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class_6 | 1 | Patterns in Mathematics | ncert_books/class_6/Ganita_Prakash/fegp101.pdf | 1
1.1 What is Mathematics?
Mathematics is, in large part, the search for patterns, and for
the explanations as to why those patterns exist.
Such patterns indeed exist all around us—in nature, in
our homes and schools, and in the motion of the sun, moon,
and stars. They occur in everything that we do and see, from
sho... |
class_6 | 2 | Lines and Angles | ncert_books/class_6/Ganita_Prakash/fegp102.pdf | 2
In this chapter, we will explore some of the most basic ideas of
geometry including points, lines, rays, line segments and angles.
These ideas form the building blocks of ‘plane geometry’, and will
help us in understanding more advanced topics in geometry such as
the construction and analysis of different shapes.
M... |
class_6 | 3 | Number Play | ncert_books/class_6/Ganita_Prakash/fegp103.pdf | 3 NUMBER PLAY
Numbers are used in different contexts and in many different ways
to organise our lives. We have used numbers to count, and have
applied the basic operations of addition, subtraction, multiplication
and division on them, to solve problems related to our daily lives.
In this chapter, we will continue this... |
class_6 | 4 | Data Handling and Presentation | ncert_books/class_6/Ganita_Prakash/fegp104.pdf | 4
If you ask your classmates about their favourite colours, you will
get a list of colours. This list is an example of data. Similarly, if you
measure the weight of each student in your class, you would get a
collection of measures of weight—again data.
Any collection of facts, numbers, measures, observations or other... |
class_6 | 5 | Prime Time | ncert_books/class_6/Ganita_Prakash/fegp105.pdf | 5 PRIME TIME
5.1 Common Multiples and Common Factors
Children sit in a circle and play a game of numbers.
One of the children starts by saying ‘1’. The second
player says ‘2’, and so on. But when it is the turn of 3, 6,
9, … (multiples of 3), the player should say ‘idli’ instead
of the number. When it is the turn of ... |
class_6 | 6 | Perimeter and Area | ncert_books/class_6/Ganita_Prakash/fegp106.pdf | 6 PERIMETER AND AREA
6.1 Perimeter
Do you remember what the perimeter of a closed plane figure is?
Let us refresh our understanding!
The perimeter of any closed plane figure is the distance covered
along its boundary when you go around it once. For a polygon, i.e.,
a closed plane figure made up of line segments, the ... |
class_6 | 7 | Fractions | ncert_books/class_6/Ganita_Prakash/fegp107.pdf | 7
Recall that when some whole number of things are shared
equally among some number of people, fractions tell us how
much each share is.
Shabnam: Do you remember, if one roti is divided equally
Mukta: Each child will get half a roti.
Shabnam: The fraction ‘one half’ is written as
Mukta: If one roti is equally shar... |
class_6 | 8 | Playing with Constructions | ncert_books/class_6/Ganita_Prakash/fegp108.pdf | 8
8.1 Artwork
Observe the following figures and try drawing them freehand.
Playing with
Constructions
Chapter 8_Playing with Constructions.indd 187 13-08-2024 16:38:44
Reprint 2025-26
Fig. 8.1
Ganita Prakash | Grade 6
Now, arm yourself with a ruler and a compass. Let us explore if
we can draw these figures with... |
class_6 | 9 | Symmetry | ncert_books/class_6/Ganita_Prakash/fegp109.pdf | 9
Look around you—you may find many objects that catch your
attention. Some such things are shown below:
SYMMETRY
Flower Butterfly
Chapter 9_Symmetry.indd 217 13-08-2024 17:05:22
There is something beautiful about the pictures above.
The flower looks the same from many different angles. What
about the butterfly? N... |
class_6 | 10 | The Other Side of Zero | ncert_books/class_6/Ganita_Prakash/fegp110.pdf | 10
More and more numbers!
Recall that the very first numbers we learned about in the study of
mathematics were the counting numbers 1, 2, 3, 4, …
Then we learned that there are even more numbers! For example,
there is the number 0 (zero), representing nothing, which comes before
1. The number 0 has a very important hi... |
class_7 | 1 | Large Numbers Around Us | ncert_books/class_7/gegp1dd/gegp101.pdf | 1
1.1 A Lakh Varieties!
Eshwarappa is a farmer in Chintamani,
a town in Karnataka. He visits the
market regularly to buy seeds for his
rice field. During one such visit he
overheard a conversation between
Ramanna and Lakshmamma. Ramanna
said, “Earlier our country had about a
lakh varieties of rice. Farmers used to
pr... |
class_7 | 2 | Arithmetic Expressions | ncert_books/class_7/gegp1dd/gegp102.pdf | 2
2.1 Simple Expressions
You may have seen mathematical phrases like 13 + 2, 20 – 4, 12 × 5, and
18 ÷ 3. Such phrases are called arithmetic expressions.
Every arithmetic expression has a value which is the number it
evaluates to. For example, the value of the expression 13 + 2 is 15. This
expression can be read as ‘1... |
class_7 | 3 | A Peek Beyond the Point | ncert_books/class_7/gegp1dd/gegp103.pdf | 3
3.1 The Need for Smaller Units
Sonu’s mother was fixing a
toy. She was trying to join
two pieces with the help of
a screw. Sonu was
watching his mother with
great curiosity. His mother
was unable to join the
pieces. Sonu asked why.
His mother said that the
screw was not of the
right size.
She brought another screw ... |
class_7 | 4 | Expressions using Letter-Numbers | ncert_books/class_7/gegp1dd/gegp104.pdf | 4
4.1 The Notion of Letter-Numbers
In this chapter we shall look at a concise way of expressing mathematical
relations and patterns. We shall see how this helps us in thinking about
these relationships and patterns, and in explaining why they may hold
true.
Example 1: Shabnam is 3 years older than Aftab. When Aftab’... |
class_7 | 5 | Parallel and Intersecting Lines | ncert_books/class_7/gegp1dd/gegp105.pdf | 5
5.1 Across the Line
Take a piece of square paper and fold it in different ways. Now, on
the creases formed by the folds, draw lines using a pencil and a scale.
You will notice different lines on the paper. Take any pair of lines and
observe their relationship with each other. Do they meet? If they do
not meet withi... |
class_7 | 6 | Number Play | ncert_books/class_7/gegp1dd/gegp106.pdf | 6 NUMBER PLAY
6.1 Numbers Tell us Things
What do the numbers in the figure below tell us?
Remember the children from the Grade 6 textbook of mathematics?
Now, they call out numbers using a different rule.
What do you think these numbers mean?
Chapter-6.indd 127 4/12/2025 11:59:04 AM
The children rearrange themsel... |
class_7 | 7 | A Tale of Three Intersecting Lines | ncert_books/class_7/gegp1dd/gegp107.pdf | 7
A triangle is the most basic closed shape. As we know, it consists of:
Observe the symbol used to denote a triangle and how the triangles
are named using their vertices. While naming a triangle, the vertices
can come in any order.
The three sides meeting at the corners give rise to three angles that
we call the ang... |
class_7 | 8 | Working with Fractions | ncert_books/class_7/gegp1dd/gegp108.pdf | 8
8.1 Multiplication of Fractions
Aaron walks 3 kilometres in 1 hour.
How far can he walk in 5 hours?
This is a simple question. We know
that to find the distance, we need to
find the product of 5 and 3, i.e., we
multiply 5 and 3.
Distance covered in 1 hour = 3 km.
Therefore,
Distance covered in 5 hours
= 3 + 3 + ... |
class_7 | 9 | Geometric Twins | ncert_books/class_7/gegp1dd2/gegp201.pdf | 1
1.1 Geometric Twins
The symbol on this signboard needs to be recreated on another board.
How do we do it?
One way is to trace the outline of this symbol on tracing paper to
reconstruct the figure. But this is difficult for big symbols. What else can
we do?
GEOMETRIC
TWINS
Chapter-1.indd 1 10/9/2025 11:44:10 AM... |
class_7 | 10 | operations with integers | ncert_books/class_7/gegp1dd2/gegp202.pdf | OPERATIONS
WITH INTEGERS
2
2.1 A Quick Recap of Integers
Rakesh’s Puzzle: A Number Game
Rakesh gives you a challenge.
“I have thought of two numbers”, he says.
“Their sum is 25, and their difference is 11.”
Can you tell me the two numbers?
You don’t need to use any formulas. Just try different pairs of numbers
a... |
class_7 | 11 | finding common ground | ncert_books/class_7/gegp1dd2/gegp203.pdf | 3
3.1 The Greatest of All
Sameeksha is building her new house. The main room of the house is 12
ft by 16 ft. She feels that the room would look nice if the floor is covered
with square tiles of the same size. She also wants to use as few tiles as
possible, and for the length of the tile to be a whole number of feet. ... |
class_7 | 12 | another peek beyond the point | ncert_books/class_7/gegp1dd2/gegp204.pdf | 4
4.1 A Quick Recap of Decimals
Recall that decimals are the natural extension of the Indian place value
system to represent decimal fractions (
1
10,
1
100,
1
1000, and so on) and
their sums.
For example, 27.53 refers to a quantity that has:
• 2 Tens
We have already learned how to multiply and divide fractions. In... |
class_7 | 13 | connecting the dots... | ncert_books/class_7/gegp1dd2/gegp205.pdf | 5
Jemimah’s batting has
been very consistent over
the past year. We can
expect a century from her
in tomorrow’s match.
5.1 Of Questions and Statements
Your teacher tells you that they are meeting two of their childhood
friends this evening. One is 5 feet tall and the other is 6 feet tall. What is
your guess as to ea... |
class_7 | 14 | constructions and tilings | ncert_books/class_7/gegp1dd2/gegp206.pdf | 6
6.1 Geometric Constructions
Eyes
Do you recall the ‘Eyes’ construction we did in Grade 6?
Of course, eyes can be drawn freehand, but we wanted to construct
them so that the lower arc and upper arc of each eye look symmetrical.
We relied on our spatial estimation to determine the two centres, A
and B (see the figu... |
class_7 | 15 | finding the unknown | ncert_books/class_7/gegp1dd2/gegp207.pdf | 7
7.1 Find the Unknowns
Unknown Weights
We have a weighing scale that behaves as follows. The numbers represent
same units of weight:
Find the unknown weights in the following cases:
FINDING THE
UNKNOWN
Chapter-7.indd 164 10/9/2025 3:19:51 PM
Fig. 7.1 Fig. 7.2 Fig. 7.3
Discuss the answers with your classmates. ... |
class_8 | 1 | A square and a cube | ncert_books/class_8/hegp1dd/hegp101.pdf | 1 A SQUARE AND A CUBE
Queen Ratnamanjuri had a will written that described her fortune of
ratnas (precious stones) and also included a puzzle. Her son Khoisnam
and their 99 relatives were invited to the reading of her will. She wanted
to leave all of her ratnas to her son, but she knew that if she did so, all
their re... |
class_8 | 2 | power play | ncert_books/class_8/hegp1dd/hegp102.pdf | 2 POWER PLAY
2.1 Experiencing the Power Play ...
An Impossible Venture!
Take a sheet of paper, as large a sheet as you can find. Fold it once. Fold
it again, and again.
How many times can you fold it over and over?
Estu says “I heard that a sheet of paper can’t be folded more than
7 times”.
Roxie replies “What if w... |
class_8 | 3 | a story of numbers | ncert_books/class_8/hegp1dd/hegp103.pdf | 3
A STORY OF
NUMBERS
3.1 Reema’s Curiosity
One lazy afternoon, Reema was flipping through an old book when—
whoosh!—a piece of paper slipped out and floated to the floor. She
picked it up and stared at the strange symbols all over it. “What is this?”
she wondered.
She ran to her father, holding the paper as if it wer... |
class_8 | 4 | quadrilaterals | ncert_books/class_8/hegp1dd/hegp104.pdf | 4 QUADRILATERALS
In this chapter, we will study some interesting types of four-sided
figures and solve problems based on them. Such figures are commonly
known as quadrilaterals. The word ‘quadrilateral’ is derived from Latin
words — quadri meaning four, and latus referring to sides.
Observe the following figures.
(i... |
class_8 | 5 | number play | ncert_books/class_8/hegp1dd/hegp105.pdf | 5 NUMBER PLAY
5.1 Is This a Multiple Of?
Sum of Consecutive Numbers
Anshu is exploring sums of consecutive numbers. He has written the
following—
Now, he is wondering—
• “Can I write every natural number as a sum of consecutive
numbers?”
7 = 3 + 4
10 = 1 + 2 + 3 + 4
12 = 3 + 4 + 5
15 = 7 + 8
= 4 + 5 + 6
= 1 + 2... |
class_8 | 6 | we distribute, yet things multiply | ncert_books/class_8/hegp1dd/hegp106.pdf | 6
We have seen how algebra makes use of letter symbols to write general
statements about patterns and relations in a compact manner. Algebra
can also be used to justify or prove claims and conjectures (like the many
properties you saw in the previous chapter) and to solve problems of
various kinds.
Distributivity is a... |
class_8 | 7 | propositional reasoning - 1 | ncert_books/class_8/hegp1dd/hegp107.pdf | 7
7.1 Observing Similarity in Change
We are all familiar with digital images. We often change the size and
orientation of these images to suit our needs. Observe the set of images
below—
PROPORTIONAL
REASONING-1
Image A
Image B Image C
Chapter 7.indd 159 10-07-2025 15:14:27
We can see that all the images are of ... |
class_8 | 8 | fractions in disguise | ncert_books/class_8/hegp2dd/hegp201.pdf | 1
1.1 Fractions as Percentages
You might have heard statements like, “Mega Sale — up to 50% off!” or
“Hiya scored 83% in her board exams”.
Do you know what the symbol ʻ%ʼ means?
This symbol is read as per cent.
The word ‘per cent’ is derived from the Latin phrase ‘per centum’,
meaning ‘by the hundred’ or ‘out of hund... |
class_8 | 9 | the baudhayana-pythagoras theorem | ncert_books/class_8/hegp2dd/hegp202.pdf | 2
2.1 Doubling a Square
In Baudhāyana’s Śulba-Sūtra (c. 800 BCE), Baudhāyana considers the
following question:
How can one construct a square having double the area
of a given square?
A first guess might be to simply double the length of
each side of the square. Will this new square have double
the area of the orig... |
class_8 | 10 | proportional reasoning-2 | ncert_books/class_8/hegp2dd/hegp203.pdf | 3
3.1 Proportionality—A Quick Recap
In an earlier chapter, we explored proportional relationships between
quantities and we used the ratio notation to represent such relationships.
When two or more related quantities change by the same factor, we call
that relationship a proportional relationship. For example, idli b... |
class_8 | 11 | exploring some geometrics themes | ncert_books/class_8/hegp2dd/hegp204.pdf | 4
In this chapter, we will explore two geometric themes. We will study
fractals which are self-similar shapes. They exhibit the same or similar
pattern over and over again — but at smaller and smaller scales. We will
then look at different ways of visualising solids.
4.1 Fractals
One of the most beautiful examples o... |
class_8 | 12 | tales by dots and lines | ncert_books/class_8/hegp2dd/hegp205.pdf | 5
5.1 The Balancing Act
Last year, we learnt about the mean and median. Recall that the mean of
some data is the sum of all the values divided by the number of values in
the data. The median is the middle value when the data is sorted.
We shall try to understand the mean and median from a different
perspective and se... |
class_8 | 13 | algebra play | ncert_books/class_8/hegp2dd/hegp206.pdf | ALGEBRA PLAY
6
6.1 Algebra Play
Over the last two years, we have used algebra to model different
situations. We have learned how to solve algebraic equations and find
the values of unknown letter-numbers. Let’s now have some fun with
algebra. We shall investigate tricks and puzzles, and explain why they
work using al... |
class_8 | 14 | area | ncert_books/class_8/hegp2dd/hegp207.pdf | 7
7.1 Rectangle and Squares
How many different ways can you divide a square into 4 parts of equal
area?
One can actually think of infinitely many such ways! Consider a
division, such as —
and alter each part as follows.
AREA
07_Chapter 7.indd 148 20-12-2025 16:59:54
In each part, the area is compressed along one ... |
class_9 | 1 | Number Systems | ncert_books/class_9/iemh1dd/iemh101.pdf | NUMBER SYSTEMS 1
NUMBER SYSTEMS
1.1 Introduction
In your earlier classes, you have learnt about the number line and how to represent
various types of numbers on it (see Fig. 1.1).
positive direction. As far as your eyes can see, there are numbers, numbers and
numbers!
Just imagine you start from zero and go on wal... |
class_9 | 2 | Polynomials | ncert_books/class_9/iemh1dd/iemh102.pdf | POLYNOMIALS 25
POLYNOMIALS
2.1 Introduction
You have studied algebraic expressions, their addition, subtraction, multiplication and
division in earlier classes. You also have studied how to factorise some algebraic
expressions. You may recall the algebraic identities :
and x
2
– y
2
= (x + y) (x – y)
and their u... |
class_9 | 3 | Coordinate Geometry | ncert_books/class_9/iemh1dd/iemh103.pdf | COORDINATE GEOMETRY
LEWIS CARROLL, The Hunting of the Snark
3.1 Introduction
You have already studied how to locate a point on a number line. You also know how
to describe the position of a point on the line. There are many other situations, in which
to find a point we are required to describe its position with refer... |
class_9 | 4 | Linear Equations in Two Variables | ncert_books/class_9/iemh1dd/iemh104.pdf | LINEAR EQUATIONS IN TWO VARIABLES 55
LINEAR EQUATIONS IN TWO VARIABLES
4.1 Introduction
In earlier classes, you have studied linear equations in one variable. Can you write
down a linear equation in one variable? You may say that x + 1 = 0, x + 2 = 0 and
2 y + 3 = 0 are examples of linear equations in one variable. ... |
class_9 | 5 | Introduction to Euclid's Geometry | ncert_books/class_9/iemh1dd/iemh105.pdf | 60 MATHEMATICS
INTRODUCTION TO EUCLID’S GEOMETRY
5.1 Introduction
The word ‘geometry’ comes form the Greek words ‘geo’, meaning the ‘earth’,
and ‘metrein’, meaning ‘to measure’. Geometry appears to have originated from
the need for measuring land. This branch of mathematics was studied in various
forms in every anci... |
class_9 | 6 | Lines and Angles | ncert_books/class_9/iemh1dd/iemh106.pdf | LINES AND ANGLES
6.1 Introduction
In Chapter 5, you have studied that a minimum of two points are required to draw a
line. You have also studied some axioms and, with the help of these axioms, you
proved some other statements. In this chapter, you will study the properties of the
angles formed when two lines intersec... |
class_9 | 7 | Triangles | ncert_books/class_9/iemh1dd/iemh107.pdf | TRIANGLES
7.1 Introduction
You have studied about triangles and their various properties in your earlier classes.
You know that a closed figure formed by three intersecting lines is called a triangle.
(‘Tri’ means ‘three’). A triangle has three sides, three angles and three vertices. For
example, in triangle ABC, den... |
class_9 | 8 | Quadrilaterals | ncert_books/class_9/iemh1dd/iemh108.pdf | 104 MATHEMATICS
QUADRILATERALS
8.1 Properties of a Parallelogram
You have already studied quadrilaterals and their types in Class VIII. A quadrilateral
has four sides, four angles and four vertices. A parallelogram is a quadrilateral in
which both pairs of opposite sides are parallel.
Let us perform an activity.
C... |
class_9 | 9 | Circles | ncert_books/class_9/iemh1dd/iemh109.pdf | 116 MATHEMATICS
CIRCLES
9.1 Angle Subtended by a Chord at a Point
You have already studied about circles and its parts in Class VI.
Take a line segment PQ and a point R not on the line containing PQ. Join PR and QR
(see Fig. 9.1). Then ∠ PRQ is called the angle subtended by the line segment PQ at
the point R. What ... |
class_9 | 10 | Heron's Formula | ncert_books/class_9/iemh1dd/iemh110.pdf | HERON’S FORMULA
10.1Area of a Triangle — by Heron’s Formula
We know that the area of triangle when its height is given, is
1
2
× base × height. Now
suppose that we know the lengths of the sides of a scalene triangle and not the height.
Can you still find its area? For instance, you have a triangular park whose side... |
class_9 | 11 | Surface Areas and Volumes | ncert_books/class_9/iemh1dd/iemh111.pdf | SURFACE AREAS AND VOLUMES 137
SURFACE AREAS AND VOLUMES
11.1 Surface Area of a Right Circular Cone
We have already studied the surface areas of cube, cuboid and cylinder. We will now
study the surface area of cone.
So far, we have been generating solids by stacking up congruent figures. Incidentally,
such figures a... |
class_9 | 12 | Statistics | ncert_books/class_9/iemh1dd/iemh112.pdf | STATISTICS 151
STATISTICS
12.1 Graphical Representation of Data
The representation of data by tables has already been discussed. Now let us turn our
attention to another representation of data, i.e., the graphical representation. It is well
said that one picture is better than a thousand words. Usually comparisons a... |
class_10 | 1 | Real Numbers | ncert_books/class_10/jemh1dd/jemh101.pdf | 1.1 Introduction
In Class IX, you began your exploration of the world of real numbers and encountered
irrational numbers. We continue our discussion on real numbers in this chapter. We
begin with very important properties of positive integers in Sections 1.2, namely the
Euclid’s division algorithm and the Fundamental ... |
class_10 | 2 | Polynomials | ncert_books/class_10/jemh1dd/jemh102.pdf | 10 MATHEMATICS
2.1 Introduction
In Class IX, you have studied polynomials in one variable and their degrees. Recall
that if p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of
the polynomial p(x). For example, 4x + 2 is a polynomial in the variable x of
degree 1, 2y2
– 3y + 4 is a poly... |
class_10 | 3 | Pair of Linear Equations in Two Variables | ncert_books/class_10/jemh1dd/jemh103.pdf | 24 MATHEMATICS
3.1 Introduction
You must have come across situations like the one given below :
and play Hoopla (a game in which you throw a ring on the items kept in a stall, and if
the ring covers any object completely, you get it). The number of times she played
Hoopla is half the number of rides she had on the G... |
class_10 | 4 | Quadratic Equations | ncert_books/class_10/jemh1dd/jemh104.pdf | 38 MATHEMATICS
4.1 Introduction
In Chapter 2, you have studied different types of polynomials. One type was the
quadratic polynomial of the form ax2
+ bx + c, a 0. When we equate this polynomial
to zero, we get a quadratic equation. Quadratic equations come up when we deal with
many real-life situations. For instan... |
class_10 | 5 | Arithmetic Progressions | ncert_books/class_10/jemh1dd/jemh105.pdf | 5.1 Introduction
You must have observed that in nature, many things follow a certain pattern, such as
the petals of a sunflower, the holes of a honeycomb, the grains on a maize cob, the
spirals on a pineapple and on a pine cone, etc.
examples are :
ARITHMETIC PROGRESSIONS 49
(i) Reena applied for a job and got sele... |
class_10 | 6 | Triangles | ncert_books/class_10/jemh1dd/jemh106.pdf | 6.1 Introduction
You are familiar with triangles and many of their properties from your earlier classes.
In Class IX, you have studied congruence of triangles in detail. Recall that two figures
are said to be congruent, if they have the same shape and the same size. In this
chapter, we shall study about those figures ... |
class_10 | 7 | Coordinate Geometry | ncert_books/class_10/jemh1dd/jemh107.pdf | 7.1 Introduction
In Class IX, you have studied that to locate the position of a point on a plane, we
require a pair of coordinate axes. The distance of a point from the y-axis is called its
x-coordinate, or abscissa. The distance of a point from the x-axis is called its
y-coordinate, or ordinate. The coordinates of a ... |
class_10 | 8 | Introduction to Trigonometry | ncert_books/class_10/jemh1dd/jemh108.pdf | 8.1 Introduction
You have already studied about triangles, and in particular, right triangles, in your
earlier classes. Let us take some examples from our surroundings where right triangles
can be imagined to be formed. For instance :
INTRODUCTION TO TRIGONOMETRY 113
1. Suppose the students of a school are
visiting ... |
class_10 | 9 | Some Applications of Trigonometry | ncert_books/class_10/jemh1dd/jemh109.pdf | SOME APPLICATIONS OF TRIGONOMETRY 133
9.1 Heights and Distances
In the previous chapter, you have studied about trigonometric ratios. In this chapter,
you will be studying about some ways in which trigonometry is used in the life around
you.
Let us consider Fig. 8.1 of prvious chapter, which is redrawn below in Fig. ... |
class_10 | 10 | Circles | ncert_books/class_10/jemh1dd/jemh110.pdf | 10.1 Introduction
You have studied in Class IX that a circle is a collection of all points in a plane
which are at a constant distance (radius) from a fixed point (centre). You have
also studied various terms related to a circle like chord, segment, sector, arc etc.
Let us now examine the different situations that can... |
class_10 | 11 | Areas Related to Circles | ncert_books/class_10/jemh1dd/jemh111.pdf | 154 MATHEMATICS
11.1 Areas of Sector and Segment of a Circle
You have already come across the terms sector and
segment of a circle in your earlier classes. Recall
that the portion (or part) of the circular region enclosed
by two radii and the corresponding arc is called a
sector of the circle and the portion (or part... |
class_10 | 12 | Surface Areas and Volumes | ncert_books/class_10/jemh1dd/jemh112.pdf | SURFACE AREAS AND VOLUMES 161
12.1 Introduction
From Class IX, you are familiar with some of the solids like cuboid, cone, cylinder, and
sphere (see Fig. 12.1). You have also learnt how to find their surface areas and volumes.
SURFACE AREAS AND
VOLUMES
12
of two or more of the basic solids as shown above.
You mus... |
class_10 | 13 | Statistics | ncert_books/class_10/jemh1dd/jemh113.pdf | 13.1 Introduction
In Class IX, you have studied the classification of given data into ungrouped as well as
grouped frequency distributions. You have also learnt to represent the data pictorially
in the form of various graphs such as bar graphs, histograms (including those of varying
widths) and frequency polygons. In ... |
class_10 | 14 | Probability | ncert_books/class_10/jemh1dd/jemh114.pdf | 202 MATHEMATICS
14.1 Probability — ATheoretical Approach
Let us consider the following situation :
When we speak of a coin, we assume it to be ‘fair’, that is, it is symmetrical so
that there is no reason for it to come down more often on one side than the other.
We call this property of the coin as being ‘unbiased’... |
class_11 | 1 | Sets | ncert_books/class_11/kemh1dd/kemh101.pdf | 1.1 Introduction
The concept of set serves as a fundamental part of the
present day mathematics. Today this concept is being used
in almost every branch of mathematics. Sets are used to
define the concepts of relations and functions. The study of
geometry, sequences, probability, etc. requires the knowledge
of sets.
T... |
class_11 | 2 | Relations and Functions | ncert_books/class_11/kemh1dd/kemh102.pdf | 2.1 Introduction
Much of mathematics is about finding a pattern – a
recognisable link between quantities that change. In our
daily life, we come across many patterns that characterise
relations such as brother and sister, father and son, teacher
and student. In mathematics also, we come across many
relations such as nu... |
class_11 | 3 | Trigonometric Functions | ncert_books/class_11/kemh1dd/kemh103.pdf | 3.1 Introduction
The word ‘trigonometry’ is derived from the Greek words
‘trigon’ and ‘metron’ and it means ‘measuring the sides of
a triangle’. The subject was originally developed to solve
geometric problems involving triangles. It was studied by
sea captains for navigation, surveyor to map out the new
lands, by engi... |
class_11 | 4 | Complex Numbers and Quadratic Equations | ncert_books/class_11/kemh1dd/kemh104.pdf | 76 MATHEMATICS
4.1 Introduction
In earlier classes, we have studied linear equations in one
and two variables and quadratic equations in one variable.
We have seen that the equation x
2
+ 1 = 0 has no real
solution as x
2
+ 1 = 0 gives x
2
= – 1 and square of every
real number is non-negative. So, we need to exten... |
class_11 | 5 | Linear Inequalities | ncert_books/class_11/kemh1dd/kemh105.pdf | 5.1 Introduction
In earlier classes, we have studied equations in one variable and two variables and also
solved some statement problems by translating them in the form of equations. Now a
natural question arises: ‘Is it always possible to translate a statement problem in the
form of an equation? For example, the heigh... |
class_11 | 6 | Permutations and Combinations | ncert_books/class_11/kemh1dd/kemh106.pdf | 6.1 Introduction
Suppose you have a suitcase with a number lock. The number
lock has 4 wheels each labelled with 10 digits from 0 to 9.
The lock can be opened if 4 specific digits are arranged in a
particular sequence with no repetition. Some how, you have
forgotten this specific sequence of digits. You remember only
... |
class_11 | 7 | Binomial Theorem | ncert_books/class_11/kemh1dd/kemh107.pdf | 126 MATHEMATICS
7.1 Introduction
In earlier classes, we have learnt how to find the squares
and cubes of binomials like a + b and a – b. Using them, we
could evaluate the numerical values of numbers like
(98)2
= (100 – 2)2
, (999)3
= (1000 – 1)3
, etc. However, for
higher powers like (98)5
, (101)6
, etc., the calcu... |
class_11 | 8 | Sequences and Series | ncert_books/class_11/kemh1dd/kemh108.pdf | 8.1 Introduction
In mathematics, the word, “sequence” is used in much the
same way as it is in ordinary English. When we say that a
collection of objects is listed in a sequence, we usually mean
that the collection is ordered in such a way that it has an
identified first member, second member, third member and
so on. ... |
class_11 | 9 | Straight Lines | ncert_books/class_11/kemh1dd/kemh109.pdf | 9.1 Introduction
We are familiar with two-dimensional coordinate geometry
from earlier classes. Mainly, it is a combination of algebra
and geometry. A systematic study of geometry by the use
of algebra was first carried out by celebrated French
philosopher and mathematician René Descartes, in his book
‘La Géométry, pu... |
class_11 | 10 | Conic Sections | ncert_books/class_11/kemh1dd/kemh110.pdf | 176 MATHEMATICS
10.1 Introduction
In the preceding Chapter 10, we have studied various forms
of the equations of a line. In this Chapter, we shall study
about some other curves, viz., circles, ellipses, parabolas
and hyperbolas. The names parabola and hyperbola are
given by Apollonius. These curves are in fact, known... |
class_11 | 11 | Introduction to Three Dimensional Geometry | ncert_books/class_11/kemh1dd/kemh111.pdf | 208 MATHEMATICS
11.1 Introduction
You may recall that to locate the position of a point in a
plane, we need two intersecting mutually perpendicular lines
in the plane. These lines are called the coordinate axes
and the two numbers are called the coordinates of the
point with respect to the axes. In actual life, we do... |
class_11 | 12 | Limits and Derivatives | ncert_books/class_11/kemh1dd/kemh112.pdf | 12.1 Introduction
This chapter is an introduction to Calculus. Calculus is that
branch of mathematics which mainly deals with the study
of change in the value of a function as the points in the
domain change. First, we give an intuitive idea of derivative
(without actually defining it). Then we give a naive definition... |
class_11 | 13 | Statistics | ncert_books/class_11/kemh1dd/kemh113.pdf | 13.1 Introduction
We know that statistics deals with data collected for specific
purposes. We can make decisions about the data by
analysing and interpreting it. In earlier classes, we have
studied methods of representing data graphically and in
tabular form. This representation reveals certain salient
features or char... |
class_11 | 14 | Probability | ncert_books/class_11/kemh1dd/kemh114.pdf | 14.1 Event
We have studied about random experiment and sample space associated with an
experiment. The sample space serves as an universal set for all questions concerned
with the experiment.
Consider the experiment of tossing a coin two times. An associated sample space
is S = {HH, HT, TH, TT}.
Now suppose that we ar... |
class_12 | 1 | Relations and Functions | ncert_books/class_12/lemh1dd/lemh101.pdf | 1.1 Introduction
Recall that the notion of relations and functions, domain,
co-domain and range have been introduced in Class XI
along with different types of specific real valued functions
and their graphs. The concept of the term ‘relation’ in
mathematics has been drawn from the meaning of relation
in English languag... |
class_12 | 2 | Inverse Trigonometric Functions | ncert_books/class_12/lemh1dd/lemh102.pdf | 18 MATHEMATICS
2.1 Introduction
In Chapter 1, we have studied that the inverse of a function
f, denoted by f
–1, exists if f is one-one and onto. There are
many functions which are not one-one, onto or both and
hence we can not talk of their inverses. In Class XI, we
studied that trigonometric functions are not one-o... |
class_12 | 3 | Matrices | ncert_books/class_12/lemh1dd/lemh103.pdf | 34 MATHEMATICS
3.1 Introduction
The knowledge of matrices is necessary in various branches of mathematics. Matrices
are one of the most powerful tools in mathematics. This mathematical tool simplifies
our work to a great extent when compared with other straight forward methods. The
evolution of concept of matrices is... |
class_12 | 4 | Determinants | ncert_books/class_12/lemh1dd/lemh104.pdf | 76 MATHEMATICS
v All Mathematical truths are relative and conditional. — C.P. STEINMETZ v
4.1 Introduction
In the previous chapter, we have studied about matrices
and algebra of matrices. We have also learnt that a system
of algebraic equations can be expressed in the form of
matrices. This means, a system of linear ... |
class_12 | 5 | Continuity and Differentiability | ncert_books/class_12/lemh1dd/lemh105.pdf | 5.1 Introduction
This chapter is essentially a continuation of our study of
differentiation of functions in Class XI. We had learnt to
differentiate certain functions like polynomial functions and
trigonometric functions. In this chapter, we introduce the
very important concepts of continuity, differentiability and
rel... |
class_12 | 6 | Application of Derivatives | ncert_books/class_12/lemh1dd/lemh106.pdf | 6.1 Introduction
In Chapter 5, we have learnt how to find derivative of composite functions, inverse
trigonometric functions, implicit functions, exponential functions and logarithmic functions.
In this chapter, we will study applications of the derivative in various disciplines, e.g., in
engineering, science, social ... |
class_12 | 7 | Integrals | ncert_books/class_12/lemh2dd/lemh201.pdf | 7.1 Introduction
Differential Calculus is centred on the concept of the
derivative. The original motivation for the derivative was
the problem of defining tangent lines to the graphs of
functions and calculating the slope of such lines. Integral
Calculus is motivated by the problem of defining and
calculating the area... |
class_12 | 8 | Application of Integrals | ncert_books/class_12/lemh2dd/lemh202.pdf | 292 MATHEMATICS
8.1 Introduction
In geometry, we have learnt formulae to calculate areas
of various geometrical figures including triangles,
rectangles, trapezias and circles. Such formulae are
fundamental in the applications of mathematics to many
real life problems. The formulae of elementary geometry
allow us to ca... |
class_12 | 9 | Differential Equations | ncert_books/class_12/lemh2dd/lemh203.pdf | 9.1 Introduction
In Class XI and in Chapter 5 of the present book, we
discussed how to differentiate a given function f with respect
to an independent variable, i.e., how to find f ′(x) for a given
function f at each x in its domain of definition. Further, in
the chapter on Integral Calculus, we discussed how to find
... |
class_12 | 10 | Vector Algebra | ncert_books/class_12/lemh2dd/lemh204.pdf | 10.1 Introduction
In our day to day life, we come across many queries such
as – What is your height? How should a football player hit
the ball to give a pass to another player of his team? Observe
that a possible answer to the first query may be 1.6 meters,
a quantity that involves only one value (magnitude) which
is ... |
class_12 | 11 | Three Dimensional Geometry | ncert_books/class_12/lemh2dd/lemh205.pdf | 11.1 Introduction
In Class XI, while studying Analytical Geometry in two
dimensions, and the introduction to three dimensional
geometry, we confined to the Cartesian methods only. In
the previous chapter of this book, we have studied some
basic concepts of vectors. We will now use vector algebra
to three dimensional g... |
class_12 | 12 | Linear Programming | ncert_books/class_12/lemh2dd/lemh206.pdf | 394 MATHEMATICS
12.1 Introduction
In earlier classes, we have discussed systems of linear
equations and their applications in day to day problems. In
Class XI, we have studied linear inequalities and systems
of linear inequalities in two variables and their solutions by
graphical method. Many applications in mathemat... |
class_12 | 13 | Probability | ncert_books/class_12/lemh2dd/lemh207.pdf | 406 MATHEMATICS
13.1 Introduction
In earlier Classes, we have studied the probability as a
measure of uncertainty of events in a random experiment.
We discussed the axiomatic approach formulated by
Russian Mathematician, A.N. Kolmogorov (1903-1987)
and treated probability as a function of outcomes of the
experiment. W... |
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