www.kalvisolai.

com 12/9/2015

CENTURY FOUNDATION MATRIC

HIGHER SECONDARY SCHOOL,
TIRUPUR.

STD X
MATHEMATICS
UNIT WISE
FORMULAE AND DEFINITIONS

By,
D.Ponnaiyan. M.Sc., B.Ed.,
P.G.T in Mathematics
Phone no: 9944200642
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1. SETS AND FUNCTIONS

Commutative Property:
AB=BA

(Set union)

AB=BA

(Set intersection)

Associative Property:
A (B C) = (A B) C

(Set union)

A (B C) = (A B) C

(Set intersection)

Distributive Property:
A (B C) = (A B) (A C)

(Intersection distributes over union)

A (B C) = (A B) (A C)

(Union distributes over intersection)

De Morgans laws for set difference:

A \ (B C) = (A \ B) (A \ C)
A \ (B C) = (A \ B) (A \ C)

De Morgans laws for complementation:

(A B) = A B
(A B) = A B

Cardinality of sets:
n (A B) = n (A) + n (B) n (A B)
n (A B C) = n (A) + n (B) + n (C) n (A B) n (B C) n (A C) + n (A B C)

Function:
Domain: The set A is called the domain of the function.
Co-Domain: The set B is called the co-domain of the function.
One-One function: (Injective function)
The function is called anone-one function if it takes different elements of
A into different elements of B. In other words f is one-one if no element in B is
associated with more than one element in A.

Onto function: (Surjective function)

The function is called an onto function if every element in B has a
pre-image in A. This is same as saying that B is the range of f.

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One-One and onto function: (Bijective function) (One-One correspondence)

The function is called a one-one and onto function if f maps distinct
elements of A into distinct images in B and every element in B is an image of some
element in A.

Constant function:
The function is said to be a constant function if every element of A has the
same image in B. The range of a constant function is a singleton set.

Identity function:
An identity function maps each element of A into itself.

Note:
A set is a collection of well defined objects. The objects in a set are called elements or members
of that set.
Every function can be represented by a graph.
If every vertical line intersects a graph in at most one point, then the graph represents a function.

*****
2. SEQUENCES AND SERIES OF REAL NUMBERS
Arithmetic Progression: (A.P.)
A.P. is, a, a + d, a + 2d, a + 3d, . . .
General term, tn= a + (n-1) d
l a
Number of terms, n =
+1
d

Three consecutive terms, a d, a, a + d.

Four consecutive terms, a 3d, a - d, a + d, a + 3d.

Arithmetic Series:
When is given, Sn =

n
a l
2

When is not given, Sn =

n
2a n 1d
2
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Geometric Progression: (G.P.)

2

G.P. is a, ar, ar , ar , . . .
General term, tn= a r

n-1

Three consecutive terms,

Four consecutive terms,

, a , ar.

a
,
r3

, ar, ar3.

Amount (or Appreciation), A = P 1

100
r

Depreciation, A = P 1

100

Geometric Series:
If r = 1, Sn = na.

If r 1, Sn =

a r n 1
a 1 rn
(or) Sn =
r 1
1 r

If n = , Sn =

a
1 r

Special Series:
The sum of the first n natural numbers,

( + 1)

k 1

The sum of the first n odd natural numbers,

2k 1 = n

k 1

l 1
The sum of the first n odd natural numbers (when l is given), 1 + 3 + 5 + . . . + l =

The sum of squares of first n natural numbers,

k =
k 1

+ 1 (2 + 1)
6

nn 1
The sum of cubes of the first n natural numbers, k
2
k 1
n

Note:
A sequence of real numbers is an arrangement or a list of real numbers in a specific order.
In a sequence, if the terms are same then the sequence is called constant sequence.
For example, 1, 1, 1, 1, 1, 1.
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A functionis not necessarily a sequence.

The sequence given by F1 = F2 = 1 and Fn = Fn-1 + Fn-2, n = 3, 4, is called the Fibonacci sequence.
Its terms are listed as 1, 1, 2, 3, 5, 8, 13, 21, 34, . . .
An expression of addition of terms of a sequence is called a series.
The sum of infinite number of positive numbers may give a finite value.
The French mathematician Carl Fredrick Gauss (C.F.Gauss) is known as
Prince of Mathematics.

*****
3. ALGEBRA
The Basic Relationship Between
Zeros and Coefficients of a
Quadratic Polynomial
Sum of zeros, + =
Product of zeros,

Relationship Between Roots and

Coefficients of a Quadratic Equation
Sum of roots, + =

Product of roots, =

A quadratic polynomial is,

A quadratic equation is,

p () = ( + ) + .

2 ( + ) + = 0.

Relation Between LCM and GCD:

f (x) g (x) = LCM [f (x), g (x)] GCD [f (x), g (x)]

Rational Expressions:
a2 b 2

(a + b) (a - b)

a3 + b3

(a + b) (a2 ab + b2)

a3 - b3

(a - b) (a2 + ab + b2)

a4 + a2 b2 + b4

(a2 + ab + b2) (a2 ab + b2)

(a3 + b3) (a3 - b3)

a6 b 6

Quadratic Formula:
=

b b 2 4ac
2a

Time =

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Nature of Roots:
= b2 4ac
If > 0 (Positive answer), then the roots are Real and unequal.
If = 0, then the roots are Real and equal.
If < 0 (Negative Answer), then No real roots.(Imaginary roots)

Relationship Between Roots and Coefficients of a Quadratic Equation:

= 2 4

2 + 2

= ( + )2 - 2

2 - 2

= ( + ) 2 4

3 + 3

= ( + )3 - 3 ( + )

3 - 3

= ( - )3 + 3 ( - )

4 + 4

= [( + )2 - 2]2 2()2

4 - 4

= ( + ) ( - ) (2 + 2)

Note:
A system of linear equations a1x +b1y = c1, a2x + b2y = c2 in two variables is said to be,
Consistent if at least one pair of values of x and y satisfies both equations.
Inconsistent if there are no values of x and y that satisfy both equations.
The system of equations a1x +b1y - c1 = 0, a2x + b2y - c2 = 0,
If a1b2 b1a2 0 or

a1 b1
, then the system of equations has a unique solution.
a 2 b2

If

a1
b
c
= 1 = 1 , then the system of equations has infinitely many solutions.
a2
b2 c 2

If

a1 b1 c1
=
, then the system of equations has no solution.
a 2 b2 c 2

A set of finite number of linear equations in two variables x and y is called a system of linear
equations in x and y. Such a system is also called simultaneous equations.
Eliminating one of the variables first and then solving a system is called method of elimination.

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The following arrow diagram helps us very much to apply the method of cross multiplication in
solving a1x +b1y + c1 = 0, a2x + b2y + c2 = 0.

b1

c1

a1

b1

b2

c2

a2

b2

The method of synthetic division is introduced by Paolo Ruffin in 1809.

For any polynomial p( ), = a is zero if and only if p(a) = 0.
( - a) is a factor for p( ) if and only if p(a) = 0.
( - 1) is a factor of p( ) if and only if the sum of coefficients of p( ) is 0.
( + 1) is a factor of p( ) if and only if sum of the coefficients of even powers of , including
constant is equal to sum of the coefficients of odd powers of .
There are infinitely many quadratic equations with the same roots.
A quadratic equation can be solved by,
The method of factorization
The method of completing square
Using a quadratic formula.

*****
4. MATRICES
Order or Dimension of a Matrices: If a matrix A has m rows and n columns, then we
say that the order of A is m n.

Types of Matrices:
Row Matrix: A matrix is said to be a row matrix if it has only one row. A row matrix is also
called as row vector. For example, A = 5 0 3

Column Matrix: A matrix is said to be a column matrix if it has only one column. It is also
5
called as column vector. For example, B = 0

3

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Square Matrix: A matrix in which the number of rows and the number of columns are equal is
1 2 2
said to be a square matrix. For example, C = 2 3 3

3 4 0

Diagonal Matrix: A square matrix in which all the elements above and below the leading
3 0 0
diagonal are equal to zero, is called a diagonal matrix. For example, D = 0 0 0

0 0 1

Scalar Matrix: A diagonal matrix in which all the elements along the leading diagonal are
7 0 0
equal to a non-zero constant is called a scalar matrix. For example, E = 0 7 0

0 0 7

Unit Matrix: A diagonal matrix in which all the leading diagonal entries are 1 is called a
1 0
unit matrix. A unit matrix of order n is denoted by In. For example, I2 =

0 1

Null Matrix or Zero-matrix: A matrix is said to be a null matrix or zero-matrix if each

0 0
of its elements is zero. It is denoted by O. For example, O =

0 0

Transpose of a Matrix: The transpose of a matrix A is obtained by interchanging rows and

1 5
1 7
columns of the matrix A and it is denoted by AT. For example, if A =
, then AT =

7 4
5 4

Note:
Equality of Matrices: Two matrices A = [aij]m n and B = [bij]m n are said to be equal if ,
They are of the same order and
Each element of A is equal to the corresponding element of B.
Properties of Matrix Addition:
Matrix addition is commutative. That is A + B = B + A.
Matrix addition is associative. That is A + (B + C) = (A + B) + C.
Existence of additive identity. That is A + O = O + A = A, where O is the null matrix.
Existence of additive inverse. That is A + (-A) = (-A) + A = O, where A is
the additive inverse of A.
Properties of Matrix Multiplication:
Matrix multiplication is not commutative in general. That is AB BA.
Matrix multiplication is always associative. That is (AB) C = A (BC).
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Matrix multiplication is distributive over addition. That is,

(i) A (B + C) = AB + AC
(ii) (A + B) C = AC + BC.
Existence of multiplicative identity. That is AI = IA =A.
Existence of multiplicative inverse. That is AB = BA = I.
Reversal law for transpose of matrices. That is (AB)T = BT AT.
A matrix is a rectangular array of numbers in rows and columns enclosed within square brackets or
parenthesis.
A unit matrix is also called an identity matrix with respect to multiplication.
Every unit matrix is clearly scalar matrix. However a scalar matrix need not be a unit matrix.
A zero-matrix need not be a square matrix.
A matrix does not change if the zero-matrix of some order is added to it or subtracted from it.
The additive inverse of a matrix is its negative matrix and it is unique.
Multiplication of two diagonal matrices of same order is commutative.

2 3
Some of the square matrices like
do not have multiplicative inverses.
4 6
If B is the multiplicative inverse of A, Then A is the multiplicative inverse of B.
If multiplicative inverse of a square matrix exists, then it is unique.
Addition or subtraction of two matrices are possible only when they are of same order.
If A is a matrix of order m n and B is a matrix of order n p, then the product matrix AB is
defined and is of order m p.
(AT)T = A, (A + B)T = AT + BT.
If AB = O, it is not necessary that A = O or B = O. That is product of two non-zero matrices
may be a zero matrix.

*****
5. COORDINATE GEOMETRY
Section Formula:

Midpoint:

Centroid:

Internally:

x x 2 y1 y 2
,
M(x, y) = M 1

2
2
x x 2 x 3 y1 y 2 y 3
,
G(x, y) = G 1

3
3

lx mx1 ly 2 my1
,
P(x,y) = P 2

lm
lm
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lx mx1 ly 2 my1
Externally: P(x,y) = P 2
,

l m
l m

Distance: (length) d =

x2 x1 2 y 2 y1 2 units.

Property of a Parallelogram: The diagonals of a parallelogram bisect each other.

Area of a Triangle = 1 {(x1y2 + x2y3 + x3y1) (x2y1 + x3y2 + x1y3)} sq.units.

2

Area of the Quadrilateral =

1
{(x1y2 + x2y3 + x3y4 +x4y1) (x2y1 + x3y2 + x4y3+ x1y4)} sq.units.
2

Slope (or Gradient):

When angle of inclination is given, m = tan .
When two points are given, m =

y 2 y1
x 2 x1

When straight line(ax + by + c = 0) is given, m =

a
b

c

b
If two non-vertical straight lines are parallel, then their slopes are equal. That is m1 = m2.

When straight line(ax + by + c = 0) is given, y-intercept =

If two straight lines are perpendicular, then m1 m2= -1.

Property of Rhombus: The diagonals of a rhombus bisect each other at right angle.

Equation of a straight Line:

Slope-Point Form:
Two-Points Form:

y y1 = m (x x1)
x x1
y y1
=
x 2 x1
y 2 y1

Slope-Intercept Form: y = mx + c.
x y
Two Intercepts Form:
= 1.
a b

General Form of Equation of a Straight Line:

Equation of a line parallel to the line ax + by + c = 0 is ax + by + k =0.
Equation of a line perpendicular to the line ax + by + c = 0 is bx - ay + k =0.
The point of intersection is obtained by solving the given two equations

Note:
The ratios obtained by equating x-coordinates and by equating y-coordinates are same only
when the three points are collinear.
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l
is positive.
m
l
If a point divides the line segment externally in the ratio l : m , then is negative.
m
If is the angle of inclination of a straight line l ,then

If a point divides the line segment internally in the ratio l : m , then

0 180
For horizontal lines, = 0 or 180
For vertical lines, = 90
The slope of x-axis or straight lines parallel to x-axis is zero.
The slope of y-axis or straight lines parallel to y-axis is not defined.
If is acute, then the slope is positive.
If is obtuse, then the slope is negative.
The equation of x-axis is y = 0.
The equation of y-axis is x = 0.
The equation of a straight line parallel to x-axis is y = k, where k is a constant.
If k > 0, then the straight line lies above the x-axis.
If k < 0, then the straight line lies below the x-axis.
If k = 0, then the straight line is nothing but the x-axis.
The equation of a straight line parallel to y-axis is x = c, where c is a constant.
c > 0, then the straight line lies to the right y-axis.
If c < 0, then the straight line lies to the left of y-axis.
If c = 0, then the straight line is nothing but the y-axis.
The general form of equation of a straight line is px + qy + r = 0.
Two straight lines a1x +b1y + c1 = 0 and a2x + b2y + c2 = 0, where the coefficients are non-zero,
Are parallel if and only if

a1
b
= 1
a2
b2

Are perpendicular if and only if a1a2 + b1b2 = 0.

Three points A(x1, y1), B(x2, y2) and C(x3, y3) are collinear if and only if
The area of triangle is equal to zero.
Slope of AB = Slope of BC.

*****
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6. GEOMETRY

Basic Proportionality Theorem or Thales Theorem: If a straight line is drawn parallel

to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.

Converse of Thales Theorem: If a straight line divides any two sides of a triangle in the same
ratio, then the line must be parallel to the third side.

Angle Bisector Theorem: The internal (external) bisector of an angle of a triangle divides the
opposite side internally (externally) in the ratio of the corresponding sides containing the angle.

Converse of Angle Bisector Theorem: If a straight line through one vertex of a triangle
divides the opposite side internally (externally) in the ratio of the other two sides, then the line
bisects the angle internally (externally) at the vertex.

Criteria for Similarity of Triangles:

AA (Angle-Angle) Similarity.
SSS (Side-Side-Side) Similarity.
SAS (Side-Angle-Side) Similarity.

Similarity Results:
The ratio of the areas of two similar triangles is equal to the ratio of the squares of their
corresponding sides.
If a perpendicular is drawn from the vertex of a right angled triangle to its hypotenuse, then the
triangles on each side of the perpendicular are similar to the whole triangle.
If two triangles are similar, then the ratio of the corresponding sides is equal to the ratio of their
corresponding altitudes.
If two triangles are similar, then the ratio of the corresponding sides is equal to the ratio of the
corresponding perimeters.

Pythagoras Theorem (Baudhayan Theorem): In a right angled triangle, the square of

the hypotenuse is equal to the sum of the squares of the other two sides.

`BC2 = AB2 + AC2

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Converse of Pythagoras Theorem: In a triangle, if the square of one side is equal to the
sum of the squares of the other two sides, then the angle opposite to the first side is a right angle.

Tangent-Chord Theorem: If from the point of contact of tangent (of a circle), a chord is
drawn, then the angleswhich the chord makes with the tangent line are equal respectively to the
angles formed by the chord in the corresponding alternate segments.

Converse of Tangent-Chord Theorem: If in a circle, through one end of a chord,

a straight line is drawn making an angle equal to the angle in the alternate segment, then
the straight line is a tangent to the circle.
a c
ab cd
is called componendo rule.
,then

b d
b
d

Componendo Rule: If

Tangent and Point of Contact: A straight line which touches a circle one point is
called a tangent to the circle and the point at which it touches the circle is called its point of contact.

Circles and Tangents Results:

A tangent at any point on a circle is perpendicular to the radius through the point of contact.
Only one tangent can be drawn at any point on a circle. However, from an exterior point of
a circle two tangents can be drawn to the circle.
The lengths of the two tangents drawn from an exterior point to a circle are equal.
If two circles touch each other, then the point of contact of the circles lies on the line
joining the centres.
If two circles touch externally, the distance between their centres is equal to the sum of their radii.
If two circles touch internally, the distance between their centres is equal to the difference
of their radii.
If AB and CD are two chords of a circle which intersect each other internally at P or intersect
each other externally at P, then the proof is PA PB = PC PD.

*****
7. TRIGONOMETRY
Trigonometric Identities:
sin2 + cos2 = 1

sec2 - tan2 = 1

cosec2 - cot2 = 1

sin2 = 1 - cos2

sec2 = 1 + tan2

cosec2 = 1 + cot2

cos2 = 1 - sin2

tan2 = sec2 1

cot2 = cosec2 - 1

cosec . sin = 1

sec . cos = 1

cot . tan = 1

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sin =

1
cos ec

cosec =

1
sin

cos =

1
sec

sec =

1
cos

tan =

sin
cos

cot =

cos
sin

tan =

1
cot

cot =

1
tan

sin (90 - ) = cos

cos (90 - ) = sin

cosec (90 - ) = sec

sec (90 - ) = cosec

tan (90 - ) = cot

cot (90 - ) = tan

Heights and Distances:

Hypotenuse Side
Opposite Side

Adjacent Side

sin =
cos =

tan =

{(Old Henry) (And His) (Old Aunty)}

(Henry Old)

(Old Henry)

cosec =

(And His)

sec =

(His And)

(Old Aunty)

cot =

(Aunty Old)

Trigonometric Ratios of Some Special Angles:

Ratio

30

45

60

90

sin

1
2

3
2

cos

3
2

1
2

tan

1
3

2
2

not
defined()
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Componendo and Dividendo Rule:

If

a c

b d

, then

ab cd

a b cd

2 = 1.414
3 = 1.732

*****
8. MENSURATION
Cylinder:
If the base of a cylinder is not circular then it is called oblique cylinder.
If the base is circular but not perpendicular to the axis of the cylinder, then the cylinder is called
circular cylinder.
If the axis is perpendicular to the circular base, then the cylinder is called right circular
cylinder.

Cone:

If the base of a cone is not circular, then it is called oblique cone.

If the base of a cone is circular then, it is called circular cone.

If the vertex is directly above the centre of the circular base, then it is a right circular cone.

Right Angled Triangle

Cone

Base
2

Radius

Height (h)

Height (h)

Sector
Radius ()
Arc length (L)

Area (
r2)

360

Slant height, =

Cone

Slant height ()
Perimeter of the base (2r)

C.S.A ()

h2 r 2

Height, h =

l2 r2

Radius, r =

l 2 h2

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Frustum of a Cone:
Let us consider a right circular solid cone and cut it into two solids so as to obtain a smaller
right circular cone. The other portion of the cone is called frustum of the cone.
In other words, if a solid right circular cone is sliced with a plane parallel to its base, the part
of the cone containing the base is called a frustum of the cone. Thus a frustum has two
circular discs, one at the bottom and the other at the top of it.
The Latin word frustum means piece cut off and its plural is frusta.

Sphere:
If a circular disc is rotated about one of its diameter, the solid thus generated is called sphere.
Thus sphere is a 3 dimensional object which has surface area and volume.

Volume:
Volume is literally the amount of space filled .

S.No

Solid Name

L.S.A (or)
C.S.A
(sq.units)

T.S.A
(sq.units)

Volume
(cu.units)

2rh

2r (h+r)

r2h

2h (R + r)

2 (R + r) (R r + h)

h (R2 r2)

r ( + r)

-------

-- - - - - -

4 r2

-------

-------

-------

2 r2

3 r2

2 (R2 + r2)

(3R2 + r2)

1. Cylinder
2.
3.
4.
5.
6.
7.
8.

Hollow Cylinder
Cone
Frustum
Sphere
Hollow Sphere
Hemisphere
Hollow Hemisphere

1
3
1
3

h (R2 + r2 + Rr)
4
3
4
3

3
2
3

2rh

10. External surface area of the cylinder (Hollow)

2Rh

11. Inner surface area of the sphere (Hollow)

2 r2

12. Outer surface area of the sphere (Hollow)

2 R2

13. Volume of water discharged

14. Conversions

r3

(R3 - r3)
2

Internal surface area of the cylinder (Hollow)

9.

r2h

r3

(R3 - r3)

Cross section area ( r2) Speed Time

1 m3
= 1000 litres
3
1 d.m
= 1 litre
1000 cm3 = 1 litre
1000 litres = 1

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King Henrys Daughter Makes Delicious Choco Mittai

*****
9. PRACTICAL GEOMETRY
Two tangents can be drawn to a circle from an external point.
Diameters subtend 90 on the circumference of a circle.

*****
10. GRAPHS
Quadratic Graphs:
Let us consider the quadratic polynomial, y ax 2 bx c ,
If a > 0, then the curve is open upward.
If a < 0, then the curve is open downward.

Special Graphs:

Direct variation, y k x
Indirect variation, y

k
x

*****
11. STATISTICS
Measures of dispersion:
Measures of dispersion give an idea about the scatteredness of the data of the distribution.
Range (R), Quartile Deviation (Q. D), Mean Deviation (M. D) and Standard Deviation (S. D) are the
measures of dispersion.

Range:
Range is the simplest measure of dispersion. Range of a set of numbers is the difference
between the largest and the smallest items of the set.
Range = Largest value Smallest value
Range = L S.

The coefficient of range = +

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Standard Deviation ():

Data

Direct Method

Ungrouped

x
n

Grouped

-------

Actual Mean
Method

n
d= xx

fd
f

Assumed Mean
Method

n
n

d= x A

fd
f

Coefficient of variation, C.V =

d
c

n
n

x A
d=
c

fd

Standard Deviation of the first n natural numbers is, =

Step Deviation
Method

fd
f

fd
c

n2 1
12

100

Note:
A better way to measure dispersion is to square the differences between each data and the mean
before averaging them. This measure of dispersion is known as the Variance and the positive
square root of the Variance is known as the Standard Deviation.
The term Standard Deviation was first used by Karl Pearson in 1894 as a replacement
of the term mean error used by Gauss.
xx 0

x nx

x nx

Assumed mean method and step deviation method are just simplified forms of direct method.
S.D of a collection of data remains unchanged when each value is added or subtracted by a
constant.
S.D of a collection of data gets multiplied or divided by the quantity k , if each item is
multiplied or divided by k .
The S.D of any n successive terms of an A.P. with common difference d is, = d

The S.D of any n consecutive even(or odd) integers is, = 2

n2 1
12

n2 1
12

Variance is the square of Standard deviation.

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The coefficient of variation helps us to compare the consistency of two or more

collections of data.
When the C.V is more, the given data is less consistent.
When the C.V is less, the given data is more consistent.
C.V is also called as a Relative Standard Deviation.

*****
12. PROBABILITY

Experiment:
Any process of observation or measurement is called an experiment. For example,
tossing a coin.

Random experiment:
A random experiment is one in which the exact outcome cannot be predicted before
conducting the experiment. However, one can list out all possible outcomes of the experiment.

Sample space:

The set of all possible outcomes of a random experiment is called its sample space
and it is denoted by the letter S.
Trail: Each repetition of the experiment is called a trail.
Event: A subset of the sample space S is called an event.
Equally likely events:
Two or more events are said to be equally likely, if each one of them has an equal
chance of occurrence.
Mutually exclusive events:
Two or more events are said to be mutually exclusive events, if the occurrence
of one event prevents the occurrence of other events. In other words, if A and B are mutually
exclusive events, then A B = .
Complementary events:
Let E be an event of a random experiment and S be its sample space. The set
containing all the other outcomes which are not in E but in the sample space is called the
complimentary event of E. It is denoted by E . Thus, E = S - E. Note that E and E are mutually
exclusive events.
Exhaustive events: Events E1, E2,, En are exhaustive events if their union is the sample space S.
Sure event:
The sample space of a random experiment is called sure or certain event as any one
of its elements will surely occur in any trail of the experiment. The probability of the sure event is 1.
That is P(S) = 1.
Impossible event:
An event which will not occur on any account is called an impossible event.
It is denoted by . The probability of an impossible event is 0. That is P() = 0.

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Favourable outcomes:
The outcomes corresponding to the occurrence of the desired event are called
favourable outcomes of the event. For example, if E is an event of getting an odd number in
rolling a die, then the outcomes 1, 3, 5 are favourable to the event E.

Probability:
n( A)
. This is not applicable if the number of possible outcomes is infinite and the
n( S )
outcomes are not equally likely.
The probability of an event A lies between 0 and 1, both exclusive. That is 0 P(A) 1.
P( A) 1 P( A)

P( A)

Addition theorem on probability:

P (A B) = P (A) + P (B) P (A B).
The event A B is said to occur if the event A occurs or the event B occurs or both A and B
occur simultaneously.
The event A B is said to occur if both the events A and B occur simultaneously.
If A and B are mutually exclusive events, then A B = . Thus,
P(A B) = P(A) + P(B).
P(A B) = 0.
A B is same as A \ B in the language of set theory.

Important sample spaces:

Tossing one coin, S = {H, T}
n (S) = 2
Tossing two coins, S = {HH, HT, TH, TT}
n (S) = 4
Tossing three coins, S = {HHH, HHT, HTH, THH, HTT, TTH, THT, TTT}
n (S) = 8
Rolling one die, S = {1, 2, 3, 4, 5, 6}
n (S) = 6
Rolling two dice, S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
n (S) = 36
Rolling three dice, n (S) = 216. (Sample space is not necessary).

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A leap year,
S = {(Sun, Mon), (Mon, Tue), (Tue, Wed), (Wed, Thu), (Thu, Fri), (Fri, Sat), (Sat, Sun)}
n (S) = 7
A non leap year,
S = {Sun, Mon, Tue, Wed, Thu, Fri, Sat}
n (S) = 7
Playing cards, S = {52 cards}
n (S) = 52

Spade
(Black)

Hearts
(Red)

Clavor
(Black)

Diamond
(Red)

A
2
3
4
5
6
7
8
9
10
J
Q
K

A
2
3
4
5
6
7
8
9
10
J
Q
K

A
2
3
4
5
6
7
8
9
10
J
Q
K

A
2
3
4
5
6
7
8
9
10
J
Q
K

Red colour cards

= 26

Black colour cards = 26

Spade cards

= 13

Hearts cards

= 13

Clavor cards

= 13

Diamond cards

= 13

Number cards

= 04

(each number)

*****
~~~ Yes. Now we are 100% knowledgeable
students in Mathematics ~~~
Mathematics is the alphabet with which
God has written the Universe

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