UNIT 3
SQUARE-SQUARE
ROOT AND
CUBE-CUBE ROOT
(A)
Main Concepts and Results
A natural number is called a perfect square if it is the square of
some natural number.
i.e., if m = n2, then m is a perfect square where m and n are natural
numbers.
A natural n

PLAYING
WITH
NUMBERS
249
CHAPTER
Playing with Numbers
16
16.1 Introduction
You have studied various types of numbers such as natural numbers, whole numbers,
integers and rational numbers. You have also studied a number of interesting properties
about them

Overview of Safety Instrumented
Systems
IDC Technologies
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Overview of Safety Instrumented Systems
First Edition
2012 IDC Technologies &

EXPONENTS
AND
POWERS
193
CHAPTER
Exponents and Powers
12
12.1 Introduction
Do you know?
Mass of earth is 5,970,000,000,000, 000, 000, 000, 000 kg. We have
already learnt in earlier class how to write such large numbers more
conveniently using exponents, a

UNIT 2
DATA HANDLING
(A)
Main Concepts and Results
The information collected in term of numbers is called data.
Data are represented graphically to have a quick glance on them.
Data available in an unorganised form are called raw data.
The number of t

SQUARES
AND
SQUARE ROOTS
Squares and Square
Roots
6.1 Introduction
You know that the area of a square = side side (where side means the length of
a side). Study the following table.
Side of a square (in cm)
Area of the square (in cm2)
1
1 1 = 1 = 12
2
2 2

UNIT 6
VISUALISING SOLID
SHAPES
(A) Main
Concepts
and
Results
3D shapes/objects are those which do not lie completely in a
plane.
3D objects have different views from different positions.
A solid is a polyhedron if it is made up of only polygonal faces

UNIT 9
COMPARING QUANTITIES
(A)
Main
Concepts
and
Results
Discount is a reduction given on marked price.
Discount = Marked Price Sale Price (S.P.)
Discount can be calculated when discount percentage is given.
Discount = Discount % of Marked Price
Addit

UNIT 10
DIRECT AND INVERSE
PROPORTIONS
(A)
Main
Concepts
and
Results
Two quantities x and y are said to be in direct proportion if they
increase or decrease together in such a manner that the ratio of
x
= k where
y
k is a positive number, if x and y are

COMPARING QUANTITIES
117
CHAPTER
Comparing Quantities
8
8.1 Recalling Ratios and Percentages
We know, ratio means comparing two quantities.
A basket has two types of fruits, say, 20 apples and 5 oranges.
Then, the ratio of the number of oranges to the num

ALGEBRAIC EXPRESSIONS
AND IDENTITIES
Algebraic Expressions
and Identities
9.1 What are Expressions?
In earlier classes, we have already become familiar with what algebraic expressions
(or simply expressions) are. Examples of expressions are:
x + 3, 2y 5,

UNIT 5
UNDERSTANDING
QUADRILATERALS AND
PRACTICAL GEOMETRY
(A)
Main
Concepts
and
Results
A simple closed curve made up of only line segments is called a
polygon.
A diagonal of a polygon is a line segment connecting two nonconsecutive vertices.
A convex

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Efficiency
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3
Our commitment
At the heart of our
strategy is a simple
and powerful idea:
using natural resources
much more productively
- efficiently - is both
profitable and better
for the environment.
5
Content
Challenges
6-7 . Ener

LINEAR EQUATIONS
IN
ONE VARIABLE
21
Linear Equations in
One Variable
2.1 Introduction
In the earlier classes, you have come across several algebraic expressions and equations.
Some examples of expressions we have so far worked with are:
5x, 2x 3, 3x + y,

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Biomolecules
David Bakewell
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Micro- and Nano- Transport of Biomolecules
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CUBES
AND
CUBE ROOTS
109
CHAPTER
Cubes and Cube Roots
7
7.1 Introduction
This is a story about one of Indias great mathematical geniuses, S. Ramanujan. Once
another famous mathematician Prof. G.H. Hardy came to visit him in a taxi whose number
was 1729. W

Class XII
Chapter 5 Continuity and Differentiability
Maths
Exercise 5.1
Question 1:
Prove that the function
is continuous at
Answer
Therefore, f is continuous at x = 0
Therefore, f is continuous at x = 3
Therefore, f is continuous at x = 5
Question 2:
Exa

MENSURATION
169
CHAPTER
Mensuration
11
11.1 Introduction
We have learnt that for a closed plane figure, the perimeter is the distance around its
boundary and its area is the region covered by it. We found the area and perimeter of
various plane figures su

Christopher C. Tisdell
Engineering Mathematics:
YouTube Workbook
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2
Engineering Mathematics: YouTube Workbook
2nd edition
2013 Christopher C. Tisdell & bookboon.com
ISBN 978-87-403-0522-7
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UNIT 11
MENSURATION
(A)
Main
Concepts
and
Results
Length of boundary of a simple closed figure is known as perimeter.
Area is the measure of region enclosed in a simple closed curve.
Perimeter of a rectangle = 2 (length + breadth).
Area of a rectangle

UNIT 8
EXPONENTS AND POWERS
(A)
Main Concepts and Results
Exponential notation is a powerful way to express repeated
multiplication of the same number. Specifically, powers of 10
express very large and very small numbers in a manner which is
convenient t

FACTORISATION
217
CHAPTER
Factorisation
14
14.1 Introduction
14.1.1 Factors of natural numbers
You will remember what you learnt about factors in Class VI. Let us take a natural number,
say 30, and write it as a product of other natural numbers, say
30 =

DIRECT
AND INVERSE
Direct and Inverse
Proportions
PROPORTIONS
201
CHAPTER
13
13.1 Introduction
Mohan prepares tea for himself and his sister. He uses 300 mL of
water, 2 spoons of sugar, 1 spoon of tea leaves and 50 mL of milk.
How much quantity of each it

Class XII
Chapter 1 Relations and Functions
Maths
Exercise 1.1
Question 1:
Determine whether each of the following relations are reflexive, symmetric and
transitive:
(i)Relation R in the set A = cfw_1, 2, 313, 14 defined as
R = cfw_(x, y): 3x y = 0
(ii) R

Large Scale Data Handling in
Biology
Karol Kozak
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Karol Kozak
Large Scale Data Handling in Biology
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2
Large Scale Data Handling in Biology
1st edition
2010 Karol Kozak & bookboon.com
ISBN 978-87-7

Dr. Wasif Naeem
Concepts in Electric Circuits
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2
Concepts in Electric Circuits
2009 Dr. Wasif Naeem & Ventus Publishing ApS
ISBN 978-87-7681-499-1
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3
Concepts in Electric Circuits
Co

RATIONAL NUMBERS
1
CHAPTER
1
Rational Numbers
1.1 Introduction
In Mathematics, we frequently come across simple equations to be solved. For example,
the equation
x + 2 = 13
(1)
is solved when x = 11, because this value of x satisfies the given equation. T

UNDERSTANDING QUADRILATERALS
Understanding
Quadrilaterals
3.1 Introduction
You know that the paper is a model for a plane surface. When you join a number of
points without lifting a pencil from the paper (and without retracing any portion of the
drawing o

UNIT 1
RATIONAL
(A)
Main
Concepts
and
NUMBERS
Results
p
, where p and q are
q
integers and q 0, is called a rational number.
A number that can be expressed in the form
p
is said
q
to be in the lowest form or simplest form if p and q have no common
factor

Class XII
Chapter 7 Integrals
Maths
Exercise 7.1
Question 1:
sin 2x
Answer
The anti derivative of sin 2x is a function of x whose derivative is sin 2x.It is known that,
Therefore, the anti derivative of
Question 2:
Cos 3x
Answer
The anti derivative of cos