SESSION: 2011-2012
RESOURCE MATERIAL
CLASS: XI
SUBJECT: MATHEMATICS
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STUDY MATERIAL PREPRATION TEAM 2011-12
PATRON
 Sh. Suresh Kumar Sharma Deputy Commissioner NVS (RO)Chd.
CO-ORDINATOR(club incharge)

Sh. Rajinder Kumar Verma Principal JNV Dhanansu Distt. Ludhiana
PARTICIPANTS
 Mr. Chander Sarup
PGT (Maths) JNV Ludhiana
 Mr. Rajiv Kumar
PGT (Maths) JNV Firozepur
 Mr. Sunil Kumar
PGT (Maths) JNV Gurdaspur
 Mr. Ashok Kumar
PGT (Maths) JNV Bilaspur
 Mr. A.S.Bhullar
PGT (Maths) JNV Hoshiarpur
 Mr. S.D.Lakhanpal
PGT (Maths) JNV Una
 Mr. V.K Bahukhandi PGT (Maths) JNV Sirmour
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SIMPLIFIED
MATERIAL
FORM
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1.SETS
Sets are the basic structures on which the concepts of higher mathematics are developed.
Sets: Well defined collection of distinct elements. E.g.
Set of Vowels V = V  a, e, i, o, u
Note that all the elements of the set possess a common property.
Mathematically to represent that a is an element of V we write as a  V .
Representation of Sets:
a) Roaster Form: In this form all the elements are written separelty within the curly brackets.
b) Set Builder Form: In set builder form the common property among the elements is highlighted.
Example: Let A be the set of all prime numbers less than 20.
Roaster Form: A  2,3,5,7,11,13,17,19
Set builder form: A  x : x is a prime number , x  20
Types of sets:
a) Finite and Infinite Sets: A set is said to be finite if it consist of only finite number of elements
otherwise it is infinite.
Example: A  2,3,5,7,11,13,17,19; finite set
N  Set of natural numbers. ;infinite set
b) Empty Set: A set having no elements.
c) Singleton Set: A set having only one element is called a singleton set.
Equality of two sets: Two sets are equal if they have exactly the same elements.
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Order of a finite set: Number of elements of a set represents its order.
Intervals: Intervals represents the set of real numbers between two real numbers.
Types:
a) Open Interval a, b : The set of all real number s between a and b.
b) Closed Interval a, b : The set of all real numbers between and b including and b.
c) Open closed interval a, b : The set of all real numbers between a and b including b.
d) Closed open Interval a, b : The set of all real numbers between a and b including a.
Subsets: Let A and B are sets. Then A is said to be subset of B if all its elements are also in B. Note
that B is said to be superset of A.
Example: Let N be the set of all natural numbers and R be the set of all real numbers. Then N is a
subset of R, mathematically N  R.
Power Set: Set of all subsets of a set is said to be its power set.
A  1,2,3
P A   , A, 
1 , 2, 3, 1,2, 1,3, 2,3.
Note that number of elements in the power set  2 n , where n is the no. of elements in set A.
Universal Set: If there are some set under consideration then the superset of all these sets is called
the Universal Set and it is denoted by U,X and E.
Euler-Venn Diagram: This concept was basically developed John Venn and Euler.
In these diagrams universal sets are represented by enclosed rectangular region,
subsets by enclosed circular region and elements by small dots.
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Operation on sets:
a) Union of two sets: Let A and B be two sets. Then A B is the set containing all elements of A
and B and repeated elements are taken once.
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Properties of Union:
i.
ii.
A  B  B  A.
 A  B   C  A  B  C 
iii.
A    A.
iv.
v.
A  A  A.
U  A U.
Commutative law
Associative law
 is the identity element
Idempotent law
Law of U
d) Intersection of two sets: The set of all the common elements of two sets it called intersection
A  B of two sets.
Properties of Intersection:
i.
ii.
A  B  B  A.
 A  B   C  A  B  C 
iii.
A    A, U  A  A
Laws of  and U
iv.
v.
A  A  A.
A  B  C    A  B    A  C 
Idempotent law
Distributive Law
Commutative law
Associative law
Disjoint sets:
Two sets are disjoint if there intersection is empty set i.e. A  B  .
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e) Difference of two sets:
Let A and B be two sets. Then difference A  B is the set of all those elements of A which are not
in B.
e) Complement of Set: Let U be the universal set and A be its subset.Then complement of A is
the set of all those element of U which are not in A.
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Example: Sets and odd numbers are complements to each other.
Properties of complement of sets:
i.
ii.
iii.
iv.
A  A  U
A  A  
A  B  A  B
A  A
  U
Complement laws
A  B  A  B
De Morgan’s Law
U   .
Practical Problems related to union and intersection of sets:
Concept of set can be used to solve various problems related to our daily life. Note that interms of
notation of sets:
or means

and means
niether A nor B means

A  B
Formulae:
1) n A  B   n A  nB   nA  B.
2) n A  B  C   n A  nB   nC   n A  B   n A  C   nB  C   n A  B  C 
3) n A  n U   n A.
4) n  A  B   n U   n  A  B .
…………………………………
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2. RELATIONS AND FUNCTIONS
Functions have same importance in higher mathematics as numbers in lower class mathematics.
Cartesian product:
Let A and B are two non empty sets. The Cartesian product is the set defined as:
A  B  a ,b  : a  A, b  B 
Note that if number of elements in A and B are m and n respectively then number of elements in the
Cartesian product is m  n.
Relation:
Relation between two sets is a subset of Cartesian product. This subset is derived by describing a
relationship between the 1st element and 2nd element of the ordered pairs. The set of all 1st elements
is the domain and set of all 2nd elements is the range of the Relation.
Let R : A  B is a relation from A to B. B is called the co domain of R.
Note that
range codomain.
Example: Let A  1,2 ,3,4 and B  3,4 ,5 ,6 ,8 R  x , y  : y  x  2 , x  A, y  B
Note that total number of relation defined from set A to set B is 2 mn , where n A  m , nB   n .
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Functions:
Let A and B are two nonempty sets. A function from A to B is a relation or formula which
associates each element of A with unique element of B. Mathematically,
y  f x , x  A , y  B .
Here x is known pre image and y is called image. Set A is called domain of function f and set of
all images is called Range of function f . Set B is called co domain of function f .
Real and real valued functions: A function which has either R or one of its subset as its range is
called a real valued function. If domain is also R, it is called real function.
Types of real and real valued functions:
1) Constant Function: A function f : R  R f x   c , x  R ,c is cons tan t is called constant
function.
Domain D f  R
Range R f  c.
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2) Identity Function: A function f : R  R f x   x , x  R is called Identity function.
Domain D f  R
Range R f  R .
3) Polynomial Function: A function
f : R  R f x   a0 x n  a1 x n1  a2 x n2  .......... an1` x  an , x  R and a0  0 ,a1 ,a 2 ......a n 1 ,a n are
real constants, is called polynomial function of degree n.
4) Rational function: A function f defined as f  x  
g x 
,h  x   0 where f  x  and h  x  are
hx 
polynomial functions, is called rational function.
5) Modulus function (absolute value function): A function f : R  R f x   x , x  R where
x , x  0
x 
 x ,x  0
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Domain D f  R
Range R f  0 , .
1 , x  0

6) Signum function: A function f : R  R f x   0 , x  0 , x  R is called signum function.
 1 ,x  0

Domain D f  R
Range R f  1,0 ,1.
7) Greatest Integer function:
A function f : R  R f x   x  x  R where x  assumes the value greatest integer, less than or
equal to x, is called greatest integer function.
Domain D f  R
Range R f  Z .
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Algebra of real functions: Let f : X  R and g : X  R , X  R
1) Addition and Subtraction:  f  g x   f x   g x , x  X .
2) Multiplication by a scalar: f x   f x , x  X .
3) Multiplication of functions:  fgx   f x  g x , x  X .
f 
f x 
, g  x   0 x  X .
4) Quotient of two real functions:  x  
g x 
g
…………………….
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3.TRIGONOMETRIC FUNCTIONS
If in a circle of radius r , an arc length l subtends an angle of  radians then l  r
MEASUREMENT OF AN ANGLE
i)
Sexagesimal System : In this system an anle is measured in degrees,minutes and
1
seconds.One degree is
th of a complete rotation i.e one complete revolution = 3600
360
ii)
Circular System: In this system the angle is measured in radians.
A radian is an angle subtended at the centre of a circle by an arc whose length is equal to the
radius
RELATION BETWWN DEGREE MEASURE AND RADIAN MEASURE
Radian Measure 

180
Degree Measure = 
 Degree Measure
180

 Radian Measure
TRIGONOMETRIC IDENTITIES
1. sin 2   cos 2   1
2. 1  tan 2   sec 2 
3. 1  cot 2   cos ec 2
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TRIGONOMETRIC RATIOS OF ALLIED ANGLES
I (90   )
II (90   )
III (180   )
IV (  )
sin 
 ve
 ve
 ve
 ve
cos
 ve
 ve
 ve
 ve
tan 
 ve
 ve
 ve
 ve
cot 
 ve
 ve
 ve
 ve
sec 
 ve
 ve
 ve
 ve
cos ec
 ve
 ve
 ve
 ve
Quadrants 
T.ratios 
TRIGONOMETRIC RATIOS OF MULTIPLE AND SUBMULTIPLE ANGLES
1. sin 2 A  2 sin A cos A 
2 tan A
1  tan 2 A
2. cos 2 A  cos 2 A  sin 2 A  1  2 sin 2 A  2 cos 2 A  1 
3. tan 2 A 
1  tan 2 A
1  tan 2 A
2 tan A
1  tan 2 A
4. sin 3 A  3 sin A  4 sin 3 A
5. cos 3 A  4 cos 3 A  3 cos A
6. tan 3 A 
3 tan A  tan 3 A
1  3 tan 2 A
TRIGONOMETRIC RATIOS OF COMPOUND ANGLES
1. sin( A  B)  sin A cos B  cos A sin B
2. sin( A  B)  sin A cos B  cos A sin B
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3. cos( A  B)  cos A cos B  sin A sin B
4. cos( A  B)  cos A cos B  sin A sin B
5. tan( A  B) 
tan A  tan B
1  tan A tan B
6. tan( A  B) 
tan A  tan B
1  tan A tan B
SOME OTHER IMPORTANT FORMULAE
C  D C  D
1. sin C  sin D  2 sin 
 cos

 2   2 
C  D C  D
2. sin C  sin D  2 cos
 sin 

 2   2 
C  D C  D
3. cos C  cos D  2 cos
 cos

 2   2 
C  D C  D
4. cos C  cos D  2 sin 
 sin 

 2   2 
5. sin( A  B)  sin( A  B)  2 sin A cos B
6. sin( A  B)  sin( A  B)  2 cos A sin B
7. cos( A  B)  cos( A  B)  2 cos A cos B
8. cos( A  B)  cos( A  B)  2 sin A sin B
SOLUTION OF TRIGONOMETRIC SOLUTIONS
1.The general solution of the equation sin   sin  is given by
  n  (1) n 
2. The general solution of the equation cos   cos  is given by
  2n  
3. The general solution of the equation tan   tan  is given by
  n  
In all the above solutions n Z
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4.PRINCIPLE OF MATHEMATICAL INDUCTION
In deduction method, if a statement is to be proven valid deductive steps are derived and approved
may or may not be established i.e. deduction is the application of the general case to a particular
case.In contrast to deduction in induction we move from general to particular cases .In algebra,
there are certain results or statements that are formulated in terms of n where n is a positive integer.
To prove such statements, the well suited principle which is used based on specific techniques is
known as the principle of mathematical induction.
Working Rule :
Suppose there is a given statement P(n) involving the natural number n such that
(i) the statement is true for n=1.
(ii) the statement is true for n = k (where k is a positive integer.) Then the statement is also true for
n= k+1.i.e. truth of P(k) implies the truth of P(k+1) .Then P(n) is true for all natural number n .
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5.COMPLEX NUMBERS AND QUADRATIC EQUATIONS
NEED FOR COMPLEX NUMBERS
The equations of the form 𝑥 2 + 1 = 0, 𝑥 2 + 4 = 0, are not solvable in the set of R i.e there is no real
number whose square is negatie .Euler was the first mathematyician to introduce the symbol 𝑖 (iota)
for the square root of -1 i.e a solution of 𝑥 2 + 1 = 0 with the property 𝑖 2 = −1.
INTEGRAL POWERS OF IOTA
We have 𝑖 = √−1
𝑖 2 = −1
𝑖 3 = 𝑖 2 × 𝑖 = −𝑖
𝑖 4 = (𝑖 2 )2 = 1
IMAGINARY QUANTITIES
The square root of a negative real number is called an imaginary quantity or an imaginary number .
If a, and b are positive real numbers then √−𝑎 × √−𝑏 = −√𝑎𝑏
For any two real numbers √𝑎 × √𝑏 = √𝑎𝑏 is true only when atleast one of a and b is either positive
or zero.
In other words √𝑎 × √𝑏 = √𝑎𝑏 is not valid if a and b both are negative.
COMPLEX NUMBERS
If a and b are two real numbers then a number of the form 𝑎 + 𝑖𝑏 is called a complex number where
a is known as the real part and b is known as the imaginary part of the complex number.
Equality of complex number
Two complex number 𝑎 + 𝑖𝑏 and 𝑐 + 𝑖𝑑 are equal if and only if a = b and c = d.
ADDITION OF COMPLEX NUMBERS
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Let 𝑧1 = 𝑎1 + 𝑖𝑏1 and 𝑧2 = 𝑎2 + 𝑖𝑏2 be two complex numbers. Then their sum 𝑧1 + 𝑧2 is defined as
the complex number (𝑎1 + 𝑎2 ) + 𝑖(𝑏1 + 𝑏2 ) .
DIFFERENCE OF COMPLEX NUMBERS :
The difference of two complex numbers z1  a  ib and z 2  c  id is given by
z1  z 2  (a  c)  i(b  d )
MULTIPLICATION OF TWO COMPLEX NUMBERS
The difference of two complex numbers z1  a  ib and z 2  c  id is given by
z1  z 2  (ac  bd )  i(ad  bc)
THE MODULUS AND CONJUGATE OF A COMPLEX NUMBER
If z  a  ib be a complex number then the modulus of z denoted by z is defined as a non negative
real number
a 2  b 2 ie z  a 2  b 2 and the conjugate of the complex number z is given by
z  a  ib
If z  2  5i ,then z  2 2  (5) 2  29
POLAR FORM OF A COMPLEX NUMBER
The polar form of a complex number z  x  iy is z  r (cos   i sin  ) where r  a 2  b 2 (the
modulus of z) and cos  
that
x
y
, sin   ,where  is known as the argument of z.The value of  such
r
r
      is called the principal argument of z.
SOLUTION OF A QUADRATIC EQUATION
The solution of a quadratic equation ax 2  bx  c  0 with real coefficient a , b ,c ,and a  0 is given by
 b  bb 2  4ac
x
2a
A polynomial equation has atleast one root and a polynomial equation of degree n has n roots.
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6.LINEAR INEQUALITIES
INEQUATIONS : -
A statement involving variables and the sign of inequality viz,> , < , or ,  is called an inequation or an
inequality.
An equation may contain one or more variables. Also, it may be linear or quadratic or cubic etc.e.g
i) 3x  2  0
ii) 3x  2  0
iii) x 2  3x  2  0
iv) x 2  5 x  4  0
LINEAR INEQUATION IN ONE VARIABLE :
Let a be a non zero real numbr and x be a variable.Then inequations of the form
ax  b  0, ax  b  0, ax  b  0, ax  b  0 are known as linear equation in one variable.
LINEAR INEQUATION IN TWO VARIABLE :
Let a and b be non-zero real numbers and x, y be variable.Then inequations opf the form
ax  by  c, ax  by  c, ax  by  c, ax  by  c are known as linear equations in two variables.
QUADRATIC INEQUATIONS :
Let a be a non-zero real number.Then an equation of the form
ax 2  bx  c  0, ax 2  bx  c  0, ax 2  bx  c  0, ax 2  bx  c  0 are known as quadratic inequations.
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SOME POINTS TO BE TAKEN CARE OF
1.If a be a positive real number, then x  a  a  x  a
2.If a be a positive real number, then x  a  a  x  a
3. If a be a positive real number, then x  a  x  a or x  a
4. If a be a positive real number, then x  a  x  a or x  a
5. Let r be a positive real number and a be a fixed real number .Then,
i) x  a  r  a  r  x  a  r
ii) x  a  r  a  r  x  a  r
iii) x  a  r  x  a  r or x  a  r
iv) x  a  r  x  a  r or x  a  r
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7.PERMUTATIONS AND COMBINATIONS
FACTORIAL NOTATION:
The continued product of first n natural numbers is called the n factorial and denoted by n !
n! n  (n  1)  (n  2)  (n  3)..............................................4  3  2  1.
ZERO FACTORIAL:
Factorial zero has no meaning. But conventionally we define it as 0 ! = 1
FUNDAMENTAL PRINCIPLE OF COUNTING :
If an event can occur in m different ways , following which another event can occur in n different
ways, then the totoal number of occurance of the events in the given order is m  n .
PERMUTATION:
A permutation is an arrangement in a definite order of a number of objects taken some or all at a
time.
If we have to find the number of permutations of n different things taken r at a time or find the
n!
value of P (n, r ) . It is given by P(n, r ) 
n  r!
The number of permutations of n different things, taken r at a time, where repetitions is allowed is
given by n r .
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PERMUTATION WHEN ALL THE OBJECTS ARE NOT DISTINCT:
The number of permutations of n objects, when p1 objects are of one kind, p 2 are of another kind
,……….. pk are of k th kind and the rest ,if any,are of different kind than the number of
permutations are given by
n!
.
p1! p 2 !................ p K !
COMBINATIONS:
The number of combinations of n different things taken r at a time is denoted by C n, r  and is
given by
C n, r  
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n!
r!n  r!
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8.BINOMIAL THEOREM
BINOMIAL THEOREM
An expression containing two terms is called a Binomial Binomial Theorem for any positive integer
n is given by
( x  a) n  n C 0 x n  n C1 x n 1 a  n C 2 x n 2 a 2  n C3 x n 3 a 3  ............................. n C n a n .
Where x and a are any two numbers.
GENERAL TERM IN THE EXPANSION OF ( x  a) n
General term in the expansion of ( x  a) n is given by
Tr 1  n Cr x nr a r
NUMBER OF TERMS IN THE EXPANSION OF ( x  a) n
The number of terms in the expansion of ( x  a) n is always (n + 1)
MIDDLE TERM OR TERMS IN THE EXPANSION OF ( x  a) n
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1. If the exponent n is even in the power of Binomial ( x  a) n , then the number of terms in the
n 
expansion is odd, so there is only one middle term and it is   1th term.
2 
2. If the exponent n is odd in the power of Binomial ( x  a) n , then the number of terms in the
 n 1
 n  3
expansion is even ,so there are two middle terms and these are 
th and 
th
 2 
 2 
terms.
BINOMIAL COEFFICIENTS
In the expansion (1  x) n  n C 0  n C1 x n  n C 2 x 2  n C 3 x 3  ............................. n C n x n . the coefficients
of various terms i.e. n C 0 , n C1 , n C 2 , n C3 ,..........................., n C n . are called binomial coefficients. These
are briefly written as C0 , C1 , C2 , C3 ,..........................., Cn .
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9.SEQUENCES AND SERIES
A sequence is an arrangement of number in definite order according to some rule. Or in other words
A sequence is a function whose domain is the set N of natural numbers.
A sequence containing a finite number of terms is called a finite sequence. A sequence is called
infinte if it is not finite.
ARITHMETIC PROGRESSION(A.P) :
A sequence is cllled an Arithmetic progression iff the difference of any term from its preceding
term is constant.This constant is usually denoted by ‘d’.
Thus a1 , a 2 , a3 , a 4 ,........................................., a n is in A.P iff a k 1  a k  d
e.g. the sequence 3,7,11,15,19,…………………….,35 is in A.P
GENERAL TERM OF AN A.P.:
If T1 , T2 , T3 ,........................................., Tn denote the 1st,2nd,3rd,…………………nth, term of the
A.P.having first term ‘a’ and common difference ‘d’ then the general term is given by
Tr 1  a  (n  1)r
SUM OF FIRST n TERMS OF AN A.P
If ‘a’ and ‘d’ denotes the first term and common difference of an A.P then the sum of first n terms
a1 , a 2 , a3 , a 4 ,........................................., a n of an A.P is given by
Sn 
n
2a  (n  1)d 
2
If the last term of an A.P is known say ‘l’ in that case we can also find the sum by applying the
formula
Sn 
n
a  l 
2
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ARITHMATIC MEAN :
The arithmetic mean of any two numbers ‘a’ and ‘b’ is given by A.M . 
ab
2
GEOMETRIC PROGRESSION:
A sequence is cllled an Geometric Progression iff the ratio of any term to its preceding term is
constant.This constant is usually denoted by ‘r’and is called common ratio.
Thus a1 , a 2 , a3 , a 4 ,........................................., a n is in G.P iff
e.g the sequence
a k 1
r
ak
1
,1,2,4,8,16..........................,512 is in G.P with common ratio 2
2
GENERAL TERM OF AN G.P.:
If T1 , T2 , T3 ,........................................., Tn denote the 1st,2nd,3rd,…………………nth, term of a
G.P.having first term ‘a’ and common ratio ‘r’ then the general term is given by
Tn  ar n 1
SUM OF FIRST n TERMS OF AN G.P:
If ‘a’ and ‘r’ denotes the first term and common ratio of an G.P then the sum of first n terms
a1 , a 2 , a3 , a 4 ,........................................., a n of an G.P is given by
Sn 
a(r n  1)
if
r 1
r  1 and
Sn 
a(r n  1)
if
r 1
r 1
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GEOMETRIC MEAN :
The geometric mean of any two positive numbers ‘a’ and ‘b’ is given by
ab
SUM TO INFINITY OF AN G.P
The sum to infinity of an G.P. a, ar , ar 2 , ar 3 , ar 4 ,................................ is given by
S 
a
1 r
SUM OF FIRST n NATURAL NUMBERS
1  2  3  4  5  ......................................................  n 
n(n  1)
2
SUM OF THE SQUARES FIRST n NATURAL NUMBERS
12  2 2  3 2  4 2  5 2  ......................................................  n 2 
n(n  1)( 2n  1)
6
SUM OF THE CUBES FIRST n NATURAL NUMBERS
 n(n  1) 
1  2  3  4  5  ......................................................  n  

 2 
3
3
3
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3
3
2
3
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10.STRAIGHT LINES
SLOPE OF A LINE :
The angle (say)  made by the line l with positive direction of x-axis and measured anti clockwise
is called the inclination of the line.The slope this line is given by tan  .
If the line passed through the points ( x1 ,y1 ) and (x2, y2) then the slope of the line is given by
m
y 2  y1
x 2  x1
ANGLE BETWEEN TWO LINES
The angle between the lines having slopes m1 and m2 is given by
tan   
m2  m1
1  m1 m2
i)
Two lines are parallel only if their slopes are equal.
ii)
Two lines are perpendicular if product of their slopes is  1 .
EQUATION OF A LINE PARALLEL TO X-AXIS
The equation of a line parallel to x-axis at a distance ‘b’from it is y = b
EQUATION OF A LINE PARALLEL TO Y-AXIS
The equation of a line parallel to x-axis at a distance ‘a’from it is x = a
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SLOPE INTERCEPT FORM OF A LINE
The equation of a line with slope m and making an intercept c on y-axis is y  mx  c
POINT SLOPE FORM OF A LINE
The equation of a line passing through the point ( x1 , y1 ) and has the slope m is given by
y  y1  m( x  x1 )
TWO POINT FORM OF A LINE
The equation of a line passing through the point ( x1 , y1 ) and ( x2 , y 2 ) is given by
 y  y1 
( x  x1 )
y  y1   2
 x 2  x1 
TWO INTERCEPT FORM OF A LINE
The equation of a line which cuts off intercepts a and b respectively from the x and y-axis is given
by
x y
 1
a b
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NORMAL FORM OF A LINE
The equation of a straight line upon which the length of the perpendicular from the origin is p and
this perpendicular makes ;an angle  with x-axis is x cos   y sin   p .
DISTANCE OF A LINE FROM A POINT
The distance of a line Ax  By  C  0 from a point ( x1 , y1 ) is given by
d
Ax  By  C
A2  B 2
DISTANCE BETWEEN TWO PARALLEL LINES
The distance between two parallel lines Ax  By  C1  0 and Ax  By  C2  0 is given by
d
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C1  C 2
A2  B 2
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11.CONIC SECTION
Curves which are obtained as intersection of plane with a double napped right circular cone are
known as conic sections (conics)
Upper
Plane
Lower
Cone
CIRCLE:
A circle is the set of all points which are equidistant from a fixed point in the plane.
The Equation of circle, when centre is at (0, 0) and radius r
x2 + y2 = r2
The equation of circle, when centre is at (h, k) and r radius r
(x-h)2 + (y-k)2 = r2
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PARABOLA:
Parabola is set of all points in a plane that are equidistant from a fixed line and a fixed point (not on
the line) in the plane .
fixed line is called directrix and the, fixed point ‘ focus’ The standard equations of the parabola are
y 2  4ax
y 2  4ax
x 2  4ay
x 2  4ay
LATUS RECTUM
The Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola ,through
the focus and whose end points lie on the parabola.
PARABOLA:
y 2  4ax
y 2  4ax
x 2  4ay
x 2  4ay
0,0
(0,0)
(0,0)
(0,0)
Coordinates of the
foci
(a,0)
(a,0)
(0, a )
(0,a)
Equation of
directrices
x  a
xa
y  a
ya
y0
y0
x0
x0
4a
4a
4a
4a
Coordinates of the
vertices
Equation of axis
Length of latus
rectum
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ELLIPSE
Ellipse is the set of points in a plane , the sum of whose distances from two fixed points in the
plane is a constant.
Two fixed points are foci.The equation of ellipse with length of manor axis as 2a and minor axis 2b
is given
x2 y2

 1,
a2 b2
If 2a ,2b and c represent the length of major axis ,length of minor axis and the distance of the focus
from the centre of the ellipse.Then we have the relation
c2  a2  b2
Eccentricity: Ir is the ratio of distance from the centre of the ellipse to one of the foci and to one of
the vertices of the ellipse,
For ellipse : e 
c
, 0< e<1
a
Length of latus rectum =
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2b 2
a
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ELLIPSE
x2 y2

 1, a  b
a2 b2
x2 y2

 1, b  a
a2 b2
Coordinates of the centre
0,0
(0,0)
Coordinates of the vertices
( a,0)
(0,b)
Coordinates of the foci
( ae,0)
(0,be)
Length of major axis
2a
2b
Length of minor axis
2b
2a
Equation of major axis
y0
x0
Equation of minor axis
x0
y0
Equation of directrices
Eccentricity
Length of latus rectum
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x
e  1
2b 2
a
a
e
b2
a2
y
e  1
b
e
a2
b2
2a 2
b
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HYPERBOLA:
Is the set of all points in a plane ,the difference of whose distances from two fixed points in
the plane is a constant.
x2 y2

1
a2 b2
If 2a ,2b and c represent the length of transverse axis ,length of conjugate axis and the distance of
the focus from the centre of the hyperbola.Then we have the relation
c2  a2  b2
x2 y2

1
a2 b2
y2
x2

1
b2
a2
Coordinates of the centre
0,0
(0,0)
Coordinates of the vertices
( a,0)
(0,b)
Coordinates of the foci
( ae,0)
(0,be)
Length of ttransverse axis
2a
2b
Length of conjugate axis
2b
2a
Equation of transverse axis
y0
x0
Equation of conjugate axis
x0
y0
Equation of directrices
x
Eccentricity
b2
e  1 2
a
a2
e  1 2
b
Length of latus rectum
2b 2
a
2a 2
b
a
e
y
b
e
Eccentricity e>1
a hyperbola in which a = b is called equilateral hyperbola
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12.THREE DIMENSIONAL GEOMETRY
COORDINATE SYSTEM FOR 3-D:
To locate the position of lowest tip of electric bulb in a room or the position of central tip of the
ceiling fan in room, we require two perpendicular distances of the point to be located from
perpendicular walls of the room but also the height of the point from the floor of the room. Three
numbers representing the perpendicular distances of the point from three mutually perpendicular
planes. These three numbers representing three distances are called the coordinates of the point with
references to three coordinates planes. So point in space has three coordinates
Z
Y
O
X’
X
Y’
Z’
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The plane XOY, YOZ, ZOX is called XY-plane, YZ- plane and ZX-plane.
In this way whole of the space is divided into eight parts known as octants.
Numbered I ,II,III,……….VIII in anti-clockwise direction. The following table give the sign in
eight octants
Octant→
OXYZ
OX’YZ
OX’Y’Z OXY’Z
OXYZ’
OX’YZ’ OX’Y’Z’ OXY’Z’
x
+
-
-
+
+
-
-
+
y
+
+
-
-
+
+
-
-
z
+
+
+
+
-
-
-
-
Coordinated↓
DISTANCE FORMULA:
Distance between the two points P(x1, y1,z1) and Q(x2, y2,z2)
( x 2  x1 ) 2  ( y 2  y1 ) 2  ( z 2  z1 ) 2
SECTION FORMULA:
For internal division :The coordinates of the point which divides the join of the points ( x1 , y1 , z1 )
and ( x2 , y 2 , z 3 ) in the ratio of m1 : m2
(
m1 x2  m2 x1 m1 y 2  m2 y1 m1 z 2  m2 z1
,
,
)
m1  m2
m1  m2
m1  m2
For external division
(
m1 x2  m2 x1 m1 y 2  m2 y1 m1 z 2  m2 z1
,
,
)
m1  m2
m1  m2
m1  m2
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13.LIMITS AND DERIVATIVES
Introduction
Consider the function f(x) =
x2  4
.
x2
Clearly, this function is defined for all except x = 2 as it assumes the form
However, if x  2, then
F(x) =
0
at x=2.
0
( x  2)( x  2)
= x+2.
x2
The following table exhibits the values of F(x) at points which are close to 2 on its two sides viz.
left and right on the real line .
X
1.4
1.5
1.6
1.7
1.8
1.9
1.99 2
2.01 2.1
2.2
2.3
2.4
2.5
2.6
F(x) 3.4
3.5
3.6
3.7
3.8
3.9
3.99 N.D. 4.01 4.1
0
( 0)
4.2
4.3
4.4
4.5
4.6
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
f(x)
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Left Hand Limit (LHL)
Right Hand Limit (RHL)
It is evident from the above table and the graph
of f(x) that as x increases and comes closer to 2
from left hand side of 2 ,
the values of f(x) increase and come closer to 4 .
It is evident from the above table and the graph
of f(x) that as x decreases and comes closer to 2
from right hand side of 2 , the values of f(x)
decrease and come closer to 4 .
This is interpreted as :
This is interpreted as :
When x approaches to 2 from its left hand side ,
the function f(x) tends to the limit 4.
If we use the notation ‘x  2 ’ to denote ‘ x
tends to 2 from left hand side ‘ ,

When x approaches to 2 from its right hand side
, the function f(x) tends to the limit 4.

If we use the notation ‘x  2 ’ to denote ‘ x
tends to 2 from right hand side ‘ ,
the above statement can be restated as ‘x  2 ,

the above statement can be restated as’ x  2
,f(x)  4’ or
lim f ( x)  4
lim f ( x)  4
x2
means that
as x tends to 2 from left hand side ,
f(x) is tending to 4.
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f(x)  4 or
x2
Or Left hand limit of f(x) at x=2 is 4.
Thus,

lim f ( x)  4
x2 
Or Right hand limit of f(x) at x=2 is 4.
Thus ,
lim f ( x)  4
x2 
means that
as x tends to 2 from right hand side ,
f(x) is tending to 4.
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Algorithm to Evaluate of LHL and RHL
Steps
1
LHL
Write
RHL
lim f ( x)
Write
xa 

Put x = a-h and replace x  a by
lim f ( x)
xa 

Put x = a+h and replace x  a by
2
h  0 to obtain
Simplify
3
lim f (a  h)
h 0
lim f (a  h)
h 0
by using the
h  0 to obtain
Simplify
lim f (a  h)
h 0
lim f (a  h)
h 0
by using the
formula for the given function.
formula for the given function.
The value obtain in step 3 is the LHL
The value obtain in step 3 is the
of f(x) at x=a.
RHL of f(x) at x=a.
4
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Summary
lim f ( x )
x a
1.
Exists
 lim f ( x)
x a
=
2. For a function f(x) and a real number a ,
(i)
lim f ( x )
xa
lim f ( x)  f (a)
x a 
.
lim f ( x) and f(a) may
x a
not be same:
exists but f(a) ( the value of f(x) at x= a ) may
not exists.
e.g. Consider f(x) =
lim f ( x )
x 3
x2  9
x3
.
= 6 , but f(3) does not exists.
(ii) The value f(a) exists but
lim f ( x) does not exists.
x a
e.g. Consider f(x) =
x .
lim f ( x )
x 3
does not exist but f(3) = 3
lim f ( x )
(iii)
x a
and f(a) both exist but are unequal.
e.g. Consider function f(x) defined by
f(x) =
x2  4
,x2

 x2
3, x  2

It can be seen that
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lim f ( x) = 4 , but f(2) = 3
x 2
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(iv)
lim f ( x) and f(a) both exist and are equal
x a
e.g. Consider function f(x) defined by f(x) =
It can be seen that
x2  4
,x2

 x2
4, x  2

lim f ( x) = 4 = f(2)
x 2
3. Algebra of Limits :
(i)
lim kf ( x) = k lim f ( x )
(ii)
lim ( f  g )( x )
(iii)
lim ( fg )( x) = lim f ( x )
(iv)
lim f ( x )
xa
f
lim
( x ) lim g ( x )
x a g
= xa
x a
xa
xa
x a
=
xa
xa
x a
lim g ( x)
x a
lim g ( x )
lim { f ( x)} g ( x )
(v) x a
lim f ( x )  lim g ( x)
{lim f ( x)} x  a
=
x a
4. Commit to memory:
(A) If angle x is measured in radians, then
sin x
1
x 0
x
(i)
sin( x  a )
lim
1
xa
x

a
(iii)
lim
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tan x
1
x 0
x
(ii)
tan( x  a )
lim
1
xa
x

a
(iv)
lim
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(B)
(i)
xn  an
lim
 na n 1
x a
xa
(iii)
log( 1  x)
1
x 0
x
lim
(ii)
ax 1
lim
 log e a, a  0, a  1
x 0
x
ex 1
lim
1
x 0
x
(iv)
5. Evaluation of Algebraic Limits:
(i) Direct substitution Method
(ii) Factorization Method
Consider the following limit :
lim
xa
f ( x)
f ( x)
g ( x) . If by substituting x=a, g ( x)
reduces to the form
0
0
,
then (x-a) isa factor of f(x) and g(x) both . So , we first factorize f(x) and g(x) and then cancel out
the common factor to evaluate the limit.
(iii) Rationalization Method
This is particularly used when either the numerator or denominator or both involve expression
consisting of square roots and substituting the value of x ,the rational expression takes the form
0 
,
0 
etc.
(iv) By using some standard Limits
e.g.
xn  an
lim
 na n1 where n  Q.
x a x  a
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(v) Method of evaluation of algebraic limits at infinity.
f ( x)
g ( x)
Write down the given expression in the form of a rational function , i.e.
, if it is not so. If k
is the highest power of x in numerator and denominator both, then divide each term in numerator
k
and denominator by x . Use the results
lim
x 
c
` 0
xn
and
lim c  c
x 
,where n>0.
Derivative
Derivative at a point
(a, b) and let c  (a, b). Then,
Let f(x) be real valued function defined on an open interval
lim
f(x) is said to be differential or derivative at x=c, iff
xc
f ( x) f (c)
xc
This limit is called the derivative or differentiation of f(x) at x=c and is denoted by f’( c) or Df(
c) or
d

f
(
x
)
 dx

x c
Thus ,we have f’( c) =
Hence ,
lim
x c
x c
f ( x ) f (c )
xc
f ( x ) f (c )
xc
 F’( c) = lim
h0
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lim
=
lim
x c
f (c  h) f (c)
h
=
, provided that the limit exists.
f ( x ) f (c )
xc
lim
h0
f (c  h) f (c)
h
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Geometrical interpretation of derivative at a point
The derivative of a function f(x) at point x=c is the slop of the tangent to the curve y = f(x) at
the point (c, f( c ) ).
f’(c) = tan  = slope of tangent on f(x) at Point(c, f(c)).
Commit to Memory:
1.
d
( f ( x ))
dx
measures the rate of change of f(x) with respect to x .
2. Some standard derivatives:
(i)
d
(k )  0
dx
(ii)
d
( x n )  nx n 1
dx
(iii)
d
(a x )  a x log e a,
dx
(iv)
d
(e x )  e x
dx
(v)
d
1
(log e x) 
dx
x
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a>0 , a  1.
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(vi)
d
1
(log a x) 
dx
x log e a
(vii)
d
(sin x)  cos x
dx
(viii)
d
(cos x)   sin x
dx
(ix)
d
(tan x )  sec 2 x
dx
(x)
d
(cot x)   cos ec 2 x
dx
d
(sec x)  sec x tan x
(xi)
dx
(xii)
d
(cos ecx)   cos ecx cot x
dx
(xiii)
d
x
dx
=
x
x
3. Algebra of Derivatives:
If f(x) and g(x) are differentiable functions , then
(i)
d
d
d
{ f ( x)  g ( x)} 
f ( x) 
g ( x)
dx
dx
dx
(ii)
d
d
d
{ f ( x)  g ( x)}  g ( x) 
f ( x)  f ( x) 
g ( x)
dx
dx
dx
(iii)
d  f ( x) 


dx  g ( x) 
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g ( x) 
d
d
f ( x)  f ( x) 
g ( x)
dx
dx
g ( x)2
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14.MATHEMATICAL REASONING
Sentence
Assertive Sentenc
Imperative Sentence
Exclamatory
Sentence
Interrogative
Sentence
(declaration)
(request or
command)
(express feelings)
(questions)
Mars support life.
Please bring me a
cup of tea.
How big is the whale
fish!
What is your age?
But here, we will be discussing a specific type of sentence which will called as Statements or
Propositions.
A statement or a
proposition is an assertive
(or declarative) sentence
which is either true or false
but not both
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Summary
1.A sentence is called a mathematically acceptable statement or simply a statement if it is either
true or false but not both.
2. The denial of a statement is called negation of the statement. The negation of a statement p is
denoted by ”~p” and is read as “ not p”.
Negation of any statement p is formed by writing
“ It is not the case that ….”
Or “ It is false that..” before p
Or , if possible, - by inserting in p the world “not”.
3. Simple statement: Any statement whose truth value does not
another statement is said to be a simple statement.
explicitly depend on
4.Compound Statement: A statement is called a compound statement if it is made up of two or
more simple statements.
The simple statements are called components statements of
the compound statements .
Compound statements are obtained by using connecting words like “and “, “or”, etc. and
phrases “ if then”,
“ only if “, “if and only if” ,” There exists”, “for all” etc.
Let p and q be two statements , truth table is given below
(A)
p
q
p and q (p  q)
T
T
T
T
F
F
F
T
F
F
F
F
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(B)
p
q
p or q (p  q)
T
T
T
T
F
T
F
T
T
F
F
F
(C)
p
~p
T
F
F
T
A sentence wit “If p ,then q” can be written in the following ways:
p implies q (denoted by p  q)
p is sufficient condition for q.
q is necessary condition for p.
p only if q
~q implies ~p.
The contrapositive of the statement ‘p  q’ is the statement ‘~q  ~p’.
The converse of the statement ‘p  q’ is the statement ‘q  p’.
The inverse of the statement ‘p  q’ is the statement ‘~p  ~q’.
‘For all’ or ‘ For every’ is called Universal Quantifier.
‘There exists’ is called Existential Quantifier.
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Negation of statements:
Statement
Negation
p
It is not the case that ….
It is false that………….
By inserting in p the world “not”.
p or q
p and q
~p and ~q
~p or ~ q
A statement is said to valid or invalid according as it is true or false.
True (valid)
Condition
p and q
Both p and q are true
p or q
P is false  q is true
Q is false  p is true.
p is true  q is true.(Direct Method)
If p then q.
q is false  p is false(Contrapositive
Method)
p is true and q is false leads us to
contradiction.(Contradiction Method)
p is true  q is true
And
q is true  p is true.
p if and only if q
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15.STATISTICS
Measures of Dispersion:
The dispersion or scatter of data is measured on the basis of the observations and the types of the
measures of central tendency. Range, Quartile Deviation, Mean Deviation, Standard Deviation are
the measures of dispersion.
Range
The range of the data gives us the idea about width of the data i.e in how much numerical value the
data is varied.
Range = Maximum Value – Minimum Value
Quartile Deviations ( QD ):
QD gives us the values which divide the observations into four equal parts; these are denoted by
Q1, Q2, Q3 and Q4. The formula to find QD is given by:
N
r    cf
4
Qr  l   
.h
f
Where Qr = rth Quartile deviation.
l = lower limit of model class interval {model class interval is the interval corresponding to
N
 r 
 4  }.
the least cf
N = Sum of frequencies.
cf = cf of class preceding the model class interval.
f = frequency of model class.
h = class size of model class.
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MEAN DEVIATION ABOUT MEAN ( MD ):
Here deviation means the differences of observations from mean/ arithmetic mean/ X . As we know
 x
that
i

x 0
. Therefore mean of deviations will always gives us the value zero, hence there is
no use to find it. To measure the dispersion from central values say ‘a’ (that may be mean, median
etc), we ignore the sign of deviation by taking its absolute value.
Hence mean deviation about mean is given by :
n
x
i 1
(i)
i
a
n
M.D. ( x ) =
MEAN DEVIATION ABOUT MEDIAN ( MD ):
If M is the median of the data x1 , x 2 , x3 , x 4 ,.................................., x n be the n observations with
frequencies f1 , f 2 , f 3 , f 4 ,......................., f n then the mean deviation about median is given by
n
f
(ii)
M.D. (M) =
i 1
i
n
f
i 1
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xi  a
i
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VARIANCE [VAR(X)]:
As we have seen that in MD we have used the absolute values of the deviations from central values
as it may becomes zero. Therefore to overcome from this drawback we make the squares of the
deviations then we use it to find the dispersion. In this way we not ignoring the signs of deviations
and hence it will give us perfect measure of dispersion. This measure of dispersion is called
Variance and written as Var (X), and is given by
 x
n
(i) Var ( X ) 
i 1
i
X

2
n
Where x1 , x 2 , x3 , x 4 ,.................................., x n are the n observations x is the mean of the data
If discrete frequency distribution is given that time
 f x
n
(ii) Var ( X ) 
i 1
i
i
X

2
n
f
i 1
i
Where here x1 , x 2 , x3 , x 4 ,.................................., x n are the n observations with frequencies
f1 , f 2 , f 3 , f 4 ,......................., f n
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Standard Deviation [SD]:
The positive square root of variance is known as SD and is written as
 x
n

(i) x =
i 1
X
i
 x . Hence we have:

2
 Var ( X )
n
Here x = mean of the data
If discrete frequency distribution is given then
 f x
n
(ii)  x 
i
i 1
i
X
i 1
2
 Var ( X ) ,
n
f

i
(iii)There is an formula to find SD where don’t use mean but the direct observations. The formula
is given by:
1
x 
N
n
N
i 1
 n

f i xi    f i xi 
 i 1

2
2
iv) We can also find S.D by short cut method in this case
2
 n

x A
N  f i yi    f i yi 
yi  i
i 1
 i 1
 ; where
h and A is the assumed mean & h is the HCF
of the deviations of observations from A.
h
x 
N
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n
2
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ANALYSIS OF FREQUENCY DISTRIBUTIONS:
To analyze two or more group of data we compare coefficients of variation denoted by C.V. The
C.V. of series X is written as C.V.(X) and is given by;
C.V ( X ) 
x
X
.100.where X  0 .
In Case C.V.(X) > C.V.(Y), then we say that the series X is more variable than the other. On the
other hand we say that the series Y is more stable/ consistent than the other.
If both the series have same means then the series having greater SD is said to be more variable and
other is said to be more consistent.
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16.PROBABILITY
RANDOM EXPERIMENTWhen we perform an activity about an uncertain things, whose results are somehow known to us, is
called an ‘Experiment’.
OUTCOMESThe possible happenings of an experiment are known as outcomes.
SAMPLE SPACEThe collection of all the possible outcomes of an experiment is known as sample space written by
letter ‘S’.
EVENTAny subset of the sample space is known as ‘Event’, denoted by capital letters e.g. A, B, C, X, Y &
Z etc.
FAVORABLE OUTCOMES OF AN EVENTThe elements belongs to the event A, are known as favorable outcomes. The number of favorable
outcomes is given by n(A).
OCCURRENCE OF AN EVENTIf the actual outcome of the experiment is one of the elements of the event A, then we say that event
A has occurred.
Otherwise we say the event A has not occurred.
TYPES OF EVENTSImpossible Event.: An event associated with a random experiment is called an impossible ivwent if
it never occurs whenever the experiment is performed.
e.g. In the experiment of rolling a die coming up of 7 is an impossible event.
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Sure Event: An event associated with a random experiment is called an sure event if it always
occurs whenever the experiment is performed.
e.g Turning up of less than 6 in the throw of a dice.
Simple Event:
If an event has only one sample point of the Sample Space ,it is called a simple event.
Compound Event
If an event has more than one sample point , it is called a compound event.
Mutually Exclusive Events:
Two events are said to be Mutually Exclusive Events if the occurrence of any one of them
exclude the occurance of the other event.In this case the sets describing the event are disjoint.
Exhaustive Event
If E1 , E 2 , E3 , E 4 ,........................, E n are n events of the sample space S and
E1  E2  E3  E4 ........................  En  S than the events are said to be exhaustive events.
Mutually Exclusive and Exhaustive Events:
Events E1 , E 2 , E3 , E 4 ,........................, E n are mutually exclusive and exhaustive if
E1  E2  E3  E4 ........................  En  S and E i  E j  Ø
Algebra of Events-
Complementary Event:
For every event A there corresponds another event A’ called the complementary event of A or not
A which is obtained by subtracting A from the Sample space of the experiment i.e
A'  S  A
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The Events ‘A or B’:
The event A or B is denoted by the union of two sets A  B .
The Event ‘A and B’:
The event A and B is denoted by the intersection of two sets A  B .
The Event ‘ A but not B’:
The event A but not B is denoted by the intersection of two sets A  B  A  B' .
Axiomatic Approach to ProbabilityLet S be the sample space of a random experiment. The probability of an event E written as P(E), is
a real valued function whose domain is the power set of S and range is the interval [0, 1] satisfying
the following axioms
For any event E, P(E)  0.
P(S) = 1.
If E and F are mutually exclusive events, then P( E  F )  P( E )  P( F ).
From the above definition of probability we follow that, if S =
(i)
(ii)
0  P(i )  1
for each
1 , 2 ,...........n , then we
have-
i  S .
P(1 )  P(2 )  ..........  P(n )  1
(iii)
For any event
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Probability of an Event -
The probability of an event E is given by division of favorable outcomes by total possible
outcomes. Therefore P( E ) 
n( E )
.
n( S )
Equally Likes Outcomes –
 ,  ,...........n , then wi’s are said to be equally likely outcomes if and only if
If S = 1 2
P ( i ) 
1
n
, for all i’s.
Probability of the event ‘not A’ –
The probability of the event ‘not A’ is given by P (not A) = 1 – P (A).
Probability of the event ‘A or B’ –
The probability of the event ‘A or B’ i.e. P ( A  B ) is given by
P( A  B)  P( A)  P( B)  P( A  B)
And if A and B are mutually exclusive event then P( A  B)  P( A)  P( B) .
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