CBSE Class 12 Mathematics formulas

Kendriya Vidyalaya
SESSION 2020-21
CLASS 12 MATHEMATICS
FORMULAS
Submitted to:- Submitted by:-

1. Relations and
Functions
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Empty relation is the relation R in X given by R =   X × X.
Universal relation is the relation R in X given by R = X × X.
Reflexive relation R in X is a relation with (a, a)  R  a  X.
Symmetric relation R in X is a relation satisfying (a, b)  R implies (b, a)  R.
Transitive relation R in X is a relation satisfying (a, b)  R and (b, c)  R
implies that (a, c) R.
Equivalence relation R in X is a relation which is reflexive, symmetric and
transitive.
Equivalence class [a] containing a  X for an equivalence relation R in X is
the subset of X containing all elements b related toa.
A function f : X  Y is one-one (or injective) if
f (x ) = f (x )  x = x  x , x  X.
1 2 1 2 1 2
A function f : X  Y is onto (or surjective) if given any y  Y,  x  X such
that f (x) = y.
A function f : X  Y is one-one and onto (or bijective), if f is both one-one
and onto.
The composition of functions f : A  B and g : B  C is the function
gof : A  C given by gof (x) = g(f (x))  x  A.
A function f : X  Y is invertible if  g : Y  X such that gof = IX and
fog = IY.
A function f : X  Y is invertible if and only if f is one-one and onto.
◆ Given a finite set X, a function f : X  X is one-one (respectively onto) if and only if f
is onto (respectively one-one). This is the characteristic property of a finite set. This
is not true for infinite set.
◆ A binary operation  on a set A is a function  from A × A to A.
◆ An element e  X is the identity element for binary operation  : X × X  X, if a  e
= a = e  a  a  X.
◆ An element a  X is invertible for binary operation  : X × X  X, if there exists b
 X such that a  b = e = b  a where, e is the identity for the binary operation .
The element b is called inverse of a and is denoted by a–1.
◆ An operation  on X is commutative if a  b = b  a  a, b in X.
◆ An operation  on X is associative if (a  b)  c = a  (b  c) a, b, c in X.

2.INVERSE TRIGONOMETRIC
FUNCTIONS
◆ The domains and ranges (principal value branches) of inverse trigonometric
functionsare given in the followingtable:
Functions Domain Range
(Principal Value Branches)
y = sin–1 x [–1, 1]
  
 2
,
2
y = cos–1 x [–1, 1] [0, ]
y = cosec–1 x R – (–1,1)
  
2
,
2 
– {0}
y = sec–1 x R – (–1, 1)

[0, ] – { }
2
y = tan–1 x R


,

 
 2 2 
y = cot–1 x R (0, )
◆ sin–1x should not be confused with (sin x)–1. In fact (sin x)–1 =
1
sinx
and
similarlyforother trigonometricfunctions.
◆ The value of an inverse trigonometric functions which lies in its principal
value branch is called the principal value of that inverse trigonometric
functions.
For suitable values of domain, wehave
◆
◆
y = sin–1 x  x = sin y
sin (sin–1 x) = x
◆
◆
x = sin y  y = sin–1 x
sin–1 (sin x) = x
◆
1
sin–1 = cosec–1 x ◆ cos–1 (–x) =  – cos–1 x
◆ cos–1
x
1
x
= sec–1
x ◆ cot–1
(–x) =  – cot–1
x
◆
1
x
tan–1
= cot–1
x ◆ sec–1
(–x) =  – sec–1
x

sin–1 (–x) = – sin–1 x tan–1 (–x) = – tan–1 x
◆
◆ 2

tan–1 x + cot–1 x =
◆
◆ cosec–1 (–x) = –cosec–1 x
◆ tan–1x + tan–1y = tan–1
x  y
1 xy ◆ 2

sin–1 x + cos–1 x =
◆ tan–1
x + tan–1
y = tan–1
x  y
1 xy ◆ 2

cosec–1
x + sec–1
x =
◆ 2tan–1 x = sin–1
2x
1 x2 = cos–1
1 x2
1 x2 ◆ 2tan–1x = tan–1
2
2x
1 x
 x  y 
xy
◆ tan–1x + tan–1y =   tan–1

1  , xy>1; x, y> 0

◆ A matrix is an ordered rectangular array of numbers or functions.
◆ A matrix having m rows and n columns is called a matrix of order m × n.
◆ [aij]m × 1 is a column matrix.
◆ [aij]1 × n is a row matrix.
◆ An m × n matrix is a square matrix if m = n.
◆ A = [aij]m × m is a diagonal matrix if aij = 0, when i  j.
3.Matrices
A = [aij]n × n is a scalar matrix if aij = 0, when i  j, aij = k, (k is some
constant), when i = j.
A = [aij]n × n is an identity matrix, if aij = 1, when i = j, aij = 0, when i  j. A
zero matrix has all its elements as zero.
A = [aij] = [bij] = B if (i) A and B are of same order, (ii) aij = bij for all
possible values of i and j.
kA = k[aij]m × n =[k(aij)]m × n
– A = (–1)A
A – B = A + (–1) B
A + B = B + A
(A + B) + C = A + (B + C), where A, B and C are of same order.
k(A + B) = kA + kB, where A and B are of same order, k is constant.
(k + l ) A = kA + lA, where k and l are constant.
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◆
◆
◆ If A = [aij]m× n and B = [bjk]n × p , thenAB = C = [cik]m × p, where cik  ∑ aij bjk
j1
(i) A(BC) = (AB)C, (ii) A(B + C) = AB + AC, (iii) (A + B)C = AC + BC
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◆
If A = [a ] , then A
or AT = [a ]
ij m × n ji n × m
(i) (
A

)
= A, (ii) (kA)= kA
,(iii) (A + B
)
= A
+ B

, (iv) (AB)= B

A
A is a symmetric
matrix if A
=A.
A is a skew symmetric matrix if A
= – A.
Any square matrix can be represented as the sum of a symmetric and a
skew symmetric matrix.
Elementary operations of a matrix are asfollows:
(i) Ri
 Rj
or Ci
 Cj
(ii) Ri  kRi or Ci  kCi
(iii) Ri  Ri + kRj or Ci  Ci + kCj
If A and B are two square matrices such that AB = BA = I, then B is the
inverse matrix of A and is denoted by A–1 and A is the inverse ofB.
Inverse of a square matrix, if it exists, is unique.
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4.Determinants
◆ Determinant of a matrix A = [a11]1× 1 is given by | a11| = a11

a a
◆ Determinant of a matrix A 
a11
 21 22 
a12 
is given by
21 22
a11 a12
A 
a a
= a a – a a
11 22 12 21

◆ Determinantof a matrix A a2 2

a3
a1 b1 c1
b2

b3 c3 
1
c is given by (expandingalongR )
2 2 2 2
3 3 3 3 3 3
b c a c a b
2 2 2 1
b c 1
a 1
a
3 3 3
c b
a b c
a1 b1 c1
A  a b c  a 2
 b 2
 c
For any square matrix A, the |A| satisfy following properties.
◆ |A|= |A|, where A
= transpose of A.
◆ If we interchange any two rows (or columns), then sign of determinant
changes.
◆ If any two rows or any two columns are identical or proportional, then value
of determinant iszero.
◆ If we multiplyeach elementof a row or a column of a determinantby constant
k, then value of determinant is multiplied by k.
◆ Multiplying a determinant by k means multiply elements of only one row
(or one column) by k.
◆If A [a ] , then k .A k3
A
◆
ij 33
◆ If elements of a row or a column in a determinant can be expressed as sum
of two or more elements, then the given determinant can be expressed as
sum of two or more determinants.
If to each element of a row or a column of a determinant the equimultiples of
corresponding elements of other rows or columns are added, then value of
determinant remains same.

◆ Area of a triangle with vertices (x1
, y1
), (x2
, y2
) and (x3
, y3
) is given by
2 2
2
x3 y3
x1 y1 1
 
1
x y 1
1
Minor of an element aij of the determinant of matrix A is the determinant
ij
obtained by deleting ith
row and jth
column and denoted by M .
Cofactor of a of given by A = (– 1)i + j
M
ij ij ij
◆
◆
◆ Valueof determinantof a matrixAis obtainedby sum of productof elements
of a row (or a column) with corresponding cofactors. Forexample,
A = a11 A11 + a12 A12 + a13 A13.
◆ If elements of one row (or column) are multiplied with cofactors of elements
of any other row (or column), then their sum is zero. For example, a11 A21 + a12
A + a A = 0
◆ a a
22 13 23
a11 a12 a13 
,
If A  a
 21 22 23
a31 a32 a33
12 22
A11 A21 A31 
 32 
A13 A23 A33 
ij
then adj A  A A A  , where A is
cofactor of aij
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◆
A (adj A) = (adj A) A = |A| I, where A is square matrix of order n.
A square matrix A is said to be singular or non-singular according as
|A| = 0 or |A|  0.
If AB = BA = I, where B is square matrix, then B is called inverse of A.
Also A–1 = B or B–1 = A and hence (A–1)–1 = A.
A square matrixAhas inverse if and only ifA is non-singular.
A–1

1
(adj A)
A
◆
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◆ If a1 x + b1 y + c1 z = d1 a2 x
+ b2 y + c2 z = d2 a3 x +
b3 y + c3 z = d3,
b3
then these equations can be written as A X = B, where
a1 b1 c1  x d1 
b c ,X= y and B= d 
A a 2 2   
z
 2
a3 c3 
 2 
d3

◆ Unique solution of equation AX = B is given by X = A–1 B, where A  0.
◆ A system of equation is consistent or inconsistent according as its solution
exists or not.
◆ For a square matrixA in matrix equationAX = B
(i) |A|  0, there exists unique solution
(ii) |A| = 0 and (adj A) B  0, then there exists no solution
(iii)|A| = 0 and (adj A) B = 0, then system may or may not be consistent.

.
5.Continuity and
Differentiability
◆ A real valued function is continuous at a point in its domain if the limit of the
function at that point equals the value of the function at that point. A function
is continuous if it is continuous on the whole of its domin.
◆ Sum, difference, product and quotient of continuous functions are continuous.
i.e., if f and g are continuous functions,then
(f ± g) (x) = f (x) ± g (x) is continuous.
(f . g) (x) = f (x) . g (x) is continuous.
g(x)
f
(x) 
f (x)
(wherever g (x)  0) is continuous.
g
◆ Every differentiable function is continuous, but the converse is not true.
◆ Chain rule is rule to differentiate composites of functions. If f = v o u, t = u (x)
and if both
dt
and
dv
existthen
dx dt
df

dv

dt
dx dt dx
◆ Following are some of the standard derivatives (in appropriate domains):

6. Application of
Derivatives
◆ If a quantity y varies with another quantity x, satisfying some rule y  f (x),

Alternatively, if f (x) 0 for each x in (a, b)
(b) decreasing on (a,b) if
x < x in (a, b)  f (x )  f (x ) for all x , x  (a, b).
1 2 1 2 1 2
(c) constant in (a, b), if f (x) = c for all x  (a, b), where c is a constant.
◆ The equation of the tangent at (x0
, y0
) to the curve y = f (x) is given by
y  y  dy 
dx
0 0
(x0 ,y0 )
(x  x )
◆ dx 0 0
dy
If does not exist at the point ( x , y ) , then the tangent at this point is
parallel to the y-axis and its equation is x = x0.
◆ 0
0
dxx x
If tangent to a curve y = f (x) at x = x is parallel to x-axis, then
dy 
  0.
◆ Equation of the normal to the curve y = f (x) at a point ( x0 , y0 ) is given by
1
0 0
dx (x , y )
y  y0 
dy
(x x0 )
◆ dx 0 0 0
dy
If at the point ( x , y ) is zero, then equation of the normal is x = x.
dx 0 0
◆ If
dy
at the point ( x , y ) does not exist, then the normal is parallel tox-axis
and its equation is y = y0.
◆ Let y = f (x), x be a small increment in x and y be the increment in y
corresponding to the increment in x, i.e., y = f (x + x) – f (x). Then dy
givenby
dy  f (x)dxor dy  

dy 
x .
◆
 dx
isa good approximationofy when dx  x is relativelysmalland we denote
it by dy  y.
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.

◆ First Derivative Test Let f be a function defined on an open interval I. Let
f be continuous at a critical point c in I. Then
(ii)
(iii)
(i) If f (x)changes sign from positive to negative as x increases through c,
i.e., if f (x)> 0 at every point sufficiently close to and to the left of c,
and f (x) < 0 at every point sufficiently close to and to the right of c,
then c is a point of local maxima.
If f (x)changes sign from negative to positive as x increases through c,
i.e., if f (x)< 0 at every point sufficiently close to and to the left of c,
and f (x) > 0 at every point sufficiently close to and to the right of c,
then c is a point of local minima.
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. Infact, such a point
is called point of inflexion.
◆
◆
Second Derivative Test Let f be a function defined on an interval I and
c  I. 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
The values f (c) is local maximum value of f .
(ii) x = c is a point of local minima if f (c)= 0 and f (c) > 0
In this case, f (c) is local minimum value of f.
(iii) The test fails if f (c)= 0 and f (c) = 0.
In this case, we go back to the first derivative test and find whether c is
a pointof maxima,minimaor a pointof inflexion.
Working rulefor findingabsolutemaxima and/orabsoluteminima
Step 1: Find all critical points of f in the interval, i.e., find points x where
either f (x)= 0 or f is not differentiable.
Step 2:Take the end points of theinterval.
Step 3:At all these points (listed in Step 1 and 2), calculate the values of f .
Step 4: Identify the maximum and minimum values of f out of the values
calculated in Step 3. This maximum value will be the absolute maximum
value of f and the minimum value will be the absolute minimum value of f .

Some standard integrals
(i) n
x dx
xn1
n1
 C , n  – 1. Particularly, dx  x C

(ii) (iii)
(iv) (v)
sin x dx  – cosx  C
cosec2
x dx  – cot x  C
(vi)
(vii)
cos x dx  sin x  C
sec2
x dx  tan x  C
sec x tan x dx  secx  C
cosec x cot x dx  – cosec x  C (viii)
1 x2

dx
 sin1
x  C
(ix)
1 x2

dx
 cos1
x  C (x)
1 x2

dx
 tan  1
x  C
(xi)
1 x2

dx
  cot 1
x C (xii) ex
dx  ex
C
(xiii)
x ax
loga
 C
 a dx  (xiv)  sec 1
x  C
(xv) 
dx
 cosec 1
x  C (xvi)
dx
x x2
1
1
x
dx  log| x |  C



x x2
1
Integration by partial fractions
Recall that a rational function is ratio of two polynomials of the form P(x) ,
Q(x)
where P(x) and Q (x) are polynomials in x and Q (x)  0. If degree of the
polynomial P (x) is greater than the degree of the polynomial Q (x), then we
may divide P (x) by Q (x) so that
P(x)
 T (x) 
P1(x)
, where T(x) is a
Q(x) Q(x)
polynomial in x and degree of P1 (x) is less than the degree of Q(x). T(x)
being polynomial can be easily integrated.
P (x)
1
Q(x)
can be integratedby

expressing
P1(x)
Q(x)
as the sum of partial fractions of the following type:
1. =
A B

x  a x b
, a  b
2. 2
px q
(x  a) (x b)
px q
(x a)
= 2
A B

x  a (x a)
3. =
A

B

C
x  a x  b x c
4. =
5.
px2
 qx  r
(x  a) (x  b) (x  c)
px2
 qx  r
(x  a)2
(x b)
px2
 qx  r
(x  a) (x2
 bx  c)
=
A

B

C
x  a (x  a)2
x b
A

Bx +C
x  a x2
 bx c
where x2 + bx + c can not be factorised further.
Integration by substitution
A change in the variableof integrationoften reduces an integralto one of the
fundamentalintegrals. The method in which we change the variable to some
othervariableiscalledthemethod ofsubstitution.Whentheintegrandinvolves
some trigonometricfunctions, we use some well known identitiesto find the
integrals. Using substitution technique, we obtain the following standard
integrals.
(ii) cot x dx  log sin x  C
(i)
(iii)
(iv)
tan x dx  log secx  C
sec x dx  log secx  tan x  C
cosecx dx  log cosecx  cot x  C
Integrals of some special functions
(i)  log
dx 1 x  a
C
x2
 a2
2a x a

(ii)  log
dx 1 a  x
 C
a2
 x2
2a a x
 (iii)
x
a
1
 tan C
dx 1
x2
 a2
a


(iv)
x2
a2
dx
 log x  x2
 a2
 C (v)
a
a2
x2
dx
 sin 1 x
C
(vi)
x2
a2
dx
 log |x  x2
 a2
| C
Integration by parts
1
For given functions f and f , we have
2
, i.e., the
integral of the product of two functions = first function × integral of the
second function – integral of {differential coefficient of the first function ×
integral of the second function}. Care must be taken in choosing the first
function and the second function. Obviously, we must take that function as
the second function whose integral is well known to us.
ex
[ f (x)  f 
(x)] dx  ex
f (x) dx  C
Some special types of integrals
2 2 2 2 2 2
x a2
x  a dx  x a  log x  x  a C
2 2
(i)
(ii)
2 2 2 2 2 2
x a2
x  a dx  x a  log x  x  a C
2 2
(iii) 2 2 2 2
x a2
1 x
a  x dx  a x  sin C
2 2 a
(iv) Integrals of the types
ax2
 bx c ax2
 bx c
dx
or
dx
can be
transformed into standard form by expressing
b c b 2
c b2
x  
ax2
+ bx + c = a x2
  a x 
a a 2a a 4a2
px  qdx
or
px q dx
ax  bx c
(v) Integrals of the types 2
ax2
 bx  c
can be

b
x
x
transformed into standard form byexpressing
px  q  A
d
(ax2
 bx  c)  B  A (2ax  b)  B , whereAand B are
dx
determined by comparing coefficients on both sides.
We have defined  a
f (x) dx as the area of the region bounded by thecurve
y = f (x), a  x  b, the x-axis and the ordinates x = a and x = b. Let x be a
given point in [a, b]. Then  a
f (x) dx represents the Area function A (x).
This concept of area functionleads to the FundamentalTheorems of Integral
Calculus.
First fundamental theorem of integral calculus
Let the area function be defined by A(x) =  a
f (x) dx for all x  a, where
the function f is assumed to be continuous on [a, b].ThenA
(x) = f (x) for all
x  [a, b].
Second fundamental theorem of integral calculus
Let f be a continuous function of x defined on the closed interval [a, b] and
dx
d
let F be another functionsuch that F(x)  f (x) for all x in the domain of
 
b b
a
a

f, then f (x) dx  F(x) C  F (b)  F (a) .
This is called the definite integral of f over the range [a, b], where a and b
are called the limits of integration, a being the lower limit and b the
upperlimit.

8.Application of Integrals
b b
d d
The area of the region bounded by the curve y = f (x), x-axis and the lines
x = a and x = b (b > a) is given by the formula: Area   a
ydx  a
f (x)dx .
The area of the region bounded by the curve x =  (y), y-axis and the lines
y = c, y = d is given by the formula: Area c
xdy  c
 (y)dy .
The area of the region enclosed between two curves y = f (x), y = g (x) and
the lines x = a, x = b is given by theformula,
 
b
a

Area f (x)  g(x) dx , where, f(x)  g (x) in [a, b]
If f (x)  g (x) in [a, c] and f (x)  g (x) in [c, b], a < c < b, then
 
c b
a c
Area  f (x) g(x) dx  g(
   x)  f (x)dx.

dx dy
dx
 g (x, y) where, f (x, y) and g(x, y) are homogenous
An equation involving derivatives of the dependent variable with respect to
independent variable (variables) is known as a differential equation.
Order of a differential equation is the order of the highest order derivative
occurringin the differentialequation.
Degree of a differential equation is defined if it is a polynomial equation in its
derivatives.
Degree (when defined) of a differential equation is the highest power (positive
integer only) of the highest order derivative in it.
A function which satisfies the given differential equation is called its solution.
The solution which contains as many arbitrary constants as the order of the
differential equation is called a general solution and the solution free from
arbitraryconstants is called particularsolution.
To form a differential equation from a given function we differentiate the
function successively as many times as the number of arbitrary constants in
the given function and then eliminate the arbitrary constants.
Variable separable method is used to solve such an equation in which variables
can be separated completely i.e. terms containing y should remain with dy
and terms containing x should remain withdx.
A differential equation which can be expressed in the form
dy
 f (x, y) or
functions of degree zero is called a homogeneous differential equation.
A differentialequationofthe form
dy
+Py  Q , where P and Q are constants
dx
or functions of x only is called a first order linear differential equation.
9. Differential Equations

10.Vector Algebra
Positionvectorof a pointP(x, y, z) is given as ,andits
magnitude by x2
 y2
 z2 .
The scalar components of a vector are its direction ratios, and represent its
projections along the respective axes.
The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of
any vector are related as:
l 
a
, m 
b
, n 
c
r r r

, then their cross product is
If  is the angle between two vectors
given as
where nˆ is a unitvectorperpendicularto the planecontaining .Such
that form right handed system of coordinate axes.
If we have two vectors , given in component form as
and  any scalar,
then = (a1  b1)iˆ (a2  b2 ) ˆj (a3  b3) kˆ;
 = (a1 )iˆ (a2 ) ˆj (a3 )kˆ;
= a1b1 a2b2  a3b3 ;
and =
ˆj kˆ
iˆ
a1 b1 c1 .
a2 b2 c2

12. Linear
Programming
A linear programming problemis one that is concerned with finding the optimal
value (maximum or minimum) of a linear function of several variables (called
objective function) subject to the conditions that the variables are
non-negative and satisfy a set of linear inequalities (called linear constraints).
Variables are sometimes called decision variables and are non-negative.
A few important linearprogramming problems are:
(i) Dietproblems
(ii) Manufacturingproblems
(iii) Transportationproblems
The common region determined by all the constraints including the non-negative
constraints x  0, y  0 of a linear programming problem is called the feasible
region (or solution region) for the problem.
Points within and on the boundary of the feasible region represent feasible
solutions of the constraints.
Any point outside the feasible region is an infeasible solution.

Any point in the feasible region that gives the optimal value (maximum or
minimum) of the objective function is called an optimal solution.
The following Theorems are fundamental in solving linear programming
problems:
Theorem 1 Let R be the feasible region (convex polygon) for a linear
programming problem and let Z = ax + by be the objective function. When Z
has an optimal value (maximum or minimum), where the variables x and y
are subject to constraints described by linear inequalities, this optimal value
must occur at a corner point (vertex) of the feasible region.
Theorem 2 Let R be the feasible region for a linear programming problem,
and let Z = ax + by be the objective function. If R is bounded, then the
objective function Z has both a maximum and a minimum value on R and
each of these occurs at a corner point (vertex) of R.
If the feasible region is unbounded, then a maximum or a minimum may not
exist. However, if it exists, it must occur at a corner point of R.
Corner point method for solving a linear programming problem. The method
comprisesof the following steps:
(i) Findthefeasibleregionofthelinearprogrammingproblemanddetermine
its corner points (vertices).
(ii) Evaluate the objective function Z = ax + by at each corner point. Let M
and m respectively be the largest and smallest values at thesepoints.
(iii) If thefeasibleregionisbounded,M andm respectivelyarethemaximum
and minimumvaluesof theobjectivefunction.
If the feasible region is unbounded, then
(i) M is the maximum value of the objective function, if the open half plane
determined by ax + by > M has no point in common with the feasible
region. Otherwise, the objective function has no maximum value.
(ii) m is the minimum value of the objective function, if the open half plane
determined by ax + by < m has no point in common with the feasible
region. Otherwise, the objective function has no minimum value.
If two corner points of the feasible region are both optimal solutions of the
same type, i.e., both produce the same maximum or minimum, then any point
on the line segment joining these two points is also an optimal solution of the
same type.

13. Probability
The salient features of the chapter are –
Theconditionalprobabilityof an eventE, given theoccurrenceof the eventF
is given by P(E | F) 
P(E F) , P(F)  0
P(F)
0  P (E|F)  1, P (E|F)= 1 – P (E|F)
P ((E  F)|G) = P (E|G) + P (F|G) – P ((E  F)|G)
P (E  F) = P (E) P (F|E), P (E)  0
P (E  F) = P (F) P (E|F), P (F)  0
If E and F are independent, then
P (E  F) = P (E) P (F)
P (E|F) = P (E), P (F)  0
P (F|E) = P (F), P(E)  0
Theorem of total probability
Let {E1, E2, ...,En) be a partition of a sample space and suppose that each of
E1
, E2
, ..., En
has nonzero probability. Let A be any event associated with S,
then
P(A) = P(E1) P (A|E1) + P (E2) P (A|E2) + ... + P (En) P(A|En)
Bayes' theorem If E , E , ..., E are events which constitute a partition of
1 2 n
sample space S, i.e. E , E , ..., E are pairwise disjoint and E E ... E = S
1 2 n 1 2 n
andA be any event with nonzero probability, then
P(Ei )P(A|Ei )
i n
P(E | A) 
P(E j )P(A|E j )
j1
A random variable is a real valued function whose domain is the sample
space of a random experiment.
The probability distribution ofa random variableX is the system of numbers
X : x1
P(X) : p1
x
2
p2
...
...
x
n
pn
n
where, pi 0, pi 1, i 1,2,...,n
i1

Let X be a random variable whose possible values x , x , x , ..., x occur with
1 2 3 n
1 2 3 n
probabilities p , p , p , ... p respectively. The mean of X, denoted by , is
n
the number  xi pi .
i1
The mean of a randomvariableX is alsocalledtheexpectationof X, denoted
by E (X).
Let X be a random variable whose possible values x1, x2, ..., xn occur with
1 2 n
probabilities p(x ), p(x ), ..., p(x ) respectively.
Let  = E(X) be the mean of X. The variance of X, denoted by Var (X) or
x
2
 , is defined as
2
or equivalently  = E (X –) 2
x
The non-negativenumber
is called the standard deviation of the random variable X.
Var (X) = E (X2) – [E(X)]2
Trials of a random experiment are called Bernoulli trials, if they satisfy the
following conditions:
(i) There should be a finite number of trials.
(ii) The trials should be independent.
(iii) Each trial has exactly two outcomes : success or failure.
(iv) The probability of success remains the same in eachtrial.
For Binomial distribution B (n, p), P (X = x) = nC q n–x px, x = 0, 1,..., n
x
(q = 1 – p)

CBSE Class 12 Mathematics formulas

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