Mathematics Important Results And Formulae Chapter - I Applications Of

Std XII

MATHEMATICS IMPORTANT RESULTS AND FORMULAE CHAPTER - I APPLICATIONS OF MATRICES AND DETERMINANTS

Adjoint : adj A is the transpose of the cofactor matrix [A ij] of A i.e., adj A = [Aij]T Result : A (adj A) = (adj A) A = |A| I n where A is a square matrix of order n and I n is the identity matrix of order n. a

eg : A =  c 

b  d 

−b   a  

 d

adj A =  − c 

Inverse : A–1 =

1 |A|

(adj A) where A is a non-singular matrix.

Properties : 1. Reversal law for inverses : If A, B are any two non-singular matrices of the same order, then AB is also non – singular and (AB)–1 = B–1 A–1. 2. Reversal law for transposes : (AB)T = BT AT. 3. (AT)–1 = (A–1)T where A is any non-singular matrix. Matrix inversion method : If a system of linear equations is given in matrix form as AX=B, where | A|≠ 0, then X=A-1B. Rank of a matrix : It is the order of any highest non-zero minor of A. i.e. ρ(A) =r if (a) A has atleast one minor of order r which does not vanish and (b) every minor of higher order is zero. Equivalent matrices have the same rank. The rank of a matrix in echelon form is equal to the number of non-zero rows of the matrix.

1

Std : XII

Applications of Matrices and Determinants Important Results and Formulae

Consistency of a system of linear equations :  A system of non-homogeneous equations is called consistent if it has atleast one solution.  The system is called inconsistent if it has no solution.  A system of homogeneous linear equations is consistent, as it always has the trivial solution.  A consistent system may have one or more solutions. If it has only one solution it is called a unique solution. Elementary transformations on a matrix : 4. Ri ↔ Rj (Ci ↔ Cj)

→ kRi (Ci → kCi) 6. Ri → Ri + kRj (Ci → Ci + k Cj) 5. Ri

Echelon form of a matrix : A matrix A of order m x n is said to be in echelon form or triangular form if a) Every row of A which has all its entries 0 occurs below every row which has a non – zero entry. b) The first non – zero entry in each non – zero row is 1. c) The number of zeros before the first non – zero element in a row is less than the number of such zeros in the next row. Note: The result holds even if the non-zero entry in each non-zero row is

other than 1.

Cramer’s rule method or Determinant method : If a system of non – homogeneous linear equations is given in matrix form as AX = B and if det A ≠ 0, then the system has a unique solution, given by det(A1) det(A2) det(An) x1 = det(A) , x2 = det(A) , . . . . ., xn = det(A) where Aj is the matrix obtained by replacing the entries in the jth column of A by the entries in the matrix B. Cramer’s rule is applicable only when ∆ ≠ 0.

2

Std : XII

Applications of Matrices and Determinants Important Results and Formulae

If ∆ = 0, then the given system may be consistent or inconsistent. a) If ∆ = 0 and atleast one of ∆x, ∆y, ∆z is non-zero then the system is inconsistent. b) If ∆ = ∆x = ∆y = ∆z = 0 and atleast one minor of order 2 is non-zero, then the system is consistent and has infinitely many solutions. c)

If ∆ = ∆x = ∆y = ∆z = 0 and all 2 x 2 minors are also zero, but atleast one element is non-zero, then the system is consistent and has infinitely many solutions.

d) If ∆ = ∆x = ∆y = ∆z = 0, all 2 x 2 minors of ∆ = 0 and atleast one 2 x 2 minor of ∆x or ∆y or ∆z is non – zero then the system is inconsistent. Consistency by rank method : A system of non – homogeneous linear equations AX = B is consistent if and only if A and the augmented matrix [A, B] have the same rank. a) If ρ (A) ≠ ρ [A, B], then the system is inconsistent and it has no solution. b) If ρ (A) = ρ [A, B] = n, where n is the number of unknowns, then the system is consistent and there is a unique solution. c) If ρ (A) = ρ [A, B] < n, then the system is consistent, but has infinitely many solutions. x---x---x---x---x

3

Std : XII

Vector Algebra

CHAPTER – II

Important Results and Formulae

VECTOR ALGEBRA

1. Angle between two vectors :

b

If θ is the numerical measure of the

0

angle between two vectors, then 0 ≤ θ

b

θ aa

≤ .

θ

0

a

2. Scalar product or dot product : →

→

→

a . b

→

→

→

= | a || b | cos θ = ab cos θ, where θ is the angle between a and b →

→

→

→

3. Scalar produc is commutative : i.e. a . b = b . a 4. Projection of a on b =. || →

→

→

→

→

b on a = . ||

Projection of

→

→

→

∧

→

. b = a . b

→

→

∧

= a .b

= || →

5. If θ = 0º, a . b = ab ; If θ = , 6. If

→

= a. ||

→

a . b = − ab

→

→

→

a . b = 0, then a is a zero vector or b

is a zero vector or a and

→

b

are perpendicular. →

→

→

→

7. a . b = 0, if and only if a ⊥ b →

→

→

→2

8. a . a = │ a │2 = a →

→

→

9. i . i

→

→

= j . j

→

= a2 where | a | = a →

=k . k

→

→

→

= 1 and

→

→

→

i . j = j. k

→

→ →

10. If m is any scalar, then ( m a ) . b = m( a . b ) =

→

= k . i =0 →

→

a . (m b )

11. Scalar product is distributive over addition. i.e.,

→

→

→

→

→

→

→

→

→

→

a . (b +c ) = a . b +a . c

→

→

→

→

a . (b − c ) = a . b − a . c →

12. For any two vectors a

→

→

→

→ →

and b , ( a + b )2 = a2 + 2 a . b + b2 →

→

→ →

( a + b )2 = a2 + 2 a . b + b2 →

→

→

→

( a – b ).( a + b ) = a2 – b2 4 k

Std : XII

Vector Algebra →

13. If

→

→

→

= a1 i + a2 j + a3 k

a

Important Results and Formulae

→

→

→

→

→

→

b = b1 i + b2 j + b3 k then a . b

;

= a1b1 +

a2b2 + a3b3 →

→

14. If θ is the angle between a and b then θ = cos

.

-1

Gives exact position of the angle θ

|||| →

→

15. For any two vectors a and →

→

→

→

→

→

→

b , | a + b | ≤ |a | + |b |

→

→

→

16. | a + b |2 + | a − b |2 = 2 ( | a |2 + | b |2) →

→

17. a || b →

→ ⇒ → a = λb

→

→

where λ is any non – zero scalar

→

18. If r = a i + b j + c k then D.C’s are →

→

→

→

→

   b c  a  →, →, → r r r      →

→

→ →

19. | a + b + c |2 = a2 + b2 + c2 + 2 ( a . b + b . c + c . a ) 20. Cosine formula : In ∆ABC,

cos C =

cosA =

b2 + c2 − a2 2bc

; cos B =

c2 + a2 − b2 2ca

a2 + b2 − c2 2ab

21. Projection formula : a = bcosC + c cosB, b = acosC + ccosA, c = acosB + bcosA →

→

22. If F

denotes the force and d

denotes the displacement then work done by

the →

→

force = F . d

→

→

→

→

→ → →

→

23. For any vector r , r = ( r . i ) i 24. sin →

θ

∧

∧

= ½ | a − b |; 2 →

→

→

cos

→ →

→

+ (r . j ) j + (r .k )k

θ

∧

θ

∧

∧

∧

∧

∧

= ½ | a + b |; tan =| a − b |/| a + b | 2 2

∧

∧

25. a x b = | a || b | sin θ n where n is a unit vector perpendicular to both →

→

→

→

∧

a and b such that a , b , n form a right handed system.

→

→

→

→

26. a x b = vector area of a parallelogram with a and b as adjacent sides.

5

Std : XII →

Vector Algebra →

→

27. a . b →

→

Important Results and Formulae

→

= | a | (Projection of b on a )

→

→

→

→

28. a x b ≠ b x a →

→

→

→

; a x b = − (a x b )

→

→

→

→

29. If a and b are collinear or parallel then a x b = 0 →

→

→

→

30. a x a = 0 for any non – zero vector a 31.

→

→

i xi

=

→

x j = k x k

→

;

j

→

→

→

→

→

→

→

→

→

i x j = k

→

→

= 0 →

j x i

=− k

→

→

→

→

→

i x k=− j →

32. m a x b

→

k x j =− i

k x i = j ; →

→

→

j x k = i ;

→

→

→

→

→

= m (a x b ) = a x mb

33. Distributivity of vector product over vector addition →

→

→

→

→

→

→

a x (b + c ) = a x b + a x c →

→

→

→

→

→

→

(b + c ) x a = b x a + c x a →

→

→

→

→

→

34. If a = a1 i + a2 j + a3 k ; b = b1 i →

→

then

→

a x b

=

i a1 b1

→

j a2 b2

→

→

+ b2 j + b3 k

→

k a3 b3

35. Angle between two vectors : sin θ = x (Gives acute angle only) x ∧ |||| 36. Unit vectors perpendicular to two given vectors : n = ± | x| →

→

37. Vectors of magnitude μ (normal to the plane containing a and b ) is ± μ x | x| → → 38. Area of quad. ABCD is ½ | AC x BD | →

→

perpendicular to both a and b

→

→

→

→

39. Area of a parallelogram is ½ | d x d | where d and d are the diagonals. 1 2 1 2 →

→

→

40. If a , b , c are the position vectors of the vertices A, B, C of a ∆ABC, then the →

→

→

→

→

→

area of ∆ABC is ½ | a x b + b x c + c x a | 6

Std : XII

Vector Algebra

Important Results and Formulae →

→

→

41. Condition for the points A, B, C whose p.v. are a , b , c to be collinear is →

→

→

→

→

→

→

a x b +b x c + c x a = 0

→

→

42. Magnitude of the moment = | r x F | →

→

→

→

Moment (or) Torque of force F about O is M = r x F →

→

→

→

43. Area of a parallelogram with adjacent sides a and b is | a x b | →

→

→

→

44. Vector area of a parallelogram with adjacent sides a and b is a x b →

→

→

→

45. Area of a triangle with sides a and b is ½ | a x b | →

→

→

→

46. Vector area of a triangle with sides a and b is ½ ( a x b ) →

→

→

→

→

→

47. Area of a ∆ABC is ½ | AB x AC | or ½ | BC x BA | or ½ | CA x CB | →

48. If

→

→

→

→

a x b = 0 then (i) a is a zero vector and b is any vector →

(ii)

→

(iii) →

→

→

→

49. | a x b |2 + ( a . b ) →

→

→

→

2

→

b is a zero vector and a is any vector →

a and b are parallel (collinear) →

→

= | a |2 | b |2

→

→

→

→

→

→

50. a x ( b + c ) + b x ( c + a ) + c x ( b + a ) = 0 →

→

→

→

51. If a x b = c x d

→

and

→

→

→

a x c= b x d

then

→

→

→

→

a − d and b – c

are parallel. →

→

→

→

→

→

52. ( a x b ) . c

or

→

→

→

a . ( b x c ) is a scalar triple product.

→

53. ( a x b ) . c = [ a

→

b

→

c]

54. Geometrical interpretation of scalar triple product →

→

→

(a x b ) . c →

represents the volume of the parallelepiped whose co-terminus

→

→

edges a , b , c form a right handed system of vectors. 55. Properties : →

→

→

( a x b). c

→

→

→

→

→

→

= ( b x c ). a = ( c x a ) . b (Cyclic order) 7

Std : XII

Vector Algebra →

→

→

→

→

→

[a

b

[a

a

→

[λ a

→

→

c ] = − [b →

→

→

→

→

b ] = [a

b

→

→

c ] = λ [a

b

→

→

c ]= −[ c

a

→

c ] = [a

→

Important Results and Formulae

→

b

b

→

→

a ] = −[ a

→

c

→

b ]

→

a]=0

b

→

c ]

56. a , b , c are coplanar ⇔ [ a , b , c ]= 0 for three non-zero, non-collinear →

→

→

→

→

→

→

→

vectors →

57. If [ a , b , c ] = 0 then →

→

→

i) atleast one of the vectors a , b , c →

→

is a zero vector.

→

ii) any two of the vectors a , b , c are parallel. →

→

→

iii) a , b , c are coplanar a1 a2 a3 → → → 58. [ a , b , c ] =b b2 b3 1 c1 c2 c3

→

→

→

→

if a = a1 i + a2 j + a3 k →

→

→

;

→

b = b1 i + b2 j + b3 k ; →

→

→

→

c = c 1 i + c2 j + c3 k

→

→

→

59. Volume of a parallelepiped = [ a , b , c ] →

→

→

→

→

60. For any three vectors a , b , c →

→

→

[a + b →

→

→

b +c

→

61. ( a x b ) x c

→

→

→

c + a ] = 2 [a , b , c ]

→

and

→

→

→

→

a x ( b x c ) are called vector triple products of a , b

→

, c

It is a linear combination of those two vectors which are within brackets. →

→

→

= (a . c )b − (b . c )a

→

→

→

is perpendicular to c and lies in the plane which contains a ,

62. ( a x b ) x c

63. ( a x b ) x c

→

→

→

→

→

→

→

b

→

→

→

→

→

→

→ →

→

64. a x ( b x c ) = ( a . c ) b − ( a . b ) c →

→

→

→

65. ( a x b ) . ( c x d ) = .

.

. . 8

→

→

Std : XII

Vector Algebra

Important Results and Formulae

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→ →

→

66. ( a x b ) x ( c x d ) = [ a b d ] c − [ a b c ] d

→

(a x b ) x (c x d ) = [ a c d ] b − [ b c d ] a →

67. If four vectors →

→

→

→

a , b , c , d

→

→

→

→

→

are coplanar then ( a x b ) x ( c x d ) = 0

→

→

68. If a , b be lie on one plane and

→

c , d lie on another plane, these planes

are →

→

→

→

perpendicular then ( a x b ) . ( c x d ) = 0 →

→

→

→

→

→

→

69. [ a x b , b x c , c x a ] = [ a

→

→

2 c ]

b

70. Equation of a straight line passing through a given point and parallel to a given →

vector is

→

→

r = a +t v

a) Non – parametric vector equation : b) Cartesian form :

→

→

→

→

r x v = ax v

( x − x1 ) ( y − y1 ) ( z − z1 ) = = l m n

71. Equation of a straight line passing through the origin and parallel to a vector →

is r

→

=t v

→

→

→

a) Non – parametric vector form : r x v b) Cartesian form :

= 0

x y z = = l m n

72. Equation of a straight line passing through two given points is →

→

→

→

→

r = a + t ( b − a ) or

→

→

r = ( 1 – t) a + t b →

→

→

→

→

a) Non – parametric vector form : ( r − a ) x ( r − b ) = 0 b) Cartesian form :

( x − x1 ) ( y − y1 ) ( z − z1 ) = = ( x 2 − x1 ) ( y 2 − y1 ) ( z 2 − z1 )

73. x2 – x1, y2 – y1, z2 – z1 are the d.r.s of the line joining the points (x1, y1, z1) and (x2, y2, z2). 74. Angle between two lines: θ = cos-1

.

|||| 9

Std : XII Vector Algebra Important Results and Formulae 75. The shortest distance between two intersecting lines is zero. →

76. The distance between two parallel lines r

→

→

→

→

→

= a1 + t u , r = a 2 + s u

│ x (–)│ is d = ││ →

77. The distance between the skew lines r │(–) d=

→

→

→

→

→

= a1 + t u , r = a 2 + s v

is

│ │x│ →

→

→

78. Condition for two lines to intersect is [ ( a 2 – a 1 ) u x2 – x1 y2 – y1 z2 – z1 l1 m1 n1 =0 l2 m2 n2

→

v ] = 0 or

79. If three points are collinear then their p.v.s are coplanar, but the converse need not be true. 80. Equation of a plane passing through a given point and perpendicular to a vector →

is r

→

→

→

. n = a . n

→

→

→

(or) ( r − a ). n = 0

Cartesian form : a ( x – x1) + b ( y – y1) + c ( z – z1) = 0 81. The vector equation of a plane passing through the origin and perpendicular to → → → the vector n is r . n = 0 82. Equation of the plane when distance from the origin and unit normal is given is →

∧

r. n = p

Cartesian form : l x + my + nz = p

→

83. If n is a normal vector but not a unit vector then the vector equation of the →

→

plane is r . n = q q 84. The length of the perpendicular from origin to this plane is ││ 85. Equation of the plane passing through a given point and parallel to two given →

→

→

vectors is r = a + s u + t v x – x1 y – y1 Cartesian form : l1 m1 l2 m2

z – z1 n1 n2 10

=0

Std : XII

Vector Algebra

Important Results and Formulae

Non – parametric vector equation : →

→

[ r − a

→

u

→

→

→

→

v ] = 0 or [ r u

→

→

v ] = [a

→

v ]

u

86. Equation of the plane passing through two given points and parallel to a given →

→

→

→

→

vector is r = a + s ( b − a ) + t v

→

Non – parametric vector equation : x – x1 y – y1 Cartesian form : x – x y2 – y1 2 1 l m

→

(OR)

→

→

→

r = (1 – s) a + s b + t v

→

→

[r − a z – z1 z2 – z1 n

→

→

b− a

v ]=0

=0

87. Equation of the plane passing through three given non – collinear points is →

→

→

→

r = ( 1 – s – t) a + s b + t c

→

→

→

→

Non – parametric vector equation : [ r − a b − c x – x1 y – y1 z – z1 x – x1 y2 – y1 z2 – z1 Cartesian form : 2 =0 x3 – x1 y3 – y1 z3 – z1

→

→

c− a ]=0

88. Equation of a plane passing through the line of intersection of two given planes →

→

→

→

( r . n 1 − q1) + λ ( r . n − q2) = 0 2 Cartesian form : (a1x + b1y + c1z + d1) + λ (a2x + b2y + c2z + d2) = 0 89. The distance between a point (x1, y1, z1) and a plane ax + by + cz + d = 0 is ax1 + by1 + cz1 + d √ a2 + b2 + c2 d 90. The distance between the origin and a plane ax + by + cz + d = 0 is √a2 + b2 + c2 91. The distance between two parallel planes ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 is

d1 – d2

√a2 + b2 + c2 92. Equation of a plane which contain two given lines (passing through two given →

lines) r

→

→

→

= a1 + t u →

[ r − a1

→

u

and

→

→

→

→

r = a2+ s v →

→

v ] = 0 and [ r − a 2 11

is →

u

→

v]=0

Std : XII Cartesian form : x – x1 y – y1 l1 m1 l2 m2

Vector Algebra z – z1 n1 n2

Important Results and Formulae

x – x2 l1 = 0 (OR) l 2

93. Angle between two given planes : θ = cos−1 94. If two planes are

y – y2 m1 m2

z – z2 n1 n2

=0

. ||||

→

→

a) perpendicular then n . n = 0 1 2 →

→

b) parallel then n = λ n where λ is a scalar. 1 2 →

→

→

→

→

95. Angle between a line r = a + t b and a plane r . n = q is θ = sin-1

.

|||| → 96. Vector equation of the sphere whose p.v. of centre is c and radius is a is →

→

→

|r − c|=| a| 97. Vector equation of the sphere whose centre is origin and radius →

is

→

| r | = |a | 98. General equation of a sphere is x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0 99. Centre = (−u, −v, −w) ; radius =

u 2 +v 2 +w 2 - d

100. Vector equation of the sphere when the extremities of the diameter being given : →

→

→

→

(r − a).(r - b)=0 101. Cartesian form : (x – x1) (x – x2) + ( y – y1) ( y – y2) + (z – z1) (z – z2) = 0 x---x---x---x---x

12

Std : XII

Complex Numbers

Important Results and Formulae

CHAPTER – III COMPLEX NUMBERS C = { a + ib / i2 = − 1, a , b

∈ R }.

If z = a + i b, then Re(z) = a; Im(z) = b a + i b and c + id are equal if and only if a = c and b = d The negative of z = a + i b is denoted by –z and is defined by –z = (−a)+ i (−b) z1 + z2 = (a+ i b) + (c + i d) = (a + c) + i (b + d) z1 – z2 = (a + i b) – ( c + i d) = (a – c) + i ( b – d) z1z2 = ( a + i b) (c + i d) = (ac – bd) + i (ad + bc) z1 z2

= a + ib c + id

ac +=bd c2 + d2

bc+−iad c 2 + d2

The conjugate of z = a + i b is denoted by z and is defined by z = a – i b z z = (a + ib) (a – ib) = a2 + b2, which is a non-negative real number. z

=zz

If z is real (ie., b = 0) then z = z If z = z then z is real. z+ 2

If z = a + i b, z = a – i b then Re(z) = z1 + z 2 = z1 + z 2

()

zn = z

n

;

z1 − z 2 = z1 − z 2

z

z− ; Im(z) = 2i  z

; z1 z 2 = z1 ⋅ z 2 ;  1  = 1 , z 2 ≠ 0  z2  z2

If z = a + ib is a complex number then the representation

z = a + ib = (a, b) is called the ordered pair representation. i = (0, 1) The set of complex numbers is not ordered.We cannot say z1 < z2 or z1 > z2. We can say only z1 = z2 or z1 ≠ z2. If z = a + i b then |z| = √a2 + b2 |z| = √ zz ; Re(z) ≤ |z| ; Im(z) ≤ |z| |z1z2| = |z1||z2|; |z1z2. . .. .zn| = |z1||z2| . . .. . .|zn| |z | z1 = 1 where z2 ≠0 | z2 | z2 Triangle Inequality : | z1 + z2| ≤ |z1| + |z2| |z1 – z2 | ≤ |z1| + |z2| |z1 + z2 + . . . . . zn| ≤ |z1|+ |z2| + . . . . |zn|

13

Std : XII |z1 – z2| ≥ |z1| - |z2|

Complex Numbers

Important Results and Formulae

z = x +i y = r(cosθ + isinθ) is called the modulus and amplitude form (polar form) of the complex number z. |z| = r = √x2+y2 ; θ = tan-1(y/x) Principal value : For an arbitrary complex no. z ≠0, the principal value of arg z is defined to be the unique value of z that satisfies –  < arg z ≤  For z = 0, arg(z) is indeterminate. For any z1, z2

(a) |z1. z2 |= | z1||z2| (b) arg(z1 z2) = arg(z1) + arg(z2)

|z1 z2 . . . . . zn| = |z1||z2| . . .. |zn| Arg (z1 z2 . . .. zn) = arg(z1) + arg(z2) + . . .. . + arg(zn) z z1 | z1 | and arg 1 = arg(z1) – arg(z2) where z2 ≠0. = z2 z2 | z2 | Euler’s formula : eiθ = cosθ + isinθ If z ≠0 then z = r (cosθ+ i sinθ) = reiθ, is called the exponential form of the complex number z. If z = eiθ then z-1 =

1 z

= e−iθ ; z = e−iθ ; zn = einθ

If z1 = r1eiθ1, z2= r2eiθ2 then z1 z2 = (r1r2)ei(θ1 + θ2) ,

;

│z│= 1 iff z = z

z1  r1  i(θ – θ ) =  e 1 2 z 2  r2 

If |z – z1| = |z – z2| then the locus of z is the perpendicular bisector of the line joining the two points z1 and z2. If (a1 + ib1) (a2 + ib2) . .. . (an + ibn) = A + iB then i) (a12 + b12) (a22 + b22) . . . . (an2 + bn2) = A2 + B2 b b b B ii) tan−1 1 + tan−1 2 + . . . .. + tan−1 n = k + tan−1 A a2 a1 an General rule for determining the argument θ Let z = x + i y, x, y

∈

II θ=π−α

R | y| (i) If z = x + i y then θ = tan-1 |x| (ii) If z = x – i y then θ = −tan−1| y | |x| (iii) If z = −x – i y then θ = −  + tan|-1y | |x| | y| (iv) If z = −x + i y then θ =  − tan-1 |x| 14

I θ=α

θ=−π+α

θ=−α

III

IV

, ,k

∈

Z

Std : XII arg z = − arg z

Complex Numbers

Important Results and Formulae

The distance between the complex numbers z1 and z2 is d = | z1 – z2 | The effect of multiplication of a complex number z by ei is the rotation of z counter clockwise about the origin O through an angle .

The effect of multiplication of a complex number z by i = ei/2 is the rotation of z counter clockwise about the origin O through an angle

π 2

.

|z1 + z2 + z3 |≤ |z1| + |z2| + |z3| For any polynomial equation P(x) = 0 with real coefficients, imaginary (complex) roots occur in conjugate pairs. De Moivre’s theorem : For any rational number n, cosnθ + isin nθ is the value or one of the values of (cosθ + i sinθ)n. a) (cosθ + i sinθ)n = cosnθ + i sin nθ b) (cosθ + i sinθ)−n = cosnθ – i sin nθ c) (cosθ – i sinθ)n = cosnθ – i sin nθ d) (sinθ + i cosθ)n = cosn(

π 2

– θ) + i sin n(

π 2

– θ)

(OR)

= i n (cosθ – i sinθ)n A number ω is called an nth root of a complex no. z if ωn = z and we write ω = z1/n Working Rule to find the nth roots of a complex no. : a) Write the given complex no. in polar form. b) Add 2k to the argument. c) Apply DeMoivre’s theorem. d) Put k = 0,1, . . . . . upto n – 1 2π

4π

a) The nth roots of unity are e˚, eni , ein , . . . . ei 2π

ie., 1, ω, ω2, . . . .. ωn-1 where ω = eni b) The roots are in G.P. c) 1 + ω + ω2+ . . . . + ωn–1 = 0 d) 1. ω . ω2 . . . . . . ωn–1 = (−1)n-1 1 2

2 15

2(n-1) n



Std : XII Complex Numbers a) Cube roots of unity are 1, − ±i

Important Results and Formulae

b) They are the vertices of an equilateral triangle all lying on the unit circle. c) They are in G.P. d) ω3 = 1; 1 + ω + ω2 = 0 Note : ω =cis (

2π ) 3

a) Fourth roots of unity are 1, i, −1, −i b) They are the vertices of a square all lying on the unit circle. c) If ω = i then the roots are 1, ω, ω2, ω3 d) ω4 = 1 ; 1 + ω + ω2 + ω3 = 0 Note : ω =cis (

π 2

)

π , cis2π , cis3π , cis4π , cis5π 3 3 3 3 3 b) They are the vertices of a hexagon all lying on the unit circle. a) The sixth roots of unity are 1, cis c) They are in G.P. π cis d) ω6=1 ;1+ω+ω2+ω3+ω4+ω5 = 0 where ω= 3 x---x---x---x---x

16

CHAPTER – IV ANALYTICAL GEOMETRY General Equation of a conic is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 p M The conic is a parabola if e = 1 ℓl The conic is an ellipse if e < 1 F The conic is a hyperbola if e > 1

FP = e PM

Parabola : Focus : The fixed point used to draw the parabola is called the focus. Directrix : The fixed line used to draw a parabola is called the directrix of the parabola. Axis : The axis of the parabola is the axis of symmetry. Vertex : The point of intersection of the parabola and its axis is called its vertex. Focal distance : The focal distance is the distance between a point on the parabola and its focus. Focal chord : A chord which passes through the focus of the parabola is called the focal chord of the parabola. Latus rectum : It is a focal chord perpendicular to the axis of the parabola. y

o c u s

2

x

r i g

Y

h t w

a

2

=

r d (sO ) p e

- 4 n

a

x

e

f t w

a

r d

s )

y

r t e

d

x

x

A

i r e c t r i x

=

4

p e

n

17

L

L a t u s r e c t u m

a

v e

x is

Focus

d

( O

n

v e

A x is

x

a

y S

Focus

p e

4

x

S d ir e c tr ix

v e rte x

d ire c trix

( O

=

o c u s

L a t u s r e c t u m

L a t u s r e c t u m

2

Y

F

x

v e rte x

F S

A x is

A x is

y

S

Types of parabolas :

i r e c t r i x

r t e x

x

L a t u s r e c t u m

y u

p

w

a

r d

s )

( O

p e

n

d

o

w

n

w

a r d

s )

Std : XII Results : Eqn

Analytical Geometry

Type

y =4ax

Diagram / End pts of LR

Eqn. of Axis

Vertex Eqn

directri

y

Open

2

Focus

Important Results and Formulae

(a, 0)

L

Lengt

of LR

h

of

x x = −a

y =0

(0,0)

x =a

LR 4a

x =a

y =0

(0,0)

x=−a

4a

y=−a

x=0

(0,0)

y=a

4a

y=a

x=0

(0,0)

y=−a

4a

rightwards x

L ' L L

y =−4a

Open

2

x

= ' =

leftwards

( a , 2 a ( a , - 2

) a

)

(−a,0)

y L

x

L '

x2=4ay

L L

Open

= ' =

( - a , 2 a ) ( - a , - 2 a )

(0,a)

upwards

L L

x =−4a

Open

y

downwards

2

= ' =

( 2 a , ( - 2 a

a ,

) a

)

(0,−a)

L L

= ' =

( + 2 ( - 2

a a

, ,

- a - a

) )

Vertex other than origin : Let v = (h, k) a) (y−k)2=4a(x − h) (Open rightwards) leftwards)

18

b) (y − k)2 = −4a(x −h) (Open

Std : XII Analytical Geometry c) (x−h)2=4a(y − k) (Open upwards) d)

(x

Important Results and Formulae −h)2=−4a(y − k) (Open

downwards) x2 y2 + = 1, a > b a2 b2

Ellipse : Equation

Diagram

x2 y2 + = 1; a > b b2 a2

where b2=a2(1 −e2)

where b2=a2(1−e2)

y

x a

2 2

w

y + b h

2

= 1

2 2

e r e

;

a

>

b

b

=

a

L 2 M

Z

d

i r e

c t r ix

A 2

2

( yI

B

e

)

L 1

F

L 1 '

P

'

M

x b

w

2 2

+

h

2

y a

2

e

;

=1 2

r e

a b

2

> =

b a

B

1

L 1 B

' 2

( I

-

e

)

C

x

(0,0)C F 1 A Z X AA’ BB’ L 1 ' 2 ' 2a B ' 2b y=0 x=0 A = (a,0) A’ = (−a, 0) B = (0,b) B’ = (0,−b) A, A’ x =a/e ; x = −a/e F1=(ae, 0) ; F2=(−ae,0) x = ±ae 2b2/a L1=(ae,b2/a); L1’=(ae,−b2/a)

(0,0) L 2 ' L 2 F 2 AA’ BB’ A ' d ir e c t r i x 2a Z ' 2b x=0 y=0 A = (0,a) A’=(0,−a) B = (b, 0) B’ = (−b, 0) A, A’ y = ±a/e F1=(0, ae) ; F2 = (0, −ae) y = ±ae 2b2/a L1=(b2/a,ae);L1’=(−b2/a,ae)

End points of LR

L2=(−ae,b2/a);L2’=(−ae,

L2=(b2/a,−ae);L2’=(−b2/a,

Eccentricity CA, CA’ CB, CB’ CZ, CZ’ Z Z’

−b2/a) e = √1 - (b2/a2) CA = CA’ = a CB = CB’ = b CZ = CZ’ = a/e Z Z’ = 2a/e

−ae) e = √1− (b2/a2) CA = CA’ = a CB = CB’ = b CZ = CZ’ = a/e Z Z’ = 2a/e

2

d i r e c t r ix

d ir e c tr ix

Centre A ' Z ' F Major axis Minor axis L Length of major axis Length of minor axis Equation of major axis Equation of minor axis Endpoints of major axis End points of minor axis Vertices Directrices Foci Equation of LR Length of LR

-

F1F2 = 2ae ; A A’ = 2a ; CF = ae Centre = M.P of F1F2 or M.P of A A’ The equation of the ellipse with centre (h, k) and (x−h)2 (y−k)2 + 2 a b2 19

=1

Std : XII Analytical Geometry Important Results and Formulae (i) Major axis along x axis and minor axis along y axis is (ii)Major axis along y axis and minor axis along x axis is(x−h)2 (y−k)2 + 2 b a2 Relation between eccentricity, semi major axis and semi minor axis :

=1

b2 = a2(1 − e2) The locus of a point which moves such that the sum of its distances from two fixed points in a plane is a constant, is an ellipse. Hyperbola : Equation

Centre Vertices Foci Transverse axis Length of transverse axis Eqn. of transverse axis Conjugate axis Length of conjugate axis Eqn. of conjugate axis Length of LR Eqn. of LR Eqn. of directrices Eccentricity Endpoints of LR

x2 y2 − =1 a2 b2

y2 x2 − =1 a2 b2

Where b2=a2(e2−1) C(0,0) A(a,0), A’(−a,0) F1=(ae,0) , F2(−ae,0) AA’ 2a y=0 BB’ 2b x=0 2b2/a x = ±ae x = ±a/e

Where b2=a2(e2−1) C = (0,0) A(0,a), A’(0, −a) F1(0,ae) , F2(0,−ae) AA’ 2a x=0 BB’ 2b y=0 2b2/a y = ±ae y = ±a/e

e = √ 1 + (b2/a2) L1 = (ae, b2/a)

e = √ 1 + (b2/a2) L1 = (b2/a, ae)

L1’ = (ae, −b2/a)

L1’ = (−b2/a, ae)

L2 = (−ae, b2/a)

L2 = (b2/a, −ae)

L2’ = (−ae, −b2/a)

L2’ = (−b2/a, −ae) b

R

e la t io n D

b

2

=

2

a

=

( e

-

( e

L 1 '

F

1 )

A Z

B

B

C A

'

Z

'

B L 2 '

Z

'

-

L 1

1

L 1

L 2

2

2

a y

y

F

2

b e et w, ae , e bn

ia g r a m

2

2

A

F

1

'

C

Z

A

x

' L 2

L 2 '

L 1 '

F

20

x

B

'

2

1 )

Std : XII Analytical Geometry F1F2 = 2ae, C = M.P of F1F2 or M.P of AA’

Important Results and Formulae

Distance between directrices = 2a / e A point moves such that the difference of its distances from two fixed points in a plane is a constant. The locus of this point is a hyperbola and this difference is equal to the length of the transverse axis. Parametric form of conics : Conic

Parametric

Parameter

Range

Any point on the

1

Parabola

Equations x = at2, y = 2at

t

conic −∞ < t < (at2, 2at )or ‘t’

2

Ellipse

x = acosθ

θ

∞ 0 ≤ θ ≤ 2

(acosθ , bsinθ)

Hyperbola

y = bsinθ x = a secθ

0 ≤ θ ≤ 2

or ‘θ’ (asecθ,

3

θ

y = b tanθ Chords, tangents and normals : Cartesian form a) Eqn. of chord

btanθ)

or ‘θ’

Parabola

Ellipse b2(x +x ) y−y1=−2 1 2 a (y1 + y2)

4a joining (x1, y1) & y−y1= y + y (x - x1) 1 2 (x2, y2) b)Eqn. of tangent at (x1,y1) /Eqn. of yy1 = 2a(x+x1) chord of contact

xx1 a2

+

a2x x1

b2y − y1

yy1 b2

Hyperbola (x-x1)

b2(x1+x2) a2(y1 + y2)

y−y1=

=1

xx1 a2

−

a2x x1

b2y + y1

yy1 b2

(x-x1)

=1

of tangents from (x1, y1) c) Eqn. of normal xy1+2ay=x1y1+2a at (x1, y1) Parametric form a) Eqn. of chord

y1 Parabola Chord joining and t2 is y(t1+t2) 2at1t2

=

t1

2

=a −b

Ellipse Chord joining

and θ2 is 2x+ x cos (θ1+θ2) a 2 y (θ1+θ2) b sin 2 = cos

21

(θ1−θ2) 2

2

θ1 +

=a2+b2

Hyperbola Chord joining and θ2 is x cos (θ1−θ2) a 2 y (θ1+θ2) b sin 2 = cos

(θ1+θ2) 2

θ1 −

Std : XII

Analytical Geometry

b) Eqn. of tangent

At ‘t’ is yt = x + at

2

c) Eqn. of normal

y + xt = 2at + at3

d) No. of tangents e)No. of normals

2 3

Important Results and Formulae

At θ is x secθ − At θ is x cosθ + a a y y b sinθ = 1 b tanθ = 1 by ax by − ax + sinθ cosθ tanθ secθ =a2−b2 =a2+b2 2 4

2 4

Eqn. of tangent at (x1, y1) is obtained from the equation of the curve by y + y1 xy1+yx1 x + x1 replacing x2 by xx1, y2 by yy1, xy by , x by and y by 2 2 2 Parabola a) Condition that y=mx+c may be a c =

a m

Ellipse

Hyperbola

c2=a2m2+b2

c2=a2m2−b2

+b2 a2m , √a2m2+b2 √a2m2+b2

−a2m , √a m2−b2

tangent to b) Point of contact

c) Eqn of tangent

a , 2a 2 m m y = mx +

a m

y

=

mx

2

−b2 √a2m2−b2

± y = mx ±√a2m2−b2

√a2m2+b2

The point of intersection of the tangents at t1’ and ‘t2’ to the parabola y2=4ax is (at1t2, a(t1+t2)) The normal at ‘t1’ on the parabola y2=4ax meets the parabola again at ‘t2’ then 2 t2= − t1 +t 1 If ‘t1’ and ‘t2’ are the extremities of any focal chord of the parabola y2 = 4ax then t1t2 = −1 Result a) The chord of contact

of

Parabola

Ellipse

Hyperbola

Focus

Corresponding

Corresponding

focus

focus

tangents from any

22

Std : XII

Analytical Geometry

point

on

Important Results and Formulae

the

directrix of . . . passes through its b) The condition that l x + my+n = 0

may

be

am2 = l n

a2 l2 + b2m2 = n2

a

tangent to . . . c) The condition that l x +my+n =

al3 + 2alm2+

0 may be a normal

m2n=0

to d)

The

locus

foot

a2 l2 − b2m2 = n2

b2 + m2

a2 l2

(a2−b2)2 = n2

b2 a2 − m2 l2

(a2+b2)2 = n2

of x = 0

the circle

the circle

of

x2 + y2 = a2

x2 + y2 = a2

perpendicular

(auxiliary circle)

from a focus to a tangent to e) The locus of the

x = −a

x2 + y2 = a2 + b2

x2 + y2 = a2 − b2

point of

(the directrix)

(director circle)

(director circle)

intersection of perpendicular tangents to Asymptotes : An asymptote to a curve is the tangent to the curve such that the point of contact is at infinity. The asymptote touches the curve at +∞ and −∞. Equation of the asymptotes to the hyperbola

x2 y2 − =1 a2 b2

is

x2 y2 − =0 a2 b2

The equations of two asymptotes to the hyperbola are y=

b −b x y x and y = x. (ie.,) − a a a a

= 0 and

x y + =0 a a

The asymptotes pass through the centre (0, 0) of the hyperbola. The slopes of asymptotes are

b −b and a a

23

.

Std : XII

Analytical Geometry

Important Results and Formulae

b  a 

Angle between the asymptotes is 2α = 2 tan-1   or 2α = 2sec-1(e) The standard equation of hyperbola and combined equation of asymptotes differs only by a constant. If l1 = 0, l2 = 0 are the separate equations of asymptotes, then the combined equation of the asymptotes is l1l2=0 and the equation of the corresponding hyperbola is of the form l1l2=k, where k is a constant. Combined equation of asymptotes is nothing but pair of straight lines. 2√h2−ab Therefore the angle between the asymptotes is tanθ = a+b A hyperbola is said to be a rectangular hyperbola if its asymptotes are at right angles. The angle between the asymptotes of the rectangular hyperbola is 90˚. Equation of rectangular hyperbola is x2 − y2 = a2 The combined equation of the asymptotes of the rectangular hyperbola x2 − y2 = a2 is x2 − y2 = 0 The separate equations of the asymptotes of a rectangular hyperbola x2 −y2 = a2 are x + y = 0 and x − y = 0 The standard equation of the rectangular hyperbola whose asymptotes a2 are the coordinate axis is xy = c2 where c2 = 2 The combined equation of the asymptotes of the rectangular hyperbola xy = c2 is xy = 0 The separate equations of the asymptotes of a rectangular hyperbola xy = c2 is x = 0 and y = 0 The eccentricity of the rectangular hyperbola is e =

2

Vertices of RH x2 − y2 = a2 are A(a, 0) A’ (−a, 0) −a Vertices of RH xy = c2 are A = ,a ,a A’ = √2 √2 √2 Foci of the RH x2 − y2 = a2 are (a 2 , 0), (−a 2 , 0)

−a √2

Foci of the RH xy = c2 are (a, a), (−a, −a) If centre of RH is at (h, k) and the asymptotes are parallel to x and y-axis then the general form of standard RH is (x −h) (y − k) = c2 c The parametric equation of RH xy = c2 are x = ct, y = where t is any t ct

c t

24

Std : XII

Analytical Geometry

Important Results and Formulae

non - zero real no. Any point on RH is

,

and is referred as ‘t’.

Equation of tangent at (x1, y1) to the RH xy = c2 is xy1 + yx1 = 2c2 Equation of the tangent at ‘t’ to the RH xy = c2 is x + yt2 = 2ct Equation of normal at (x1, y1) to the RH xy = c2 is xx1 − yy1 = x12 −y12 c ct3 Equation of normal at ‘t’ to the RH xy = c2 is y − xt2 = − t Two tangents and four normals can be drawn from a point to the RH. The foot of the perpendicular from a focus of a hyperbola on an asymptote lies on the corresponding directrix. The condition that the line lx + my + n = 0 may be a tangent to the rectangular hyperbola xy = c2 is 4c2 l m = n2 If the normal to the rectangular hyperbola xy = c2 at ‘t1’ meets the curve again at ‘t2’ then t13t2 = −1. The area of a triangle formed by the tangent at any point on the R.H xy = c2 is 2c2. Length of LR of the RH xy = c2 where c2 = Length of transverse axis of RH xy = c2 is 2a.

a2 is 2a 2

If F1 and F2 are the foci of the ellipse, then F1P + F2P = 2a, for any point P on the ellipse. If F1 and F2 are the foci of the hyperbola, then F1P~F2P =2a x---x---x---x---x

25

CHAPTER – V

DIFFERENTIAL CALCULUS APPLICATIONS - I

Velocity is the rate of displacement. If the displacement at time t is s then ds velocity v = dt Acceleration is the rate of change of velocity. If v is the velocity for time t 2 dv then the acceleration is a = d 2s = dt dt At t = 0, velocity is initial. When the particle comes to rest, velocity v = 0 When the acceleration is zero, velocity is uniform. dr dA dV Rate of change of radius = , area = , volume = dt dt dt Area of a triangle = ½ x base x height Area of a ∆ABC = ½ bc sinθ b2 = c2 + a2 − 2ac cosθ (cosine formula) Volume of the cone V = 1/3 r2h dy dx Equation of tangent to the curve y = f(x) at (x1, y1) is y − y1 = m(x −x1) Equation of normal to the curve y = f(x) at (x1, y1) is y − y1 = − 1 (x − x1) m m1 − m2 Angle between the two curves is tan θ = 1+m m 1 2 The condition for the two curves to intersect orthogonally is m1m2 = −1 The slope of the tangent to the curve y = f(x) at the point (x, y) is

The condition for the two curves to touch each other is m1 = m2. A continuous graph y = f(x) is such that f ’ (x)

→ ∞ as x → x1 at (x1, y1) then

y = f(x) has a vertical tangent x = x1. The condition for the curves ax2 + by2 = 1 and cx2 + dy2 = 1 to cut orthogonally 1− 1 1− 1= is b d a c Rolle’s theorem : Let f be a real valued function that satisfies the following three conditions. (i) f is defined and continuous on the closed interval [a, b] (ii) f is differentiable on (a, b) (iii) f(a) = f(b) then there exists atleast one point c Є (a, b ) such that f ‘ (c) = 0 f(x) = tan x is not continuous in [0, ] as tan x

→ ∞ at x = /2.

f(x) = |x| is not differentiable in (−1, 1) since f ' (0) does not exist.

26

Std : XII

Differential Calculus Applications - I

Important Results and Formulae If a and b are two roots of a polynomial equation f(x) = 0, then Rolle’s theorem says that there is atleast one root c between a and b for f ‘ (x) = 0 Mean value theorem (Law of the mean due to Lagrange) Let f(x) be a real valued function that satisfies the following conditions. (i) f(x) is continuous on [a, b] (ii) f(x) is differentiable on (a, b) then there exists atleast one point c Є (a, b) such that f ' (c) f(b) = − f(a) b−a The law of the mean can also be put in the form f(a + h) = f(a) + hf ' (a + θ h), 0 < θ < 1 Lagrange’s law of the mean is a particular case of Cauchy law of the mean. Rolle’s theorem is a particular case of Lagranges law of mean. In the law of mean, the value of ‘θ’ satisfies the condition 0 < θ < 1 Indeterminate forms : 0 Lt f(x) = f(a) = 0 x a g(x) g(a)

is called indeterminate form.

The other indeterminate forms are ∞ , 0 , ∞ , 1∞, ∞˚, 0˚, ∞ −∞, 0.∞, 0 0 ∞ ∞ 0

etc

^ l ’ Hopital’s rule : Let f and g be two functions such that lim f(x) = 0 and lim g(x) = 0 and f’(c) and x c x c f(x) g’(c) exists, and g’(c) ≠ 0 then lim x c gx) ^ l’ Hopital’s rule cannot be applied to

= limf '(c) x c g ' (c)

x+1 x + 3as x

→ 0 because f(x) = x + 1 and

g(x) = x + 3 are not in the indeterminate form as x → 0 Composite Function theorem : If lim g(x) = b and f is continuous at x = b then lim f(g(x)) = f ( lim g(x)) x a x a x a a) lim x 0

sin x x

=1

b) lim xsinx = 1 x 0+

c) lim xx = 1 x 0+ d) lim (cot x)sinx = 1 x 0 27

Std : XII

Differential Calculus Applications - I Important Results and Formulae

e) lim ( tanx)cosx = 1 x 0−

h) lim xn−an x a x−a

f) lim (cosx)1/x = 1 x 0

i) lim x ∞

log e x x

sin−1 x g) lim x x 0

j) lim x 0

x tanx = 1

=1

= nan −1 =0

Condition for increasing f’(x) ≥ 0 Condition for decreasing f’(x) ≤ 0 Condition for strictly increasing f’(x) > 0 Condition for strictly decreasing f’(x) < 0 A function f is called increasing on an interval I if f(x1) ≤ f(x2) whenever x1 < x2 in I. A function f is called decreasing on an interval I if f(x1) ≥ f(x2) whenever x1 < x2 in I. A function that is increasing or decreasing on I is called monotonic on I. Every constant function is an increasing function. Every identity function is an increasing function. The function f(x) = sinx is not an increasing function on R but f(x) = sinx is increasing on [0, /2]. The function f(x) = sinx is decreasing in [/2, ] If f is increasing then −f is decreasing. Each constant function is both increasing and decreasing. f(x) = x3 is strictly increasing. If a function changes its signs at different points of a region (interval) then the function is not monotonic in that region. So, to prove the non - monotonicity of a function it is enough to prove that f ' has different signs at different points. If a real valued differentiable function y = f(x) defined on an open interval I is dy≥ 0 increasing then dx If f is a differentiable function defined on an interval I with positive derivative then f is strictly increasing on I.

28

Std : XII

Differential Calculus Applications - I Important Results and Formulae

To find the functions which are increasing or decreasing on the interval [a, b] Steps : (i) Let y = f(x) (ii) Find f’(x) (iii) If f’(x) ≥ 0 for a ≤ x ≤ b then f(x) is increasing on [a, b] (if f’(x) > 0 then f(x) is strictly increasing ) If f’(x) ≤ 0 for a ≤ x ≤ b then f(x) is decreasing on [a, b]. (if f’(x) < 0 then f(x) is strictly decreasing ) To prove the functions which are not monotonic in the interval [a, b] given Steps : 1. Find f’(x) 2. f’(x) = 0 3. Find the values of x 4. Find the intervals. 5. Find f’ in each interval. If f ’ is of different signs at a and b. Then f is not monotonic on [a, b]. To find the intervals on which f is increasing or decreasing. Steps : 1. Find f’(x) 2. f’(x) = 0 3. Find the values of x. 4. Find the intervals. 5. In each interval, find f’ 6. Find the interval of strictly increasing / decreasing and interval of increasing/ decreasing. Absolute maximum / minimum : A function f has an absolute maximum (minimum) at c if f(c) ≥ f(x) (f(c) ≤ f(x)) for

all x in D, where D is the domain of f. The number f(c) is called the maximum (minimum) value of f on D.

Extreme value : The maximum and minimum values of f are called extreme values of f.

29

Std : XII

Differential Calculus Applications - I Important Results and Formulae

Local maximum : A function f has a local maximum (or relative maximum) at c if there is an open interval I containing c such that f(c) ≥ f(x) for all x in I. Local minimum : A function f has a local minimum (or relative minimum) at c if there is an open interval I containing c such that f(c) ≤ f(x) for all x in I. Fermat’s theorem : If f has a local extremum (maximum or minimum) at c and if f’(c) exists then f’(c) = 0 The Extreme value theorem : If f is continuous on [a, b] then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some number c and d in [a, b]. The function f(x) = x2 has minimum value at x = 0. The function f(x) = x3 has no extrema. A critical number of a function f is a number c in the domain of f such that either f’(c) = 0 or f’(c) does not exist. Stationary points are critical numbers c in the domain of f, for which f’(c) = 0 All the critical numbers are stationary numbers. To find the critical nos. and stationary points. Steps : 1. Find f’(x) 2. f’(x) = 0 (or f’(x) does not exist) 3. Find the values of x. Eg., Let f ’(x) = 0 ⇒

x = a, b and f ’(x) does not exist for x = c

The critical nos. are a, b, c and

Stationary points are (a, f(a)) (b, f(b))

To find absolute maximum and absolute minimum of f on the given interval [a,b] Steps

1. f is continuous in [a, b] 2. Find f’(x). 3. f’(x) = 0 4. Find the values of x Є[a, b], say x = c, d (critical points) 5. Find f(a), f(b), f(c), f(d) Absolute maximum = largest of these values. Absolute minimum = Smallest of these values. Note : If there is no critical point, find the values of f(a), f(b)

30

Std : XII

Differential Calculus Applications - I

Important Results and Formulae To find local maximum and minimum values of the function. Steps : 1. Find f ’(x). 2. f ’(x) = 0 3. Find the values of x

∈

[a, b] say x = c, d

4. Find f” (x) 5. Find f”(c), f”(d) 6. If at x=c, f”(x)<0 then the function attains the local maximum at x=c If at x=d, f”(x) > 0 then the function attains the local minimum at x=d Local maximum = f(c) Local minimum = f(d) Note : (1) If f ''(x) = 0 at x = c, then the second derivative test gives no information about the extreme nature of f at x = c and hence extreme values are not known. Note : (2) Consider f(θ) = sin2θ, [0, ] f’(θ) = sin2θ ; f”(θ) = 2cos2θ f’(θ) = 0 ⇒

θ = 0, /2, 

Critical numbers are 0, π/2, π At θ = /2 f '' (θ) = −2 < 0 Local maximum attains at θ = /2 and Local maximum = f(/2) = 1 At θ = 0, , the local maximum/minimum do not exist since they are end points of the interval. Practical problems involving maximum and minimum values Steps : 1. Draw a diagram. 2. Find the required function f(x) 3. Find f’(x) 4. f’(x) = 0 5. Find the values of x say a. 6. Find f '' (x) 7. Find f ''(a) 8. If f '' (a) > 0 then f is minimum at x = a. If f '' (a) < 0 then f is 31

Std : XII

Differential Calculus Applications - I Important Results and Formulae maximum at x = a. 9. maximum / minimum = f(a)

First Derivative test : (i) If f’(c) = 0 or is discontinuous and f '(x) > 0 for x < c and f’(x) < 0 for x > c, then

f(x) has a maximum at x = c.

(ii) If f '(c) = 0 or is discontinuous and f '(x) < 0 for x < c and f '(x) > 0 for x > c, then f(x) has a minimum at x = c. Second Derivative test : Let f ''(x) be continuous in an open interval that contains the point x = c. (i) If f '(c) = 0 and f ''(c) > 0, then f(x) has a local minimum at x = c. (ii) If f '(c) = 0 and f ''(c) < 0, then f(x) has a local maximum at x = c. Concavity and Convexity : If the graph of f(x) lies above all its tangents on an interval I, then the curve is said to be concave upward (or convex downward) on I. If the graph of f(x) lies below all its tangents on an interval I, then the curve is said to be concave downward (or convex upward ) on I. Test for concavity and convexity : Let f(x) be twice differentiable on an interval I. (i) If f ''(x) > 0 for all x Є I, then the graph of f(x) is concave upward (convex downward) on I. (ii) If f ''(x) < 0 for all x Є I, then the graph of f(x) is concave downward (convex upward) Points of inflection : A point P on a curve is called a point of inflection if the curve changes from concave upward (convex downward) to concave downward (convex upward) or from concave downward (convex upward) to concave upward (convex downward) at P. The point that separates the convex part of a continuous curve from the concave part is called the point of inflection of the curve. Theorem :

32

Std : XII

Differential Calculus Applications - I

Important Results and Formulae For a curve y = f(x), if f ''(xo) = 0 or f ''(xo) does not exist and if f ''(x) changes sign when passing through x = xo then x = xo is called a point of inflection. If x = xo is a root of even order for the equation f '(x) = 0 then x = x o is a point of inflection. If xo is the x - coordinate of the point of inflection of a curve y = f(x) then (second derivate exists) f ''(xo) = 0. Steps to find the intervals of concavity and the points of inflection. i) find f '(x), f ''(x) ii) f ''(x) = 0 iii) Find the values of x say a iv) Find the intervals i.e., ( −∞, a) , (a, ∞) v) Find f '' in each interval vi) If f '' < 0 in (−∞, a) then f is concave downward on (−∞, a) If f '' > 0 in (a, ∞) then f is concave upward on (a, ∞) vii) The point of inflection = (a, f(a)) Criteria for the points of inflection y = f(x) has one of its stationary point x = c as an inflection point a) If f ''(c) = 0 or not defined

b) f '''(c) ≠ 0 when f '''(c) exists. x---x---x---x---x

33

CHAPTER – VI

DIFFERENTIAL CALCULUS APPLICATIONS - II

Let y = f(x) be a differentiable function, then the differentials dx and dy are related as dy = f '(x) dx. (dx ≈ ∆x) ∆x is the absolute error in x. ∆y is the absolute error in y. ∆x , ∆y are relative errors in x and y respectively. x y ∆x Percentage error in x = x 100 x Percentage error in y = ∆y x 100 y f (x + ∆x) ≈ y + dy where y = f(x), dx = ∆x Curve Tracing : a) Domain, Extent, Intercepts and origin : (DEIO) i) Domain of a function y = f(x) is determined by the values of x for which the function is defined. ii) Horizontal (vertical) extent of the curve is determined by the intervals of x(y)

for which the curve exists.

iii) x = 0 yields the y-intercept and y = 0 yields the x - intercept. iv) If (0, 0) satisfies the given equation then the curve will pass through the origin. b) Symmetry : The curve f(x,y) is symmetrical about i) the x axis if its equation is unaltered when y is replaced by −y. i.e f (x,y ) = f(x,–y) ii) the y axis if its equation is unaltered when x is replaced by −x . i.e f(x,y) = f(–x,y) iii) the origin if it is unaltered when x is replaced by −x and y is replaced by −y simultaneously. i.e f (x,y ) = f(–x,–y) iv) the line y = x if its equation is unchanged when x and y are replaced by y

and x. i.e f (x,y ) = f(y,x) v) the line y = −x if its equation is unchanged when x and y are replaced

by

−y and −x. i.e f (x,y ) = f (–y,–x)

34

Std : XII

Differential Calculus Applications - II Important Results and Formulae

c) Asymptotes : (parallel to co-ordinate axes only) In the equation of the curve when y(x) is replaced by k(c), if the equation of the curve reduces to the form x = ±∞ (y = ±∞) then y = c (x = k) is an asymptote parallel to x axis ( y-axis) d) Monotonicity : Determine the intervals of x for which the curve is decreasing or increasing using the first derivative test. e) Special points : Determine the intervals of concavity and inflection points using the first and

second derivate test.

Partial Differentiation :

∂u ∂u Given u = f(x, y) then the first order partial derivates are and∂y ∂x ∂u ∂u x For f = u (x, y, z), ∂u , ∂y , ∂z are called first order partial derivaties. ∂x x x 2 2 Second order partial derivaties are ∂ u2 , ∂ u2 ,... ∂x ∂y 2 2 ∂u ∂u ∂2u ∂x∂y ∂y∂x ∂z2 , , . . . . x x x ∂2u ∂2u If u (x, y) and its first order partial derivatives are continuous then ∂x∂y = ∂y∂x x x Chain rule of two variables : If u = f(x, y) is differentiable and x and y are differentiable functions of t then du ∂f= ∂f + dy . dx. u is a differentiable function of t and dt ∂x ∂y dt dt Chain rule of three variables : If u = f(x, y, z) is differentiable and x, y, z are then u is a differentiable function ofdut and ∂f =dx dt ∂x dt Chain rule for partial derivaties : If w = f(u, v) ; u = g(x, y) ; v = h(x, y) then ∂w ∂w ∂w ∂v ∂u ∂x = ∂u . ∂x + ∂v . ∂x ∂w ∂y =

∂w ∂u ∂u . ∂y

∂w + ∂v

∂v . ∂y

35

differentiable functions of t, ∂f dz . ∂f +dy . + . ∂z dt ∂y dt

Std : XII

Differential Calculus Applications - II Important Results and Formulae

Homogeneous Function : A function u = f(x, y) is called a homogeneous function of degree n, if f(tx, ty) = tn f(x, y) Euler’s theorem : If f(x, y) is a homogeneous function of degree n, then ∂f x ∂x x---x---x---x---x

36

∂f +y ∂y

= nf

Std : XII

Differential Calculus Applications - II Important Results and Formulae

37

CHAPTER – VII INTEGRAL CALCULUS AND ITS APPLICATIONS First fundamental theorem of calculus :

x If f(x) is a continuous function and F(x) = ∫ f(t) dt, then F’(x) = f(x) a Second fundamental theorem of calculus : b If f(x) is a continuous function with domain a ≤ x ≤ b, then ∫ f(x) dx = F(b) − a F(a) where F is any anti - derivative of f. Properties of definite integrals : b b i) ∫ f(x) dx = ∫ f(y) dy a a b b ii) ∫ f(x) dx = − ∫ f(x) dx a a b b iii) ∫ f(x) dx = ∫ f(a + b − x) dx a a a a iv) ∫ f(x) dx = ∫ f(a − x ) dx 0 0 b b c v) ∫ f(x) dx = ∫ f(x) dx + ∫ f(x) dx for a < c < b a c a 2a a a vi) ∫ f(x) dx = ∫ f(x) dx + ∫ f(2a − x) dx 0 0 0 2a a vii) ∫ f(x) dx = 2 ∫ f(x) dx if f(2a − x) = f(x) 0 0 =0 if f(2a − x) = −f(x) a a viii) ∫ f(x) dx = 2 ∫ f(x) dx if f is an even function. −a 0 a ∫ f(x) dx = 0 −a

if f is an odd function.

Reduction Formulae : a) In = ∫ sinnx dx = ─ b) In = ∫ cosnx dx =

1 n

1 n

sinn-1x cosx +

n─1 n

cosn-1x sinx +n─1 n

38

In – 2 In – 2

Std : XII

Integral calculus and its applications Important Results and Formulae n─3 n─5 n─1 2 .1 when n is odd . n– 2 . ..... n– 4 π/2 n π/2 3 n─3 c) ∫ sinnx dx = ∫ cosnx dx = n─1 n─5 0 . n– 2. n– 4. . . . 1 . π2 when n is even 0 n 2

Bernoulli’s formula : ∫ u dv = uv – u'v1 + u''v2 – u'''v3 + . . . . Where u', u'', u''' are successive derivatives of u and v 1, v2, v3, . . . are repeated integrals of v. Gamma Integral : If n is a positive integer, then ∞ n ∫ xn e─ax dx = 0 an+1 The area of the region above x – axis bounded by the curve y = f(x), the b b ordinates x = a, x = b and x – axis is Area = ∫ f(x)a dx (or) ∫ y dx a The area of the region below x – axis bounded by the curve y = f(x), the b b ordinates x = a, x = b and x – axisa is Area = ∫ ─ f(x) dx (or) ∫ ─y dx a The area of the region to the right of y – axis bounded by the curve x = f(y), d d the lines y = c, y = d and y – axis is Areac = ∫ f(y) dy (or) ∫ x dy c The area of the region to the left of y – axis bounded by the curve x = f(y), the d d lines y = c, y = d and y – axis is Area = ∫ ─ f(y)dy (or) ∫ ─ x dy c c General Area Principle : Let f and g be two continuous curves with f lying above g, then the area R b between f and g from x = a to x = b is given by R = ∫ (f – g) dx a Volume of the solid generated by revolving the area bounded by the curve b y = f(x), the lines x = a, x = b and the x – axis, about the x – axis is a∫ π y2 dx Volume of the solid generated by revolving the area bounded by the curve d x = g(y), the lines y = c, y = d and the y – axis, about the y axis is ∫ π x2 dy c b

 The length of the curve y = f(x) from x = a to x = b is L = ∫ 1 +  a

b

The length of the curve x = g(y) from y = c to y = d is L = ∫ a

39

2

dy   dx  dx 

 dx 1 +  dy 

2

   dx 

Std : XII Integral calculus and its applications Important Results and Formulae If x = f(t), y = g(t) are the parametric equations of the curve then t2

L=

∫

t1

2

2

 dx   dy    +   dt  dt   dt 

Surface area of the solid obtained by rotating the area bounded by y = f(x), 2

b

dy x = a, x = b and the x – axis about the x – axis is S = ∫ 2πy 1 +   dx  dx  a

Surface area of the solid obtained by rotating the area bounded by x = g(y),  dx y = c, y = d and the y – axis about the y – axis is S = ∫ 2πx 1 +   dy c d

2

   dy 

If x = f(t) ; y = g(t) are the parametric equations of the curve then its surface area generated when t ranges from t1 t2 is t2

2

2

 dx   dy  S = ∫ 2πy   +   dt  dt   dt  t1

x---x---x---x---x

40

Std : XII

Integral calculus and its applications

Important Results and Formulae

CHAPTER – VIII DIFFERENTIAL EQUATIONS A differential equation is an equation which involves derivatives. Differential equations are of two types (i)

Ordinary Differential Equations

(ii)

Partial Differential Equations

An ordinary differential equation is a differential equation in which a single independent variable enters either explicitly or implicitly. A partial differential equation is a differential equation involving partial derivatives w.r. to two or more independent variables. The order of a differential equation is the order of the highest order derivative appearing on it. The degree of a differential equation is the degree of the highest derivative when the differential coefficients are made free from radicals and fractions as far as the derivatives are concerned. Differential equations can be formed by differentiating ordinary equations and eliminating the arbitrary constants. In deriving the differential equation of a family of curves, the minimum number of differentiations are to be made to eliminate the parameters. A solution of a differential equation is any relation between the variables free from derivatives which satisfies the differential equation. A first order differential equation is of 3 types : (i) Variable Separable (ii) Homogeneous (iii) Linear In variable separable, the variables x and y are separated and by integrating both sides, the solution is obtained. (a) To solve the homogeneous differential equation which is of the form dy = f dx

y x

(or)

dy = dx

f1(y/x) f2(y/x)

Put y = vx so that dy = v + x dv dx dx (b) To solve the homogeneous differential equation which is of the form dx = f dy

x y

(or)

dx = dy

f1(x/y) f2(x/y)

Put x = vy so that dx = v + y dv dy dy Then the resulting equation can be solved by variable separable method.

41

Std : XII Integral calculus and its applications 13. A linear equation in y is of the form

Important Results and Formulae

dy + Py = Q where P and Q are functions of x dx ∫pdx

Integrating factor (I.F) = e Solution is y (I.F) = ∫ Q (I.F) dx + c A linear equation in x is of the form dx + Px = Q where P and Q are functions of y, then dy

e log x = x e- log x = 1/x

∫pdy

Integrating factor (I.F) = e Solution is x (I.F) = ∫ Q (I.F) dy + c

The general form of second order linear equations with constant coefficients is of the form 2 dy a d y2 + b dx dx function of

+ cy = Q(x) where a, b, c are constants and Q(x) is a

x only. The characteristic equation of the differential equation 2 a d y2 + b dy + cy = Q(x) is ap2 + bp + c = 0 dx dx Complementary function (C.F) a) If the roots of the characteristic equation are real and distinct say λ1 and λ2 then C.F = Aeλ x + Beλ x 1

2

b) If the roots of the characteristic equation are real and equal say λ then C.F = (Ax + B)eλx c) If the roots of the characteristic equation are imaginary say λ = α ± i β then C.F = eαx [ A cosβx + B sin βx] Particular Integral (P.I) a) When Q(x) = eαx where α is constant (substitute D = α) 1 1 P.I = e αx = f(α) if f(α) ≠ 0 (Here f(D) = aD2 + bD + c) f(D) 1 If f(α) = 0, then D - a

e

αx

=xe

αx

42

1 , (D – a)2

αx

e

x2 2=

eαx

Std : XII b)

Integral calculus and its applications Important Results and Formulae When Q(x) = cos ax or sin ax where a is a constant (substitute D2 = –

a2) P.I =

1 f(D2)

P.I =

1 D + a2

cos ax =

P.I =

1 D + a2

–x sin ax = 2a

2

2

1 P.I = f(D2)

cos ax =

sin ax =

cos ax f( – a)2

if f(–a2) ≠ 0

x 2a

sin ax if f(–a2) = 0 cos ax if f(–a2) = 0

sin ax f(– a2)

if f(–a2) ≠ 0

c) When Q(x) = x or x2 Take P.I as Co + C1x if f(x) = x and Co + C1x + C2x2 if f(x) = x2 Take y = Co + C1x or y = Co + C1x + C2x2 according as f(x) = x or x2 By substituting y value and comparing like terms, we can find Co, C1, C2. General Solution y = C.F + P.I Note : 1. ( 1 – x)–1 = 1 + x + x2 + x3 + x4 +…. 2. ( 1 + x)–1 = 1 – x + x2 – x3 + …. 3. ( 1 – x )–2 = 1 + 2x + 3x2 + …. 4. (1 + x )–2 = 1 – 2x + 3x2 – 4x3 +…. x---x---x---x---x

43

CHAPTER – IX(A)

DISCRETE MATHEMATICS

A statement or a proposition is a sentence which is either true or false but not both. A statement which is both true and false simultaneously is not a statement, rather it is a paradox. The true or falsity of a statement is called its truth value which is denoted by T, F respectively. A statement is said to be simple if it cannot be broken into two or more statements. A statement is called compound, if it is a combination of two or more simple statements. Basic logical connectives are (i) conjunction (ii) disjunction and (iii) negation. They are denoted by the symbols ‘ ∧ ’ , ‘ ∨ ’ and ‘~’ respectively. Conjunction of p, q is p

∧

q. Disjunction of p, q is p

∨

q. Negation of p is ~p.

Three basic connectives are conjunction which corresponds to the word ‘and’, disjunction which corresponds to the word ‘or’, negation which corresponds to the word ‘not’. If the compound statement is made up of n sub–statements, then its truth table will contain 2n rows. Truth table for p∧ q

~p p ~ p T F F T

p

q

T

T

T

p p

∧

q

∨

q p

∨

p

q

T

T

T

T

F

F

T

F

T

F

T

F

F

T

T

F

F

F

F

F

F

q

~ ( ~ p) = p, for any statement p. Two compound statements A and B are called logically equivalent, written as

A ≡ B, if they have identical last columns in their truth tables. ‘If p then q’ is called a conditional statement, written as p → q, read as ‘p implies q’ The compound statement p → q and q → p is called a bi – conditional statement, written as p ↔ q, read as p if and only if q. Truth table for p p

q

p q

T

T

T

T

F

F

F

T

T

F

F

T

→ q

Truth table for p ↔ q

→

p

q

p↔q

T

T

T

T

F

F

F

T

F

F

F

T

A logical statement is said to be a tautology if the last column of its truth table contains only T. It is said to be a contradiction if the last column of its truth table contains only F. x---x---x---x---x

CHAPTER – IX(B)

GROUPS

If * is a binary operation on S then a, b ∈ S ⇒

∈ S where S is a non – empty set. If a, b ∈ S ⇒ a * b ∈ S then S is closed under *. a*b

This property is known as ‘closure axiom’ or ‘closure property’. List of symbols : N – The set of all natural numbers. Z – The set of all integers. W – The set of all non – negative integers (whole nos.) E – The set of all even integers. O – The set of all odd integers. Q – The set of all rational numbers. R – The set of all real numbers. C – The set of all complex numbers. Q – { 0 } – The set of all non – zero rational numbers R – { 0 } – The set of all non – zero real numbers. C – { 0 } – The set of all non – zero complex numbers. Number systems Operations + – . ÷

N

Z

Q

R

C

Q–{0}

R–{0}

/

C– {0}

b Not b b Not b

b b b Not

b b b Not b

b b b Not b

b b b Not b

Not b Not b b b

Not b Not b b b

Not b Not b b b

b Note : b

→ binary

Definition of a group : A non – empty set G, together with an operation *, namely (G, *) is called a group, if it satisfies the following axioms.

Page 47 Std : XII Groups (i) Closure axiom : If a, b ∈ G then a * b (ii) Associative axiom : For all a, b, c

∈

∈

Important Results and Formulae G

G

(a * b ) * c = a * ( b * c) (iii) Identity axiom : For all a

∈

G, there exists an element e

∈

∈

G, there exists an element a-1

G such that

a*e=e*a=a (iv) Inverse axiom : For all a a– 1 * a = a * a

-1

∈

G such

=e

∈ G is called the identity element for G. a-1 ∈ G is called the inverse of a ∈ G.

Note : e

Commutative Property : A binary operation * on a set S is called commutative if a * b = b * a, V a, b

∈

S. Abelian group : A group satisfying the commutative property is called an abelian group or a commutative group. Non – Abelian group : A group does not satisfying the commutative property is called an non – abelian group. Order of a group : The order of a group (G, *) is the number of distinct elements in G. Finite group : If the number of elements is finite then (G, *) is called a finite group. Infinite group : If the number of elements is infinite, then (G, *) is called an infinite group. Semi – group : A non – empty set G with an operation said to be a semi – group if it satisfies the following axioms : (i) closure axiom : If a, b ∈ G then a * b (ii) Associative axiom : For all a, b, c Monoid :

∈

∈

G.

G, (a * b) * c = a * (b * c)

Page 48 Std : XII Groups Important Results and Formulae A non – empty set G with an operation * is said to be a monoid if it satisfies the following axioms : (i) Closure axiom : If a, b

∈

G then a * b

(ii) Associative axiom : For all a, b, c (iii) Identity axiom : For all a

∈

∈

∈

G.

G, (a * b) * c = a * (b * c)

G, there exists an element e

∈

G such that

a*e=e*a=a (N,+) is a semi – group but not a monoid. (Z, +) is a monoid, (Z, .) is a monoid. (Z, .) is not a group. Modulo operation : There are two types of modulo operation. (i)

Addition modulo n

(ii)

Multiplication modulo n, where n is a positive integer.

Division algorithm : Let a, b

∈

Z with b ≠ 0. Dividing a by b, we get a quotient q and

a non – negative remainder r < |b|, then a = qb + r, where 0 ≤ r < | b |. Addition modulo n (+n)

∈

Let a, b

Z and n be a fixed positive integer. We define addition modulo n

by a + nb = r where r is the least non – negative remainder when a + b is divided by n (0 ≤ r < n) Multiplication modulo n ( . n) Let a, b

∈

Z and n be a fixed positive integer.

We define multiplication

modulo n by a . nb = r, 0 ≤ r < n, where r is the least non – negative remainder when ab is divided by n. Congruence modulo n : Let a, b

∈

Z and n be a fixed positive integer. We say that a is congruent to b

modulo n ⇔ a – b is divisible by n. Symbolically, we write a ≡ b (mod n). Congruence Classes modulo n : Let a

∈

Z and n be a fixed positive integer. Let [a] be the collection of all the

numbers which are congruent to a modulo n.This set [a] is called the congruence class modulo n or residue class modulo n.

Page 49 Groups Z / x ≡ a(mod n)}

Std : XII [a] = { x

∈ = { x ∈ Z / x = a + kn, k ∈

Important Results and Formulae

Z}

Zn = { [0], [1], . . . .[n – 1]} Operations on Congruence classes : (i) Addition : Let [ a ], [ b ]

∈

Zn

[ a ] + n[ b ] = [a + b] if a + b < n = [ r ] if a + b ≥ n Where r is the least non – negative remainder when a + b is divided by n. (ii) Multiplication : Let [ a ], [ b ]

∈

Zn

[ a ] . n [ b ] = [ ab ] if ab < n = [ r ] if ab ≥ n Where r is the least non – negative remainder when ab is divided by n. Order of an element : Let G be a group and a

∈ G.

The order of ‘a’ is defined as the least positive

integer n such that an = e, e is the identity element. If no such positive integer exists, then a is said to be of infinite order. The order of a is denoted by O(a). For any group G, identity element is the only element of order one. The identity element of a group is unique. The inverse of each element of a group is unique. Cancellation laws : Let G be a group. Then for all a, b, c ∈ G (i) a * b = a * c ⇒

⇒

(ii) b * a = c * a

b = c (left cancellation law) b = c (right cancellation law)

In a group G, (a-1)-1 = a for every a Reversal law :

∈

G

Let G be a group, a, b

For any element a If a

∈

∈

∈

G then (a * b )−1 = b−1 * a−1

G, O ( a ) = O ( a−1)

G has order n, then am = e if and only if n divides m where m

∈

Z.

If G is of even order then there is an element a ≠ e in G such that a2 = e (i.e., a = a

)

−1

If a = a−1 (i.e., a2 = e) for each a

∈

G, then G is abelian.

Std : XII If O(G) ≤ 4 then G is abelian.

Page 50 Groups

(a * b)2 = a2 * b2 for all a, b ∈ G ⇔ G is abelian x---x---x---x---x

Important Results and Formulae

CHAPTER – X PROBABILITY DISTRIBUTIONS Random variable : A random variable is a real valued function whose domain is the sample space of a random experiment taking values on R. Types of random variable : a)

Discrete random variable

b)

Continuous random variable

Discrete random variable : A random variable which takes only a finite or countable number of values is called a discrete random variable. Continuous random variable : A random variable X is called continuous if it can take all possible values between certain given limits. Probability mass function : If X is a discrete random variable, p(x) is called probability mass function. p(x) is non – negative for all real x and ∑ p(xi) = 1 Cumulative Distribution function or Distribution function : The function F(x) = p(x ≤ x) = ∑ p(xi), – ∞ < x < ∞ xi ≤ x

Properties of Distribution Function : a) F(x) is a non – decreasing function of x. b) 0 ≤ F(x) ≤ 1, −∞ < x < ∞ c) F(−∞) = lim F(x) = 0 x−∞ x

d) F (∞) x+∞ = limx F(x) = 1 e) p(X = xn) = F(xn) – F(xn–1) If a1, a2, . . . . . am, a, b1, b2, . . . . . bn be the values of the discrete random variable X in ascending order then c)

p(x ≤ a) = p(x = a1) + p(x = a2) + . . . . . + p(x = am) + p(x = a)

d)

p(x < a) = p(x = a1) + p(x = a2) + . . . . .+ p(x = am)

e)

p(x ≥ a) = 1 – p(x < a)

f)

p(x > a) = 1 – p(x ≤ a)

g)

p(x ≤ a) = 1 – p(x > a)

h)

p(x < a) = 1 – p(x ≥ a)

i)

p(a < x < b) = p(x=b1) + p(x = b2) + . . . . + p(x=bn)

j)

p(a ≤ x ≤ b) = p(x = a) + p(x = b 1) + . . . .+ p(x = b n)+p(x = b)

Probability Density Function (p.d.f) We denote it by f(x) and it has the following properties : (i) The probability that X lies between the values a and b b is p(a ≤ x ≤ b) = ∫ f(x) dx a (ii) f(x) is non – negative for all real x. ∞ (iii) −∞ ∫ f(x) dx = 1 a Probability at a point : For continuous random variable, the probability at a point is a always zero. a i.e., p(X = a) = ∫ f(x) dx = 0 For discrete random variable, p(X=a) is not zero for some fixed ‘a’. Probability in an interval :

b p(a ≤ x ≤ b) = p(a < x < b) = ∫ f(x) dx a Cumulative distribution function :

x Let X be a continuous random variable then F(x)= p(X ≤ x) = ∫ f(t) −∞ dt,−∞< x < ∞ where f(t) is the p.d.f of X at t is called the cumulative distribution function of X.

Properties of distribution function : (c.d.f) k)

F(x) is a non – decreasing function of x.

l)

0 ≤ F(x) ≤ 1 , −∞ < x < ∞ x -∞ F(−∞) = lim ∫ f(x) dx = ∫ f(x) dx = 0 -∞ x-∞ x −∞ ∞ x F(∞) = lim ∫ f(x) dx = ∫ f(x) dx = 1 x+∞ x −∞ -∞

m) n) o) p)

For any real constants a and b (a ≤ b), p(a ≤ x ≤ b) = F(b) – F(a) d dx f(x) = [F(x)] i.e., f(x) = F ′ (x)

Mathematical Expectation : Expectation of a discrete random variable :

Let X be a discrete random variable assuming the values x1, x2, . . . . . xn with respective probabilities p1, p2, . . . . , pn n

n

E(X) = p1x1 + p2x2 + . . . . . + pnxn = ∑ pi xi where ∑ pi = 1 i=1

E(X) = X where X is the arithmetic mean.

i=1

If φ(X) is a function of the random variable X, then E[φ(X)] = ∑ φ(x) p (X = x) Properties of Expectation : q)

E(c) = c where c is a constant

r)

E(cX)= c E(x)

s)

E(ax + b) = aE(x) + b

t)

E(ax – b) = aE(x) – b

Moments : Expected value of a function of a random variable X is used in the calculation of moments. We have two types : u)

Moments about the origin.

v)

Moments about the mean.

Moments about the origin : Let X be a discrete random variable. For any positive integer, we define the rth moment about the origin by

μr ' = E(xr) = ∑ pi xir a) First moment = μ1' = E(X) = ∑ pixi = X b) Second moment = μ2'=E(X2) = ∑ pixi2 Moments about the mean (or central moments) Let X be a discrete random variable with mean X and a positive integer r. The rth central moment of X is defined by n

μr = E(X – X ) = ∑ pi (xi – x )r r

i=1

a)

First central moment = μ1 = E(X – X ) = 0

b)

Second central moment = μ2 = E(X – X )2 = E(X2) – [E(X)]2 This is called the variance of X, denoted by var(X) var(X) = μ2 = E(X2) – [E(X)]2 Properties of variance : a) var(X ± c) = var(X) where c is a constant

b) var(aX) = a2var(X) c) var(c) = 0 for any constant c. Expectation of a continuous random variable : Let X be a continuous random variable with probability density ∞ function f(x). -∞ E(X) = ∫ x f(x) dx ∞ E(X2) = ∫ x2 f(x) dx -∞ var(X) = E(X2) – [E(X)]2 Theoretical Distributions : It is a distribution of a random variable whose values are distributed according to a definite probability law, which can be expressed mathematically. w)

Binomial distribution

x)

Poisson distribution

y)

Normal distribution

The first two distributions are discrete probability distributions and the third one is a continuous probability distribution. Binomial distribution : It can be used under the following conditions : z)

Any trial, result in a success or a failure.

aa)

There are finite no. of trials which are independent.

bb)

The probability of success is the same in each trial.

Definition : A random variable X is said to follow the binomial distribution if its probability mass function is given by x n–x P(X = x) = p(x) =nCx p q , x = 0, 1, 2, . . .n 0 , otherwise Constants : Mean = np Variance = npq Standard deviation =

npq

Note : Mean is always greater than the variance. 0 ≤ p ≤ 1, q = 1 – p Poisson distribution : It can be regarded as the limiting case of the binomial distribution under the following conditions :

→ ∞

cc)

n, the number of trials is very large i.e.,n

dd)

p, the probability of success in each trial is very small i.e., p

→0 np is finite say λ, a positive real no.

ee)

Note : It applies to the rare events. Definition : A random variable X is said to have a poisson distribution if the probability mass function of X is given by e–λ . λx x!

P(X = x) =

, x = 0, 1, 2, . . . . for some λ > 0.

Note : For a poisson distribution, mean = variance = λ ; λ is called the parameter. Normal distribution : Definition : A continuous random variable X is said to follow a normal distribution with parameters μ and σ (or μ and σ2) if the probability function is given by f(x) =

1

σ 2π

e

−1 x −µ    2 σ 

2

, − ∞ < x < ∞, − ∞ < μ < ∞, σ > 0

Constants : Mean = μ ; variance = σ2 ; S.D = σ Properties of a Normal distribution : The normal curve is bell shaped. It is symmetrical about the line X = μ Mean = Median = Mode = μ The height of the normal curve is maximum at x = μ. The maximum height = 1 σ √2π The normal curve is unimodal. The normal curve is asymptotic to the base line. The points of inflection are at x = μ ± σ .

∞

)

Since the curve is symetrical about x = μ, the skewness is 0. Area property : p(μ – σ < x < μ + σ)

= 0.6826

p(μ – 2σ < x < μ + 2σ) = 0.9544 p(μ – 3σ < x < μ + 3σ) = 0.9973 (x)

It is a close approximation to the binomial distribution, when the no. of trials n is very large and the probability of success p is close to ½

i.e. neither p nor q is so small.

(xi) It is also a limiting form of the poisson distribution. i.e., as λ → ∞, poisson distribution tends to the normal distribution. Standard normal distribution : A random variable X is called a standard normal variate if its mean is zero and its standard deviation is unity. Note : f(x) =

Z=

x–μ σ

1

σ 2π

e

−1 x −µ    2 σ 

2

, − ∞ < x < ∞.

is called a standard normal variate. x---x---x---x---x