Class 11 Applied mathematics Formula sheet_c88b01d2-a305-4e82-8384-9a47d968e190.pdf
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Dec 23, 2023
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About This Presentation
Class 11 applied math
Slide Content
Applied mathematics
class 11
Formula sheet
class 11
Formula sheet
NUMBERS
BINARY NUMBER
Computer receives, stores and processes the information (or data) using two digits ‘0’ and ‘1’
called binary digits or bits. Any number using two digits 0 and 1 is called binary number.
In computers and other electronic device we use binary number system. It consists of two digits
0 and 1. So the base of this number system is 2.
DECIMAL NUMBER SYSTEM
In this system we have ten digits i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Hence the base of this system is 10.
Let us consider the number 1532.
As the digit 1 occupies thousands place, the place value of the digit 1 = 1 x 10
3
= 1000
Similarly, the place value of digit 5 = 5 x 10
2
= 5 x 100 = 500,
The place value of digit 3 = 3 x 10
1
= 30
The place value of digit 2 = 2 x 10
0
= 2
So, 1532 = 1 x 10
3
+ 5 x 10
2
+ 3 x 10
1
+ 2 x 10
0
= 1532
In decimal system, we represent the number 1532 as (1532)
10
Computer receives, stores and processes the information (or data) using two digits ‘0’ and ‘1’
called binary digits or bits. Any number using two digits 0 and 1 is called binary number.
In computers and other electronic device we use binary number system. It consists of two digits
0 and 1. So the base of this number system is 2.
DECIMAL NUMBER SYSTEM
In this system we have ten digits i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Hence the base of this system is 10.
Let us consider the number 1532.
As the digit 1 occupies thousands place, the place value of the digit 1 = 1 x 10
3
= 1000
Similarly, the place value of digit 5 = 5 x 10
2
= 5 x 100 = 500,
The place value of digit 3 = 3 x 10
1
= 30
The place value of digit 2 = 2 x 10
0
= 2
So, 1532 = 1 x 10
3
+ 5 x 10
2
+ 3 x 10
1
+ 2 x 10
0
= 1532
In decimal system, we represent the number 1532 as (1532)
10
BINARY NUMBER SYSTEM
In this system we have two digits i.e. 0 and 1. Hence, the base of this system is 2.
Let us consider a binary number 11100 which can be written in decimal system as 1 x 2
4
+ 1 x 2
3
+1 x 2
2
+1 x 2
1
+1 x 2
0
= 16 + 8 + 4 + 0 + 0 = 28
The binary number 11101 in binary system is represented as (11100)
2
So (11100)
2 = (28)
10
CONVERSION OF DECIMAL NUMBER TO BINARY NUMBER
To convert a given decimal number to a binary number proceed as under .
Step 1. Divide the given decimal number by 2 and write down the reminder.
Step 2. Again divide the quotient obtained in step 1 by 2 and again write down the remainder.
Step 3. Repeat the step 2 again and again till you get the quotient as 1.
Step 4. Write the last quotient (i.e.1) and remainder in the reverse order (i.e. from bottom to the
top).
In this system we have two digits i.e. 0 and 1. Hence, the base of this system is 2.
Let us consider a binary number 11100 which can be written in decimal system as 1 x 2
4
+ 1 x 2
3
+1 x 2
2
+1 x 2
1
+1 x 2
0
= 16 + 8 + 4 + 0 + 0 = 28
The binary number 11101 in binary system is represented as (11100)
2
So (11100)
2 = (28)
10
CONVERSION OF DECIMAL NUMBER TO BINARY NUMBER
To convert a given decimal number to a binary number proceed as under .
Step 1. Divide the given decimal number by 2 and write down the reminder.
Step 2. Again divide the quotient obtained in step 1 by 2 and again write down the remainder.
Step 3. Repeat the step 2 again and again till you get the quotient as 1.
Step 4. Write the last quotient (i.e.1) and remainder in the reverse order (i.e. from bottom to the
top).
BINARY ADDITION
Binary addition is done in the same way as in decimal system.
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 and 1 carry over to next left columm.
BINARY SUBTRACTION
Binary subtraction is done in the same way as in decimal system.
0 –0 = 0
1 –0 = 1
1 –1 = 0
0 –1 = 1 with a borrow of 1 from the next left column.
Binary addition is done in the same way as in decimal system.
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 and 1 carry over to next left columm.
BINARY SUBTRACTION
Binary subtraction is done in the same way as in decimal system.
0 –0 = 0
1 –0 = 1
1 –1 = 0
0 –1 = 1 with a borrow of 1 from the next left column.
BINARY MULTIPLICATION
Binary multiplication is done in the same way as in decimal system.
0 x 0 = 0
1 x 0 = 1
1 x 0 = 0
1 x 1 = 1
BINARY DIVISION
Binary division is done in the same way as in decimal system. Like decimal system division by 0 is meaningless in
binary system also. So, there are only two possible rules of binary division. These two rules are given below:
0 ÷1 = 0
1 ÷1 = 1
Binary multiplication is done in the same way as in decimal system.
0 x 0 = 0
1 x 0 = 1
1 x 0 = 0
1 x 1 = 1
BINARY DIVISION
Binary division is done in the same way as in decimal system. Like decimal system division by 0 is meaningless in
binary system also. So, there are only two possible rules of binary division. These two rules are given below:
0 ÷1 = 0
1 ÷1 = 1
INDICES AND
LAGORITHMS
LAGORITHMS
LAWS OF EXPONENTES FOR REAL NUMBERS
Laws of exponents for real numbers are :
If a, b are positive real numbers and m, n are rational numbers, then the following results hold:
(i)a
m
. a
n
= a
m + n
(ii) (a
m
)
n
= a
mn
(iii)
�
�
�
�
=a
m-n
(iv) a
m
. b
m
= (ab)
m
(v)
�
�
�
=
�
�
�
�
(vi) a
-n
=
1
�
�
(vii) a
n
= b
n,
n≠ 0 ⟹a = b (viii) a
m
= a
n
m= n provided a≠1. (ix) a
0
= 1, a ≠ 0
(x)
�
�=�
Τ
1
�
Laws of exponents for real numbers are :
If a, b are positive real numbers and m, n are rational numbers, then the following results hold:
(i)a
m
. a
n
= a
m + n
(ii) (a
m
)
n
= a
mn
(iii)
�
�
�
�
=a
m-n
(iv) a
m
. b
m
= (ab)
m
(v)
�
�
�
=
�
�
�
�
(vi) a
-n
=
1
�
�
(vii) a
n
= b
n,
n≠ 0 ⟹a = b (viii) a
m
= a
n
m= n provided a≠1. (ix) a
0
= 1, a ≠ 0
(x)
�
�=�
Τ
1
�
logarithms
Definition. If a is any positive real number (except 1), n is any rational number and a
n
= b, then n is called algorithm
of b to the base a. It is written as log
ab(read as log of b to the base of a). Thus,
a
n
= b if and only if log
ab= n
a
n
= b is called the exponential form and log
ab= n is called logarithmic form
For example:
•5
4
= 625 ∴ log
5625 = 4
•7
0
= 1, ∴ log
71 = 0
•(10)
-2
= ∴ log
10(0.01) = -2
Important Points :
•(i) log
a1 = 0 (ii) log
aa= 1
where a is any positive real number (except 1).
•Logarithms to the base 10 are called common logarithms.
•log
ax= log
ay= x=y
•If no base is given, the base is always taken as 10.
For example, log 2 = log
102.
Definition. If a is any positive real number (except 1), n is any rational number and a
n
= b, then n is called algorithm
of b to the base a. It is written as log
ab(read as log of b to the base of a). Thus,
a
n
= b if and only if log
ab= n
a
n
= b is called the exponential form and log
ab= n is called logarithmic form
For example:
•5
4
= 625 ∴ log
5625 = 4
•7
0
= 1, ∴ log
71 = 0
•(10)
-2
= ∴ log
10(0.01) = -2
Important Points :
•(i) log
a1 = 0 (ii) log
aa= 1
where a is any positive real number (except 1).
•Logarithms to the base 10 are called common logarithms.
•log
ax= log
ay= x=y
•If no base is given, the base is always taken as 10.
For example, log 2 = log
102.
standard law of Logarithms
1.log
bmn= log
bm + log
bn
2.log
b(
�
�
)= log
bm –log
bn
3.log
bm
p
= plog
bm
4.log
a1 = 0.
5.log
bb = 1
6.log
bb
x
= x
7.�
���
??????�
=�
8.log
bm= log
am x log
ba
9.log
ba log
ab= 1
10.log
bm=
1
�����
11.Log
bm= -log
b
1
�
1.log
bmn= log
bm + log
bn
2.log
b(
�
�
)= log
bm –log
bn
3.log
bm
p
= plog
bm
4.log
a1 = 0.
5.log
bb = 1
6.log
bb
x
= x
7.�
���
??????�
=�
8.log
bm= log
am x log
ba
9.log
ba log
ab= 1
10.log
bm=
1
�����
11.Log
bm= -log
b
1
�
Number Standard form Characteristic
37 3.7 x 10
1
1
4623 4.623 x 10
3
3
6.21 6.21 x 10
0
0
0.8 8 x 10
-1
-1 = ത1
0.07 7 x 10
-2
-2 = ത2
Number Mantissa Logarithm
Log 4594 = (……… 6623) 3.6623
Log 459.4 = (……… 6623) 2.6623
Log 45.94 = (……… 6623) 1.6623
Log 4.594 = (……… 6623) 0.6623
Log .4594 = (……… 6623) ത1.6623
Logarithm Tables:
The logarithm of a number consists of two parts, the whole part or the integral part is called the characteristic and the decimal
part is called the mantissa where the former can be known by mere inspection, the latter has to be obtained from the logarithm
tables.
Characteristic:
The characteristic of the logarithm of any number greater than 1 is positive and is one less than the number of digits to theleft
of the decimal point in the given number. The characteristic of the logarithm of any number less than one (1) is negative and
numerically one more than the number of zeros to the right of the decimal point. If there is no zero then obviously it will be –1.
The following table will illustrate it.
Mantissa
The mantissa is the fractional part of the logarithm of a given number.
Number Mantissa Logarithm
Log 4594 = (……… 6623) = 3.6623
Log 459.4 = (……… 6623) = 2.6623
Log 45.94 = (……… 6623) = 1.6623
Log 4.594 = (……… 6623) = 0.6623
Log .4594 = (……… 6623) = 1 .6623
37 3.7 x 10
1
1
4623 4.623 x 10
3
3
6.21 6.21 x 10
0
0
0.8 8 x 10
-1
-1 = ത1
0.07 7 x 10
-2
-2 = ത2
Number Mantissa Logarithm
Log 4594 = (……… 6623) 3.6623
Log 459.4 = (……… 6623) 2.6623
Log 45.94 = (……… 6623) 1.6623
Log 4.594 = (……… 6623) 0.6623
Log .4594 = (……… 6623) ത1.6623
Logarithm Tables:
The logarithm of a number consists of two parts, the whole part or the integral part is called the characteristic and the decimal
part is called the mantissa where the former can be known by mere inspection, the latter has to be obtained from the logarithm
tables.
Characteristic:
The characteristic of the logarithm of any number greater than 1 is positive and is one less than the number of digits to theleft
of the decimal point in the given number. The characteristic of the logarithm of any number less than one (1) is negative and
numerically one more than the number of zeros to the right of the decimal point. If there is no zero then obviously it will be –1.
The following table will illustrate it.
Mantissa
The mantissa is the fractional part of the logarithm of a given number.
Number Mantissa Logarithm
Log 4594 = (……… 6623) = 3.6623
Log 459.4 = (……… 6623) = 2.6623
Log 45.94 = (……… 6623) = 1.6623
Log 4.594 = (……… 6623) = 0.6623
Log .4594 = (……… 6623) = 1 .6623
Antilogarithms
If x is the logarithm of a given number n with a given base then n is called the antilogarithm (antilog) of x to that base.
This can be expressed as follows:
If log
a
n = x then n = antilog x
For example, if log 61720 = 4.7904 then 61720 = antilog 4.7904
If x is the logarithm of a given number n with a given base then n is called the antilogarithm (antilog) of x to that base.
This can be expressed as follows:
If log
a
n = x then n = antilog x
For example, if log 61720 = 4.7904 then 61720 = antilog 4.7904
Quantitative Aptitude
Average ::
Average is also known as mean
Average =
����������������??????��
������������������������??????��
ҧ�=
�1+�2+�3……��
�
Important points :
1.If x is added to each observation then average increases by x
2.If x is subtracted to each observation then average decreased by x
Weighted Average ::
ҧ�
�=
�1�1+�2�2+�3�3……����
�1+�2+�3……..��
Important points :
1.If all the weight are equal then weighted average is same as the simple average
2.Weighted average is more accurate than the simple average.
Average speed and average velocity::
Average speed =
������??????���������������
������??????�������
Average velocity =
������??????����������
������??????�������
Average is also known as mean
Average =
����������������??????��
������������������������??????��
ҧ�=
�1+�2+�3……��
�
Important points :
1.If x is added to each observation then average increases by x
2.If x is subtracted to each observation then average decreased by x
Weighted Average ::
ҧ�
�=
�1�1+�2�2+�3�3……����
�1+�2+�3……..��
Important points :
1.If all the weight are equal then weighted average is same as the simple average
2.Weighted average is more accurate than the simple average.
Average speed and average velocity::
Average speed =
������??????���������������
������??????�������
Average velocity =
������??????����������
������??????�������
CALENDAR
A chart showing days, weeks and months of a particular year is called calendar. It has two types of calendar years.
1.Leap year: It has 366 days.
2.Non-leap or common or ordinary year. It has 365 days.
3.A year is divided into twelve months.
4.A month is divided into weeks. A week has 7 days.
5.The international is to start the week on Monday.
6.Numbers of week in a year is 52
To determine whether a year is a leap year:
1.Any year which is not divisible by 4 is not a leap year eg.2001, 2002, 2003, 2005 etc.
2.Any year which is divisible by 4 but not divisible by 100 is a leap year eg.1904, 1908, 1912 etc.
3.Any year which is divisible by 400 is a leap year eg.1200, 1600, 2000 etc. These century years are called century leap years.
4.Any year which is divisible by 100 but not divisible by 400 is not a leap year eg.1700, 1800,1900 etc.
Odd Days
1. in a Non leap year there are 1 odd days
2. in a leap year there are 2 odd days.
3. In a century ie. 100 years, there are 5 odd days.
4. In 200 years, there are 3 odd days.
5. In 300 years, there are 1 odd day.
6. In 400 years, 0 odd days.
A chart showing days, weeks and months of a particular year is called calendar. It has two types of calendar years.
1.Leap year: It has 366 days.
2.Non-leap or common or ordinary year. It has 365 days.
3.A year is divided into twelve months.
4.A month is divided into weeks. A week has 7 days.
5.The international is to start the week on Monday.
6.Numbers of week in a year is 52
To determine whether a year is a leap year:
1.Any year which is not divisible by 4 is not a leap year eg.2001, 2002, 2003, 2005 etc.
2.Any year which is divisible by 4 but not divisible by 100 is a leap year eg.1904, 1908, 1912 etc.
3.Any year which is divisible by 400 is a leap year eg.1200, 1600, 2000 etc. These century years are called century leap years.
4.Any year which is divisible by 100 but not divisible by 400 is not a leap year eg.1700, 1800,1900 etc.
Odd Days
1. in a Non leap year there are 1 odd days
2. in a leap year there are 2 odd days.
3. In a century ie. 100 years, there are 5 odd days.
4. In 200 years, there are 3 odd days.
5. In 300 years, there are 1 odd day.
6. In 400 years, 0 odd days.
CLOCK
1. Clock has two hands. The shorter hand is the hour hand and the longer hand is the minute hand.
2. The face of the clock called dial is a complete circle having 360°. The dial is divided into 12 equal parts. These parts are
marked as 1, 2, 3,..., 11, 12.
3. The angle between two consecutive numbers is 30°.
4. Angle turned by an hour hand in 1 hour = 30°.
4. Angle turned by an hour hand in 1 minute = 0.5°.
5. Angle turned by minute hand in 1 minute = 6°.
6. the relative angular speed of minute hand with respect to hour hand = 6°-0.5°= 5.5°per minute.
Relation between time and angle between the hands of clock
1.Angle turned by hour hand in H hours 30°x H (30H)°
2.Angle turned by hour hand in M minutes = (½ M)°
3.Angle turned by hour hand in H hours M minutes = (30 H + ½ M)°.
4.Angle turned by minute hand in H hours M minutes = (0x H+6M)°= 6 M°.
5.Angle between hands of clock at H hours M minutes = (30 H + ½ M)°-6 M°= (30 H –
11
2
M)°
(Value of A is always taken as positive, negative sign is ignored)
Some Facts about Clock
1.Both the hands of clock meet (coincide) 11 times in 12 hours.
2.Both the hands meet after every
12??????60
11
minutes i.e. 65
5
11
min. The meeting take place at 12:00, 1:5
5
11
,2:10
10
11
... and so on.
3.Both the hands of clock are opposite to each other (at 180°) 11 times in 12 hours.
4.Both the hands of clock are at right angles (at 90°) to each other 22 times in 12 hours.
5.5. Any angle other than 0 and 180°between the two hands is made 22 times in 12 hours.
1. Clock has two hands. The shorter hand is the hour hand and the longer hand is the minute hand.
2. The face of the clock called dial is a complete circle having 360°. The dial is divided into 12 equal parts. These parts are
marked as 1, 2, 3,..., 11, 12.
3. The angle between two consecutive numbers is 30°.
4. Angle turned by an hour hand in 1 hour = 30°.
4. Angle turned by an hour hand in 1 minute = 0.5°.
5. Angle turned by minute hand in 1 minute = 6°.
6. the relative angular speed of minute hand with respect to hour hand = 6°-0.5°= 5.5°per minute.
Relation between time and angle between the hands of clock
1.Angle turned by hour hand in H hours 30°x H (30H)°
2.Angle turned by hour hand in M minutes = (½ M)°
3.Angle turned by hour hand in H hours M minutes = (30 H + ½ M)°.
4.Angle turned by minute hand in H hours M minutes = (0x H+6M)°= 6 M°.
5.Angle between hands of clock at H hours M minutes = (30 H + ½ M)°-6 M°= (30 H –
11
2
M)°
(Value of A is always taken as positive, negative sign is ignored)
Some Facts about Clock
1.Both the hands of clock meet (coincide) 11 times in 12 hours.
2.Both the hands meet after every
12??????60
11
minutes i.e. 65
5
11
min. The meeting take place at 12:00, 1:5
5
11
,2:10
10
11
... and so on.
3.Both the hands of clock are opposite to each other (at 180°) 11 times in 12 hours.
4.Both the hands of clock are at right angles (at 90°) to each other 22 times in 12 hours.
5.5. Any angle other than 0 and 180°between the two hands is made 22 times in 12 hours.
Time and work::
1.1 day’s work =
1
�����������������������ℎ�����
2.N����������������������������ℎ�����=
1
1���
′
�����
3.�??????������??????�������������??????�����=
������������
1���
′
�����
4.Remuneration is in proportion of work done .
Time and Distance ::
Speed of a moving body is defined as the distance covered by it in unit time
1.Distance = Speed x Time
2.Speed =
�??????������
�??????��
3.Time =
�??????������
�����
4.When distance is constant, then speed α
1
�??????��
i.e. Speed x Time = Constant ⟹S₁₂T₁ = S₂T₂
5.When time is constant, then distance αspeed i.e.
�??????������
�����
= Constant ⟹
�1
�1
=
�2
�2
6.When speed is constant, then distance αtime i.e.
�??????������
�??????��
= Constant ⟹
�1
�1
=
�2
�2
1.1 day’s work =
1
�����������������������ℎ�����
2.N����������������������������ℎ�����=
1
1���
′
�����
3.�??????������??????�������������??????�����=
������������
1���
′
�����
4.Remuneration is in proportion of work done .
Time and Distance ::
Speed of a moving body is defined as the distance covered by it in unit time
1.Distance = Speed x Time
2.Speed =
�??????������
�??????��
3.Time =
�??????������
�����
4.When distance is constant, then speed α
1
�??????��
i.e. Speed x Time = Constant ⟹S₁₂T₁ = S₂T₂
5.When time is constant, then distance αspeed i.e.
�??????������
�����
= Constant ⟹
�1
�1
=
�2
�2
6.When speed is constant, then distance αtime i.e.
�??????������
�??????��
= Constant ⟹
�1
�1
=
�2
�2
Average speed ::
Average speed =
������??????���������������
������??????�������
=
�1+�2+�3+⋯..+��
�1+�2+�3+⋯….+��
When travelling time is same
Average speed =
�+�
2
When distance travelled is same
Average speed =
2��
�+�
RELATIVE SPEED
Relative speed is defined as the speed of a moving object with respect to another object.
1.When two objects are moving in the same direction, then their relative speed is difference of their speeds.
2.When two objects are moving in the opposite direction, then their relative speed is the sum of their speeds.
3.When one object is stationary and other object is moving, then their relative speed is the speed of the moving object
Average speed =
������??????���������������
������??????�������
=
�1+�2+�3+⋯..+��
�1+�2+�3+⋯….+��
When travelling time is same
Average speed =
�+�
2
When distance travelled is same
Average speed =
2��
�+�
RELATIVE SPEED
Relative speed is defined as the speed of a moving object with respect to another object.
1.When two objects are moving in the same direction, then their relative speed is difference of their speeds.
2.When two objects are moving in the opposite direction, then their relative speed is the sum of their speeds.
3.When one object is stationary and other object is moving, then their relative speed is the speed of the moving object
SEATING ARRANGEMENT
In seating arrangement, we have to arrange a group of persons fulfilling the given conditions.
1.Linear arrangement: In this type of arrangement we have to arrange the group of persons in a line fulfilling the
given conditions.
2.Circular arrangement: In this type of arrangement we have to arrange the group of persons around a circular
table fulfilling the given conditions.
3.Arrangement around a square table: In this type of arrangement we have to arrange the group of persons
around a square table fulfilling the given conditions.
In seating arrangement, we have to arrange a group of persons fulfilling the given conditions.
1.Linear arrangement: In this type of arrangement we have to arrange the group of persons in a line fulfilling the
given conditions.
2.Circular arrangement: In this type of arrangement we have to arrange the group of persons around a circular
table fulfilling the given conditions.
3.Arrangement around a square table: In this type of arrangement we have to arrange the group of persons
around a square table fulfilling the given conditions.
Mensuration
2-Dimensional Figures 3-Dimensional Figures
As the name suggests, a 2D shape means that it will have only 2
dimensions which are length and breadth.
Here, a 3D shape means that this figure will have 3 dimensions, which
are length, breadth, and height.
For 2D shapes, we can calculate 2 things i.e. area and perimeter.
For 3D shapes, we can calculate their volume, total surface area, and
curved/lateral surface area.
2D shapes are flat as they do not have depth and also, these cannot
be held physically because of the lack of depth.
3D shapes contain a depth so they can be held physically and are not
flat like 2D shapes.
Example: Square, Rectangle, Triangle, Circle, etc. Example: Cone, Cylinder, Sphere, Cube, Prism, Pyramid, etc.
Differences between 2-Dimensional and 3-Dimensional Figures
There are two types of figures in geometry, one is 2-Dimensional and 3-Dimensional figures. Check out the below
table to know and understand the 2-Dimensional and 3-Dimensional figures and the differences between them.
As the name suggests, a 2D shape means that it will have only 2
dimensions which are length and breadth.
Here, a 3D shape means that this figure will have 3 dimensions, which
are length, breadth, and height.
For 2D shapes, we can calculate 2 things i.e. area and perimeter.
For 3D shapes, we can calculate their volume, total surface area, and
curved/lateral surface area.
2D shapes are flat as they do not have depth and also, these cannot
be held physically because of the lack of depth.
3D shapes contain a depth so they can be held physically and are not
flat like 2D shapes.
Example: Square, Rectangle, Triangle, Circle, etc. Example: Cone, Cylinder, Sphere, Cube, Prism, Pyramid, etc.
Differences between 2-Dimensional and 3-Dimensional Figures
There are two types of figures in geometry, one is 2-Dimensional and 3-Dimensional figures. Check out the below
table to know and understand the 2-Dimensional and 3-Dimensional figures and the differences between them.
Important Terms Related to Mensuration Formula
●Perimeter:This is measured in units such as m, cm, etcand it is the measure of or sum of the continuous length
of the boundary of a figure.
●Area:This is measured in square units such as m², cm², etcand it is the surface enclosed in a figure.
●Volume: This is measured in cubic units such as m³, cm³, etc, and is nothing but the space occupied by an object.
●Curved/Lateral Surface area:This is measured in square units such as m², cm², etcand it is the area of the
curved surface in a figure.
●Total Surface area:This is measured in square units such as m², cm², etcand it is the area of the total surface in
a figure including the top and bottom portions.
●Perimeter:This is measured in units such as m, cm, etcand it is the measure of or sum of the continuous length
of the boundary of a figure.
●Area:This is measured in square units such as m², cm², etcand it is the surface enclosed in a figure.
●Volume: This is measured in cubic units such as m³, cm³, etc, and is nothing but the space occupied by an object.
●Curved/Lateral Surface area:This is measured in square units such as m², cm², etcand it is the area of the
curved surface in a figure.
●Total Surface area:This is measured in square units such as m², cm², etcand it is the area of the total surface in
a figure including the top and bottom portions.
Mensuration Formulas for 2-Dimensional Figures
Shape Area Perimeter
Circle πr²
2 π r
Square (side)²
4 ×side
Rectangle length ×breadth 2 (length + breadth)
Scalene Triangle
√[s(s−a)(s−b)(s−c),Where, s = (a+b+c)/2
a+b+c (sum of sides)
Isosceles Triangle ½ ×base ×height
2a + b (sum of sides)
Equilateral Triangle (√3/4) ×(side)² 3 ×side
Right Angled Triangle ½ ×base ×hypotenuse A + B + hypotenuse, where the
hypotenuse is √A²+B²
Parallelogram base ×height 2(l+b)
Rhombus ½ ×diagonal1 ×diagonal2
4 ×side
Trapezium ½ h(sum of parallel sides) a+b+c+d(sum of all sides)
Shape Area Perimeter
Circle πr²
2 π r
Square (side)²
4 ×side
Rectangle length ×breadth 2 (length + breadth)
Scalene Triangle
√[s(s−a)(s−b)(s−c),Where, s = (a+b+c)/2
a+b+c (sum of sides)
Isosceles Triangle ½ ×base ×height
2a + b (sum of sides)
Equilateral Triangle (√3/4) ×(side)² 3 ×side
Right Angled Triangle ½ ×base ×hypotenuse A + B + hypotenuse, where the
hypotenuse is √A²+B²
Parallelogram base ×height 2(l+b)
Rhombus ½ ×diagonal1 ×diagonal2
4 ×side
Trapezium ½ h(sum of parallel sides) a+b+c+d(sum of all sides)
Area Perimeter
Circular Ring π (R² -r² )
Semicircle ½ πr² ( π + 2)r
Quadrant ¼ πr² ( π/2 + 2)r
Sector of circle
•Length of arc = θ360 x 2πr
•Perimeter of minor sector = θ360 x 2πr+2r
•Area of minor sector = θ360 x πr²
Segment of circle
Perimeter minor segment = = θ360 x 2πr + length of chord
Area of segment = θ360 x πr2-area of corresponding triangle
Circular Ring π (R² -r² )
Semicircle ½ πr² ( π + 2)r
Quadrant ¼ πr² ( π/2 + 2)r
Sector of circle
•Length of arc = θ360 x 2πr
•Perimeter of minor sector = θ360 x 2πr+2r
•Area of minor sector = θ360 x πr²
Segment of circle
Perimeter minor segment = = θ360 x 2πr + length of chord
Area of segment = θ360 x πr2-area of corresponding triangle
Mensuration Formulas for 3-Dimensional Figures
Shape Area
Curved Surface Area
(CSA)/
Lateral Surface Area
(LSA)
Total Surface Area (TSA)
Cone (1/3) π r² h π r l πr (r + l)
Cube (side)³ 4 (side)² 6 (side)²
Cuboid
length ×breadth ×
height
2 height (length +
breadth)
2 (lb +bh +hl)
Cylinder π r² h 2π r h 2πrh + 2πr²
Hemisphere (2/3) π r³ 2 π r² 3 π r²
Sphere 4/3πr³ 4πr² 4πr²
Shape Area
Curved Surface Area
(CSA)/
Lateral Surface Area
(LSA)
Total Surface Area (TSA)
Cone (1/3) π r² h π r l πr (r + l)
Cube (side)³ 4 (side)² 6 (side)²
Cuboid
length ×breadth ×
height
2 height (length +
breadth)
2 (lb +bh +hl)
Cylinder π r² h 2π r h 2πrh + 2πr²
Hemisphere (2/3) π r³ 2 π r² 3 π r²
Sphere 4/3πr³ 4πr² 4πr²
SETS and
Relations
Relations
Defination:
Asetisdefinedtobeacollectionofwell-defineddistinctobjects.Thiscollectionmaybelistedordescribed.Eachobjectis
calledanelementoftheset.Weusuallydenotesetsbycapitallettersandtheirelementsbysmallletters.
Example:A={a,e,i,o,u}.
Representationofsets:
1.RosterorBracesform.
A={a,e,i,o,u}.
2.set-BuilderorAlgebraicformorRuleMethod.
A=thesetofvowelsinthealphabet
Importantpoints:
1.Asetmaycontaineitherafiniteoraninfinitenumberofmembersorelements.
2.Ifanyelementsrepeatthenitssufficienttowriteitonce.
Asetisdefinedtobeacollectionofwell-defineddistinctobjects.Thiscollectionmaybelistedordescribed.Eachobjectis
calledanelementoftheset.Weusuallydenotesetsbycapitallettersandtheirelementsbysmallletters.
Example:A={a,e,i,o,u}.
Representationofsets:
1.RosterorBracesform.
A={a,e,i,o,u}.
2.set-BuilderorAlgebraicformorRuleMethod.
A=thesetofvowelsinthealphabet
Importantpoints:
1.Asetmaycontaineitherafiniteoraninfinitenumberofmembersorelements.
2.Ifanyelementsrepeatthenitssufficienttowriteitonce.
Some standard sets of numbers
1.Natural numbers. The set of natural (or counting) numbers is denoted by N.
2.Whole numbers. The set of whole numbers is denoted by W.
3.Integers. The set of all integers is denoted by I or Z.
4.Even integers. The set of even integers is denoted by E.
5.Odd integers. The set of odd integers is denoted by O.
6.Rational numbers. The set of rational numbers is denoted by Q.
7.Real numbers. The set of real numbers is denoted by R.
8.Irrational numbers. The set of irrational numbers is denoted by T.
9.the set of all real numbers that are not rational.
10.Positive rational numbers. The set of positive rational numbers is denoted by �
+
11.Positive real numbers. The set of positive real numbers is denoted by �
+
1.Natural numbers. The set of natural (or counting) numbers is denoted by N.
2.Whole numbers. The set of whole numbers is denoted by W.
3.Integers. The set of all integers is denoted by I or Z.
4.Even integers. The set of even integers is denoted by E.
5.Odd integers. The set of odd integers is denoted by O.
6.Rational numbers. The set of rational numbers is denoted by Q.
7.Real numbers. The set of real numbers is denoted by R.
8.Irrational numbers. The set of irrational numbers is denoted by T.
9.the set of all real numbers that are not rational.
10.Positive rational numbers. The set of positive rational numbers is denoted by �
+
11.Positive real numbers. The set of positive real numbers is denoted by �
+
Kind of sets
1. Empty set. A set which does not contain any element is called the empty set or the null set or the void set.
There is only one such set. It is denoted by φor { }
2. Singleton set. A set that contains only one element is called a singleton (or unit) set.
3. Finite set. A set that contains a limited (definite) number of different elements is called a finite set.
4. Infinite set. A set that contains an unlimited number of different elements is called an infinite set.
5. Equivalent sets. Two (finite) sets A and B are called equivalent if they have the same number of elements. Thus
two finite sets A and B are equivalent, written as A ⟷B (read as A is equivalent to B), if n(A)=(B).
6. Equal sets. Two sets A and B are said to be equal if they have exactly the same elements. We write it as A = B.
Thus AB if every member of A is a member of B and every member of B is a member of A.
Note::
All infinite sets cannot be written in the roster form. For example, the set of real numbers cannot be written in
this form because the elements of this set do not follow any pattern.
2. Two finite equal set are always equivalent but two equivalent sets may not be equal.
Cardinal number (or order) of a finite set
The number of different elements in a finite set A is called the cardinal number (or order) of A, and it is denoted
by n(A) on O (A)
The cardinal number of the empty set is zero and the cardinal number of a singleton set is one.
The cardinal number of an infinite set is never defined
1. Empty set. A set which does not contain any element is called the empty set or the null set or the void set.
There is only one such set. It is denoted by φor { }
2. Singleton set. A set that contains only one element is called a singleton (or unit) set.
3. Finite set. A set that contains a limited (definite) number of different elements is called a finite set.
4. Infinite set. A set that contains an unlimited number of different elements is called an infinite set.
5. Equivalent sets. Two (finite) sets A and B are called equivalent if they have the same number of elements. Thus
two finite sets A and B are equivalent, written as A ⟷B (read as A is equivalent to B), if n(A)=(B).
6. Equal sets. Two sets A and B are said to be equal if they have exactly the same elements. We write it as A = B.
Thus AB if every member of A is a member of B and every member of B is a member of A.
Note::
All infinite sets cannot be written in the roster form. For example, the set of real numbers cannot be written in
this form because the elements of this set do not follow any pattern.
2. Two finite equal set are always equivalent but two equivalent sets may not be equal.
Cardinal number (or order) of a finite set
The number of different elements in a finite set A is called the cardinal number (or order) of A, and it is denoted
by n(A) on O (A)
The cardinal number of the empty set is zero and the cardinal number of a singleton set is one.
The cardinal number of an infinite set is never defined
Subset
Let A, B be any two sets, then A is called a subset of B if every member of A is also a member of B. We write it as
A⊂B (read as 'A is a subset of B' or 'A is contained in B’).
If A is contained in B, we may also say that B contains A or B is a superset of A. We write it as B⊃A (read as 'B
contains A' or 'Bi a superset of A’)
Proper subset.
Let A be any set and B be a non-empty set, then A is called a proper subset of B if every member of A is also a
member of B and there exists atleastone element in B which is not a member of A. If A is a proper subset of B, we
write it as A⊂B, A≠B.
Important points :
1.A ⊂A i.e. every set is a subset of itself, but not a proper subset. A subset which is not a proper subset is called
an improper subset.
2.Every set has only one improper subset.
3.the empty set is a subset of every set.
4.4. Empty set is a proper subset of every set except itself.
5.If A is a set with n (A)=m, then the number of subsets of A= 2
�
and the number of proper subsets of A = 2
�
-1
Let A, B be any two sets, then A is called a subset of B if every member of A is also a member of B. We write it as
A⊂B (read as 'A is a subset of B' or 'A is contained in B’).
If A is contained in B, we may also say that B contains A or B is a superset of A. We write it as B⊃A (read as 'B
contains A' or 'Bi a superset of A’)
Proper subset.
Let A be any set and B be a non-empty set, then A is called a proper subset of B if every member of A is also a
member of B and there exists atleastone element in B which is not a member of A. If A is a proper subset of B, we
write it as A⊂B, A≠B.
Important points :
1.A ⊂A i.e. every set is a subset of itself, but not a proper subset. A subset which is not a proper subset is called
an improper subset.
2.Every set has only one improper subset.
3.the empty set is a subset of every set.
4.4. Empty set is a proper subset of every set except itself.
5.If A is a set with n (A)=m, then the number of subsets of A= 2
�
and the number of proper subsets of A = 2
�
-1
Power set
The set formed by all the subsets of a given set A is called the power set of A, it is denoted by P(A)
The number of elements in a power set is 2
�
. n(P(A))= 2
�
Universal set
A set that contains all the elements under consideration in a given problem is called universal set. It is denoted by
or U. It is a kind of "parent set. Every set under discussion is a subset of universal set.
Subsets of real numbers
We know some standard sets of numbers. These sets are subsets of the set of real numbers. It is easy to see that:
N ⊂W ⊂l ⊂Q ⊂R , T ⊂R,
The set of all real numbers lying between a and b is said to form an open interval written as (a, b)
The set of all real numbers lying between a and b and including the numbers a and b is said to form a closed
interval. It is denoted by [a, b]
The set of all real numbers lying between a and b, and including the number b is said to form an open-closed
interval. This interval is open on the left but closed on the right, it is denoted by (a, b]
The set of all real numbers lying between a and b, and including the number a is said to form an open-closed
interval. This interval is closed on the left but open on the right, it is denoted by [a, b]
The set formed by all the subsets of a given set A is called the power set of A, it is denoted by P(A)
The number of elements in a power set is 2
�
. n(P(A))= 2
�
Universal set
A set that contains all the elements under consideration in a given problem is called universal set. It is denoted by
or U. It is a kind of "parent set. Every set under discussion is a subset of universal set.
Subsets of real numbers
We know some standard sets of numbers. These sets are subsets of the set of real numbers. It is easy to see that:
N ⊂W ⊂l ⊂Q ⊂R , T ⊂R,
The set of all real numbers lying between a and b is said to form an open interval written as (a, b)
The set of all real numbers lying between a and b and including the numbers a and b is said to form a closed
interval. It is denoted by [a, b]
The set of all real numbers lying between a and b, and including the number b is said to form an open-closed
interval. This interval is open on the left but closed on the right, it is denoted by (a, b]
The set of all real numbers lying between a and b, and including the number a is said to form an open-closed
interval. This interval is closed on the left but open on the right, it is denoted by [a, b]
VENN DIAGRAMS
Most of the ideas about sets and the various relationships between them can be visualisedby means of
geometric figures known as Venn diagrams ( Venn-Euler diagrams). Usually, the universal set ⋃is denoted by a
rectangle and its subsets by closed curves with in the rectangle, such as circles, ovals (ellipse) etc.
OPERATIONS ON SET
1.Union of two sets. The union of two sets A and B written as AUB (read as 'A union B), is the set consisting of
all the elements which belong to A or B or both.
2.Intersection of two sets. The intersection of two sets A and B, written as A∩B (read as 'A intersection B'), is
the set consisting of all elements which belong to both A and B.
3.Two sets A and B are called disjoint if A∩B = φ; otherwise, they are called joint or overlapping sets.
4.Difference of two sets. Let A, B be two sets, then A-B is the set consisting of all the elements which belong
to A but do not belong to B.
5.Symmetric difference of two sets. The symmetric difference of two sets A and B, denoted by AB, is defined
as A △B = (A-B) U (B-A).
6.Complement of a set. Let U be the universal set and A be any set then the complement of A, denoted by A'
or �
�
or ҧ�, is the set consisting of all the elements of U which do not belong to A.
△
Most of the ideas about sets and the various relationships between them can be visualisedby means of
geometric figures known as Venn diagrams ( Venn-Euler diagrams). Usually, the universal set ⋃is denoted by a
rectangle and its subsets by closed curves with in the rectangle, such as circles, ovals (ellipse) etc.
OPERATIONS ON SET
1.Union of two sets. The union of two sets A and B written as AUB (read as 'A union B), is the set consisting of
all the elements which belong to A or B or both.
2.Intersection of two sets. The intersection of two sets A and B, written as A∩B (read as 'A intersection B'), is
the set consisting of all elements which belong to both A and B.
3.Two sets A and B are called disjoint if A∩B = φ; otherwise, they are called joint or overlapping sets.
4.Difference of two sets. Let A, B be two sets, then A-B is the set consisting of all the elements which belong
to A but do not belong to B.
5.Symmetric difference of two sets. The symmetric difference of two sets A and B, denoted by AB, is defined
as A △B = (A-B) U (B-A).
6.Complement of a set. Let U be the universal set and A be any set then the complement of A, denoted by A'
or �
�
or ҧ�, is the set consisting of all the elements of U which do not belong to A.
△
Some basic results about cardinal number
1.(AUB) = n(A)+n(B)-n(A∩B).
2.if A∩B = φ, then n(AUB) = n(A) + n (B)
3.n(A-B) = (AUB)-n(B) = n(A)-(A∩�)
4.n (B-A)= n(AUB)-n(A) = n(B)-n(A∩B)
5.n(AUB)= n(A-B)+n(B-A) + n(A∩B)
6.n(A') = n(??????)-n(A)
7.(AUBUC) = n(A) + n(B) + n(C)-n(A∩B)-n (B∩C)-n(A∩C)+(A∩B∩C).
1.(AUB) = n(A)+n(B)-n(A∩B).
2.if A∩B = φ, then n(AUB) = n(A) + n (B)
3.n(A-B) = (AUB)-n(B) = n(A)-(A∩�)
4.n (B-A)= n(AUB)-n(A) = n(B)-n(A∩B)
5.n(AUB)= n(A-B)+n(B-A) + n(A∩B)
6.n(A') = n(??????)-n(A)
7.(AUBUC) = n(A) + n(B) + n(C)-n(A∩B)-n (B∩C)-n(A∩C)+(A∩B∩C).
ORDERED PAIR
1. An ordered pair is a pair of objects taken in a specific order. An ordered pair is written by listing its two members
in a specific order, separating them b comma and enclosing the pair in parentheses. In the ordered pair (a, b), a is
called the first mem(or component) and b is called the second member (or component).
2. Equality of ordered pairs. Two ordered pairs (a, b) and (c, d) are called equal, written(a, b) = (c, d), iffa=c and b
=d.
3. The ordered pairs (a, b) and (b, a) are different unless a = b.
4. The two components of an ordered pair may be equal.
CARTESIAN PRODUCT OF TWO SETS
A and B be any two non-empty sets, then the set of all ordered pairs (a, b) for all a ϵA and bϵB is called the
cartesian product of A and B. It is written as A x B (read as 'A cross B’).
1.A x B ≠ B x A unless A = B.
2.A x B = φwhen one or both of A, B are empty sets.
3.A x B ≠ φiffA ≠ φand B ≠ φ.
4.If A and B are (non-empty) finite sets, then n(A x B)= n(A) x n(B) and n(A x B) = (B x A).
5.If A and B are non-empty sets and either A or B is an infinite set, then A x B is also infinite set.
1. An ordered pair is a pair of objects taken in a specific order. An ordered pair is written by listing its two members
in a specific order, separating them b comma and enclosing the pair in parentheses. In the ordered pair (a, b), a is
called the first mem(or component) and b is called the second member (or component).
2. Equality of ordered pairs. Two ordered pairs (a, b) and (c, d) are called equal, written(a, b) = (c, d), iffa=c and b
=d.
3. The ordered pairs (a, b) and (b, a) are different unless a = b.
4. The two components of an ordered pair may be equal.
CARTESIAN PRODUCT OF TWO SETS
A and B be any two non-empty sets, then the set of all ordered pairs (a, b) for all a ϵA and bϵB is called the
cartesian product of A and B. It is written as A x B (read as 'A cross B’).
1.A x B ≠ B x A unless A = B.
2.A x B = φwhen one or both of A, B are empty sets.
3.A x B ≠ φiffA ≠ φand B ≠ φ.
4.If A and B are (non-empty) finite sets, then n(A x B)= n(A) x n(B) and n(A x B) = (B x A).
5.If A and B are non-empty sets and either A or B is an infinite set, then A x B is also infinite set.
Relation:
If A and B are two non empty sets then any subset of A X B is called a relation from A to B
Representation of a relation
1.Roster form. In this form, a relation is represented by the set of all ordered pairs which belong to the given
relation. For example, let A= (1, 2, 3, 4, 5) and B (1, 2, 3, 4,, 20), and let R be the relation has as its square'
from A to B, then R= ((1, 1), (2, 4), (3, 9), (4, 16)).
2.Set-builder form. In this form, the relation is represented as {(x, y): x ϵA, y ϵB, x... Y}, the blank is to be
replaced by the rule which associates x and y.
3.By arrow (ray) diagram. In this form, the relation is represented by drawing arrows (rays) from first
components to the second components of all ordered pairs which belong to the given relation.
Domain and range of a relation
Let A, B be any two (non-empty) sets and R be a relation from A to B, then the domain of the relation R, is the
set of all first components of the ordered pairs which belong to R, and the range of the relation R is the of all
second components of the ordered pairs which belong to R.
If R is a relation from A to B then B is called codomain of R
If A and B are two non empty sets then any subset of A X B is called a relation from A to B
Representation of a relation
1.Roster form. In this form, a relation is represented by the set of all ordered pairs which belong to the given
relation. For example, let A= (1, 2, 3, 4, 5) and B (1, 2, 3, 4,, 20), and let R be the relation has as its square'
from A to B, then R= ((1, 1), (2, 4), (3, 9), (4, 16)).
2.Set-builder form. In this form, the relation is represented as {(x, y): x ϵA, y ϵB, x... Y}, the blank is to be
replaced by the rule which associates x and y.
3.By arrow (ray) diagram. In this form, the relation is represented by drawing arrows (rays) from first
components to the second components of all ordered pairs which belong to the given relation.
Domain and range of a relation
Let A, B be any two (non-empty) sets and R be a relation from A to B, then the domain of the relation R, is the
set of all first components of the ordered pairs which belong to R, and the range of the relation R is the of all
second components of the ordered pairs which belong to R.
If R is a relation from A to B then B is called codomain of R
SEQUENCE AND SERIES
Sequence
An ordered collection of numbers a
1, a
2, a
3, a
4, ................., a
n, ................. is a sequence if according to some definite rule or
law, there is a definite value of an , called the term or element of the sequence, corresponding to any value of the natural
number n.
SERIES
An expression of the form a
1+ a
2+ a
3+ ….. + a
n+ ............................ which is the sum of the elements of the sequenece{ a
n} is
called a series. If the series contains a finite number of elements, it is called a finite series, otherwise called an infinite series
ARITHMETIC PROGRESSION (A. P.)
A sequence a
1 , a
2,a
3, ……, a
n is called an Arithmetic Progression (A.P.) when a
2–a
1= a
3–a
2= ….. = a
n–a
n–1. That means A. P. is a sequence
in which each term is obtained by adding a constant d to the preceding term. This constant ‘d’ is called the common difference of the A.P.
1. n
th
term (t
n) = a + ( n –1) d,where n is the position number of the term.
1.Using this formula we can get
2.50
th
term (= t
50) = a+ (50 –1) d = a + 49d
2. n
th
term from end = l -( n –1) d
3. Sum of the first n terms
∴s =
??????
�
{2a + (n-1)d}
.
An ordered collection of numbers a
1, a
2, a
3, a
4, ................., a
n, ................. is a sequence if according to some definite rule or
law, there is a definite value of an , called the term or element of the sequence, corresponding to any value of the natural
number n.
SERIES
An expression of the form a
1+ a
2+ a
3+ ….. + a
n+ ............................ which is the sum of the elements of the sequenece{ a
n} is
called a series. If the series contains a finite number of elements, it is called a finite series, otherwise called an infinite series
ARITHMETIC PROGRESSION (A. P.)
A sequence a
1 , a
2,a
3, ……, a
n is called an Arithmetic Progression (A.P.) when a
2–a
1= a
3–a
2= ….. = a
n–a
n–1. That means A. P. is a sequence
in which each term is obtained by adding a constant d to the preceding term. This constant ‘d’ is called the common difference of the A.P.
1. n
th
term (t
n) = a + ( n –1) d,where n is the position number of the term.
1.Using this formula we can get
2.50
th
term (= t
50) = a+ (50 –1) d = a + 49d
2. n
th
term from end = l -( n –1) d
3. Sum of the first n terms
∴s =
??????
�
{2a + (n-1)d}
.
4. arithmetic mean between two numbers a and b =
�+�
�
5. If the sum of the numbers is given, then in an A.P.,
(i)three numbers are taken as a-d , a, a+d
(ii)four numbers are taken as a-3d, a-d, a+d, a + 3d.
(iii)five numbers are taken as a-2d, a-d, a, a+d, a +2d.
6. Sum of 1
st
n natural or counting numbers
1.i.e. 1 + 2 + 3 + ........ + n =
??????(??????+�)
�
7. Sum of 1
st
n odd number
sum of first, n odd numbers is n
2
, i.e. 1 + 3 + 5 + ..... + ( 2n –1 ) = n
2
8. Sum of the Squares of the first, n natural nos.
1.i.e. 1
2
+ 2
2
+ 3
2
+ ........ + n
2
=
??????(??????+�)(�??????+�)
??????
9. Sum of the cube of the first, n natural nos.
1.i.e. 1
3
+ 2
3
+ 3
3
+ ........ + n
3
=
??????(??????+�)
�
�
�+�
�
5. If the sum of the numbers is given, then in an A.P.,
(i)three numbers are taken as a-d , a, a+d
(ii)four numbers are taken as a-3d, a-d, a+d, a + 3d.
(iii)five numbers are taken as a-2d, a-d, a, a+d, a +2d.
6. Sum of 1
st
n natural or counting numbers
1.i.e. 1 + 2 + 3 + ........ + n =
??????(??????+�)
�
7. Sum of 1
st
n odd number
sum of first, n odd numbers is n
2
, i.e. 1 + 3 + 5 + ..... + ( 2n –1 ) = n
2
8. Sum of the Squares of the first, n natural nos.
1.i.e. 1
2
+ 2
2
+ 3
2
+ ........ + n
2
=
??????(??????+�)(�??????+�)
??????
9. Sum of the cube of the first, n natural nos.
1.i.e. 1
3
+ 2
3
+ 3
3
+ ........ + n
3
=
??????(??????+�)
�
�
10. If the sum of an AP is denoted by S
n, then its common difference= S
n-2S
n-1
11. The sum of the terms equidistant from the beginning and the end of an AP is always same and equals to the sum of the
first and the last terms.
12. If the terms of an arithmetic progression (AP) are increased, decreased, multiplied or div by the same non-zero constant,
they remain in arithmetic progression.
13. If the anthterm of a sequence is a linear expression in n i.e.
a
n =an + b, then it is an AP with common difference a (coefficient of n)
14. If the sum of first terms of a sequence is a quadratic expression in a i.e. S
n= an
2
+ bn + c then it is an AP with common
difference 2a.
15. If an AP consist of n terms then
(i)The sum of these n terms = n x (middle term) , if n is odd
(ii)The sum of these n terms = n x (mean of 2 middle term) , if n is even
16. a
n= S
n-S
n-1
n, then its common difference= S
n-2S
n-1
11. The sum of the terms equidistant from the beginning and the end of an AP is always same and equals to the sum of the
first and the last terms.
12. If the terms of an arithmetic progression (AP) are increased, decreased, multiplied or div by the same non-zero constant,
they remain in arithmetic progression.
13. If the anthterm of a sequence is a linear expression in n i.e.
a
n =an + b, then it is an AP with common difference a (coefficient of n)
14. If the sum of first terms of a sequence is a quadratic expression in a i.e. S
n= an
2
+ bn + c then it is an AP with common
difference 2a.
15. If an AP consist of n terms then
(i)The sum of these n terms = n x (middle term) , if n is odd
(ii)The sum of these n terms = n x (mean of 2 middle term) , if n is even
16. a
n= S
n-S
n-1
ARITHMETIC mean
Given two (different) numbers a and b. We can insert a number A between them so that a, A, bare in A.P. Such a
number A is called arithmetic mean (briefly written as A.M.) of the numbers a and b. In other words, if a, A, b are in
A.P. then A is called the arithmetic mean between the numbers a and b
arithmetic mean between two numbers a and b =
�+�
�
In a similar way, we can insert as many numbers as we like between two given numbers a and b such that the
resulting sequence is an A.P . Let A1, A2, A3….. An be n numbers between a and b such that a, A1, A2, A3, An, b is
an A.P., then the numbers A1, A2, A3, A are called n arithmetic means between a and b,
sum of these n arithmetic means=A1 + A2+ …..+An =nx
�+�
�
= times the A.M. between a and b.
Important ::
If an A.P. consists of n terms, then
(i)the sum of these n terms = n x (middle term), if n is odd
(ii)the sum of these terms = n x (arithmetic mean of two middle terms), if n is even
Given two (different) numbers a and b. We can insert a number A between them so that a, A, bare in A.P. Such a
number A is called arithmetic mean (briefly written as A.M.) of the numbers a and b. In other words, if a, A, b are in
A.P. then A is called the arithmetic mean between the numbers a and b
arithmetic mean between two numbers a and b =
�+�
�
In a similar way, we can insert as many numbers as we like between two given numbers a and b such that the
resulting sequence is an A.P . Let A1, A2, A3….. An be n numbers between a and b such that a, A1, A2, A3, An, b is
an A.P., then the numbers A1, A2, A3, A are called n arithmetic means between a and b,
sum of these n arithmetic means=A1 + A2+ …..+An =nx
�+�
�
= times the A.M. between a and b.
Important ::
If an A.P. consists of n terms, then
(i)the sum of these n terms = n x (middle term), if n is odd
(ii)the sum of these terms = n x (arithmetic mean of two middle terms), if n is even
Geometric Progression(GP)
If in a sequence of terms each term is constant multiple of the proceeding term, then the sequence is called a Geometric
Progression (G.P). The constant multiplier is called the common ratio
Examples: 1) In 5, 15, 45, 135,….. common ratio is 15/5 = 3
2) In 1, 1/2, 1/4, 1/9 … common ratio is (1/2) /1 = 1/2
3) In 2, –6, 18, –54, …. common ratio is (–6) / 2 = –3
the sequence a, ar, ar
2
, ar
3
, …. ar
n-1
is generelisedform of GP
where r =
�
??????
�
??????−�
If in a sequence of terms each term is constant multiple of the proceeding term, then the sequence is called a Geometric
Progression (G.P). The constant multiplier is called the common ratio
Examples: 1) In 5, 15, 45, 135,….. common ratio is 15/5 = 3
2) In 1, 1/2, 1/4, 1/9 … common ratio is (1/2) /1 = 1/2
3) In 2, –6, 18, –54, …. common ratio is (–6) / 2 = –3
the sequence a, ar, ar
2
, ar
3
, …. ar
n-1
is generelisedform of GP
where r =
�
??????
�
??????−�
1.a
n=ar
n-1
2.nth term from end = l
1
�
�−1
3.if three numersa, b, and c are in GP then b
2
= ac
4.sum of first n terms of GP
�
�=
��
�
−1
�−1
0��
�=
�1−�
�
1−�
5. If product of numbers are given in GP then
(i). three numbers are taken
�
�
, a , ar
(ii). four numbers are taken
�
�
3
,
�
�
, a , ar
3
(iii) five numbers are taken
�
�
2
,
�
�
, a , ar, ar
2
6. sum of infinite GP
�
∞=
�
1−�
, where |r| < 1
7. geometric mean between two numbers a and b
GM = ��
8. Relationship between AM and GM
AM ≥GM
9. If a GP consist of n term ,then
(i). The product of these n terms = (middle term)
n
, if n is odd
(ii). The product of these n terms = (GM of two middle term)
n
, if n is odd
n=ar
n-1
2.nth term from end = l
1
�
�−1
3.if three numersa, b, and c are in GP then b
2
= ac
4.sum of first n terms of GP
�
�=
��
�
−1
�−1
0��
�=
�1−�
�
1−�
5. If product of numbers are given in GP then
(i). three numbers are taken
�
�
, a , ar
(ii). four numbers are taken
�
�
3
,
�
�
, a , ar
3
(iii) five numbers are taken
�
�
2
,
�
�
, a , ar, ar
2
6. sum of infinite GP
�
∞=
�
1−�
, where |r| < 1
7. geometric mean between two numbers a and b
GM = ��
8. Relationship between AM and GM
AM ≥GM
9. If a GP consist of n term ,then
(i). The product of these n terms = (middle term)
n
, if n is odd
(ii). The product of these n terms = (GM of two middle term)
n
, if n is odd
PERMUTATIONS AND
COMBINATIONS
COMBINATIONS
THE FACTORIAL
Definition: The factorial n, written as n! or ∟n , represents the product of all integers from 1 to n both inclusive. To make the
notation meaningful, when n = o, we define o! Or ∟o = 1.
Thus, n! = n (n –1) (n –2)..... ...3.2.1
FUNDAMENTAL PRINCIPLES OF COUNTING
(a) Multiplication Rule: If certain thing may be done in ‘m’ different ways and when it has been done, a second thing can be
done in ‘n ‘ different ways then total number of ways of doing both things simultaneously = m ×n.
Eg.If one can go to school by 5 different buses and then come back by 4 different buses then total number of ways of going to
and coming back from school = 5 ×4 = 20.
(b) Addition Rule: It there are two alternative jobs which can be done in ‘m’ ways and in ‘n’ ways respectively then either of
two jobs can be done in (m + n) ways.
Eg.If one wants to go school by bus where there are 5 buses or to by auto where there are 4 autos, then total number of ways
of going school = 5 + 4 = 9.
Note :-1) AND -Multiply
OR -Add
Definition: The factorial n, written as n! or ∟n , represents the product of all integers from 1 to n both inclusive. To make the
notation meaningful, when n = o, we define o! Or ∟o = 1.
Thus, n! = n (n –1) (n –2)..... ...3.2.1
FUNDAMENTAL PRINCIPLES OF COUNTING
(a) Multiplication Rule: If certain thing may be done in ‘m’ different ways and when it has been done, a second thing can be
done in ‘n ‘ different ways then total number of ways of doing both things simultaneously = m ×n.
Eg.If one can go to school by 5 different buses and then come back by 4 different buses then total number of ways of going to
and coming back from school = 5 ×4 = 20.
(b) Addition Rule: It there are two alternative jobs which can be done in ‘m’ ways and in ‘n’ ways respectively then either of
two jobs can be done in (m + n) ways.
Eg.If one wants to go school by bus where there are 5 buses or to by auto where there are 4 autos, then total number of ways
of going school = 5 + 4 = 9.
Note :-1) AND -Multiply
OR -Add
PERMUTATIONS
Definition: The ways of arranging or selecting smaller or equal number of persons or objects from a group of persons or
collection of objects with due regard being paid to the order of arrangement or selection, are called permutations.
1.
n
P
n= n!
2.
n
P
r=
??????!
??????−??????!
3. 0! = 1.
4. Number of permutations of n distinct objects taken r at a time when a particular object is not taken in any arrangement is
n–1
p
r
5. Number of permutations of r objects out of n distinct objects when a particular object is always included in any
arrangement is r.
n–1
p
r-1
6. Permutations when some of the things are alike, taken all at a time
P=
??????!
??????
�
!??????
�
!??????
�
!
7. Permutations when each thing may be repeated once, twice,...uptor times in any arrangement. = n
r
Definition: The ways of arranging or selecting smaller or equal number of persons or objects from a group of persons or
collection of objects with due regard being paid to the order of arrangement or selection, are called permutations.
1.
n
P
n= n!
2.
n
P
r=
??????!
??????−??????!
3. 0! = 1.
4. Number of permutations of n distinct objects taken r at a time when a particular object is not taken in any arrangement is
n–1
p
r
5. Number of permutations of r objects out of n distinct objects when a particular object is always included in any
arrangement is r.
n–1
p
r-1
6. Permutations when some of the things are alike, taken all at a time
P=
??????!
??????
�
!??????
�
!??????
�
!
7. Permutations when each thing may be repeated once, twice,...uptor times in any arrangement. = n
r
CIRCULAR PERMUTATIONS
if we arrange the objects along a closed curve viz., a circle, the permutations are known as circular permutations.
The number of circular permutations of n different things chosen at a time is (n–1)!.
the number of ways of arranging n persons along a round table so that no person has the same two neighbors is = ½ (n-1)!
the number of necklaces formed with n beads of different colours= ½ (n-1)!
if we arrange the objects along a closed curve viz., a circle, the permutations are known as circular permutations.
The number of circular permutations of n different things chosen at a time is (n–1)!.
the number of ways of arranging n persons along a round table so that no person has the same two neighbors is = ½ (n-1)!
the number of necklaces formed with n beads of different colours= ½ (n-1)!
COMBINATIONS
Definition : The number of ways in which smaller or equal number of things are arranged or selected from a collection of things
where the order of selection or arrangement is not important, are called combinations.
1.
n
C
r= n!/r! ( n –r )!
2.
n
C
r=
n
C
n–r
3.
n+1
C
r =
n
C
r+
n
C
r–1
4.
n
C
o= 1
5.
n
C
n= 1
6.
n+1
C
r=
n
C
r+
n
C
r-1
7.
n
P
r=
n-1
P
r+ r
n-1
P
r-1
Definition : The number of ways in which smaller or equal number of things are arranged or selected from a collection of things
where the order of selection or arrangement is not important, are called combinations.
1.
n
C
r= n!/r! ( n –r )!
2.
n
C
r=
n
C
n–r
3.
n+1
C
r =
n
C
r+
n
C
r–1
4.
n
C
o= 1
5.
n
C
n= 1
6.
n+1
C
r=
n
C
r+
n
C
r-1
7.
n
P
r=
n-1
P
r+ r
n-1
P
r-1
FUNCTIONS
Function
Any relation from X to Y in which no two different ordered pairs have the same first element is called a function. Let A and B be
two non-empty sets. Then, a rule or a correspondence f which associates to each element x of A, a unique element, denoted by
f(x) of B , is called a function or mapping from A to B and we write f : A→B
The element f(x) of B is called the image of x, while x is called the pre-image of f (x).
DOMAIN & RANGE OF A FUNCTION
Let f : A→B, then A is called the domain of f, while B is called the co-domain of f.
Equal Functions: Two functions f and g are said to be equal, written as f = g if they have the same domain and they satisfy the
condition f(x) = g(x), for all x.
Any relation from X to Y in which no two different ordered pairs have the same first element is called a function. Let A and B be
two non-empty sets. Then, a rule or a correspondence f which associates to each element x of A, a unique element, denoted by
f(x) of B , is called a function or mapping from A to B and we write f : A→B
The element f(x) of B is called the image of x, while x is called the pre-image of f (x).
DOMAIN & RANGE OF A FUNCTION
Let f : A→B, then A is called the domain of f, while B is called the co-domain of f.
Equal Functions: Two functions f and g are said to be equal, written as f = g if they have the same domain and they satisfy the
condition f(x) = g(x), for all x.
Graph of a real functions
Identity Function
Suppose the real-valued function f : R →R by y = f(x) = x for each x ∈R (i.e. the set of real numbers). Such a function is
called the identity function
Constant Function
The function f: R →R by y = f (x) = c, x ∈R where c is a constant and each x ∈R is called a constant function.
Identity Function
Suppose the real-valued function f : R →R by y = f(x) = x for each x ∈R (i.e. the set of real numbers). Such a function is
called the identity function
Constant Function
The function f: R →R by y = f (x) = c, x ∈R where c is a constant and each x ∈R is called a constant function.
Polynomial Function
A function f : R →R is said to be polynomial function if for each x in R, y = f(x) = a
0+ a
1x + a
2x
2
+ …+ a
nx
n
, where n is a
non-negative integer and a
0, a
1, a
2,…,a
n∈R.
Rational Functions
A function is of the form f(x)/g(x), where f(x) and g(x) are polynomial functions of x defined in a domain such that g(x)
≠ 0 is called a rational function.
A function f : R →R is said to be polynomial function if for each x in R, y = f(x) = a
0+ a
1x + a
2x
2
+ …+ a
nx
n
, where n is a
non-negative integer and a
0, a
1, a
2,…,a
n∈R.
Rational Functions
A function is of the form f(x)/g(x), where f(x) and g(x) are polynomial functions of x defined in a domain such that g(x)
≠ 0 is called a rational function.
Modulus Function
The function f: R →R defined by y = f(x) = |x| for each x ∈R is called the modulus function.
Signum Function
The function f: R →R defined by , f(x) = ቐ
−1,�<0
0,�=0
1,�>0
is called the signum function. The domain of this function is R and the range is the set {–1, 0, 1}.
The function f: R →R defined by y = f(x) = |x| for each x ∈R is called the modulus function.
Signum Function
The function f: R →R defined by , f(x) = ቐ
−1,�<0
0,�=0
1,�>0
is called the signum function. The domain of this function is R and the range is the set {–1, 0, 1}.
Greatest Integer Function
The function f: R →R defined by f(x) = [x], x ∈R assumes the greatest integer value, less than or equal to x. Such a
function is called the greatest integer function.
The function f: R →R defined by f(x) = [x], x ∈R assumes the greatest integer value, less than or equal to x. Such a
function is called the greatest integer function.
Exponential Function :
An exponential function is a Mathematical function in the form y = f(x) = b
x
, where “x” is a variable and “b” is a
constant which is called the base of the function such that b > 1.
Logarithmic Function :
If the inverse of the exponential function exists then we can represent the logarithmic function as given below:
Suppose b > 1 is a real number such that the logarithm of a to base b is x if b
x
= a.
The logarithm of a to base b can be written as log
ba
An exponential function is a Mathematical function in the form y = f(x) = b
x
, where “x” is a variable and “b” is a
constant which is called the base of the function such that b > 1.
Logarithmic Function :
If the inverse of the exponential function exists then we can represent the logarithmic function as given below:
Suppose b > 1 is a real number such that the logarithm of a to base b is x if b
x
= a.
The logarithm of a to base b can be written as log
ba
OPERATIONS ON REAL FUNCTIONS
The algebraic operations of addition, subtraction, multiplication and division etc. can be performed on two real valued functions
suitably in the same manner as they are performed on two real numbers
1.(f+g) (x) = f(x)+g(x), for all x ϵX.
2.(f-g) (x) = f(x)-g(x), for all x ϵX.
3.(fg) (x) = f(x) g(x), for all x ϵX.
4.(
�
�
)(x) =
�(�)
�(�)
, for all x ϵX, g(x) ≠ 0
5.(�
�
) (x) = (�(�))
�
, for all x ϵX.
6.(
1
�
)(x) =
1
�(�)
, x ϵX,f(x) ≠ 0
7.(cf)(x)=cf(x), for all x ϵX.
Composition of functions
Let f , g be two real valued functions and let D = {x : x ϵ�
�, f(x) ϵ�
�} ≠φ, then the composite 0f f and g, denoted by gof, is the
function defined by (gof) (x) = g(f(x)) with domain D.
The composite of two functions is also called the resultant of two functions or function of a function:
In particular, if �
�⊂�
��ℎ���
���=�
�
The algebraic operations of addition, subtraction, multiplication and division etc. can be performed on two real valued functions
suitably in the same manner as they are performed on two real numbers
1.(f+g) (x) = f(x)+g(x), for all x ϵX.
2.(f-g) (x) = f(x)-g(x), for all x ϵX.
3.(fg) (x) = f(x) g(x), for all x ϵX.
4.(
�
�
)(x) =
�(�)
�(�)
, for all x ϵX, g(x) ≠ 0
5.(�
�
) (x) = (�(�))
�
, for all x ϵX.
6.(
1
�
)(x) =
1
�(�)
, x ϵX,f(x) ≠ 0
7.(cf)(x)=cf(x), for all x ϵX.
Composition of functions
Let f , g be two real valued functions and let D = {x : x ϵ�
�, f(x) ϵ�
�} ≠φ, then the composite 0f f and g, denoted by gof, is the
function defined by (gof) (x) = g(f(x)) with domain D.
The composite of two functions is also called the resultant of two functions or function of a function:
In particular, if �
�⊂�
��ℎ���
���=�
�
Limit and continuity
Existence of limit
If �??????�
�⟶�
��=lim
�⟶�
−
��=lim
�⟶
�
+
�(�)then �??????�
�⟶�
����??????��
Evaluation of limits of algebraic functions
1.Method of direct substitution.
2.Method of factorization.
3.Method of rationalization.
4.Using the formulas::
(i) lim
�⟶�
�
�
−�
�
�−�
=��
�−1
(ii) lim
�⟶0
�
�
−1
�
=1
(iii) lim
�⟶0
�
�
−1
�
=����
(iv) lim
�⟶0(1+�)
1
�=�
(v)lim
�⟶0
log(1+�)
�
=1
Continuity::
A function f(x) is said to be continuous if �??????�
�⟶�
��=lim
�⟶�
−
��=lim
�⟶
�
+
�(�)then �??????�
�⟶�
����??????��
If �??????�
�⟶�
��=lim
�⟶�
−
��=lim
�⟶
�
+
�(�)then �??????�
�⟶�
����??????��
Evaluation of limits of algebraic functions
1.Method of direct substitution.
2.Method of factorization.
3.Method of rationalization.
4.Using the formulas::
(i) lim
�⟶�
�
�
−�
�
�−�
=��
�−1
(ii) lim
�⟶0
�
�
−1
�
=1
(iii) lim
�⟶0
�
�
−1
�
=����
(iv) lim
�⟶0(1+�)
1
�=�
(v)lim
�⟶0
log(1+�)
�
=1
Continuity::
A function f(x) is said to be continuous if �??????�
�⟶�
��=lim
�⟶�
−
��=lim
�⟶
�
+
�(�)then �??????�
�⟶�
����??????��
Differentiation
Differentiation
Some Basic Differntiation::
1.
�
��
( �
�
) = n �
�−1
2.
�
��
(constant) = 0
3.
�
��
(x) = 1
4.
�
��
( �) =
1
2�
5.
�
��
(
1
�
�
) = -
�
�
�+1
6.
�
��
(�
�
) = �
�
7.
�
��
(�
�
) = �
�
loga
8.
�
��
(logx) =
1
�
9.
�
��
(|x|) =
�
|�|
�≠0
Some Basic Differntiation::
1.
�
��
( �
�
) = n �
�−1
2.
�
��
(constant) = 0
3.
�
��
(x) = 1
4.
�
��
( �) =
1
2�
5.
�
��
(
1
�
�
) = -
�
�
�+1
6.
�
��
(�
�
) = �
�
7.
�
��
(�
�
) = �
�
loga
8.
�
��
(logx) =
1
�
9.
�
��
(|x|) =
�
|�|
�≠0
Rules of differentiation
1. Addition Rule
�
��
(( f(x) + g(x)) = f’(x) + g’(x)
2. Subtraction Rule
�
��
(( f(x) -g(x)) = f’(x) -g’(x)
3. Product Rule
�
��
( f(x) g(x)) = f’(x) g(x)+ f(x)g’(x)
4. Quotient Rule
�
��
(
�(�)
�(�)
) =
f’(x)g(x)−f(x)g’(x)
(�(�))
2
5. Chain Rule
�
��
(f(t)) =
�
��
f(t) .
��
��
1. Addition Rule
�
��
(( f(x) + g(x)) = f’(x) + g’(x)
2. Subtraction Rule
�
��
(( f(x) -g(x)) = f’(x) -g’(x)
3. Product Rule
�
��
( f(x) g(x)) = f’(x) g(x)+ f(x)g’(x)
4. Quotient Rule
�
��
(
�(�)
�(�)
) =
f’(x)g(x)−f(x)g’(x)
(�(�))
2
5. Chain Rule
�
��
(f(t)) =
�
��
f(t) .
��
��
First principle method
F’(x) = lim
ℎ⟶0
��+ℎ−�(�)
ℎ
Differentiation of composite Functions
��
��
=
��
��
.
��
��
Differentiation of Implicit Functions
If the variable x and y are connected so that it is not possible to express y as a function of x , then y is said to be an implicit
function of x
Differentiation of Parametric functions
When x and y are connected with some other variable then we say x is called parametric function of y and y is
called parametric function of x For ex x = f(t) and y = g(t) , here t is called the parameter
F’(x) = lim
ℎ⟶0
��+ℎ−�(�)
ℎ
Differentiation of composite Functions
��
��
=
��
��
.
��
��
Differentiation of Implicit Functions
If the variable x and y are connected so that it is not possible to express y as a function of x , then y is said to be an implicit
function of x
Differentiation of Parametric functions
When x and y are connected with some other variable then we say x is called parametric function of y and y is
called parametric function of x For ex x = f(t) and y = g(t) , here t is called the parameter
probability
Experiment: An experiment may be described as a performance that produces certain results.
Random Experiment: An experiment is defined to be random if the results of the experiment depend on chance only.
Events:The results or outcomes of a random experiment are known as events. Sometimes events may be combination of
outcomes. The events are of two types:
(i) Simple or Elementary
(ii) Composite or Compound.
(iii) Sure event
(iv) Impossible event
(v) Equally Likely Events or Mutually Symmetric Events or Equi-Probable Events:
Algebra of event :
(i)Complement of event : it is denoted by
ഥ����
′
���
�
??????�������������������ℎ��������������ℎ���ℎ���ℎ�������������??????��??????��
(ii)The event A or B: it is denoted by AUB . It is the set which contain all elements of A or B or both
(iii)The event A and B: it is denoted by A⋂B . It is the set which contain all elements of A and B both
(iv)The event A but not B: it is denoted by A-B . It is the set which contain all elements of A but not B
(i)Mutually Exclusive Events or Incompatible Events: If A⋂B = 0 then it is said to be A and B are mutually exclusive events
(i)Exhaustive Events : If union of all the event is sample space then these events are called Exhaustive events
(i)Mutually Exclusive and Exhaustive Events: If union of all the event is sample space and their intersection is null set than it
is called Mutually Exclusive and Exhaustive Events
Random Experiment: An experiment is defined to be random if the results of the experiment depend on chance only.
Events:The results or outcomes of a random experiment are known as events. Sometimes events may be combination of
outcomes. The events are of two types:
(i) Simple or Elementary
(ii) Composite or Compound.
(iii) Sure event
(iv) Impossible event
(v) Equally Likely Events or Mutually Symmetric Events or Equi-Probable Events:
Algebra of event :
(i)Complement of event : it is denoted by
ഥ����
′
���
�
??????�������������������ℎ��������������ℎ���ℎ���ℎ�������������??????��??????��
(ii)The event A or B: it is denoted by AUB . It is the set which contain all elements of A or B or both
(iii)The event A and B: it is denoted by A⋂B . It is the set which contain all elements of A and B both
(iv)The event A but not B: it is denoted by A-B . It is the set which contain all elements of A but not B
(i)Mutually Exclusive Events or Incompatible Events: If A⋂B = 0 then it is said to be A and B are mutually exclusive events
(i)Exhaustive Events : If union of all the event is sample space then these events are called Exhaustive events
(i)Mutually Exclusive and Exhaustive Events: If union of all the event is sample space and their intersection is null set than it
is called Mutually Exclusive and Exhaustive Events
AXIOMATIC OR MODERN DEFINITION OF PROBABILITY
(i) P(A) ≥0 for every A ⊆S (subset)
(ii) P(S) = 1
(iii) For any sequence of mutually exclusive events A1, A2, A3,..
P(A1 ∪A2 ∪A3 ∪….) = P(A1) + P(A2) + P(A3) +…….
Important points:
(a) The probability of an event lies between 0 and 1, both inclusive.
i.e. 0≤P(A)≤1
If P(A) = 0 -impossible event
If P(A) = 1-sure event.
(b) Non-occurrence of event A is denoted by A’ or A
C
or
P(A) + P (A’) = 1
(c) The ratio of no. of favourableevents to the no. of unfavourableevents is known as odds in favourof the event A and its
inverse ratio is known as odds against the event A.
i.e. odds in favourof A = m
A: (m –m
A)
and odds against A = (m –m
A) : m
A
(i) P(A) ≥0 for every A ⊆S (subset)
(ii) P(S) = 1
(iii) For any sequence of mutually exclusive events A1, A2, A3,..
P(A1 ∪A2 ∪A3 ∪….) = P(A1) + P(A2) + P(A3) +…….
Important points:
(a) The probability of an event lies between 0 and 1, both inclusive.
i.e. 0≤P(A)≤1
If P(A) = 0 -impossible event
If P(A) = 1-sure event.
(b) Non-occurrence of event A is denoted by A’ or A
C
or
P(A) + P (A’) = 1
(c) The ratio of no. of favourableevents to the no. of unfavourableevents is known as odds in favourof the event A and its
inverse ratio is known as odds against the event A.
i.e. odds in favourof A = m
A: (m –m
A)
and odds against A = (m –m
A) : m
A
Laws of PROBABILITY
If A and B are event associated with random experiment having sample space S and if A ⊂B then
(i)P(A) ≤P(B) (ii) P(B-A) = P(B)-P(A)
Two events A and B are mutually exclusive if P (A ∩B) = 0 or more precisely,
P (A ∪B) = P(A) + P(B)
Similarly three events A, B and C are mutually exclusive if
P (A ∪B ∪C) = P(A) + P(B) + P(C)
Two events A and B are exhaustive if
P(A ∪B) = 1
Similarly three events A, B and C are exhaustive if
P(A ∪B ∪C) = 1
Three events A, B and C are equally likely if
P(A) = P(B) = P(C)
ADDITION THEOREMS OR THEOREMS ON TOTAL PROBABILITY
(i)P (A ∪B) = P(A) + P(B) , if A and B are mutually exclusive
(ii)For any sequence of mutually exclusive events A1, A2, A3,..
P(A1 ∪A2 ∪A3 ∪….) = P(A1) + P(A2) + P(A3) +…….
P (A ∪B) = P(A) + P(B) -P (A ∩B)
If A and B are event associated with random experiment having sample space S and if A ⊂B then
(i)P(A) ≤P(B) (ii) P(B-A) = P(B)-P(A)
Two events A and B are mutually exclusive if P (A ∩B) = 0 or more precisely,
P (A ∪B) = P(A) + P(B)
Similarly three events A, B and C are mutually exclusive if
P (A ∪B ∪C) = P(A) + P(B) + P(C)
Two events A and B are exhaustive if
P(A ∪B) = 1
Similarly three events A, B and C are exhaustive if
P(A ∪B ∪C) = 1
Three events A, B and C are equally likely if
P(A) = P(B) = P(C)
ADDITION THEOREMS OR THEOREMS ON TOTAL PROBABILITY
(i)P (A ∪B) = P(A) + P(B) , if A and B are mutually exclusive
(ii)For any sequence of mutually exclusive events A1, A2, A3,..
P(A1 ∪A2 ∪A3 ∪….) = P(A1) + P(A2) + P(A3) +…….
P (A ∪B) = P(A) + P(B) -P (A ∩B)
CONDITIONAL PROBABILITY
P(B/A) =
??????(�∩�)
??????(�)
, similarly P(A/B) =
??????(�∩�)
??????(�)
Some more properties of conditional probability:
1.P(A/S) =
??????(�∩??????)
??????(??????)
= P(A)
2.P(A/A) =
??????(�∩�)
??????(�)
= 1
3.P(S/A) =
??????(�∩??????)
??????(�)
= 1
4.0 ≤P(A/B) ≤1
5.P(AUB/F) = P(A/F) + P(B/F) –P(A∩B/F)
6.P(AUB/F) = P(A/F) + P(B/F) , If A and B are disjoint sets
7.P(A’/B) = 1 –P(A/B)
Independent Event :
Two events A and B are said to be independent if P(A ∩B) = P(A) ×P(B)
if two events A and B are independent, then the following pairs of events are also independent:
(i) A and B’(ii) A’ and B (iii) A’ and B’
Theorems of Compound Probability
Theorem 1: For any two events A and B, the probability that A and B occur simultaneously is given by the product of the unconditional
probability of A and the conditional probability of B given that A has already occurred
i.e. P(A ∩B) = P(A) ×P(B/A) Provided P(A) > 0
Theorem 2: For any three events A, B and C, the probability that they occur jointly is given by
P(A ∩B ∩C) = P(A) ×P(B/A) ×P(C/(A ∩B)) Provided P(A ∩B) > 0
In the event of independence of the events
P(A ∩B) = P(A) ×P(B)
and P(A ∩B ∩C) = P(A) ×P(B) ×P(C)
P(B/A) =
??????(�∩�)
??????(�)
, similarly P(A/B) =
??????(�∩�)
??????(�)
Some more properties of conditional probability:
1.P(A/S) =
??????(�∩??????)
??????(??????)
= P(A)
2.P(A/A) =
??????(�∩�)
??????(�)
= 1
3.P(S/A) =
??????(�∩??????)
??????(�)
= 1
4.0 ≤P(A/B) ≤1
5.P(AUB/F) = P(A/F) + P(B/F) –P(A∩B/F)
6.P(AUB/F) = P(A/F) + P(B/F) , If A and B are disjoint sets
7.P(A’/B) = 1 –P(A/B)
Independent Event :
Two events A and B are said to be independent if P(A ∩B) = P(A) ×P(B)
if two events A and B are independent, then the following pairs of events are also independent:
(i) A and B’(ii) A’ and B (iii) A’ and B’
Theorems of Compound Probability
Theorem 1: For any two events A and B, the probability that A and B occur simultaneously is given by the product of the unconditional
probability of A and the conditional probability of B given that A has already occurred
i.e. P(A ∩B) = P(A) ×P(B/A) Provided P(A) > 0
Theorem 2: For any three events A, B and C, the probability that they occur jointly is given by
P(A ∩B ∩C) = P(A) ×P(B/A) ×P(C/(A ∩B)) Provided P(A ∩B) > 0
In the event of independence of the events
P(A ∩B) = P(A) ×P(B)
and P(A ∩B ∩C) = P(A) ×P(B) ×P(C)
The law of PROBABILITY
If E1 ,E2, E3 ….Enare mutually exclusive and exhaustive events associated with sample space S of a random experiment and A is
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 E1 ,E2, E3 ….Enare mutually exclusive and exhaustive events associated with a random experiment and A is any event
associated with the experiment then,
P�
??????/�=
��
??????�(�/�
??????)
∑��
??????�(�/�
??????)
, where I = 1, 2, 3, …..n
If E1 ,E2, E3 ….Enare mutually exclusive and exhaustive events associated with sample space S of a random experiment and A is
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 E1 ,E2, E3 ….Enare mutually exclusive and exhaustive events associated with a random experiment and A is any event
associated with the experiment then,
P�
??????/�=
��
??????�(�/�
??????)
∑��
??????�(�/�
??????)
, where I = 1, 2, 3, …..n
Descriptive statistics
Dispersion
The second important characteristic of a distribution is given by dispersion. Two distributions may be
identical in respect of its first important characteristic i.e. central tendency and yet they may differ on
account of scatterness.
Range
For a given set of observations, range may be defined as the difference between the largest and
smallest of observations. Thus if L and S denote the largest and smallest observations respectively
then we have
Range = L –S
The second important characteristic of a distribution is given by dispersion. Two distributions may be
identical in respect of its first important characteristic i.e. central tendency and yet they may differ on
account of scatterness.
Range
For a given set of observations, range may be defined as the difference between the largest and
smallest of observations. Thus if L and S denote the largest and smallest observations respectively
then we have
Range = L –S
Quartile:
Quartiles are the values (or observations ) which divide the data set into 4 equal parts . Thus there are 3 quartiles Q
1, Q
2
, Q
3and four Quarters .
Q
1is the lower quartile or first quartile.
Q
2 ismiddle quartile . it is same as median.
Q
3 is the upper quartile or third quartile.
Quartile of ungrouped data : Quartile of grouped data :
Q
1, =
�+1
4
�ℎ
��������??????��Q
1= L +
�
4
−�
�
x h
Q
2 = (
�+1
4
)
�ℎ
��������??????��, Q
2= L +
�
2
−�
�
x h
Q
3 = 3(
�+1
4
)
�ℎ
��������??????��Q
3= L +
3�
4
−�
�
x h
Quartile deviation is a statistic that measure the deviation in the middle of the data .It is also referred as semi-inter–
quartile range which is given by
Q. D. =
�3−�1
2
Quartiles are the values (or observations ) which divide the data set into 4 equal parts . Thus there are 3 quartiles Q
1, Q
2
, Q
3and four Quarters .
Q
1is the lower quartile or first quartile.
Q
2 ismiddle quartile . it is same as median.
Q
3 is the upper quartile or third quartile.
Quartile of ungrouped data : Quartile of grouped data :
Q
1, =
�+1
4
�ℎ
��������??????��Q
1= L +
�
4
−�
�
x h
Q
2 = (
�+1
4
)
�ℎ
��������??????��, Q
2= L +
�
2
−�
�
x h
Q
3 = 3(
�+1
4
)
�ℎ
��������??????��Q
3= L +
3�
4
−�
�
x h
Quartile deviation is a statistic that measure the deviation in the middle of the data .It is also referred as semi-inter–
quartile range which is given by
Q. D. =
�3−�1
2
Mean Deviation :
mean deviation about the mean
For ungrouped data :: MD(ҧ�) =
Σ|�
??????−ഥ�|
�
, where n is the no. of observation
For Grouped/Descretedata :: MD(ҧ�) =
Σfi|�
??????−ഥ�|
�
, where n = Σfi
And, mean deviation about the median
For ungrouped data :: MD(M) =
Σ|�
??????−�|
�
, where n is the no. of observation
For Grouped/ Descretedata :: MD(M) =
Σfi|�
??????−�|
�
, where n = Σfi
Limitations of mean deviation:
i. Mean deviation from the median is not fully reliable measure of dispersion where there is a high degree of variability, as
the median is not a representative central tendency measure in such a case.
ii. Also, as absolute value of deviations are taken, further algebraic treatment is not possible.
iii. The sum of absolute deviations from the mean is more than the sum of the absolute deviations from the median. In some
cases, this is not a reliable measure
mean deviation about the mean
For ungrouped data :: MD(ҧ�) =
Σ|�
??????−ഥ�|
�
, where n is the no. of observation
For Grouped/Descretedata :: MD(ҧ�) =
Σfi|�
??????−ഥ�|
�
, where n = Σfi
And, mean deviation about the median
For ungrouped data :: MD(M) =
Σ|�
??????−�|
�
, where n is the no. of observation
For Grouped/ Descretedata :: MD(M) =
Σfi|�
??????−�|
�
, where n = Σfi
Limitations of mean deviation:
i. Mean deviation from the median is not fully reliable measure of dispersion where there is a high degree of variability, as
the median is not a representative central tendency measure in such a case.
ii. Also, as absolute value of deviations are taken, further algebraic treatment is not possible.
iii. The sum of absolute deviations from the mean is more than the sum of the absolute deviations from the median. In some
cases, this is not a reliable measure
VARIANCE AND STANDARD DEVIATION:
The mean of squared deviations is called variance, denoted var(x) or ??????²
for ungrouped data:
Var (x) or ??????² =
Σ(�
??????−�)
2
�
0r ??????² =
Σ(�
??????)
2
�
−ҧ�
2
for grouped data:
Var (x) or ??????² =
Σ�
??????(�
??????−�)
2
�
0r ??????² =
Σ�
??????�
??????
2
�
−ҧ�
2
for ungrouped data:
Standard deviation (σ) =
Σ(�
??????−�)
2
�
or σ =
Σ(�
??????)
2
�
−ҧ�
2
For grouped data :
Standard deviation (σ) =
Σ�
??????(�
??????−�)
2
Σ�
??????
or σ =
Σ�
??????�
??????
2
Σ�
??????
−ҧ�
2
Or it can be written as, σ =
1
Σ�
??????
�Σ�
??????�
??????
2
−(Σ�
??????�
??????)
2
Deviation method for standard deviation:
σ =
Σ�
??????
2
Σ�
??????
−(
Σ�
??????
Σ�
??????
)
2
where d
i= x
i–a and a is assumed mean
Step Deviation method for standard deviation:
σ = ℎ
Σ�
??????�
??????
2
Σ�
??????
−(
Σ�
??????�
??????
Σ�
??????
)
2
where t
i=
�
??????−�
ℎ
and h is width of C.I,
The mean of squared deviations is called variance, denoted var(x) or ??????²
for ungrouped data:
Var (x) or ??????² =
Σ(�
??????−�)
2
�
0r ??????² =
Σ(�
??????)
2
�
−ҧ�
2
for grouped data:
Var (x) or ??????² =
Σ�
??????(�
??????−�)
2
�
0r ??????² =
Σ�
??????�
??????
2
�
−ҧ�
2
for ungrouped data:
Standard deviation (σ) =
Σ(�
??????−�)
2
�
or σ =
Σ(�
??????)
2
�
−ҧ�
2
For grouped data :
Standard deviation (σ) =
Σ�
??????(�
??????−�)
2
Σ�
??????
or σ =
Σ�
??????�
??????
2
Σ�
??????
−ҧ�
2
Or it can be written as, σ =
1
Σ�
??????
�Σ�
??????�
??????
2
−(Σ�
??????�
??????)
2
Deviation method for standard deviation:
σ =
Σ�
??????
2
Σ�
??????
−(
Σ�
??????
Σ�
??????
)
2
where d
i= x
i–a and a is assumed mean
Step Deviation method for standard deviation:
σ = ℎ
Σ�
??????�
??????
2
Σ�
??????
−(
Σ�
??????�
??????
Σ�
??????
)
2
where t
i=
�
??????−�
ℎ
and h is width of C.I,
Moments:
A moment is a specific quantitative measure of the shape of a frequency distribution. It describe two important
properties of frequency distribution Skewness and Kurtosis.
Moments about mean (Central moment)
For ungrouped data : ??????
�=
Σ(�
??????−ҧ�)
??????
�
, r = 0,1,2…..
For grouped data : ??????
�=
??????�
??????(�
??????−ҧ�)
??????
�
, r = 0,1,2…..
Remarks ::
1.??????
0=
??????�
??????(�
??????−ҧ�)
0
�
= 1
2.??????
1=
??????�
??????(�
??????−ҧ�)
1
�
=
Σ�
??????�
??????
�
−ҧ�
Σ�
??????
�
= ҧ�−ҧ�=0
3.??????
2=
??????�
??????(�
??????−ҧ�)
2
�
= ??????
2
(second central moment is equal to variance)
A moment is a specific quantitative measure of the shape of a frequency distribution. It describe two important
properties of frequency distribution Skewness and Kurtosis.
Moments about mean (Central moment)
For ungrouped data : ??????
�=
Σ(�
??????−ҧ�)
??????
�
, r = 0,1,2…..
For grouped data : ??????
�=
??????�
??????(�
??????−ҧ�)
??????
�
, r = 0,1,2…..
Remarks ::
1.??????
0=
??????�
??????(�
??????−ҧ�)
0
�
= 1
2.??????
1=
??????�
??????(�
??????−ҧ�)
1
�
=
Σ�
??????�
??????
�
−ҧ�
Σ�
??????
�
= ҧ�−ҧ�=0
3.??????
2=
??????�
??????(�
??????−ҧ�)
2
�
= ??????
2
(second central moment is equal to variance)
SKEWNESS
There are two types of Skewness: (1) Positive Skewness (ii) Negative Skewness
Positive skewness: In frequency curve of a positively skewed distribution, the tail on the side of the curve is longer. In
positively skewed distribution, i.e. Mean > Median > Mod
Negative skewness: In frequency curve of a negatively skewed distribution, the tail on side of the curve is longer. In
negatively skewed distribution, i.e. Mean < Median < Mode
Measures of skewness:
1. Absolute measures of skewness: Absolute measures of skewness tell us the extent of asymmetry and whether it is
positive or negative. The first absolute measure of skewness is based on difference between mean and mode(or median)
Absolute skewness = mean-mode or mean-median
Skewness is positive if mean is greater than mode (or median) and it is negative if mean is less than mode (or median).
The second absolute measure of skewness is based on quartiles.
In a symmetrical distribution Q, and Q, are equidistant from Q₂ ie. Q
3-Q
2= Q₂-Q
1, but if Q
3-Q₂ ≠ Q₂-Q
1 , then distribution
is asymmetrical. In this case the absolute measure of skewness can be measured by the following formula:
Absolute skewness = Q
3+ Q
1-2Q
2
If Q
3-Q
2> Q
2-Q
1, then skewness is positive and if Q
3-Q₂ <Q
2-Q
1, then skewness is negative.
There are two types of Skewness: (1) Positive Skewness (ii) Negative Skewness
Positive skewness: In frequency curve of a positively skewed distribution, the tail on the side of the curve is longer. In
positively skewed distribution, i.e. Mean > Median > Mod
Negative skewness: In frequency curve of a negatively skewed distribution, the tail on side of the curve is longer. In
negatively skewed distribution, i.e. Mean < Median < Mode
Measures of skewness:
1. Absolute measures of skewness: Absolute measures of skewness tell us the extent of asymmetry and whether it is
positive or negative. The first absolute measure of skewness is based on difference between mean and mode(or median)
Absolute skewness = mean-mode or mean-median
Skewness is positive if mean is greater than mode (or median) and it is negative if mean is less than mode (or median).
The second absolute measure of skewness is based on quartiles.
In a symmetrical distribution Q, and Q, are equidistant from Q₂ ie. Q
3-Q
2= Q₂-Q
1, but if Q
3-Q₂ ≠ Q₂-Q
1 , then distribution
is asymmetrical. In this case the absolute measure of skewness can be measured by the following formula:
Absolute skewness = Q
3+ Q
1-2Q
2
If Q
3-Q
2> Q
2-Q
1, then skewness is positive and if Q
3-Q₂ <Q
2-Q
1, then skewness is negative.
2. Relative measures of skewness:
(1)Karl Pearson's Coefficient of Skewness: This method is frequently used for measuring skewness. Karl Pearson's coefficient of skewness
is denoted by �
�
??????
and is defined as
�
�
??????
=
����−����
�����������??????��??????��
0r =
3(����−���??????��)
�����������??????��??????��
If �
�
??????
>0, then distribution is positively skewed.
If �
�
??????
= 0, then distribution is symmetrical.
If �
�??????
, <0, then distribution is negatively skewed.
(2) Bowley's coefficient of skewness
�
�??????
=
�3+�1−2���??????��
�
3−�
1
=
�3+�1−2�2
�
3−�
1
If �
�
??????
>0, then distribution is positively skewed.
If �
�
??????
= 0, then distribution is symmetrical.
If �
�??????
<0, then distribution is negatively skewed.
(3) Moment coefficient of skewness
�
1=±�
1=
??????
3
??????
2
3
,
generally skewness is measured in terms of �
1=
??????
3
2
??????
2
3where sign of skewness is determined by the sign of ??????
3
(1)Karl Pearson's Coefficient of Skewness: This method is frequently used for measuring skewness. Karl Pearson's coefficient of skewness
is denoted by �
�
??????
and is defined as
�
�
??????
=
����−����
�����������??????��??????��
0r =
3(����−���??????��)
�����������??????��??????��
If �
�
??????
>0, then distribution is positively skewed.
If �
�
??????
= 0, then distribution is symmetrical.
If �
�??????
, <0, then distribution is negatively skewed.
(2) Bowley's coefficient of skewness
�
�??????
=
�3+�1−2���??????��
�
3−�
1
=
�3+�1−2�2
�
3−�
1
If �
�
??????
>0, then distribution is positively skewed.
If �
�
??????
= 0, then distribution is symmetrical.
If �
�??????
<0, then distribution is negatively skewed.
(3) Moment coefficient of skewness
�
1=±�
1=
??????
3
??????
2
3
,
generally skewness is measured in terms of �
1=
??????
3
2
??????
2
3where sign of skewness is determined by the sign of ??????
3
KURTOSIS
Moment Coefficient of Kurtosis : A measure of kurtosis is based on central moments given by Karl Pearson.
So, moment coefficient of kurtosis is also known as Pearson's second beta (�
2) coefficient and is defined as
�
1=
??????
4
??????
2
2
where u
2and µ
4are second and fourth central moments respectively. The value of �
2measures the degree of
peakedness.
If �
2= 3 then it is normal curve or mesokurtic curve.
If �
2> 3 the curve is more peaked than mesokurtic curve then it is called leptokurtic curve
If �
2<3 the curve is less peaked than mesokurtic curve and is called platykuritccurve
Moment Coefficient of Kurtosis : A measure of kurtosis is based on central moments given by Karl Pearson.
So, moment coefficient of kurtosis is also known as Pearson's second beta (�
2) coefficient and is defined as
�
1=
??????
4
??????
2
2
where u
2and µ
4are second and fourth central moments respectively. The value of �
2measures the degree of
peakedness.
If �
2= 3 then it is normal curve or mesokurtic curve.
If �
2> 3 the curve is more peaked than mesokurtic curve then it is called leptokurtic curve
If �
2<3 the curve is less peaked than mesokurtic curve and is called platykuritccurve
Percentile rank
Percentile rank for ungrouped data
P
R=
�+0.5��
�
�100
where L= number of observations (scores) less than x
i
E= number of observations (scores) equal to x
i
n= total number of observations (scores)
Percentile rank for grouped data (discontinuous)
P
R=
�+0.5�
�
�100
where C = cumulative frequency preceding the score x
f= frequency of the score x,
n=sum of all the frequencies
Percentile rank for grouped data (continuous)
P
R=(C +
�
ℎ
��)
100
�
where C = cumulative frequency preceding the percentile class
f=frequency of the percentile class
x= score whose percentile rank is required -lower limit of percentile class
h = length of class interval
n = sum of all the frequencies.
Percentile rank for ungrouped data
P
R=
�+0.5��
�
�100
where L= number of observations (scores) less than x
i
E= number of observations (scores) equal to x
i
n= total number of observations (scores)
Percentile rank for grouped data (discontinuous)
P
R=
�+0.5�
�
�100
where C = cumulative frequency preceding the score x
f= frequency of the score x,
n=sum of all the frequencies
Percentile rank for grouped data (continuous)
P
R=(C +
�
ℎ
��)
100
�
where C = cumulative frequency preceding the percentile class
f=frequency of the percentile class
x= score whose percentile rank is required -lower limit of percentile class
h = length of class interval
n = sum of all the frequencies.
COVARIANCE OF X AND Y
Definition. The mean of the product of deviation scores (x-ҧ�) and (y-ത�) is called the covariance of X and Y i.e
Cov(X, Y) or C
XY=
Σ(�−ҧ�)(�−ത�)
�
or ……………….(i)
Cov(X, Y) =
1
�
[Σ��−
1
�
Σ�Σ�]………………..(ii)
If x, y are small numbers, it is easier to calculate Coy (X, Y) using formula (ii); if x-ҧ�, y-ത�are small fractionlessnumbers, it
is easy to use formula (i); in other cases, we can assume means A and B, use u= x-A, v = y-B, then
CoviX, Y) =
1
�
[Σuv−
1
�
ΣuΣv]
Definition. The mean of the product of deviation scores (x-ҧ�) and (y-ത�) is called the covariance of X and Y i.e
Cov(X, Y) or C
XY=
Σ(�−ҧ�)(�−ത�)
�
or ……………….(i)
Cov(X, Y) =
1
�
[Σ��−
1
�
Σ�Σ�]………………..(ii)
If x, y are small numbers, it is easier to calculate Coy (X, Y) using formula (ii); if x-ҧ�, y-ത�are small fractionlessnumbers, it
is easy to use formula (i); in other cases, we can assume means A and B, use u= x-A, v = y-B, then
CoviX, Y) =
1
�
[Σuv−
1
�
ΣuΣv]
KARL PEARSON'S COEFFICIENT OF CORRELATION
Though covariance is independent of the choice of the origin, it depends on the scale of measurement. To standardize it
further, we use the following formula for Karl Pearson's coefficient of correlation (sometimes called Product Moment
Correlation).
r or??????�,�=
���(�,�)
���(�)���(�)
=
���(�,�)
??????�??????�
coefficient of correlation is independent of choice of origin and scale. Note that -1 ≤r ≤ 1
if x-ҧ�, y-ത�are small fractionlessnumbers, we use
r =
Σ(�−ҧ�)(�−ത�)
Σ(�−ҧ�)
2
Σ(�−ത�)
2
if x and y are small numbers, we use
r =
Σ��−
1
??????
Σ�Σ�
Σ�
2
−
1
??????
(Σ�)
2
Σ�
2
−
1
??????
(Σy)
2
otherwise we use assumed mean A and B and u = x-A and v = y-B
r =
Σuv−
1
??????
ΣuΣv
Σ�
2
−
1
??????
(Σu)
2
Σ�
2
−
1
??????
(Σv)
2
Though covariance is independent of the choice of the origin, it depends on the scale of measurement. To standardize it
further, we use the following formula for Karl Pearson's coefficient of correlation (sometimes called Product Moment
Correlation).
r or??????�,�=
���(�,�)
���(�)���(�)
=
���(�,�)
??????�??????�
coefficient of correlation is independent of choice of origin and scale. Note that -1 ≤r ≤ 1
if x-ҧ�, y-ത�are small fractionlessnumbers, we use
r =
Σ(�−ҧ�)(�−ത�)
Σ(�−ҧ�)
2
Σ(�−ത�)
2
if x and y are small numbers, we use
r =
Σ��−
1
??????
Σ�Σ�
Σ�
2
−
1
??????
(Σ�)
2
Σ�
2
−
1
??????
(Σy)
2
otherwise we use assumed mean A and B and u = x-A and v = y-B
r =
Σuv−
1
??????
ΣuΣv
Σ�
2
−
1
??????
(Σu)
2
Σ�
2
−
1
??????
(Σv)
2
SPEARMAN'S RANK CORRELATION COEFFICIENT
Sometimes, it is difficult to give numerical values to a quality eg.honesty, beauty, intelligence etc. Sometimes, though it
may be possible to quantify the variable, we may chose to grade it in terms of ranks, by using numbers 1, 2, …., n
Assigning rank 1 to the highest (or lowest) value and rank 2 to the next highest (or next lowest) value and so on. If two
corresponding sets of values x and y are ranked in such manner, the Edward Spearman's coefficient of rank correlation,
denoted by r
rankor as r, is given by
r = 1 -
6Σ�
2
�(�
2
−1)
where d = difference between ranks of corresponding x and y
Sometimes, either series x or y may have common ranks. Then we use average rank for these items, and add a correction
to Σ�
2
to calculate r. If m items have common rank, the correction
1
12
(m
3
-m) is added to Σ�
2
. If more ties occur for ranks,
more corrections are added.
r = 1 -
6Σ�
2
+
1
12
�
3
−m+
1
12
�
3
−m…..
�(�
2
−1)
Sometimes, it is difficult to give numerical values to a quality eg.honesty, beauty, intelligence etc. Sometimes, though it
may be possible to quantify the variable, we may chose to grade it in terms of ranks, by using numbers 1, 2, …., n
Assigning rank 1 to the highest (or lowest) value and rank 2 to the next highest (or next lowest) value and so on. If two
corresponding sets of values x and y are ranked in such manner, the Edward Spearman's coefficient of rank correlation,
denoted by r
rankor as r, is given by
r = 1 -
6Σ�
2
�(�
2
−1)
where d = difference between ranks of corresponding x and y
Sometimes, either series x or y may have common ranks. Then we use average rank for these items, and add a correction
to Σ�
2
to calculate r. If m items have common rank, the correction
1
12
(m
3
-m) is added to Σ�
2
. If more ties occur for ranks,
more corrections are added.
r = 1 -
6Σ�
2
+
1
12
�
3
−m+
1
12
�
3
−m…..
�(�
2
−1)
Compound interest
and annuity
and annuity
INTEREST: It is the additional money besides the original money paid by the borrower to the money lender (bank, financial
agency or individual) in lieu of the money used by him.
PRINCIPAL: The money borrowed (or the money lent) is called principal.
AMOUNT: The sum of the principal and the interest is called amount.
Thus, amount principal + interest
RATE : It is the interest paid on 100 for a specified period.
TIME : It is the time for which the money is borrowed.
There are 2 type of interest
1. SIMPLE INTEREST : It is the interest calculated on the original money (principal) for any given time and rate
Simple Interest =
PrincipalxRatexTime
100
2. COMPOUND INTEREST
At the end of the first year (or any other fixed period), if the interest accrued is not paid to the money lender but is added to
the principal, then this amount becomes the principal for the next year (or any other fixed period) and so on. This process is
repeated until amount for the whole time is found.
The difference between the final amount and the (original) principal is called compound interest.
agency or individual) in lieu of the money used by him.
PRINCIPAL: The money borrowed (or the money lent) is called principal.
AMOUNT: The sum of the principal and the interest is called amount.
Thus, amount principal + interest
RATE : It is the interest paid on 100 for a specified period.
TIME : It is the time for which the money is borrowed.
There are 2 type of interest
1. SIMPLE INTEREST : It is the interest calculated on the original money (principal) for any given time and rate
Simple Interest =
PrincipalxRatexTime
100
2. COMPOUND INTEREST
At the end of the first year (or any other fixed period), if the interest accrued is not paid to the money lender but is added to
the principal, then this amount becomes the principal for the next year (or any other fixed period) and so on. This process is
repeated until amount for the whole time is found.
The difference between the final amount and the (original) principal is called compound interest.
Compound interest and Amount formula
1. CI= P [1+
�
100
�
−1] and A= P1+
�
100
�
2. When the rates of interest for the successive fixed periods are r1%, r2% r3% then amount
A is given by A = P(1+
�1
100
)(1+
�2
100
)……….(1+
��
100
)
3. S.I. (simple interest) and C.I. are equal for the first conversion period on the same sum and at the same rate.
4. C.I of 2nd conversion period is more than the C.I. of 1st conversion period and C.I. of 2nd conversion period -C.I. of 1st
conversion period = C.I. on the interest of the first conversion period.
5. When the total time is not a complete number of conversion periods, we consider simple interest for the last partial period.
For example, if time is 2 years 5 months and the interest is % per annum compounded annually, then
A= P [1+
�
100
2
1+
5
12
�
100
6. Equivalent, nominal and effective rates of interest Two annual rates of interest with different conversion periods are called
equivalent if they yield the same compound amount at the end of the year.
If nominal rate is % compounded p times in year, then effective rate of interest per rupee annually is 1+
�
100�
�
−1
7. Real interest rate
A real interest rate is the interest rate that takes inflation into consideration. It is also called inflation adjusted interest rate.
Real interest rate = nominal interest rate -actual inflation rate.
8. Equal installment (with compound interest)
Loan amount =P [
100
100+�
+
100
100+�
2
+
100
100+�
3
]
1. CI= P [1+
�
100
�
−1] and A= P1+
�
100
�
2. When the rates of interest for the successive fixed periods are r1%, r2% r3% then amount
A is given by A = P(1+
�1
100
)(1+
�2
100
)……….(1+
��
100
)
3. S.I. (simple interest) and C.I. are equal for the first conversion period on the same sum and at the same rate.
4. C.I of 2nd conversion period is more than the C.I. of 1st conversion period and C.I. of 2nd conversion period -C.I. of 1st
conversion period = C.I. on the interest of the first conversion period.
5. When the total time is not a complete number of conversion periods, we consider simple interest for the last partial period.
For example, if time is 2 years 5 months and the interest is % per annum compounded annually, then
A= P [1+
�
100
2
1+
5
12
�
100
6. Equivalent, nominal and effective rates of interest Two annual rates of interest with different conversion periods are called
equivalent if they yield the same compound amount at the end of the year.
If nominal rate is % compounded p times in year, then effective rate of interest per rupee annually is 1+
�
100�
�
−1
7. Real interest rate
A real interest rate is the interest rate that takes inflation into consideration. It is also called inflation adjusted interest rate.
Real interest rate = nominal interest rate -actual inflation rate.
8. Equal installment (with compound interest)
Loan amount =P [
100
100+�
+
100
100+�
2
+
100
100+�
3
]
ANNUITY
An annuity is a sequence of equal payments made at equal intervals of time.
Types of Annuities
Annuities can be classified in several ways.
Classification on the basis of payment term
(i) Annuity certain. An annuity certain is an annuity whose term begins and ends at certain fixed dates.
For example. Purchase of different products on instalments, Recurring deposits in a Bank or Post Office are the examples of
annuity certain.
(ii) Contingent annuity. A contingent annuity is an annuity whose payments continue for an uncertain period of time depending
upon the occurrence of an event the date of which cannot be accurately told in advance. For example. Purchase an insurance
policy for the marriage of his/her daughter.
(iii) Perpetual annuity or Perpetuity. A perpetual annuity is an annuity whose payment begins at a fixed date but continue
forever. For example. In college endowment funds the interest earned is used for scholarships.
Classification on the basis of time of payment
(i) Regular annuity or ordinary annuity. A regular annuity is an annuity in which payments are made at the end of each payment
period. For example. Repayment of home loan is a regular annuity.
(ii) Annuity due. An annuity due is an annuity in which payments are made at the beginning of each payment period. For
example. A recurring deposit is an annuity due (ii) Deferred annuity. A
An annuity is a sequence of equal payments made at equal intervals of time.
Types of Annuities
Annuities can be classified in several ways.
Classification on the basis of payment term
(i) Annuity certain. An annuity certain is an annuity whose term begins and ends at certain fixed dates.
For example. Purchase of different products on instalments, Recurring deposits in a Bank or Post Office are the examples of
annuity certain.
(ii) Contingent annuity. A contingent annuity is an annuity whose payments continue for an uncertain period of time depending
upon the occurrence of an event the date of which cannot be accurately told in advance. For example. Purchase an insurance
policy for the marriage of his/her daughter.
(iii) Perpetual annuity or Perpetuity. A perpetual annuity is an annuity whose payment begins at a fixed date but continue
forever. For example. In college endowment funds the interest earned is used for scholarships.
Classification on the basis of time of payment
(i) Regular annuity or ordinary annuity. A regular annuity is an annuity in which payments are made at the end of each payment
period. For example. Repayment of home loan is a regular annuity.
(ii) Annuity due. An annuity due is an annuity in which payments are made at the beginning of each payment period. For
example. A recurring deposit is an annuity due (ii) Deferred annuity. A
AMOUNT OF A REGULAR ANNUITY
The amount or future value of an annuity is the sum of all payments made and the compound interest earned on them at the
end of the term i.e. sum of compound amounts of all the payments made at the end of the term.
A = R [
(1+??????)
�
−1
??????
] =??????�
ത�/??????
The present value or capital value of an annuity is the sum of present values of all the payments.
P = R [
1−(1+??????)
−�
−1
??????
] =??????�
ത�/??????
AMOUNT AND PRESENT VALUE OF ANNUITY DUE
In annuity due payment is made at the beginning of each payment period.
The amount or future value of annuity due is sum of compound of all the payment made
A = R (1+i)[
(1+??????)
�
−1
??????
] =??????�
�+1/??????−1
The present value or capital value of annuity due is sum of present value of all the payment made
P = R (1+i) [
1−(1+??????)
−�
−1
??????
] =??????(1+�
�−1/??????)
AMOUNT AND PRESENT VALUE OF DEFERRED ANNUITY
In deferred annuity , payment start after deferring m intervals , so the first payment made at end of (m+1)th interval
A = R [
(1+??????)
�
−1
??????
] =??????�
ത�/??????
P = R (1+??????)
−�
[
1−(1+??????)
−�
??????
] =??????(�
�+�/??????−�
ഥ�/??????)
The amount or future value of an annuity is the sum of all payments made and the compound interest earned on them at the
end of the term i.e. sum of compound amounts of all the payments made at the end of the term.
A = R [
(1+??????)
�
−1
??????
] =??????�
ത�/??????
The present value or capital value of an annuity is the sum of present values of all the payments.
P = R [
1−(1+??????)
−�
−1
??????
] =??????�
ത�/??????
AMOUNT AND PRESENT VALUE OF ANNUITY DUE
In annuity due payment is made at the beginning of each payment period.
The amount or future value of annuity due is sum of compound of all the payment made
A = R (1+i)[
(1+??????)
�
−1
??????
] =??????�
�+1/??????−1
The present value or capital value of annuity due is sum of present value of all the payment made
P = R (1+i) [
1−(1+??????)
−�
−1
??????
] =??????(1+�
�−1/??????)
AMOUNT AND PRESENT VALUE OF DEFERRED ANNUITY
In deferred annuity , payment start after deferring m intervals , so the first payment made at end of (m+1)th interval
A = R [
(1+??????)
�
−1
??????
] =??????�
ത�/??????
P = R (1+??????)
−�
[
1−(1+??????)
−�
??????
] =??????(�
�+�/??????−�
ഥ�/??????)
TAXATION
The taxes can be classified in two main categories:
1. Indirect tax2. Direct tax
Indirect tax. The tax which doesn’t effects the person or group of persons directly is called indirect tax. For example, Value added tax (VAT),
excise duty, custom duty , sales tax, service tax, etc. From year 2017 all these indirect taxes are replaced by Goods and Services Tax (GST
Direct tax. The tax which is effect directly on a tax payer is called direct tax. For example income tax, property tax, gift tax etc.
GOODS AND SERVICES TAX
Goods and Services Tax (abbreviated GST) is an indirect tax Imposed on the supply of goods and services
terms related to GST
1. Dealer-person who buys goods or services for resale is known as a dealer (or trader).
2. Intra-state sales. Sales within the same state (or Union Territory) are called
intra-state sales.
3. Inter-state sales. Sales outside the state (or Union Territory) are called inter-state sales.
4. GST paid by a dealer is called Input GST and GST collected from a customer is called Output GST.
5. There are three taxes applicable under GST:
(i) Central Goods and Services Tax (CGST),
(ii) State Goods and Services Tax (SGST) or
Union Territory Goods and Services Tax (UTGST).
In intra-state sales, GST is divided equally among Centre and State Governments.
(iii) Integrated Goods and Services Tax (IGST).
IGST is imposed on inter-state sales ieon sales of goods and services outside the state. It is also imposed on import of goods and services into
India and on export of goods and services outside India. IGST goes to the Central Government.
Advantages of GST
(i) Ease of doing business (ii) Reduce tax elusion:
1. Indirect tax2. Direct tax
Indirect tax. The tax which doesn’t effects the person or group of persons directly is called indirect tax. For example, Value added tax (VAT),
excise duty, custom duty , sales tax, service tax, etc. From year 2017 all these indirect taxes are replaced by Goods and Services Tax (GST
Direct tax. The tax which is effect directly on a tax payer is called direct tax. For example income tax, property tax, gift tax etc.
GOODS AND SERVICES TAX
Goods and Services Tax (abbreviated GST) is an indirect tax Imposed on the supply of goods and services
terms related to GST
1. Dealer-person who buys goods or services for resale is known as a dealer (or trader).
2. Intra-state sales. Sales within the same state (or Union Territory) are called
intra-state sales.
3. Inter-state sales. Sales outside the state (or Union Territory) are called inter-state sales.
4. GST paid by a dealer is called Input GST and GST collected from a customer is called Output GST.
5. There are three taxes applicable under GST:
(i) Central Goods and Services Tax (CGST),
(ii) State Goods and Services Tax (SGST) or
Union Territory Goods and Services Tax (UTGST).
In intra-state sales, GST is divided equally among Centre and State Governments.
(iii) Integrated Goods and Services Tax (IGST).
IGST is imposed on inter-state sales ieon sales of goods and services outside the state. It is also imposed on import of goods and services into
India and on export of goods and services outside India. IGST goes to the Central Government.
Advantages of GST
(i) Ease of doing business (ii) Reduce tax elusion:
INCOME TAX
It is a tax imposed by central govt on the income of the individuals , companies , firm etc.
Terms related to Income tax
1.Financial Year -1st April of current year to 31
st
March of next Year.
2.Assessment year –The next year after the Financial year
3.Gross income –Total income earned from different sources in a financial year
4.Exemption and deduction
5.Taxable income –It is the income used to calculate the income tax in the given financial year. By subtracting the exemptions
and deductions allowed from the gross income we can find it
6.Income Tax Slab
7.Tax rebate under section 87A-A tax rebate under section 87A is allowed to individual tax payers a maximum amount of
12500 for taxable income upto25 lakh for Financial Year 2019-20
8. Health and Education Cess. A cessis a form of tax collected by the government for the development or welfare of Health
and Education sectors. At present 4% health and education cessis imposed on the income tax calculated.
Method to Calculate the Income Tax
(i) Find the gross income.
(ii) Subtract the exemptions HRA (if eligible) and standard deduction from the gross income.
(iii) Subtract all the deductions under different sections such as 80C, 80D, 80E, 80G etc. from the balance obtained in (if) to
obtain taxable income.
(iv) Calculate the income tax on taxable income as per the slab.
(v) Subtract tax relief under section 87A (if eligible) from the income tax obtained in (iv).
(vi) Add 4% of income tax obtained in (v) as health and education cess
It is a tax imposed by central govt on the income of the individuals , companies , firm etc.
Terms related to Income tax
1.Financial Year -1st April of current year to 31
st
March of next Year.
2.Assessment year –The next year after the Financial year
3.Gross income –Total income earned from different sources in a financial year
4.Exemption and deduction
5.Taxable income –It is the income used to calculate the income tax in the given financial year. By subtracting the exemptions
and deductions allowed from the gross income we can find it
6.Income Tax Slab
7.Tax rebate under section 87A-A tax rebate under section 87A is allowed to individual tax payers a maximum amount of
12500 for taxable income upto25 lakh for Financial Year 2019-20
8. Health and Education Cess. A cessis a form of tax collected by the government for the development or welfare of Health
and Education sectors. At present 4% health and education cessis imposed on the income tax calculated.
Method to Calculate the Income Tax
(i) Find the gross income.
(ii) Subtract the exemptions HRA (if eligible) and standard deduction from the gross income.
(iii) Subtract all the deductions under different sections such as 80C, 80D, 80E, 80G etc. from the balance obtained in (if) to
obtain taxable income.
(iv) Calculate the income tax on taxable income as per the slab.
(v) Subtract tax relief under section 87A (if eligible) from the income tax obtained in (iv).
(vi) Add 4% of income tax obtained in (v) as health and education cess
Utility Bills
UTILITY BILL
A utility bill is a detailed invoice issued and paid once a month for utilities such as electricity, water, natural gas, telephone
etc.
It bears the basic information such as account number of consumer , bill number, address, billing address, time period,
date of issue, due date, units of usage, tariff rates, surcharge, tax
TARIFF RATE
Tariff rate consists of two parts:
(i) Fixed charge. It does not depend on consumption.
(ii) Variable charge. It depends on consumption.
FIXED CHARGE
Fixed charge is a part of the utility bill which a consumer has to pay even if the consumer does not use the utility.
VARIABLE CHARGE
Variable charge has to pay according to usage of services
SURCHARGE
Surcharge is an additional fee imposed on a consumer in addition to the standard basic rates of thutility. Surcharges are
also referred to by other terms such as riders, adjustment factor, recover cost etc.
Some surcharges that appear on an electricity bill are given below:
(i) Fuel Surcharge or Fuel Adjustment Charges (FAC)
(ii) Peak-hour Surcharge
(ii) AC Surcharge
(iv) Recovery Surcharge
SERVICE CHARGE
It is an amount that is added to consumer's bill for the work of the person who comes and serves the consumer.
A utility bill is a detailed invoice issued and paid once a month for utilities such as electricity, water, natural gas, telephone
etc.
It bears the basic information such as account number of consumer , bill number, address, billing address, time period,
date of issue, due date, units of usage, tariff rates, surcharge, tax
TARIFF RATE
Tariff rate consists of two parts:
(i) Fixed charge. It does not depend on consumption.
(ii) Variable charge. It depends on consumption.
FIXED CHARGE
Fixed charge is a part of the utility bill which a consumer has to pay even if the consumer does not use the utility.
VARIABLE CHARGE
Variable charge has to pay according to usage of services
SURCHARGE
Surcharge is an additional fee imposed on a consumer in addition to the standard basic rates of thutility. Surcharges are
also referred to by other terms such as riders, adjustment factor, recover cost etc.
Some surcharges that appear on an electricity bill are given below:
(i) Fuel Surcharge or Fuel Adjustment Charges (FAC)
(ii) Peak-hour Surcharge
(ii) AC Surcharge
(iv) Recovery Surcharge
SERVICE CHARGE
It is an amount that is added to consumer's bill for the work of the person who comes and serves the consumer.
INTERPRETATION OF ELECTRICITY BILLS
Electricity bill is determined by three elements.
1. Number of units consumed 2. Tariff category of the consumer 3. Fixed charges, Surcharge and Energy tax.
Number of units consumed means 1 kWh = 1 unit
Tariff category Tariff rates are mainly divided into two categories (i) Domestic tariff (ii) Commercial tariff.
Water Charge
It is based on consumption of water. Water consumption is measured in m³ or kilolitres(KL). Water charges are based on
two components:
(i) volume of water consumed
(ii) a set of charges other than water consumption such as meter rent, water cess, license fee maintenance charge,
development charge etc. A variant which is more common is to use a mix of two Le. fixed monthly charge (or service of an
Increasing Block Tariff (IBT).
INTERPRETATION OF PIPED NATURAL GAS (PNG) BILL
Piped natural gas bill consists of three main elements.
1. Connection Fee and Security Deposit It is charged at the time of new connection. Connection fee is non-refundable
whereas security deposit is refundable.
2. Network Tariff
3. Gas Consumption Charge
Electricity bill is determined by three elements.
1. Number of units consumed 2. Tariff category of the consumer 3. Fixed charges, Surcharge and Energy tax.
Number of units consumed means 1 kWh = 1 unit
Tariff category Tariff rates are mainly divided into two categories (i) Domestic tariff (ii) Commercial tariff.
Water Charge
It is based on consumption of water. Water consumption is measured in m³ or kilolitres(KL). Water charges are based on
two components:
(i) volume of water consumed
(ii) a set of charges other than water consumption such as meter rent, water cess, license fee maintenance charge,
development charge etc. A variant which is more common is to use a mix of two Le. fixed monthly charge (or service of an
Increasing Block Tariff (IBT).
INTERPRETATION OF PIPED NATURAL GAS (PNG) BILL
Piped natural gas bill consists of three main elements.
1. Connection Fee and Security Deposit It is charged at the time of new connection. Connection fee is non-refundable
whereas security deposit is refundable.
2. Network Tariff
3. Gas Consumption Charge
Straight lines
1. Distance between two points : (�
2−�
1)
2
+(�
2−�
1)
2
2. Section formula If point divides a line segment internally in the ratio m:n then,
Point is given by (
��
2+��
1
�+�
,
��
2+��
1
�+�
)
3.Section formula If point divides a line segment externally in the ratio m:n then,
Point is given by (
��2−��1
�−�
,
��2−��1
�−�
)
4.Centroid oftriangle(
�
1+�
2+�
3
3
,
�
1+�
2+�
3
3
)
5.Areaoftriangle½ |�
1(�
2−�
3)+�
2(�
3−�
1)+�
3(�
1−�
2)|
6.If giventhreepointsarecollinearthen area of triangle is 0
7.Slope ofline=tan??????=
�2−�1
�2−�1
8.Slope ofxaxisis0andslopeofyaxisisnotdefined
9. If two lines are parallel then �
1= �
2
10.If twolinesareperpendicularthe�
1x �
2= -1
11. Angle betweentwolines,tan??????=|
�
2−�
1
1+�1�2
|
2−�
1)
2
+(�
2−�
1)
2
2. Section formula If point divides a line segment internally in the ratio m:n then,
Point is given by (
��
2+��
1
�+�
,
��
2+��
1
�+�
)
3.Section formula If point divides a line segment externally in the ratio m:n then,
Point is given by (
��2−��1
�−�
,
��2−��1
�−�
)
4.Centroid oftriangle(
�
1+�
2+�
3
3
,
�
1+�
2+�
3
3
)
5.Areaoftriangle½ |�
1(�
2−�
3)+�
2(�
3−�
1)+�
3(�
1−�
2)|
6.If giventhreepointsarecollinearthen area of triangle is 0
7.Slope ofline=tan??????=
�2−�1
�2−�1
8.Slope ofxaxisis0andslopeofyaxisisnotdefined
9. If two lines are parallel then �
1= �
2
10.If twolinesareperpendicularthe�
1x �
2= -1
11. Angle betweentwolines,tan??????=|
�
2−�
1
1+�1�2
|
Equation of lines
1. Equation of Horizontal line: y = a
2.Equation of vertical line: x = a
3.Point slope form : y-�
1=�(�−�
1)
4.Two pointform:y-�
1=
�2−�1
�2−�1
(�−�
1)
5.Interceptform:
�
�
+
�
�
=1where a and b are x and y intercept
6.Slopeinterceptform
(i) if y intercept is given , then y = mx + c
(ii) if x intercept is given , then y = m(x –d)
7.Normalform : xcosα+ ysinα= p , where αis the angle with positive direction of x axis and p is length of perpendicular
General Equation of lines
Standard form of equation of line is Ax + By + C = 0
1.Slope of line (m) =
−�
�
2. X intercept =
−�
�
, y intercept =
−�
�
3. Cosα=
�
�
�
+�
�
, sinα=
�
�
�
+�
�
p =
�
�
�
+�
�
Distance of a point from a line :
D =
|���+���+�|
�
�
+�
�
Distance between two parallel line :
D =
|��−��|
�
�
+�
�
1. Equation of Horizontal line: y = a
2.Equation of vertical line: x = a
3.Point slope form : y-�
1=�(�−�
1)
4.Two pointform:y-�
1=
�2−�1
�2−�1
(�−�
1)
5.Interceptform:
�
�
+
�
�
=1where a and b are x and y intercept
6.Slopeinterceptform
(i) if y intercept is given , then y = mx + c
(ii) if x intercept is given , then y = m(x –d)
7.Normalform : xcosα+ ysinα= p , where αis the angle with positive direction of x axis and p is length of perpendicular
General Equation of lines
Standard form of equation of line is Ax + By + C = 0
1.Slope of line (m) =
−�
�
2. X intercept =
−�
�
, y intercept =
−�
�
3. Cosα=
�
�
�
+�
�
, sinα=
�
�
�
+�
�
p =
�
�
�
+�
�
Distance of a point from a line :
D =
|���+���+�|
�
�
+�
�
Distance between two parallel line :
D =
|��−��|
�
�
+�
�
CIRCLE AND PARABOLA
CIRCLE
A circle is the set of all points in a plane, each of which is at a constant distance from a fixed point in the plane
Standard form :
Equation of a circle is simplest if its centreis at the origin.
x²+ y²= r²
Central form
Let C (h, k) be the centreof the circle and r(>0) be its radius. Let P(x, y) be a point in the plane, then P lies on the circle
(x-h)²+(y-k)²= r²,
Diameter form
Let A(x, y) and B(x, y) be the extremities of a diameter of the
(x-x
1) (x-x
2)+(y-y
1) (y-y
2)=0
General form
x
2
+ y² + 2gx + 2fy + c=0, where g = h, f = k and c = h²+k
2
-r
2
the equation x
2
+ y² +2gx+2fy + c = 0 represents a circle if g²+f
2
-c >0.
Its centreis (-g-f) and radius = g²+f
2
−c
A circle is the set of all points in a plane, each of which is at a constant distance from a fixed point in the plane
Standard form :
Equation of a circle is simplest if its centreis at the origin.
x²+ y²= r²
Central form
Let C (h, k) be the centreof the circle and r(>0) be its radius. Let P(x, y) be a point in the plane, then P lies on the circle
(x-h)²+(y-k)²= r²,
Diameter form
Let A(x, y) and B(x, y) be the extremities of a diameter of the
(x-x
1) (x-x
2)+(y-y
1) (y-y
2)=0
General form
x
2
+ y² + 2gx + 2fy + c=0, where g = h, f = k and c = h²+k
2
-r
2
the equation x
2
+ y² +2gx+2fy + c = 0 represents a circle if g²+f
2
-c >0.
Its centreis (-g-f) and radius = g²+f
2
−c
Equation Y
2
= 4ax Y
2
= -4ax X
2
= 4ay X
2
= -4ay
Axis Y = 0 Y = 0 x = 0 x = 0
Directrix X + a = 0 X -a = 0 y + a = 0 y -a = 0
Focus (a,0) (-a,0) (0, a) (0, -a)
Vertex (0,0) (0,0) (0,0) (0,0)
Length of latus rectum 4a 4a 4a 4a
Equation of latus rectum x-a=0 X+a=0 y-a=0 Y+a=0
PARABOLA
parabola is the set of all points in a plane which are equidistant from a
fixed line and a fixed point (not on the line) in the plane. The fixed line is called the directrix of the parabola
and the fixed point is called the focus of the parabola.
The line passing through the focus and perpendicular to the directrix is called the axis of parabola. The point
of intersection of parabola with its axis is called vertex of parabola
2
= 4ax Y
2
= -4ax X
2
= 4ay X
2
= -4ay
Axis Y = 0 Y = 0 x = 0 x = 0
Directrix X + a = 0 X -a = 0 y + a = 0 y -a = 0
Focus (a,0) (-a,0) (0, a) (0, -a)
Vertex (0,0) (0,0) (0,0) (0,0)
Length of latus rectum 4a 4a 4a 4a
Equation of latus rectum x-a=0 X+a=0 y-a=0 Y+a=0
PARABOLA
parabola is the set of all points in a plane which are equidistant from a
fixed line and a fixed point (not on the line) in the plane. The fixed line is called the directrix of the parabola
and the fixed point is called the focus of the parabola.
The line passing through the focus and perpendicular to the directrix is called the axis of parabola. The point
of intersection of parabola with its axis is called vertex of parabola
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