, if it satisfies following axioms. (i) Closure property We say that ∗ on S satisfies the closure property, if a ∈S, b ∈ S ⇒ a ∗ b ∈ S , ∀ a, b ∈ S

gof : A → C, such that (gof) ( x ) = g { f ( x )}, ∀ x ∈ A (i) In generally, gof ≠ fog . (ii) In generally, if gof is one-one, then f is one-one. And if gof is onto, then g is onto.

(ii) Commutative law Operation ∗ on S is said to be commutative, if a ∗ b = b ∗ a, ∀ a, b ∈ S . (iii) Associative law Operation ∗ on S is said to be associative, if (a ∗ b)∗ c = a ∗ (b ∗ c ); a, b, c ∈ S . (iv) Identity law An element e ∈ S is said to be the identity element of a binary operation ∗ on set S, if a ∗e = e ∗ a = a, ∀ a ∈ S (v) Invertible law or inverse law An element a ∈ S is said to be invertible, if there exists an element b ∈ S , such that a ∗ b = b ∗ a = e , ∀ b ∈ S. Element b is called inverse of element a.

7. Invertible Function A function f : X → Y is defined to be invertible, if there exists a function g : Y → X, such that gof = IX and fog = IY . The function g is called the inverse of f and it is denoted by f −1. Thus, f is invertible, then f must be one-one and onto and vice-versa. A

f

B

g

C

f(x)

x

g [f(x)]

gof

(i) If f : X → Y, g : Y → Z and h : Z → S are functions, then ho(gof ) = (hog)of. (ii) Let f : X → Y and g : Y → Z be two invertible functions. −1

Then, gof is also invertible with (gof )

−1

Zero is identity for the addition operation on R but it is not identity for addition operation on N.

−1

= f og .

Inverse Trigonometric Functions Trigonometric functions are not one-one and onto on their natural domains and ranges, so their inverse does not exist in all values but their inverse may exists in some interval of their domains and ranges. Thus, we can say that, inverse of trigonometric functions are defined within restricted domains of corresponding trigonometric functions. Inverse of f is denoted by f −1. Let y = f ( x ) = sin x be a function. Inverse

∴Its inverse is x = sin −1 y, i.e., sin x → sin −1 x. sin−1 x ≠ (sin x)−1

1 sin−1 x ≠ sin−1 x

1 (sin x)−1 ≠ sin x

Domain, Range and Principal Values of Inverse Trigonometric Functions Function

Domain

Range

x

[–1, 1]

− π , π 2 2

cos −1 x

[–1, 1]

[0, π ]

tan−1 x

R

− π, π 2 2

cosec −1 x

( −∞, − 1 ] ∪ [1, ∞ )

− π , π − { 0} 2 2

sec −1 x

( −∞, − 1 ] ∪ [1, ∞ )

π [0, π ] − 2

cot −1 x

R

( 0, π )

sin

−1

2

Principal value branch π π ≤ y≤ , 2 2

where y = sin−1 x

0 ≤ y ≤ π,

where y = cos −1 x

−

π π < y< , 2 2

where y = tan−1 x

−

π π ≤ y ≤ , y ≠ 0, where y = cosec −1 x 2 2

−

π 0 ≤ y ≤ π, y ≠ , 2

where y = sec −1 x

0 < y < π, where y = cot −1 x

Fast Track Revision Notes

Mathematics-XII T-ratios of Some Standard Angles

Angle (θ)

0° = 0

π 6

30 ° =

Ratio

45 ° =

π 4

60 ° =

π 3

90 ° =

sin θ

0

1 2

1 2

3 2

1

cos θ

1

3 2

1 2

1 2

0

tan θ

0

1 3

1

π 2

∞

3

Transformation of one Inverse Trigonometric Function to another Inverse Trigonometric Functions Function −1

Transformation of a Function

(i)

sin

x

cos −1 ( 1 − x2 )

x tan−1 1 − x2

1 − x2 cot −1 x

1 sec −1 1 − x2

1 cosec −1 x

(ii)

cos −1 x

sin−1 ( 1 − x2 )

1 − x2 tan−1 x

x cot −1 1 − x2

1 sec −1 x

1 cosec −1 1 − x2

(iii)

tan−1 x

x sin−1 1 + x2

1 cos −1 1 + x2

1 cot −1 x

sec −1 1 + x2

1 + x2 cosec −1 x

Properties of Inverse Trigonometric Functions (iv) cot −1 (− x ) = π − cot −1 x, x ∈ R

Property I

(v) cosec −1 (− x ) = π − cosec −1 x, x ≥ 1

π π (i) sin −1 (sin θ ) = θ, θ ∈ − , 2 2 (ii) cos

−1

(vi) sec −1 (− x ) = π − sec −1 x, x ≥ 1

(cos θ ) = θ, θ ∈[0, π ]

π π (iii) tan −1 (tan θ ) = θ, θ ∈ − , 2 2

Property III (i) sin −1 (1 / x ) = cosec −1 x, x ≥ 1 or x ≤ − 1

(iv) cot −1(cot θ ) = θ, θ ∈ (0, π ) π π (v) cosec −1 (cosec θ ) = θ, θ ∈ − , − { 0} 2 2

(ii) cos −1 (1 / x ) = sec −1 x, x ≥ 1 or x ≤ − 1

π (vi) sec −1 (sec θ ) = θ, θ ∈ [0, π ] − 2

(iii) tan

(vii) sin (sin −1 x ) = x, x ∈ [− 1, 1]

π , x ∈ [− 1, 1] 2 π (ii) tan −1 x + cot −1 x = , x ∈ R 2 π −1 −1 (iii) cosec x + sec x = , x ∈ (−∞,−1] ∪ [1, ∞ ) 2

(ix) tan (tan −1 x ) = x, x ∈ R

(i) sin −1 x + cos −1 x =

(x) cot (cot −1 x ) = x, x ∈ R (xi) cosec (cosec −1 x ) = x, x ∈ (− ∞, − 1] ∪ [1, ∞ ) (xii) sec (sec −1 x ) = x, x ∈ (− ∞, − 1] ∪ [1, ∞ )

Property II (i) sin

(ii) cos

(− x ) = − sin

−1

cot −1 x , x > 0 1 = x −1 − π + cot x, x < 0

Property IV

(viii) cos (cos −1 x ) = x, x ∈ [− 1, 1]

−1

−1

Property V −1

x, x ∈ [− 1, 1]

(i) sin −1 x + sin −1 y = sin −1[ x 1 − y 2 + y 1 − x 2 ],

(− x ) = π − cos −1 x, x ∈[− 1, 1]

if −1 ≤ x, y ≤ 1 and x 2 + y 2 ≤ 1 or

(iii) tan −1 (− x ) = − tan −1 x, x ∈ R

if xy < 0 and x 2 + y 2 > 1

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(ii) sin −1 x − sin −1 y = sin −1[ x 1 − y 2 − y 1 − x 2 ],

(vi) tan ( A − B) =

if −1 ≤ x, y ≤ 1 and x 2 + y 2 ≤ 1 or

cot A cot B − 1 cot B + cot A cot A cot B + 1 (viii) cot ( A − B) = cot B − cot A

(vii) cot ( A + B) =

if xy > 0 and x 2 + y 2 > 1 (iii) cos −1 x + cos −1 y = cos −1[ xy − 1 − x 2 1 − y 2 ], if −1 ≤ x, y ≤ 1 and x + y ≥ 0

(ix) 2 sin A cos B = sin ( A + B) + sin ( A − B)

(iv) cos −1 x − cos −1 y = cos −1[ xy + 1 − x 2 1 − y 2 ],

(x) 2 cos A cos B = sin ( A + B) − sin ( A − B)

if −1 ≤ x, y ≤ 1 and x ≤ y

(xi) 2 cos A cos B = cos ( A + B) + cos ( A − B)

x + y (v) tan −1 x + tan −1 y = tan −1 , xy < 1 1 − xy (vi) tan

−1

x − tan

−1

(xii) 2 sin A sin B = cos ( A − B) − cos ( A + B) C − D C + D (xiii) sin C + sin D = 2 sin cos 2 2

x− y y = tan , xy > −1 1 + xy −1

C − D C + D (xiv) sin C − sin D = 2 cos sin 2 2

Property VI (i) 2 sin −1 x = sin −1 (2 x 1 − x 2 ), −

C − D C + D (xv) cos C + cos D = 2 cos cos 2 2

1 1 ≤x≤ 2 2

C − D C + D (xvi) cos C − cos D = − 2 sin sin 2 2

(ii) 2 cos −1 x = cos −1 (2 x 2 − 1), 0 ≤ x ≤ 1 (iii) 2 tan

−1

x = sin

−1

tan A − tan B 1 + tan A tan B

D − C C + D = 2 sin sin 2 2

2x , | x | ≤ 1 or −1 ≤ x ≤ 1 1 + x2

(xvii) sin 2 x = 2 sin x cos x (xviii) cos 2 x = cos 2 x − sin 2 x = 1 − 2 sin 2 x = 2 cos 2 x − 1

2

1− x (iv) 2 tan −1 x = cos −1 , x ≥ 0 1 + x2

(xix) tan 2 x =

2x (v) 2 tan −1 x = tan −1 , − 1< x ≤ 1 1 − x2

2 tan x 1 − tan 2 x

(xx) 1 + cos 2 x = 2 cos 2 x; 1 − cos 2 x = 2 sin 2 x (xxi) sin 2 x =

Some Useful Trigonometric Formulae

2 tan x 1 − tan 2 x ; = x 2 cos 1 + tan 2 x 1 + tan 2 x

(xxii) sin 3 x = 3 sin x − 4 sin 3 x;

(i) sin( A + B) = sin A cos B + cos A sin B

cos 3 x = 4 cos 3 x − 3 cos x

(ii) sin( A − B) = sin A cos B − cos A sin B (iii) cos( A + B) = cos A cos B − sin A sin B

(xxiii) tan 3 x =

(iv) cos( A − B) = cos A cos B + sin A sin B tan A + tan B (v) tan( A + B) = 1 − tan A tan B

3 tan x − tan 3 x 1 − 3 tan 2 x

Matrices Matrix

If m = n, then matrix is a square matrix. A matrix is denoted by the symbol [ ]. i.e., [A] = [aij] m × n

A matrix is an ordered rectangular array of numbers or functions. The number or functions are called the elements or the entries of the matrix.

We shall consider only those matrices whose elements are real number or function taking real values.

Order of Matrix A matrix of order m × n is a11 a12 a a22 A = 21 K K a m1 am2

Types of Matrices

of the form a13 K a1n a23 K a2 n K K K am3 K amn

(i) Row matrix A matrix having only one row and many columns, is called a row matrix. (ii) Column matrix A matrix having only one column and many rows, is called a column matrix.

Its element in the ith row and jth column is aij.

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(iii) Zero matrix or null matrix If all the elements of a matrix are zero, then it is called a zero or null matrix. It is denoted by symbol O. (iv) Square matrix A matrix in which number of rows and number of columns are equal, is called a square matrix. (v) Diagonal matrix A square matrix is said to be a diagonal matrix, if all the elements lying outside the diagonal elements are zero. (vi) Scalar matrix A diagonal matrix in which all principal diagonal elements are equal, is called a scalar matrix. (vii) Unit matrix or identity matrix A square matrix having 1 (one) on its principal diagonal and 0 (zero) elsewhere, is called an identity matrix. It is denoted by symbol I. (viii) Equality of Matrix Two matrices are said to be equal, if their order is same and their corresponding elements are also equal.

Addition of Matrices Let A and B be two matrices each of order m × n. Then, the sum of matrices A + B is defined, if matrices A and B are of same order. If A = [aij ]m × n , B = [aij ]m × n Then, A + B = [aij + bij ]m × n

Properties of Addition of Matrices If A, B and C are three matrices of same order m × n, then (i) Commutative law A + B = B + A (ii) Associative law ( A + B) + C = A + (B + C ) (iii) Existence of additive identity A zero matrix (O) of order m × n (same as of A), is additive identity, if A+ O = A=O + A (iv) Existence of additive inverse If A is a square matrix, then the matrix (− A), is additive inverse, if A + ( − A) = O = ( − A) + A If A and B are not of same order, then A + B is not defined.

Difference of Matrices If A = [aij ], B = [bij ] are two matrices of the same order m × n, then difference, A − B is defined as a matrix D = [d ij ], where d ij = aij − bij, ∀i , j .

Multiplication of a Matrix by a Scalar Let A = [aij ]m × n be a matrix and k be any scalar. Then, the matrix obtained by multiplying each element of A by k is called the scalar multiple of A by k, i.e., kA = [kaij ]m × n . (i) k( A + B) = kA + kB (ii) (k + l )A = kA + lA

Multiplication of Matrices Let A = [aij ]m × n and B = [bij ]n × p be two matrices such that the number of columns of A is equal to the number of rows of B, then multiplication of A and B is denoted by AB, is given by c ij =

n

∑ aik bkj

Properties of Multiplication of Matrices (i) Associative law ( AB)C = A(BC ) (ii) Existence of multiplicative identity A ⋅ I = A = I ⋅ A, I is called multiplicative identity. (iii) Distributive law A(B + C ) = AB + AC

Transpose of a Matrix The matrix obtained by interchanging the rows and columns of a given matrix A, is callled transpose of a matrix. It is denoted by A′ or AT .

Properties of Transpose of Matrices (i) ( A + B)′ = A′ + B′ (iii) ( AB)′ = B′ A′

(ii) (kA)′ = kA′ (iv) ( A′ )′ = A

Symmetric and Skew-Symmetric Matrix A square matirx A is callled symmetric, if A′ = A. A square matrix A is called skew-symmetric, if A′ = − A.

Properties of Symmetric and Skew-symmetric Matrix (i) For any square matrix A with real number entries, A + A′ is a symmetric matrix and A − A′ is a skew-symmetric matrix. (ii) Any square matrix can be expressed as the sum of symmetric and a skew-symmetric matrix. 1 1 i.e., A = ( A + A′ ) + ( A − A′ ) 2 2 (iii) The principal diagonal elements skew-symmetric matrix are always zero.

of

a

Elementary Operations of a Matrix There are six operations (transformations) on a matrix, three of which are due to rows and three due to columns, which are known as elementary operations or transformations. (i) The interchange of any two rows or two columns Symbolically, the interchange of ith and jth rows is denoted by Ri ↔ R j and interchange of ith and jth columns is denoted by Ci ↔ C j. (ii) The multiplication of the elements of any row or column by a non-zero number Symbolically, the multiplication of each element of the ith row by k, where k ≠ 0 is denoted by Ri → kRi. The corresponding column operation is denoted by Ci → kCi. (iii) 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 Symbolically, the addition to the elements of ith row, the corresponding elements of jth row multiplied by k is denoted by Ri → Ri + kR j. The corresponding column operation is denoted by Ci → Ci + kC j.

k =1

where, c ij is the element of matrix C and C = AB. Generally, it is not commutative AB ≠ BA.

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Fast Track Revision Notes

Mathematics-XII Properties of Invertible Matrices

Invertible Matrices If A is a square matrix of order m and 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 A−1. ❖ ❖

Let A and B be two non-zero matrices of same order. (i) Uniqueness of inverse If inverse of a square matrix exists, then it is unique. (ii) AA−1 = A−1 A = I (iii) ( AB)−1 = B −1 A−1

A rectangular matrix does not posses inverse matrix. If B is an inverse of A, then A is also the inverse of B.

(iv) ( A−1 )− 1 = A (v) ( A′ )−1 = ( A−1 )′ or ( AT )−1 = ( A−1 )T where, A′ or AT is transpose of a matrix A.

Determinants Determinant

Minors and Cofactors

Every square matrix A is associated with a number, called its determinant and it is denoted by det(A) or A.

Minors Minor of an element aij of a matrix is the determinant obtained by deleting i th row and jth column. It is denoted by Mij. a11 a12 a13 If A = a21 a22 a23 , then

Expansion of Determinant of Order (2×2) a11 a12 = a11 a22 − a12 a21 a21 a22

a31 a32

Expansion of Determinant of Order (3 × 3) a11 a12 a21 a22 a31 a32

a33

Minors of A are a a M11 = 22 23 , a32 a33

a13 a23 = a11 (a22 a33 − a32 a23 ) a33

M12 =

a21 a23 , a31 a33

M13 =

a21 a22

− a12 (a21a33 − a31a23 ) + a13 (a21a32 − a31a22 ) Similarly, we can expand the above determinant corresponding to any row or column.

a31 a32

, etc.

The minor of an element of a determinant of order n (n ≥ 2 ) is a determinant of order n − 1.

Properties of Determinants (i) If the rows and columns of a determinant are interchanged, then the value of the determinant does not change. (ii) If any two rows (columns) of a determinant are interchanged, then sign of determinant changes. (iii) If any two rows (columns) of a determinant are identical, then the value of the determinant is zero. (iv) If each element of a row (column) is multiplied by a non-zero number k, then the value of the determinant is multiplied by k. By this property, we can take out any common factor from any one row or any one column of a determinant. (v) If A is a n × n matrix, then| kA| = k n |⋅ A|. (vi) If each element of any row (column) of a determinant is added k times the corresponding element of another row (column), then the value of the determinant remains unchanged. (vii) If some or all elements of a row (column) of a determinant are expressed as sum of two (more) terms, then the determinant can be expressed as sum of two (more) determinants.

6

Cofactor If Mij is the minor of an element aij, then the cofactor of aij is denoted by Cij or Aij and defined as follows Aij = Cij = (− 1)i + j Mij Cofactors of A are Cij = (− 1)i + j Mij where, i = 1, 2, 3 and j = 1, 2, 3. If elements of a row (column) are multiplied with cofactors of any other row (column), then their sum is zero.

Area of Triangle Let A( x1, y1 ), B( x2 , y2 ) and C( x3 , y3 ) be the vertices of a ∆ABC. Then, its area is given by x1 1 x2 ∆= 2 x3 =

y1 1 y2 1 y3 1

1 ⋅ x1( y2 − y3 ) + x2 ( y3 − y1 ) + x3 ( y1 − y2 ) 2

Fast Track Revision Notes

Mathematics-XII

(i) Since, area is positive quantity. So, we always take the absolute value of the determinant.

Properties of Inverse of a Square Matrix (A) (i) ( A−1 )−1 = A

(ii) If area is given, use both positive and negative values of the determinant for calculation.

(iii) ( AT )−1 = ( A−1 )T

(iv) (kA)−1 = kA−1

(v) adj ( A−1 ) = (adj A)−1

Condition of Collinearity Three points A( x1, y1 ), B( x2 , y2 ) and C( x3 , y3 ) are collinear, when ∆ = 0. x1 y1 1 x 2 y2 1 = 0 i . e ., x3

(ii) ( AB)−1 = B −1 A−1

A is said to be singular, if| A|= 0.

System of Linear Equations

y3 1

Let the system of equations be a1 x + b1 y + c1 z = d1, a2 x + b2 y + c 2 z = d 2 and a3 x + b3 y + c 3 z = d 3 Then, this system of equations can be written as AX = B d1 x a1 b1 c1 where, A = a2 b2 c 2 , X = y and B = d 2 d 3 z a3 b3 c 3

Adjoint of a Matrix The adjoint of a square matrix A is defined as the transpose of the matrix formed by cofactors. Let A = [ aij ] be a square matrix of order n, then adjoint of A, i . e ., adj A = CT , where C = [Cij ] is the cofactor matrix of A.

Properties of Adjoint of Square Matrix

A system of equations is consistent or inconsistent according as its solution exists or not. (i) For a square matrix A in matrix equation AX = B (a) | A| ≠ 0, then system of equations is consistent and has unique solution. (b) | A| = 0 and (adj A) B ≠ O, then there exists no solution, i . e ., inconsistent. (c) | A| = 0 and (adj A) B = O, then system of equations is consistent and has an infinite number of solutions. 0 (ii) When B = 0 , in such cases, we have 0 (a) | A| ≠ 0 ⇒ System has only trivial solution i.e., x = 0, y = 0 and z = 0 (b) | A| = 0 ⇒ System has infinitely many solutions.

If A and B are square matrices of order n, then (i) A(adj A) = | A| In = (adj A)A (ii) adj ( AT ) = (adj A)T (iii) | adj A| = | A|n − 1, if| A| ≠ 0 2

(iv) |adj [adj ( A)]| = | A|( n −1) ' if| A| ≠ 0 (v) (adj AB) = (adj B) (adj A)

Inverse of a Matrix A square matrix A has inverse, if and only if A is a non-singular (| A| ≠ 0 ) matrix. The inverse of A is denoted by A−1 i.e., 1 A− 1 = (adj A),| A| ≠ 0 | A|

Continuity and Differentiability Continuous Function

Some Basic Continuous Function (i) Every constant function is continuous.

A real function f is said to be continuous, if it is continuous at every point in the domain f.

(ii) Every identity function is continuous. (iii) Every polynomial function is continuous.

Continuity at a Point

Algebra of Continuous Function

Suppose f is a real valued function on a subset of the real numbers and let c be a point in the domain of f. Then, f is continuous at c, if lim f ( x ) = f (c ).

If f and g are two continuous functions in domain D, then (i) (f + g ) is continuous.

x→c

(ii) (f − g ) is continuous. (iii) cf is continuous. (iv) fg is continuous. f (v) is continuous in domain except at the points, g where, g( x ) = 0.

i.e., if f (c ) = lim f ( x ) = lim f ( x ), then f ( x ) is continuous at x→c +

x→c −

x = c . Otherwise, f ( x ) is discontinuous at x = c . Graphically, a function f( x) is said to be continuous at a point, if the graph of the function has no break point.

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Fast Track Revision Notes

Mathematics-XII

(vi) If f is continuous, then f is also continuous. (vii) Every rational function is continuous.

(xv)

d −1 , −1 < x < 1 (cos −1 x ) = dx 1 − x2

(viii) Suppose f and g are real valued functions such that (fog) is defined at c, if g is continuous at c and f is continuous at g(c ), then (fog) is continuous at c.

(xvi)

d 1 (tan −1 x ) = dx 1 + x2

(xvii)

d −1 (cot −1 x ) = dx 1 + x2

(xviii)

1 d ,| x| > 1 (sec −1 x ) = dx | x| x 2 − 1

Differentiability or Derivability A function f is said to be derivable or differentiable at x = c , if its left hand and right hand derivatives at c exist and are equal. f (a + h ) − f (a) (i) Right Hand Derivative Rf ′ (a) = lim h →0 h f (a − h ) − f (a) (ii) Left Hand Derivative Lf ′ (a) = lim h →0 −h

(xix)

Derivative of Composite Function by Chain Rule

f ( x ) is differentiable at x = a, if Rf ′(a) = L f ′(a). Otherwise, f ( x ) is not differentiable at x = a.

Let f be a real valued function which is a composite of two functions u and v, i.e., f = vou . Suppose t = u ( x ) and if both dt dv and exist, we have dx dt df dv dt = . dx dt dx

(i) Graphically, a function is not differentiable at a corner point of a curve. (ii) Every differentiable function is continuous. But a continuous function need not be differentiable.

Derivatives of Two or More Functions

Differentiation

d du dv (u ± v ) = ± dx dx dx d du dv dw (ii) (u ± v ± w ± K ) = ± ± ±K dx dx dx dx d d d (iii) [product rule] (u ⋅ v ) = u (v ) + v (u ) dx dx dx d d v (u ) − u (v ) d u dx dx [quotient rule] (iv) = 2 dx v v (i)

The process of finding derivative is called differentiation.

Derivatives of Standard Functions (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) (xiii) (xiv)

d dx d dx d dx d dx d dx d dx d dx d dx d dx d dx d dx d dx d dx

d −1 ,| x| > 1 (cosec −1 x ) = dx | x| x 2 − 1

( x n ) = nx n −1 (constant) = 0 (cx n ) = cn x n −1

(v) y = [f ( x )]g( x ) dy ⇒ = [f ( x )]g( x ) dx

(sin x ) = cos x (cos x ) = − sin x

g( x ) f ( x ) f ′ ( x ) + log f ( x )⋅ g′ ( x )

(vi) If y = f [g( x )], then

(tan x ) = sec 2 x

dy = f ′ [g( x )] g′ ( x ) dx

Useful Logarithmic Formulae

(cosec x ) = − cosec x cot x

(i) log a mn = log a m + log a n m (ii) log a = log a m − log a n n

(sec x ) = sec x tan x

(iii) log a mn = n log a m

(cot x ) = − cosec 2 x

For m > 0, n > 0, a > 0 and a ≠ 1

(iv) log a a = 1; a > 0 and a ≠ 1

(e x ) = e x

(v) log a m = log b m + log a b; a > 0, b > 0, a ≠ 1, b ≠ 1 and m>0

(a x ) = a x loge a, a > 0

(vi) log a m × log m a = 1 ; a > 0, m > 0, a ≠ 1 and m ≠ 1 1 (vii) log bm a = log b a ; a > 0, b > 0 and b ≠ 1 m log m a (viii) log b a = ; a > 0, b > 0, b ≠ 1, m > 0 and m ≠ 1 log m b

1 (loge x ) = , x > 0 x 1 , a > 0, a ≠ 1 (log a x ) = x loge a

m

1 d , −1 < x < 1 (sin −1 x ) = dx 1 − x2

(ix) alog a = m; a > 0, m > 0, a ≠ 1

8

Fast Track Revision Notes

Mathematics-XII

Rolle’s Theorem

Lagrange’s Mean Value Theorem

If a function y = f ( x ) is defined in [a, b] and (i) f ( x ) is continuous in [a, b]. (ii) f ( x ) is differentiable in (a, b) and (iii) f (a) = f (b) Then, there will be atleast one value of c ∈(a, b) such that f ′ (c ) = 0.

If a function f ( x ) is said to be defined on [a, b] and (i) continuous in [a, b] and (ii) differentiable in (a, b), then there will be atleast one f (b) − f (a) . value of c ∈(a, b) such that f ′ (c ) = b−a Lagrange’s mean value theorem is valid irrespective of whether f(a) = f(b ) or f(a) ≠ f(b ).

Application of Derivatives Rate of Change of Quantities

(iii) Area of a trapezium =

If y = f ( x ) is a function, where y is dependent variable and x dy [or f ′ ( x )] represents the is independent variable. Then, dx dy rate of change of y w.r.t. x and [or f ′ ( x0 )] dx x = x

1 (Sum of parallel sides) 2

× Perpendicular distance between them (iv) Area of a circle = πr 2 and circumference of a circle

0

= 2πr, where r is the radius of circle. 4 (v) Volume of sphere = πr 3 and surface area = 4πr 2 3 where, r is the radius of sphere.

represents the rate of change of y w.r.t. x at x = x0 . ∆y (i) Average rate of change of y w.r.t. x = ∆x dy (ii) Instantaneous rate of change of y w.r.t. x = dx dy dy / dt dx , if (iii) Related rate of change = = ≠0 dx dx / dt dt

(vi) Total surface area of a right circular cylinder = 2 πrh + 2 πr 2 Curved surface area of right circular cylinder = 2πrh

dy dy is positive, if y increases as x increases and is dx dx negative, if y decreases as x increases. Here,

and

volume = πr 2 h

where, r is the radius and h is the height of the cylinder. 1 (vii) Volume of a right circular cone = πr 2 h, 3 Curved surface area = πrl and total surface area = πr 2 + πrl where, r is the radius, h is the height and l is the slant height of the cone.

Marginal Cost Marginal cost represents the instantaneous rate of change of the total cost at any level of output. If C( x ) represents the cost function for x units produced, then marginal cost, d denoted by MC, is given by MC = {C( x )}. dx

(viii) Volume of a parallelopiped = xyz and surface area = 2( xy + yz + zx ) where, x,y and z are the dimensions of parallelopiped.

Marginal Revenue Marginal revenue represents the rate of change of total revenue with respect to the number of items sold at an instant. If R( x ) represents the revenue function for x units sold, then marginal revenue, denoted by MR, is given by d MR = { R( x )}. dx

(ix) Volume of a cube = x 3 and surface area = 6 x 2 where, x is the side of the cube. (x) Area of an equilateral triangle =

3 (Side) 2 . 4

Increasing and Decreasing Functions

Total cost = Fixed cost + Variable cost i.e., C ( x) = f(c ) + v( x)

(i) Increasing functions Let I be an open interval contained in the domain of a real valued function f. Then, f is said to be (a) increasing on I, if x1 < x2

Some Useful Results (i) Area of a square = x 2 and perimeter = 4x where, x is the side of the square.

⇒ f ( x1 ) ≤ f ( x2 ), ∀ x1, x2 ∈ I

(ii) Area of a rectangle = xy and perimeter = 2( x + y ) where, x and y are length and breadth of rectangle.

(b) strictly increasing on I, if x1 < x2 ⇒ f ( x1 ) < f ( x2 ), ∀x1, x2 ∈ I

9

Fast Track Revision Notes

Mathematics-XII

(ii) Decreasing functions Let I be an open interval contained in the domain of a real valued function f. Then, f is said to be

(a) If m1m2 = − 1, then tangents are perpendicular to each other. In this case, we say that the curves intersect each other orthogonally. This also happens, when m1 = 0 and m2 = ∞.

(a) decreasing on I, if x1 < x2 ⇒

(b) If m1 = m2 , then tangents are parallel to each other. (c) If φ = 0, then curves touch each other.

f ( x1 ) ≥ f ( x2 ), ∀ x1, x2 ∈ I

(b) strictly decreasing on I, if x1 < x2 ⇒

Approximation

f ( x1 ) > f ( x2 ), ∀x1, x2 ∈ I

Theorem Let f be differentiable on (a, b).

continuous

on

[a, b]

and

(a) If f ′ ( x ) > 0 for each x ∈(a, b), then f ( x ) is said to be increasing in [a, b] and strictly increasing in (a, b). (b) If f ′ ( x ) < 0 for each x ∈(a, b), then f ( x ) is said to be decreasing in [a, b] and strictly decreasing in (a, b). (c) If f ′ ( x ) = 0 for each x ∈(a, b), then f is said to be a constant function in [a, b]. (iii) A monotonic function f in an interval I, we mean that f is either increasing in I or decreasing in I.

Tangents and Normals A tangent is a straight line, which touches the curve y = f ( x ) dy represents the gradient function of a curve at a point. dx y = f ( x ). A normal is a straight line perpendicular to a tangent to the curve y = f ( x ) intersecting at the point of contact.

Equation of Tangent and Normal (a) Equation of tangent at P( x1, y1 ) is ( y − y1 ) dy = ( x − x1 ) dx ( x1, y1 ) (b) Equation of normal at P ( x1, y1 ) is ( y − y1 ) −1 = ( x − x1 ) dy dx (c) If a tangent line to the curve y = f ( x ) makes an angle θ with X-axis in the positive direction, then dy = Slope of tangent = tan θ dx

Angle of Intersection of Two Curves Let y = f1( x ) and y = f2 ( x ) be the two curves and φ be the angle between their tangents at the point of their intersection P ( x1, y1 ). m − m2 Then, tan φ = 1 1 + m1m2

and m2 =

f ( x + ∆x ) = f ( x ) + ∆y ⇒ f ( x + ∆x ) − f ( x ) = ∆y

Some Important Terms Absolute error The error ∆x in x is called the absolute error in x. ∆x Relative error If ∆x is an error in x, then is called the x relative error in x. ∆x Percentage error If ∆x is an error in x, then × 100 is x called percentage error in x.

Maxima and Minima

Monotonic Function

( x1, y1 )

m1 =

Also,

Let f be a real valued function and c be an interior point in the domain of f. Then, (i) Local maxima Point c is called a local maxima, if there is a h > 0 such that f (c ) ≥ f ( x ) for all x in (c − h, c + h ). (ii) Local minima Point c is called a point of local minima, if there is a h > 0 such that f (c ) ≤ f ( x ) for all x in (c − h, c + h ).

Let y = f ( x ) be a curve and P ( x1, y1 ) be a point on it. Then,

where,

Let y = f ( x ) be a function of x and ∆x be a small change in x and ∆y be the corresponding change in y. Then, ∆y = f ′ ( x ) + ε, where ε → 0 and ∆x → 0 ∆x ⇒ ∆y = f ′ ( x ) ∆x + ε ∆x dy ∆y dy ∆y = ∆x, approximately. Q ≈ ⇒ ∆x dx dx

dy for y = f1( x ) dx ( x1, y1 )

dy for y = f2 ( x ). dx ( x1, y1 )

10

A function which is either increasing or decreasing in their domain is called monotonic function. Every monotonic function assumes its maximum or minimum value at the end points of the domain of the function or every continuous function on a closed interval has a maximum and a minimum value. A monotonic function in an interval l, means that f is either increasing or decreasing in l.

Critical Point A point c in the domain of a function f at which either f ′ (c ) = 0 or f is not differentiable, is called a critical point of f. Note that, if f is continuous at c and f′ (c) = 0, then there exists a h > 0 such that f is differentiable in the interval (c − h, c + h).

Fast Track Revision Notes

Mathematics-XII

First Derivative Test

(ii) x = c is a point of local minima, if f ′ (c ) = 0 and f ′′(c ) > 0. Then, f (c ) is local minimum value of f.

(i) If f ′ ( x ) > 0 to closely left of c and f ′ ( x ) < 0 to closely right of c, then c is a point of local maxima. (ii) If f ′ ( x ) < 0 to closely left of c and f ′ ( x ) > 0 to closely right of c, then c is a point of local minima. (iii) If f ′( 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. This point is called a point of inflection.

(iii) If f ′ (c ) = 0 and f ′′(c ) = 0, then the test fails. As f is twice differentiable at c, we mean second order derivative of f exists at c.

Absolute Maxima and Absolute Minima Let f be a differentiable function on [a, b] and c be a point in [a, b] such that f ′ (c ) = 0. Then, find f (a), f (b) and f (c ). The maximum of these values gives a maxima or absolute maxima and minimum of these values gives a minima or absolute minima.

Second Derivative Test Let f be twice differentiable at c, then (i) x = c is a point of local maxima, if f ′ (c ) = 0 and f ′′(c ) < 0. Then, f (c ) is local maximum value of f.

Integrals Indefinite Integral

1

(xv)

Let F( x ) and f ( x ) be two functions connected together, such d that [F( x )] = f ( x ), then F( x ) is called integral of f ( x ) or dx indefinite integral or anti-derivative.

∫

(xvi)

∫

Thus, ∫ f ( x )dx = F( x ) + C

(xvii)

∫ 1+

where, C is an arbitrary constant.

x n +1 + C, n ≠ − 1 (n + 1 )

(i)

∫

x ndx =

(ii)

∫

1 dx = log| x| + C, x ≠ 0 x

(iii) (iv)

∫e ∫

x

(xix) (xx)

dx = cos − 1 x + C

∫ ∫

1 dx = − cot − 1 x + C 1+ x 2 dx x

x2 − 1 1

x

x2 − 1

= sec − 1 x + C dx = − cosec

−1

x+C

Properties of Indefinite Integral

ax a dx = +C log a x

(i) The process of differentiation and integration are inverse of each other. d i.e., f ( x )dx = f ( x ) and ∫ f ′( x )dx = f ( x ) + C dx ∫ where, C is any arbitrary constant.

∫ sin x dx = − cos x + C (vi) ∫ cos x dx = sin x + C (vii) ∫ tan x dx = log|sec x| + C (viii) ∫ cot x dx = log|sin x| + C (ix) ∫ sec x dx = log|sec x + tan x | + C

∫ { f ( x ) ± g ( x )} dx = ∫ f ( x )dx ± ∫ g ( x )dx (iii) ∫ K f ( x )dx = K ∫ f ( x )dx (ii)

(iv) In general, if f1, f2 , ..., fn are functions and k1, k2 , ..., kn are numbers, then

= log|tan (π / 4 + x / 2 )| + C

∫ cosec

∫ [k1 f1 ( x ) + k2 f2( x ) + ... + kn fn( x )]dx = k1 ∫ f1( x )dx + k2 ∫ f2 ( x ) dx + ... + kn ∫ fn ( x )dx + C

x dx = log|cosec x − cot x| + C

= log|tan( x / 2 )| + C

∫ sec x dx = tan x + C (xii) ∫ cosec 2 x dx = − cot x + C (xiii) ∫ sec x ⋅ tan x dx = sec x + C (xiv) ∫ cosec x cot x dx = − cosec (xi)

1 + x2

dx = e x + C

(v)

(x)

1

dx = − cos −1 x + C

dx = tan − 1 x + C x2

(xviii) ∫

Important Results

1− x

2

where, C is the constant of integration.

2

Integration by Substitution The method of reducing a given integral into one or other standard integral by changing the independent variable is called method of substitution. Thus, ∫ f ( x )dx = ∫ f { g (t )} ⋅ g′(t ) dt , if we substitute x = g (t ),

x+C

such that dx = g′ (t ) dt.

11

Fast Track Revision Notes

Mathematics-XII Some Standard Substitutions

Partial Fractions Form of the Rational Functions 1.

Form of the Partial Fractions

px ± q ,a≠b (x ± a) (x ± b) px ± q

2.

A B + (x ± a) (x ± a)2

(x ± a)2 px2 ± qx ± r

3.

A B + x±a x±b

1.

a −x

2.

a + x

3.

x −a 2

Bx + C A , where + 2 (x ± a) x ± bx ± c

px2 ± qx ± r (x ± a) (x2 ± bx ± c)

x2 ± bx ± c cannot be factorised further.

Integration by Parts

2

x = a tan θ or a cot θ

2

x = a sec θ or a cosec θ

4.

a+ x or a−x

5.

x −α or (x − α) (x − β) β−x

a−x a+ x

(i)

x

2

∫ (x2 + a2 ) (x2 + b2 ) dx

(ii)

a+ x

1

dx

(iii)

∫

(iv)

∫

(v)

∫

(vi)

∫

dx (x − a ) 2

2

dx x 2 + a2

= log x +

x 2 x x 2 − a2 dx = 2

a2 − x 2 dx =

dx

x 2 , then Put cos x = x 1 + tan2 2 x put tan = t 2

(iii)

∫ a ± b sin x

dx

x 2 , then x 1 + tan2 2 x put tan = t 2

(iv)

∫ a sin x +

(v)

∫

(vi)

∫ p sin x +

1 − tan2

Put sin x =

x 2 + a2 + C

∫

x 2 + a2 dx =

x 2

a2 log x + 2

x 2 − a2 + C

x 2 + a2 +

a2 log x + 2

b cos x

a sin x + b cos x + c dx p sin x + q cos x

a sin x + b cos x dx q cos x + r

Put a sin x + b cos x + c d =A (p sin x + q cos x) dx + B (p sin x + q cos x) Put a sin x + b cos x d =A (p sin x + q cos x + r) dx + B (p sin x + q cos x + r)

x 2 − a2 −

(vii)

a2 x sin −1 + C 2 a

2 tan

Put a = r cos θ and b = r sin θ

dx

x 2 − a2| + C

a2 − x 2 +

partial fraction of y

∫ a ± b cos x

+C

= log| x +

Let x2 = y and proceed for

(ii)

x−a

∫ ( x 2 − a2 ) = 2 a log x + a + C ∫ (a2 − x 2 ) = 2 a log a − x

Substitution

( y + a2 ) ( y + b2 )

Some Important Integrals 1

x = α cos 2 θ + β sin2 θ

Integral

If two functions are of different types, then consider the 1st function (i.e., u) which comes first in word ILATE, where I : Inverse trigonometric function e.g., sin −1 x L : Logarithmic function e.g., log x A : Algebraic function e.g., 1, x, x 2 T : Trigonometic function e.g., sin x, cos x E : Exponential function e.g., e x

dx

x = a cos 2 θ

Integration of Irrational and Trigonometric Functions

Let u and v be two differentiable functions of a single variable x, then the integral of the product of two functions is d dx = u ∫ v dx − ∫ u ∫ v dx dx ∫ uv dx I II

(i)

Substitution x = a sin θ or a cos θ

2

2

A B C + + (x ± a) (x ± b) (x ± b)2

(x ± a) (x ± b)2 4.

Expression 2

x 2 + a2 + C

12

(vii)

∫ (ax +

(viii)

∫ (ax2 +

(ix)

∫

(x)

∫

dx b) px + q

Put

px + q = t

dx

Put

px + q = t

bx + c) px + q dx

(px + q)( ax2 + bx + c ) dx (px + q) ax + b 2

2

Put px + q = Put x =

1 t

1 t

Fast Track Revision Notes

Mathematics-XII Fundamental Theorem of Calculus

Definite Integral An integral is of the form of

b

∫a

Theorem 1 Let f be a continuous function defined on the closed interval [a, b] and A( x ) be the area of function.

f ( x )dx is known as definite

x

[i . e ., A( x ) = ∫ f ( x )dx ]. Then, A′( x ) = f ( x ), for all x ∈[a, b].

integral and is given by

a

b

Theorem 2 Let f be a continuous function defined on the closed interval [a, b] and F be an anti-derivative of f.

∫a f ( x )dx = g(b) − g(a) where, a and b are lower and upper limits of an integral.

Definite Integral as a Limit of Sum

Then,

(ii)

∫a f ( x )dx = − ∫b f ( x )dx

(iii)

∫a f ( x )dx = 0

(iv)

∫a f ( x )dx = ∫a f ( x )dx + ∫c f ( x )dx, where a < c < b.

(v)

∫a f ( x )dx = ∫a f (a + b − x )dx

(vi)

∫0 f ( x )dx = ∫0 f (a − x )dx

(vii)

∫0

b

+ … + f { a + (n − 1 ) h}] nh = b − a

Some Standard Formulae 1. Σn = 1 + 2 + 3 + … + n =

3

3

3

b

a

a

b

c

b

b

a

a

2a

b

a

f ( x )dx = ∫ f ( x )dx + 0

a

∫0 f (2 a − x )dx

2 a f ( x )dx, if f (2 a − x ) = f ( x ), even f x dx ( ) = ∫0 ∫0 0, if f (2 a − x ) = − f ( x ), odd a a if f (− x ) = f ( x ), even (ix) ∫ f ( x )dx = 2 ∫0 f ( x )dx, −a 0, if f (− x ) = − f ( x ), odd

n 2 (n + 1)2 3. Σn = 1 + 2 + 3 + … + n = 4 3

b

∫a f ( x )dx = ∫a f (t )dt

b−a n

n(n + 1) 2 n(n + 1)(2 n + 1) 2 2 2 2 2 2. Σn = 1 + 2 + 3 + … + n = 6

b

(i)

h [f (a) + f (a + h ) + f (a + 2 h ) ∫a f ( x )dx = hlim →0

where,

b

Properties of Definite Integral

Let us define a continuous function f ( x ) in [a, b] divide interval into n equal sub-intervals, each of length h, so that h=

b

∫a f ( x ) dx = [F( x )]a = F(b) − F(a)

Then,

(viii)

3

2a

Application of Integrals 1.

The area enclosed by the curve y = f ( x ), the X-axis and

3.

b

the ordinates at x = a and x = b, is given by ∫ | y| dx. a

If the curve y = f ( x ) lies below the X-axis, then area bounded by the curve y = f ( x ), X-axis and the ordinates at x = a and x = b, is given by

Y A x=a X′ O

x=a X′

x=b

X

X

L dx M

Y′

4. The area enclosed by the curve x = f ( y ), the Y-axis and d

the abscissae at y = c and y = d , is given by ∫ | x| dy. c

.

x=b

O

Y′

2.

ydx .

Y

f(x) B y= y

b

∫a

y = f (x)

Generally, it may happen that some position of the curve is above X-axis and some is below the X-axis which is shown in the figure. The area A bounded by the curve y = f ( x ), X-axis and the ordinates at x = a and x = b, is given by A = | A2 | + A1.

Y B dy X′

Y

y=d C x x = f(y)

A y=c

A1 X′

D

x=b X

x=a O

X

O Y′

Y′

13

A2

Fast Track Revision Notes

5.

Mathematics-XII 7.

The area enclosed between two curves, y1 = f ( x ) and y2 = g ( x ) and the ordinates at x = a and x = b, is given by

b

∫a

y2 − y1 dx.

Area bounded by the two curves, y = f ( x ) and y = g( x ) between the ordinates at x = a and x = b, is given by c b ∫ f ( x )dx + ∫ g( x ) dx. a

c

Y

Y

y= ) f (x

y2 = g(x)

y=

) g (x

y1 = f(x) X′

x=a

O

X

x=b

X′

The area enclosed between two curves, x1 = f ( y ) and x2 = g ( y ) and the abscissae at y = c and y = d , is given by ∫

d

c

x=b

X

Curve Sketching The points given below are of great help in curve sketching.

x2 − x1 dy.

(i) If the equation of the curve contains only even powers of x, then it is symmetrical about Y-axis.

Y

(ii) If the equation of the curve contains only even powers of y, then it is symmetrical about X-axis.

x2 = g(y)

x1 = f(y)

y=d

X′

x=c

Y′

Y′

6.

O x=a

y=c

(iii) If the equation of the curve remains unchanged when x and y are interchanged, then it is symmetrical about the line y = x.

X

O

(iv) If the equation of the curve remains unchanged when x and y are replaced by − x and − y respectively, then the curve is symmetrical in opposite quadrants.

Y′

Differential Equations Differential Equation An equation containing an independent variable, dependent variable and derivative of dependent variable with respect to independent variable, is called a differential equation. The derivates are denoted by the symbols dy d 2 y d ny , 2 ,...., n or y′ , y′ ′, ..., y′ ′ ′ ... n or y1, y2 ,..., yn dx dx dx

Order of a Differential Equation The order of the highest derivative occurring in the differential equation, is called order.

Degree of a Differential Equation The power of the highest order derivative in the differential equation, is called degree.

Solution of a Differential Equation A relation between the dependent and independent variables which, when substituted in the differential equation reduces it to an identity, is called a solution.

General Solution of a Differential Equation

Particular Solution of a Differential Equation The solution obtained from the general solution given particular values to the arbitrary constants, is called a particular solution of the differential equation.

Formation of a Differential Equation An equation with independent, dependent variables involving some arbitrary constants is given, then a differential equation is obtained as follows (i) Differentiate the given equation with respect to the independent variable (say x) as many times as the number of arbitrary constants in it. (ii) Eliminate the arbitrary constants. (iii) The obtained equation is the required differential equation. The order of a differential equation representing a family of curves is same as the number of arbitrary constants present in the equations corresponding to the family of curves.

Equation in Variable Separable Form

A solution of a differential equation which contains as many arbitrary constants as the order of the differential equation, is called the general solution or primitive solution of the differential equation.

14

If the equation can be reduced into the form f ( x )dx + g( y ) dy = 0, we say that the variables have been separated. Then,

∫ f ( x ) dx + ∫ g( y ) dy = C

Fast Track Revision Notes

Mathematics-XII

Homogeneous Differential Equation A differential equation of the form

dy f ( x, y ) = dx g( x, y )

where, f ( x, y ) and g( x, y ) are homogeneous functions of x and y are of the same order. If the homogeneous differential equation is in the form dx = F( x, y), where F( x, y) is a homogeneous function dy x of degree zero, then we make substitution = v, i.e., y x = vy and we proceed further to find the general solution.

derivative is one and they do not get multiplied together, is called a linear differential equation. There are two types of linear differential equations. dy Type I Form + Py = Q, where P and Q are constants or dx functions of x. We find integrating factor (IF) = e ∫ P dx . Now, solution is y × (IF) = ∫ [Q × (IF)] dx + C. dx + Px = Q, where P and Q are constants or dy functions of y, then IF = e ∫ P dy .

Type II Form

Its solution is x × (IF) = ∫ [Q × (IF)] dy + C.

Linear Differential Equation A first order and first degree differential equation in which the degree of dependent variable and its

Vector Algebra Vector

Like Vectors

A quantity that has magnitude as well as direction, is called a vector.

The vectors which have same direction are called like vectors.

→

Unlike Vectors

Since, the length is never negative, so the notation | a | < 0 has no meaning.

The vectors which have opposite directions are called unlike vectors.

Scalar

Equal Vectors

A quantity that has magnitude only, is called scalar.

Two vectors are equal, if they have same magnitude and direction.

Magnitude of a Vector If

→

→ a = a $i + b $j + c k$ , then | a | = a2 + b2 + c 2

Negative of a Vector A vector whose magnitude is same as that of given vector but the direction is opposite is called negative vector of the

Free Vector

→

If the initial point of a vector is not specified, then it is called a free vector.

→

→

given vector. e.g., Let AB be a vector, then − AB or BA is a negative vector.

Type of Vectors

Coplanar Vectors

Zero or Null Vector

A system of vectors is said to be coplanar, if their supports are parallel to same plane.

A vector whose magnitude is zero i . e ., whose initial and final points coincide, is called a null vector or zero vector.

Addition of Vectors

Unit Vector A vector whose magnitude is one unit. The unit vector in →

$ The unit vectors the direction of a is represented by a. along X-axis, Y-axis and Z-axis are represented by $i , $j and k$ , respectively.

→

→

→

→

If

OA = a

and

AB = b, →

→

→

→

→

then a + b = OA + AB = OB B

Coinitial Vectors Two or more vectors having the same initial point are called coinitial vectors.

b

Collinear Vectors

O

The vectors which have same support are called collinear vectors.

15

a

A

Fast Track Revision Notes

Mathematics-XII

Triangle Law of Vector Addition

Section Formulae

If two vectors are represented along two sides of a triangle taken in order, then their resultant is represented by the third side taken in opposite direction i.e., from ∆ ABC, by triangle law of vector addition, we have

Let A and B be two points with position vectors a and b respectively and P be a point which divides AB

→

→

→

BC + CA = BA

→

→

→

internally in the ratio m : n. Then, position vector of P →

=

C

→

mb + n a . m+ n

If P divides AB externally in the ratio m:n. Then, position →

B

vector of P =

A

→

→

mb − n a . m−n →

Parallelogram Law of Vector Addition

If R is the mid-point of AB, then OR =

If two vectors are represented along the two adjacent sides of a parallelogram, then their resultant is represented by the diagonal of the sides. If the sides OA and OC of parallelogram

Component of a Vector

→

→

OABC represents OA and OC respectively, then we get →

→

→

→

→

→

→

OA + OC = OB or OA + AB = OB

→

[Q AB = OC]

C

B

→

→

a+ b . 2

→ If a = a1$i + a2 $j + a3k$ , we say that the scalar components →

of a along X-axis, Y-axis and Z-axis are a1, a2 and a3 , respectively.

Important Results in Component Form →

O

A

Thus, we may say that the two laws of vectors addition are equivalent to each other.

→

→

→

a+ b = b + a

[commutative law]

→ →

→

(b) For any three vectors a , b and c , →

→

→

→

→

→

→

→

[associative law] →

→

→

(d) For any vector a , a + (− a ) = 0 →

→

→

→

→

→

→

1

→

→

→

If a and b are any two vectors, then their difference a − b is →

→

defined as a + ( − b ). →

b a+

→

→

b

→

a

O

→

→

a−

→

2

3

b b b (v) If 1 = 2 = 3 = k (constant) a1 a2 a3

Difference of Vectors →

→

(iii) The vectors a and b are equal, if and only if a1 = b1, a2 = b2 and a3 = b3 →

(e) |a + b| ≤ |a + b|and |a − b| ≥ |a| − |b|

→

→ a − b = (a1 − b1 ) $i + (a2 − b2 ) $j + (a3 − b3 )k$

(iv) The multiplication of vector a by any scalar λ is → given by λ a = (λa ) $i + (λa ) $j + (λa )k$

→

The vector − a is additive inverse of a . →

→

→

→

→

The zero vector 0 is called the additive identity for the vector addition. →

→

(i) The sum (or resultant) of the vectors a and b is given by → → a + b = (a1 + b1 ) $i + (a2 + b2 ) $j + (a3 + b3 )k$ →

(c) For any vector a , we have a + 0 = 0 + a = a .

→ →

→

(ii) The difference of the vectors a and b is given by

→

a + (b + c ) = (a + b ) + c →

→

→

→

(a) For any two vectors a and b, →

Then, (a1 , a2 , a3 ) and ( b1 , b2 , b3 ) are called direction ratios of a and b, respectively.

Properties of Vector Addition →

→

If a and b are any two vectors given in the component → → form such that a = a1$i + a2 $j + a3k$ and b = b1$i + b2 $j + b3k$

−b

b

16

→

Then, vectors a and b will be collinear. (vi) If it is given that l, m and n are direction cosines of a vector, then l $i + m $j + n k$ = (cos α ) $i + (cos β ) $j + (cos γ )k$ is the unit vector in the direction of that vector, where α , β and γ are the angles which the vector makes with X, Y and Z-axes, respectively.

Fast Track Revision Notes

Mathematics-XII

Multiplication of a Vector by Scalar

If α, β and γ are the direction angles of vector → a = a1$i + a2 $j + a3k$ , then its DC’s is given as a a a cos α = 1 , cos β = 2 , cos γ = 3 → → → |a| |a| |a|

→

Let a be a given vector and λ be a scalar. Then, the →

→

product of the vector a by the scalar λ, denoted by λa , is →

called the multiplication of vector a by the scalar λ. →

→

Let a and b be any two vectors and k and m be any scalars. Then, →

→

Cross Product or Vector Product

→

(i) k a + m a = (k + m) a →

→

(ii) k (m a ) = (km) a →

→

→

→

→

(iii) k (a + b ) = k a + k b

n$ =

→

2

1

2

→

→ →

→

→

→ →

a⋅ b < 0

→ →

(iii) a ⊥ b ⇔ a ⋅ b = 0.

→

(viii) If

→ →

→

→

→

→

→

a = a1$i + a2 $j + a3k$

$i $j → → $ $ $ and b = b1i + b2 j + b3k, then a × b = a1 a2 b1 b2

→ → a⋅ b (v) Projection of a on b = → |b |

→

→

(vi) If a force F displaces a particle from a point A to a → → point B, then work done by the force = F . AB

(ix) a × b = − b × a

(vii) Properties of scalar product

(x) a × (b + c ) = a × b + a × c

→

→ →

→

(a) a ⋅ b = b⋅ a

→

→

→

→ →

(b) a ⋅ (b + c ) = a ⋅ b + a ⋅ c (c) a ⋅ a = |a | 2 = a 2 →

→

→

(d) (a + b )⋅ (a − b ) = a2 − b2 , where a and b represent →

→

the magnitude of vectors a and b. →

→

→

→

→

→

→

→

→

→

→

→

→ → → (xiii) $i × $i = 0 , $j × $j = 0 , k$ × k$ = 0

→

where, a represents magnitude of vector a . →

→

(xi) a × a = 0 (xii) $i × $j = k$ , $j × k$ = $i , k$ × $i = $j; $j × $i = −k$ , k$ × $j = − $i , $i × k$ = − $j

[commutative] → →

→

(vii) a || b ⇔ a × b = 0

→ (iv) $i ⋅ $i = $j ⋅ $j = k$ ⋅ k$ = 1 and $i ⋅ $j = $j ⋅ k$ = k$ ⋅ $i = 0

→ →

→

1 → → 1 → → (vi) Area of a ∆ ABC = | AB × AC | = | BC × BA | 2 2 1 → → = | CB × CA | 2

→ →

(ii) If θ is acute, then a ⋅ b > 0 and if θ is obtuse,

→ →

→

→

a ⋅ b =| a || b |cos θ

→ →

→

⋅

(iv) Area of a parallelogram with diagonals a and b 1 → → = | a × b| 2 1 → → (v) Area of a quadrilateral ABCD = ( AC × BD ) 2

1

→

→

→

→

(a × b )

→

(i) If θ is the angle between a and b, then

→

→

= | a × b|

Dot Product or Scalar Product

then

→

(iii) Area of a parallelogram with sides a and b

= ( x2 $i +y2 $j + z2k$ ) − ( x1$i +y1$j + z1k$ ) = ( x − x ) $i + ( y − y ) $j +( z − z ) k$ 1

→

|a × b |

P1 P2 = OP2 − OP1

2

→

(ii) A unit vector perpendicular to both a and b is given by

If P1 ( x1, y1, z1 ) and P2 ( x2 , y2 , z2 ) are any two points, then vector joining P1 and P2 is →

→

perpendicular to the plane of a and b.

Vector Joining Two Points →

→

(i) If θ is the angle between the vectors a and b, then → → → → a × b =| a|| b |sin θ n$ , where, n$ is a unit vector

→

→ →

(e) (λ a )⋅ b = λ( a ⋅ b )

17

k$ a3 b3

Fast Track Revision Notes

Mathematics-XII

Three-Dimensional Geometry Direction Cosines and Direction Ratios

two

→

→

r =a +tb

lines

cos θ =

(ii) Angle between two lines If θ is the angle between two lines with direction cosines l1, m1, n1 and l 2 , m2 , n2 , then cos θ = l1 l 2 + m1 m2 + n1 n2 . Numbers proportional to the direction cosines of a line are called the direction ratios of the line.

→

distance

→

→

between →

→

the

skew-lines

r = a 1 + λ b 1 and r = a 2 + λ b 2 is given by →

SD =

→

→

→

→

→

|(a 2 − a 1 ) ⋅ (b 1 × b 2 )| |b 1 × b 2 |

(b) The shortest distance between the parallel lines →

→

→

→

→

→

r = a 1 + λ b and r = a 2 + µ b is given by →

SD =

→

→

b × (a 2 − a 1 ) →

b

Equation of Line in Cartesian Form (i) The equation of a line passing through a point A( x1, y1, z1 ) and having direction ratios a, b and c is x − x1 y − y1 z − z1 = = a b c

where, PQ = ( x2 − x1 )2 + ( y2 − y1 )2 + ( z2 − z1 )2

(ii) If l, m and n are the direction cosines of the line, then equation of the line is x − x1 y − y1 z − z1 = = l m n

Line A straight line is the locus of the intersection of two planes. A line is uniquely determined, if

(iii) The equation of a line passing through two points A( x1, y1, z1 ) and B( x2 , y2 , z2 ) is x − x1 y − y1 z − z1 = = x2 − x1 y2 − y1 z2 − z1

(i) it passes through a given point and has given direction or (ii) it passes through two given points.

(iv) If a1, b1, c1 and a2 , b2 , c 2 are direction ratios of two lines respectively, then the angle θ between the lines is given by

Equation of Line in Vector Form (i) Equation of a line through a given point A with →

position vector a and parallel to a given vector b is

cos θ =

→

given by r = a + t b, where t is a scalar. →

$ then a, b and c are direction ratios If b = a $i + b$j + ck, of the line and conversely, if a, b and c are direction → ratios of a line, then b = a $i + b$j + ck$ will be the

or sinθ =

a1a2 + b1b2 + c1c 2 a12

+ b12 + c12 ⋅ a22 + b22 + c 22

(a1b2 − a2 b1 )2 + (b1c 2 − b2c1 )2 + (c1a2 − c 2 a1 )2 a12 + b12 + c12 a22 + b22 + c 22

(v) Two lines with direction ratios a1, b1, c1 and a2 , b2 , c 2 are (a) perpendicular, if a1a2 + b1b2 + c1c 2 = 0. a b c (b) parallel, if 1 = 1 = 1 . a2 b2 c 2

parallel to the line. (ii) The vector equation of a line passing through two →

points with position vectors a and b is given by →

then

. → | b|| d|

→

The direction cosines and direction ratios of the line segment joining P( x1, y1, z1 ) and Q( x2 , y2 , z2 ) are respectively given by x2 − x1 y2 − y1 z2 − z1 and ( x2 − x1, y2 − y1, z2 − z1 ) , , PQ PQ PQ

→

→

b⋅ d

Direction Cosines and Direction Ratios of a Line

→

→

r = c + s d,

→

(a) Shortest

(ii) In any line, direction cosines are unique but direction ratios are not unique.

→

→

(v) Shortest distance between two skew-lines The shortest distance between two skew-lines is the length of perpendicular to both the lines.

(i) If a, b and c are the direction ratios of a line, then a b , m= l= a2 + b2 + c 2 a2 + b2 + c 2 c and n = a2 + b2 + c 2

→

and

(iv) Skew-lines If two lines do not meet and not parallel, then they are known as skew-lines.

(i) We always have, l 2 + m2 + n 2 = 1.

→

→

→ →

If a line make angles α, β and γ with X-axis, Y-axis and Z-axis respectively, then l = cos α, m = cos β and n = cos γ are called the direction cosines of the line.

→

If θ is the angle between

(iii) Angle between two lines

→

r = a + t (b − a ), where t is a scalar.

18

Fast Track Revision Notes

Mathematics-XII

(vi) The shortest distance between the lines x − x1 y − y1 z − z1 = = a1 b1 c1 and

(vi) Distance of a point from a plane Distance of a point P( x1, y1, z1 ) from a plane ax + by + cz + d = 0 is given by

x − x 2 y − y 2 z − z2 is = = a2 b2 c2 x2 − x1

y2 − y1

z2 − z1

a1 a2

b1 b2

c1 c2

p=

If θ is the angle between the planes a1 x + b1 y + c1 z + d1 = 0 and a2 x + b2 y + c 2 z + d 2 = 0, then

+ (a1b2 − a2 b1 )2

cos θ =

Plane

a1a2 + b1b2 + c1c 2 a12

+ b12 + c12

a22 + b22 + c 22

If two planes are perpendicular, then a1a2 + b1b2 + c1c 2 = 0 and if they are parallel, then a1 b1 c1 . = = a2 b2 c 2

A plane is a surface such that a line segment joining any two points on it lies wholly on it. A plane is determined uniquely, if any one of the following is known.

(viii) Equation of the bisector plane The equation of the bisector plane to the planes a1 x + b1 y + c1 z + d1 = 0 and a2 x + b2 y + c 2 z + d 2 = 0 are given by

(i) The normal to the plane and its distance from the origin is given, i.e., equation of a plane in normal form. (ii) It passes through a point and is perpendicular to a given direction.

a1 x + b1 y + c1 z + d1 a12

(iii) It passes through three given non-collinear points. Now, we shall find vector and cartesian equations of the planes.

+

+ c12

b12

a2 x + b2 y + c 2 z + d 2

=±

a22 + b22 + c 22

Coplanarity of Two Lines x − x1 y − y1 z − z1 = = a1 b1 c1 x − x 2 y − y 2 z − z2 and = = a2 b2 c2 are coplanar, if and only if x2 − x1 y2 − y1 z2 − z1 a1 b1 c1 = 0

Equation of Plane in Cartesian Form

Two lines

(i) The general equation of a plane is ax + by + cz + d = 0. The direction ratios of the normal to this plane are a, b and c . (ii) If a plane cuts (intercepts) a, b and c with the coordinate axes, then the equation of the plane is x y z + + = 1. a b c

a2

b2

c2

The equation of plane containing the above lines is x − x1 y − y1 z − z1 a1 b1 c1 = 0

(iii) Equation of a plane passing through a point and perpendicular to a given vector The equation of a plane passing through a point ( x1, y1, z1 ) is

a2

a ( x − x1 ) + b( y − y1 ) + c ( z − z1 ) = 0 where, a, b and c are direction ratios of perpendicular vector.

b2

c2

Equation of Plane in Vector Form (i) Let O be the origin and n$ be a unit vector in the direction of the normal ON to the plane and let

(iv) Equation of a plane passing through three non-collinear points The equation of a plane passing through three non-collinear points A( x1, y1, z1 ), B( x2 , y2 , z2 ) and C( x3 , y3 , z3 ) is given by x − x1 y − y1 z − z1 x2 − x1 y2 − y1 z2 − z1 = 0 y3 − y1

a2 + b2 + c 2

(vii) Angle between two planes The angle between two planes is the angle between their normals.

2 2 (bc 1 2 − b2c1 ) + (c1a2 − c 2 a1 )

x3 − x1

| ax1 + by1 + cz1 + d|

→

ON = p. Then, the equation of the plane is r ⋅ n$ = p. → If r . (a $i + b$j + ck$ ) = d is the vector equation of a plane, then ax + by + cz = d is the cartesian equation of the plane, where a, b and c are the direction ratios of the normal to the plane.

z3 − z1

(v) Equation of a plane through the intersection of two planes The equation of a plane through the intersection of the planes

(ii) Equation of a plane perpendicular to a given vector and passing through a given point The equation of a →

plane passing through the point a and perpendicular

a1 x + b1 y + c1 z + d1 = 0 and a2 x + b2 y + c 2 z + d 2 = 0 is (a1 x + b1 y + c1 z + d1 ) + λ (a2 x + b2 y + c 2 z + d 2 ) = 0

→

to the given vector n is →

→

→

→

(r − a ) ⋅ n = 0

19

Fast Track Revision Notes

Mathematics-XII

(iii) Equation of plane passing through three non-collinear points The equation of plane passing through three → →

→

→

→

→

→

→

perpendicular distance of a point a from the → →

non-collinear points a , b and c is →

(vii) Distance of a point from a plane The →

plane r ⋅ n = d, where n is normal to the plane, is

→

( r − a ) ⋅ [( b − a ) × ( c − a )] = 0

→ →

a ⋅n −d

(iv) Plane passing through the intersection of two given planes The equation of plane passing through the intersection of two planes → →

→ → r ⋅ n1

→

→

→

→

→

→

(v) Two lines r = a 1 + λ b 1 and r = a 2 + λ b 2 are coplanar, if and only if

→

→

→

→

(a 2 − a 1 ) ⋅ (b 1 × b 2 ) = 0. → →

→

b⋅ n

r ⋅ n = d is cos θ =

→

→

→

→ →

→ →

. | n 1|| n 2| →

→

(ix) The angle θ between line r = a + λ b and plane

→

n1⋅ n 2

→

(viii) The length of the perpendicular from origin O to → → d the plane r ⋅ n = d is . → | n| →

→ →

(vi) Angle between the planes r ⋅ n 1 = p1 and r ⋅ n 2 = p2 is given by cos θ =

n

= d1 and r ⋅ n 2 = d 2 is

r ⋅ ( n 1 + λ n 2 ) = d1 + λ d 2 . →

→

→ →

→

→

b

→

n

Two planes are perpendicular to each other, if n 1 ⋅ n 2 = 0 →

→

→

→ →

and parallel, if n1 × n2 = 0 .

and

sin φ =

b ⋅n

→

→

, where φ = 90 °− θ.

b ⋅ n

Linear Programming Linear Programming Problem

5. Solution

A linear programming problem is one that is concerned with finding the optimal value (maximum or minimum value) of a linear function (called objective function) of several variables (say x and y called decision variable), subject to the conditions that the variables are non-negative and satisfy a set of linear inequalities (called linear constraints).

A set of values of the variables which satisfy the constraints of given linear function (objective function) of several variable, subject to the conditions that the variables are non-negative is called a solution of the linear programming problem.

Some Terms Related to LPP

Any solution to the given linear programming problem which also satisfies the non-negative restrictions of the problem is called a feasible solution. Any point outside the feasible region is called an infeasible solution.

1. Constraints The linear inequations or inequalities or restrictions on the variables of a linear programming problem are called constraints. The conditions x ≥ 0, y ≥ 0 are called non-negative restrictions.

2. Optimisation Problem A problem which seeks to maximise or minimise a linear function subject to certain constraints determined by a set of linear inequalities is called an optimisation problem. Linear programming problems are special type of optimisation problems.

3. Objective Function A linear function of two or more variables which has to be maximised or minimised under the given restrictions in the form of linear inequations (or linear constraints) is called the objective function.The variables used in the objective function are called decision variables.

4. Optimal Values The maximum or minimum value of an objective function is known as its optimal value.

20

6. Feasible Solution

7. Feasible Region The set of all feasible solutions constitutes a region which is called the feasible region. Each point in this region represents a feasible choice. The region other than feasible region is called an infeasible region.

8. Bounded Region A feasible region of a system of linear inequalities is said to be bounded, if it can be enclosed within a circle. Otherwise, it is said to be unbounded region.

9. Optimal Solution A feasible solution at which the objective function has optimal value is called the optimal solution of the linear programming problem.

10. Optimisation Technique The process of obtaining the optimal solution is called optimisation technique.

Fast Track Revision Notes

Mathematics-XII

Corner Point Method

Step III When the feasible region is bounded, M and m are the maximum and minimum values of z.

It is a graphical method to solve the LPP. The following steps are given below

Step IV In case, the feasible region is unbounded, we have

Step I Find the feasible region of the linear programming problem and determine its corner points (vertices) either by inspection or by solving the two equations of the lines intersecting at that point.

(a) M is the maximum value of z, if the open half plane determined by ax + by > M has no point in common with the feasible region. Otherwise, z has no maximum value.

Step II Evaluate the objective function z = ax + by at each corner point. Let M and m respectively denote the largest and smallest values of these points.

(b) Similarly, m is the minimum value of z, if the open half plane determined by ax + by < m has no point in common with the feasible region. Otherwise, z has no minimum value.

Probability Some Basic Definitions

Complement of an Event

1. Experiment An operation which results in some well-defined outcomes, is called an experiment.

2. Random Experiment

Let some A be an event in a sample space S, then complement of A is the set of all sample points of the space other than the sample point in A and it is denoted by A′ or A. i . e ., A′ = { n : n ∈ S , n ∉ A}

An experiment in which total outcomes are known in advance but occurrence of specific outcome can be told only after completion of the experiment, is known as a random experiment.

3. Trial

Some Basic Terms Coin

When a random experiment is repeated under identical conditions and it does not give the same result each time but may result in one of the several possible outcomes, then such experiment is called a trial.

4. Sample Space The set of all possible outcomes of a random experiment is called its sample space. It is usually denoted by S.

5. Discrete Sample Space A sample space is called a discrete sample space, if S is a finite set.

A coin has two sides, head and tail. If an event consists of more than one coin, then coins are considered as distinct, if not otherwise stated. (i) Sample space of one coin= { H, T } (ii) Sample space of two coins = {(H, T ), (T, H ), ( H, H ), (T, T )} (iii) Sample space of three coins = {( H, H, H ), ( H, H, T ), ( H, T, H ), (T, H, H ), ( H, T, T ), (T, H, T ), (T, T, H ), (T, T, T )}

Die

Event A subset of the sample space associated with a random experiment is called an event or case.

A die has six faces marked 1, 2, 3, 4, 5 and 6. If we have more than one die, then all dice are considered as distinct, if not otherwise stated. (i) Sample space of a die = {1, 2, 3, 4, 5, 6} (ii) Sample space of two dice

Type of Events Equally Likely Events The given events are said to be equally likely, if none of them is expected to occur in preference to the other.

Mutually Exclusive Events A set of events is said to be mutually exclusive, if the happening of one excludes the happening of the other i.e., if A and B are mutually exclusive, then ( A ∩ B) = φ.

Exhaustive Events A set of events is said to be exhaustive, if the performance of the experiment always results in the occurrence of atleast one of them. If E1, E2 ,… , En are exhaustive events, then E1 ∪ E2 ∪…∪ En = S .

21

(1, 1), (1, 2 ), (1, 3), (1, 4), (1, 5), (1, 6) (2, 1), (2, 2 ), (2, 3), (2, 4), (2, 5), (2, 6) (3, 1), (3, 2 ), (3, 3), (3, 4), (3, 5), (3, 6) = (4, 1), (4, 2 ), (4, 3), (4, 4), (4, 5), (4, 6) (5, 1), (5, 2 ), (5, 3), (5, 4), (5, 5), (5, 6) (6, 1), (6, 2 ), (6, 3), (6, 4), (6, 5), (6, 6)

Fast Track Revision Notes

Mathematics-XII If a set of events A1, A2 ,… , An are exhaustive, then P ( A1 ∪ A2 ∪…∪ An ) = 1.

Playing Cards A pack of playing cards has 52 cards. There are 4 suits (spade, heart, diamond and club), each having 13 cards. There are two colours, red (heart and diamond) and black (spade and club), each having 26 cards. In 13 cards of each suit, there are 3 face cards namely king, queen and jack, so there are in all 12 face cards. Also, there are 16 honour cards, 4 of each suit namely ace, king, queen and jack.

Booley’s Inequality (i) If A1, A2 ,… , An are n events associated with a random experiment, then n

(a) P (A1 ∩ A2 ∩…∩ An ) ≥ n

(b) P ( ∪ Ai ) ≤

Cards

i= 1

52

Σ P(Ai ) − (n − 1) i= 1

n

Σ P(Ai ) i= 1

(ii) If A and B are two events associated to a random experiment, then P ( A ∩ B) ≤ P( A) ≤ P ( A ∪ B) ≤ P( A) + P(B)

Colours Red 26

Black 26

(iii) If A and B are two events associated with a random experiment, then (a) P ( A ∩ B) = P(B) − P ( A ∩ B)

Heart 13

Diamond 13

Club 13

(b) P ( A ∩ B ) = P( A) − P ( A ∩ B)

Spade 13

(c) P[( A ∩ B ) ∪ ( A ∩ B)] = P( A) + P(B) (d) P ( A ∩ B ) = 1 − P ( A ∪ B)

Probability

(e) P( A ∪ B ) = 1 − P ( A ∩ B)

In a random experiment, let S be the sample space and E be the event. Then, Number of distinct elements in E n( E ) = P(E ) = Number of distinct elements in S n(S ) =

(f) P( A) = P ( A ∩ B) + P ( A ∩ B ) (g) P(B) = P ( A ∩ B) + P (B ∩ A ) (iv) (a) P (exactly one of A, B occurs) = P( A) + P(B) − 2 P( A ∩ B) = P ( A ∪ B) − P ( A ∩ B)

Number of outcomes favourable to E Number of all possible outcomes

(b) P (neither A nor B) = P (A ∩ B) = 1− P(A ∪ B)

(i) If E is an event and S is the sample space, then (a) 0 ≤ P(E ) ≤ 1

− 2P ( A ∩ B)

(v) If A, B and C are three events, then P (exactly one of A, B, C occurs)

(b) P(φ ) = 0

(c) P(S ) = 1

= P( A) + P(B) + P(C ) − 2 P ( A ∩ B)

(ii) P (E ) = 1 − P (E )

− 2 P (B ∩ C ) − 2 P ( A ∩ C ) + 3P ( A ∩ B ∩ C ) (vi) P(atleast two of A, B, C occurs)

Addition Theorem of Probability

= P( A ∩ B) + P(B ∩ C ) + P(C ∩ A) − 2 P( A ∩ B ∩ C )

(a) For two events A and B, P ( A ∪ B) = P( A) + P(B) − P ( A ∩ B)

(vii) P (exactly two of A, B, C occurs) = P( A ∩ B) + P(B ∩ C ) + P( A ∩ C ) − 3P( A ∩ B ∩ C )

If A and B are mutually exclusive events, then P ( A ∪ B) = P( A) + P(B) [for mutually exclusive, P ( A ∩ B) = 0 ]

(viii) (a) P( A ) = 1 − P( A) (b) P ( A ∪ A ) = P(S ), P (φ ) = 0

(b) For three events A, B and C, P ( A ∪ B ∪ C ) = P( A) + P(B) + P(C ) − P ( A ∩ B ) − P (B ∩ C ) − P ( A ∩ C) + P ( A ∩ B ∩ C) If A, B and C are mutually exclusive, then P ( A ∪ B ∪ C ) = P( A) + P(B) + P(C ) for mutually exclusive events, P( A ∩ B) = P(B ∩ C ) = P(C ∩ A) = P( A ∩ B ∩ C)=0

Conditional Probability Let E and F be two events associated with a random experiment. Then, probability of occurrence of event E, when the event F has already occurred, is called conditional probability of event E over F and is denoted by P(E / F ). P(E ∩ F ) , where P(F ) ≠ 0. P(E / F ) = P(F )

Properties of Conditional Probability Let A,B and C be the events of a sample space S. Then,

(c) If a set of events A1, A2 ,… , An are mutually exclusive, then A1 ∩ A2 ∩ A3 ∩…∩ An = φ. ∴ P( A1 ∪ A2 ∪ A3 ∪…∪ An ) = P( A1 ) + P( A2 ) +…+ P( An ) and P ( A1 ∩ A2 ∩ A3 ∩…∩ An ) = 0

(i) P(S / A) = P( A / A) = 1 (ii) P{( A ∪ B)/ C} = P( A / C ) + P(B / C ) − P{( A ∩ B)/ C}; P(C ) ≠ 0 (iii) P( A′ / B) = 1 − P( A / B), where A′ is complement of A.

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Fast Track Revision Notes

Mathematics-XII

Multiplication Theorem on Probability Let A and B are two events associated with a random experiment, then P( A)⋅ P(B / A), where P( A) ≠ 0 P( A ∩ B) = P(B)⋅ P( A / B), where P( A) ≠ 0 The above result is known as the multiplication rule of probability.

Probability Distribution of a Random Variable The system in which the value of a random variable are given along with their corresponding probability is called probability distribution. If X is a random variable and takes the value x1, x2 , x3 ,..., xn with respective probabilities p1, p2 , p3 ,..., pn . Then, the probability distribution of X is represented by

Multiplication Probability for more than Two Events Let E, F and G be three events of sample space S, then F G P(E ∩ F ∩ G ) = P(E )⋅ P ⋅ P E E ∩ F

x1

x2

x3

...

xn

P( X )

p1

p2

p3

...

pn

such that Σ pi = 1 If xi is one of the possible values of a random variable X, the statement X = x i is true only at some point(s) of the sample space. Hence, the probability that X takes value x i is always non-zero, i.e., P( X = x i ) ≠ 0.

Independent Events Two events A and B are said to be independent, if the occurrence or non-occurrence of one event does not affect the occurrence or non-occurrence of another event. Then, P( A ∩ B) = P( A)⋅ P(B); P( A / B) = P( A) and P(B / A) = P(B)

X

Mean and Variance of a Probability Distribution n

Mean of a probability distribution is

Partition of Sample Space

∑ xi ⋅ pi.

n

A set of events E1, E2 , ..., En is said to represent a partition of the sample space S, if it satisfies the following conditions (i) Ei ∩ E j = φ, i ≠ j , i , j = 1, 2,..., n (ii) E1 ∪ E2 ∪...∪ En = S (iii) P(Ei ) > 0, for all i = 1, 2,..., n

called expectation of X, i.e., E( X ) = ∑ xi pi. i =1

n Variance is given by V ( X ) = ∑ xi2 ⋅ pi − ∑ xi ⋅ pi i=1 i =1 n

2

n

or V ( X ) = E( X 2 ) − [E( X )]2 , where E( X 2 ) = ∑ xi2 ⋅ p( xi ) i =1

Theorem of Total Probability Let S be the sample space and E1, E2 , E3 ,..., En be n mutually exclusive and exhaustive events associated with a random experiment. If E is any event which occurs with E1, E2 , E3 ,..., En . Then, P(E ) = P(E1 )⋅ P(E / E1 ) + P(E2 )⋅ P(E / E2 ) + P(E3 )⋅ P(E / E3 ) + ... + P(En )⋅ P(E / En ) E P(E ) = ∑ P (Ei )⋅ P Ei i =1 n

or

It is also

i =1

Baye’s Theorem Let S be the sample space and E1, E2 ,..., En be n mutually exclusive and exhaustive events associated with a random experiment. If A is any event which occurs with E1, E2 , ..., En , then probability of occurrence of Ei when A occurred, P(E )P( A / Ei ) , i = 1, 2,..., n P(Ei / A) = n i

∑ P(Ei )P( A/ Ei ) i =1

Bernoulli Trials The independent trials which have only two outcomes i.e., success or failure, is called Bernoulli trial.

Conditions for Bernoulli Trials (i) (ii) (iii) (iv)

There should be a finite number of trials. The trials should be independent. Each trial has exactly two outcomes success or failure. The probability of success remains the same in each trial.

Binomial Distribution The probability distribution of number of successes in an experiment consisting n Bernoulli trials obtained by the binomial expression ( p + q )n , is called binomial distribution. This distribution can be represented by the following table 0

X

If P(E1 ) = P(E2 ) = P(E3 ) =... = P(En ), then P(E / Ei ) P(Ei / E ) = n

n

P( X ) C 0 q

1 n n

C1 q

2

n−1 1 n

p

C2 q

...r

n−2 2

n

p ... C r q

n− r

... p

r

...

n n

Cnpn

Here, P( X = r ) = n Cr q n − r pr is called the probability function of the binomial distribution. where, p = Probability of success q = Probability of failure, n = Number of trials and p + q = 1

∑ P(E / Ei ) i =1

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