Approved by the Government of Nepal, Ministry of Education, Science and Technology,
Curriculum Development Centre, Sanothimi, Bhaktapur as an Additional Learning Material
vedanta
MExcAeTl iHn EMATICS
7Book
Author
Hukum Pd. Dahal
Editor
Tara Bahadur Magar
vedanta
Vedanta Publication (P) Ltd.
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Vanasthali, Kathmandu, Nepal
+977-01-4982404, 01-4962082
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www.vedantapublication.com.np
vedanta
MExcAeTl iHn EMATICS
7Book
All rights reserved. No part of this publication may be
reproduced, copied or transmitted in any way, without
the prior written permission of the publisher.
First Edition: B.S. 2077 (2020 A. D.)
Second Edition: B.S. 2078 (2021 A. D.)
Published by:
Vedanta Publication (P) Ltd.
j]bfGt klAns;] g kf| = ln=
Vanasthali, Kathmandu, Nepal
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Preface
The series of 'Excel in Mathematics' is completely based on the contemporary pedagogical teaching
learning activities and methodologies extracted from Teachers' training, workshops, seminars and
symposia. It is an innovative and unique series in the sense that the contents of each textbooks of
the series are written and designed to fulfill the need of integrated teaching learning approaches.
Excel in Mathematics is an absolutely modified and revised edition of my three previous series:
'Elementary mathematics' (B.S. 2053), 'Maths In Action (B. S. 2059)' and 'Speedy Maths' (B. S. 2066).
Excel in Mathematics has incorporated applied constructivism. Every lesson of the whole series
is written and designed in such a manner, that makes the classes automatically constructive and
the learners actively participate in the learning process to construct knowledge themselves, rather
than just receiving ready made information from their instructors. Even the teachers will be able
to get enough opportunities to play the role of facilitators and guides shifting themselves from the
traditional methods of imposing instructions.
Each unit of Excel in Mathematics series is provided with many more worked out examples.
Worked out examples are arranged in the hierarchy of the learning objectives and they are reflective
to the corresponding exercises. Therefore, each textbook of the series itself is playing a role of a
‘Text Tutor’. There is a well balance between the verities of problems and their numbers in each
exercise of the textbooks in the series.
Clear and effective visualization of diagrammatic illustrations in the contents of each and every
unit in grades 1 to 5, and most of the units in the higher grades as per need, will be able to integrate
mathematics lab and activities with the regular processes of teaching learning mathematics
connecting to real life situations.
The learner friendly instructions given in each and every learning contents and activities during
regular learning processes will promote collaborative learning and help to develop learner-
centred classroom atmosphere.
In grades 6 to 10, the provision of ‘General section’, ‘Creative section - A’ and ‘Creative
section - B’ fulfills the coverage of overall learning objectives. For example, the problems in
‘General section’ are based on the Knowledge, understanding and skill (as per the need of the
respective unit) whereas the ‘Creative sections’ include the Higher ability problems.
The provision of ‘Classwork’ from grades 1 to 5 promotes learners in constructing knowledge,
understanding and skill themselves with the help of the effective roles of teacher as a facilitator
and a guide. Besides, teacher will have enough opportunities to judge the learning progress and
learning difficulties of the learners immediately inside the classroom. These classworks prepare
learners to achieve higher abilities in problem solving. Of course, the commencement of every
unit with 'Classwork-Exercise' may play a significant role as a 'Textual-Instructor'.
The 'project works' given at the end of each unit in grades 1 to 5 and most of the units in higher
grades provide some ideas to connect the learning of mathematics to the real life situations.
The provision of ‘Section A’ and ‘Section B’ in grades 4 and 5 provides significant opportunities
to integrate mental maths and manual maths simultaneously. Moreover, the problems in ‘Section
A’ judge the level of achievement of knowledge and understanding and diagnose the learning
difficulties of the learners.
The provision of ‘Looking back’ at the beginning of each unit in grades 1 to 8 plays an important
role of ‘placement evaluation’ which is in fact used by a teacher to judge the level of prior
knowledge and understanding of every learner to make his/her teaching learning strategies.
The socially communicative approach by language and literature in every textbook especially in
primary level of the series will play a vital role as a ‘textual-parents’ to the young learners and
help them in overcoming maths anxiety.
The Excel in Mathematics series is completely based on the latest curriculum of mathematics,
designed and developed by the Curriculum Development Centre (CDC), the Government of Nepal.
I do hope the students, teachers and even the parents will be highly benefited from the ‘Excel in
Mathematics’ series.
Constructive comments and suggestions for the further improvements of the series from the
concerned will be highly appreciated.
Acknowledgments
In making effective modification and revision in the Excel in Mathematics series from my
previous series, I’m highly grateful to the Principals, HOD, Mathematics teachers and experts,
PABSON, NPABSAN, PETSAN, ISAN, EMBOCS, NISAN and independent clusters of many
other Schools of Nepal, for providing me with opportunities to participate in workshops,
Seminars, Teachers’ training, Interaction programmes and symposia as the resource person.
Such programmes helped me a lot to investigate the teaching-learning problems and to research
the possible remedies and reflect to the series.
I’m proud of my wife Rita Rai Dahal who always encourages me to write the texts in a more
effective way so that the texts stand as useful and unique in all respects. I’m equally grateful to
my son Bishwant Dahal and my daughter Sunayana Dahal for their necessary supports during
the preparation of the series.
I’m extremely grateful to Dr. Ruth Green, a retired professor from Leeds University, England
who provided me very valuable suggestions about the effective methods of teaching-learning
mathematics and many reference materials.
Grateful thanks are due to Mr. Tara Bahadur Magar for his painstakingly editing of the series.
Moreover, I gratefully acknowledge all Mathematics Teachers throughout the country who
encouraged me and provided me the necessary feedback during the workshops/interactions
and teachers’ training programmes in order to prepare the series in this shape.
I’m profoundly grateful to the Vedanta Publication (P) Ltd. to get this series published. I would
like to thank Chairperson Mr. Suresh Kumar Regmi, Managing Director Mr. Jiwan Shrestha,
Marketing Director Mr. Manoj Kumar Regmi for their invaluable suggestions and support during
the preparation of the series.
Last but not the least, I’m heartily thankful to Mr. Pradeep Kandel, the Computer and Designing
Senior Officer of the publication house for his skill in designing the series in such an attractive
form.
Hukum Pd. Dahal
Contents Page
5-22
S.N Unit 23-33
1. Set
34-55
1.1 Set – Looking back, 1.2 Membership of a set and set notation,
1.3 Methods of describing sets, 1.4 Cardinal number of a set, 1.5 Types of 56-77
sets, 1.6 Relationships between sets, 1.7 Universal set and subset, 1.8 Set
Operations, 1.9 Venn-diagrams 78-98
2. Number System in Different Bases 99-117
2.1 Whole numbers - Looking back, 2.2 Decimal or Denary 118-127
number system, 2.3 Periods and place, 2.4 Binary number system, 128-140
2.5 Conversion of binary numbers to decimal numbers, 2.6 Conversion 141-147
of decimal numbers to binary numbers, 2.7 Quinary number
system, 2.8 Conversion of quinary numbers to decimal numbers,
2.9 Conversion of decimal numbers to quinary numbers
3. Operations on Whole Numbers
3.1 Factors and multiples - Looking back, 3.2 Highest common factor
(H. C. F.), 3.3 Finding H. C. F. by Factorization method, 3.4 Finding H.C.F. by
Division method, 3.5 Lowest common multiple ( L.C.M), 3.6 Finding L.C.M.
by factorisation method, 3.7 Finding L.C.M. by division method, 3.8 Square
and square root, 3.9 Process of finding square root, 3.10 Cube and cube root
4. Real Numbers
4.1 Integers – Looking back, 4.2 Absolute value of integers,
4.3 Operations on integers, 4.4 Sign rules of addition and subtraction
of integers, 4.5 Properties of addition of integers, 4.6 Multiplication and
division of integers, 4.7 Sign rules of multiplication and division of integers,
4.8 Properties of multiplication of integers, 4.9 Order of operations,
4.10 Rational numbers – review, 4.11 Properties of Rational numbers,
4.12 Terminating and non-terminating rational numbers, 4.13 Irrational
numbers
5. Fraction and Decimal
5.1 Fraction – Looking back, 5.2 Addition and subtraction of fraction - revision,
5.3 Multiplication of fractions, 5.4 Division of fractions, 5.5 Decimal - revision,
5.6 Terminating and non-terminating recurring decimal, 5.7 Four fundamental
operations on decimals
6. Ratio, Proportion and Unitary Method
6.1 Ratio and Proportion – Looking back, 6.2 Proportion, 6.3 Types of
proportions, 6.4 Unitary method
7. Percent and Simple Interest
7.1 Percent – Looking back, 7.2 Operations on percent, 7.3 Simple interest –
Review, 7.4 Calculation of simple interest
8. Profit and Loss
8.1 Profit and Loss – Looking back, 8.2 Profit and loss per cent,
8.3 Calculation of S.P. when C.P. and profit or loss per cent are given,
8.4 Calculation of C.P. when S.P. and profit or loss per cent are given,
8.5 Discount, 8.6 Discount per cent, 8.7 Value Added Tax (VAT)
9. Algebraic Expressions
9.1 Algebraic terms and expressions – Looking back, 9.2 Types of
algebraic expressions, 9.3 Polynomial, 9.4 Degree of polynomials,
9.5 Evaluation of algebraic expressions, 9.6 Addition and subtraction of
algebraic expressions
10. Laws of Indices 148-178
10.1 Laws of indices (or exponents), 10.2 Multiplication of algebraic
expressions, 10.3 Some special products and formulae, 10.4 Division 179-200
of algebraic expressions, 10.5 Factors and factorisation – Introduction,
10.6 Simplification of rational expressions 201-218
11. Equation, Inequality and Graph 219-231
11. 1 Open statement and equation - Looking back, 11.2 Linear equations 232-251
in one variable, 11.3 Solution to equations, 11.4 Applications of equations,
11.5 Trichotomy – Review, 11.6 Inequalities, 11.7 Replacement set and 252-258
solution set, 11.8 Graphical representation of solution sets, 11.9 Function 259-267
machine and relation between the variables x and y,
268-270
12. Coordinates 271-293
12.1 Coordinates – Looking back, 12.2 Coordinate axes and quadrants, 294-298
12.3 Finding points in all four quadrants, 12.4 Plotting points in all four quadrants, 299-305
12.5 Reflection of geometrical figures, 12.7 Rotation of geometrical figures, 306-318
12.8 Rotation of geometrical figures using coordinates, 12.9 Displacement,
318-332
13. Geometry: Angles 333-334
13.1 Angels – Looking back, 13.2 Different pairs of angles – Review,
13.3 Verification of properties of angles, 13.4 Pairs of angles made by a
transversal with parallel lines
14. Triangle, Quadrilateral and Polygon
14.1 Triangles – Looking back, 14.2 Properties of triangles,
14.3 Some special types of quadrilaterals, 14.4 Verification of properties of
special types of quadrilaterals, 14.5 Interior and exterior angles of regular
polygons
15. Congruency and Similarity
15.1 Congruent figures – Introduction, 15.2 Congruent triangles,
15.3 Conditions of congruency of triangles, 15.4 Similar triangles
16. Construction
16.1 Construction of perpendicular bisector of a line segment,
16.2 Transferring angles, 16.3 Construction of different angles,
16.4 Construction of triangles, 16.5 Construction of parallelograms,
16.6 Construction of squares, 16.7 Construction of rectangles,
16.8 Construction of rhombus, 16.9 Construction of kite
17. Circle
17.1 Circle and its different parts – review
18. Perimeter, Area and Volume
18.1 Perimeter, Area and Volume – Looking back, 18.2 Perimeter of plane
figures, 18.3 Area of plane figures, 18.4 Nets and skeleton models of regular
solids, 18.5 Area of solids, 18.6 Volume of solids
19. Symmetry, Design and Tessellation
19.1 Symmetrical and asymmetrical shapes, 19.2 Line or axis of symmetry,
19.3 Rotational symmetry, 19.4 Order of rotational symmetry, 19.5 Tessellations,
19.6 Types of tessellations
20. Scale Drawing and Bearing
20.1 Scale drawing, 20.2 Scale factor, 20.3 Bearing
21. Statistics
21.1 Statistics – Review, 21.2 Types of data and frequency table, 21.3 Cumulative
frequency table of ungrouped data, 21.4 Grouped and continuous data,
21.5 Bar graph, 21.6 Average (or Mean), 21.7 Mean or average of ungrouped
repeated data
Answers
Evaluation Model
Unit Set
1
1.1 Set – Looking back
Classroom - Exercise
1. Let's write any three members of the well defined collections inside braces.
a) A collection of High mountains of Nepal ....................................................
b) A collection of mountains of Nepal which have more than 6000 m altitude.
.........................................................................................................................
c) A collection of fruits. .....................................................................................
d) A collection of tasty fruits. ............................................................................
2. If W = {0, 1, 2, 3, 4, 5} and A = {2, 4, 6}, let's tell and write 'true' or 'false'.
a) A W = ................. b) 2 W = .................
c) 6 W = ................. d) {2, 4} W = .................
3. Let's tell and list the elements of these sets.
a) {letters of the word ‘SUCCESS’} ...................................................................
b) {x : x <10, x odd number} .......................................................................
4. The set-builder form of A = {2, 4, 6, 8} .............................................................
0 It is a collection of whole numbers less than 5. Then, it
12 definitely includes the numbers 0, 1, 2, 3 and, 4. Because the
34 statement ‘whole numbers less than 5’ defines the distinct and
distinguishable objects which are to be included in the collection.
Therefore, it is a well-defined collection and it is called a set.
On the other hand, is it possible to write the members of the collection
of tall students of your class? Is it possible to write the members of
the collection of tasty fruits? Discuss why these collection are not the
well - defined. Therefore, the collections which are not well-defined are not sets.
1.2 Membership of a set and set notation
A member or an element of a set is any one of the distinct objects that make up that
set.
For example, in a set N = {1, 2, 3, 4, 5} , the members or elements of the set N are
1, 2, 3, 4, and 5.
5 Vedanta Excel in Mathematics - Book 7
Set
The membership of a member of a set is denoted by the symbol ‘'. For example,
1N or 1{ 1, 2, 3, 4, 5}. We read it as '1 belongs to set N' or '1 is a member of set N'
or 1 is an element of set N. Similarly, 2N, 3N, 4N and 5N.
However, when any element is not a member of a given set, it is denoted by the
symbol . For example: In N = {1, 2, 3, 4, 5}, 6 N, 7 N, ... and so on.
Set notation
We denote sets by capital letters, such as, A, B, C, W, N, etc. The members of a set
are enclosed in braces { } and they are separated by commas. For example,
A = {a, e, i, o, u}, W = {0, 1, 2, 3, 4, 5}, and so on.
1.3 Methods of describing sets
We usually write the members of a set by the following four methods:
(i) Diagramatic method 23
In this method, we write the members of a set inside a circular 57
or oval diagram. A set of prime numbers less than 10 is shown
in the diagram.
(ii) Description method
In this method, we describe the common property (or properties) of the
members of a set inside the braces. For example:
N = {natural numbers less than 10}
P = {prime numbers between 10 and 20}
V = {vowels of English alphabets}, and so on.
(iii) Listing/Roster/Tabular Method
In this method, we list the members of a set inside the braces and members are
separated by commas. For example:
N = {1, 2, 3, 4, 5, 6, 7, 8, 9}
P = {11, 13, 17, 19}
V = {a, e, i, o, u}, and so on.
(iv) Set-builder/Rule Method
In this method, we use a variable such as x, y, z, p, q, etc. to represent the
members of a set and the common property (or properties) of the members is
described by the variable. For example:
We describe, N ={1, 2, 3, 4, 5, 6, 7, 8, 9}= {x : x is a natural number less than 10}
and read as: “N is the set of all values of x, such that x is a whole number less
than 10.”
Similarly, P = {11, 13, 17, 19} = {y : 10 < y < 20, y prime number}
V = {a, e, i, o, u} = {z : z is a vowel of English alphabets}
Vedanta Excel in Mathematics - Book 7 6
Set
1.4 Cardinal number of a set
Let's take a set of even numbers less than 12, i.e., E = {2, 4, 6, 8, 10}.
There are five members or elements in this set E. Therefore, the cardinal number of
the set E, denoted by n (E) = 5. Similarly, in A = {m, a, t, h}, its cardinal number
n (A) = 4. Thus, the number of members or elements contained by a set is known
as its cardinal number.
1.5 Types of sets
According to the number of elements contained by sets, there are four types of sets.
(i) Empty or null set (ii) Unit or singleton set (iii) Finite set (iv) Infinite set
(i) Empty or null set
There can be certain sets that do not contain any element at all. For example,
the sets of months with 35 days, the set of triangles with 4 sides, etc. do not
contain any element. So, the listing for these sets contains no element at all
and we call it an empty or null set.
Thus, a set containing no elements is called an empty or a null set. It is
represented by the symbol { } or I (Phi). Similarly,
if A = {natural numbers between 9 and 10}, A = { } or I and n(A) = 0.
if B = {whole numbers less than 0}, B = {} or I and n(B) = 0, and so on.
(ii) Unit or singleton set
Let's take a set A = {odd numbers between 8 and 10}.
Here, A = {9}. Thus, the set A contains exactly one element. Therefore, the
set A is called a unit or singleton set. A set containing exactly one element is
called a unit or singleton set.
Similarly,
if P = {prime and even numbers}, P = {2} and n(P) = 1
if Q = {x : 6 < x < 8, x ∈ N}, Q = {7} and n(Q) = 1, and so on.
(iii) Finite sets
Let's consider a set C = {composite numbers less than 10}.
Here, C = {4, 6, 8, 9} and we can count the number of elements of this set,
i.e. n(C) = 5. Hence, the set C is said to be a finite set.
Thus, a set containing finite number of elements is called a finite set.
Similarly,
if A = {1, 2,3, …, 20}, n(A) = 20
if B = {x : 2 < x < 31, x ∈ prime number}, n(B) = 9. It is a finite set.
7 Vedanta Excel in Mathematics - Book 7
Set
(iv) Infinite sets
Let’s take a set W = {0, 1, 2, 3, 4, 5, ...}. It is the set of whole numbers. This
set is so large that we can never finish counting its elements. It has infinite
number of elements. So, it is an infinite set.
Thus, a set containing infinite number of elements is called an infinite set.
EXERCISE 1.1
General Section -Classwork
1. Let's tick () the well-defined collections. Also, list any three members of
well-defined collections.
a) A collection of greater natural numbers. ............................
b) A collection of natural numbers less than 4. ............................
c) A collection of name of English months starting ............................
with 'J' letter.
d) A collection of favourite English months. ............................
2. If A = { 0, 1, 2, 3, 4, 5 } and P = {2, 3, 5, 7}, let's insert the correct symbol ''
or '' in the blanks..
a) 4 .................. A b) 7 .................. A c) 1 .................. P
d) 5 .................. P e) {2, 3, 5, 7}.............. A f) {2, 3, 5} .................. P
3. Let's tell and write the cardinal numbers of these sets as quickly as possible.
a) A = { 2, 3, 5, 7}, n (A) = .................
b) B = { even numbers less than 7 }, n (B) = .................
c) P = { letters of the word 'PUPIL' }, n (P) = .................
d) O = { odd numbers between 3 and 5 }, n (O) = .................
4. Let's tell and write whether these sets are empty, unit, finite, or infinite as
quickly as possible.
a) A = { odd numbers less than 100 } ...................................................
b) B = { odd numbers more than 100 } ...................................................
c) C = { composite number between 1 and 5 } ...................................................
d) D = { prime number between 7 and 10 } ..................................................
Creative section - A
5. a) Define set. Is the collection of 'tall'' students' in a class a set? Why?
b) Write four methods of describing sets. Give one examples of each method.
Vedanta Excel in Mathematics - Book 7 8
Set
c) Define cardinal number of a set with an example.
d) Write four types of sets on the basis of cardinal numbers. Write one example
of each.
6. Let's rewrite these sets in description method.
a) A = {1, 2, 3, 4} b) B = {5, 10, 15, 20, 25}
c) C = {7, 11, 13, 17, 19} d) D = {1, 2, 3, 6}
7. Let's rewrite these sets in listing method.
a) P = { prime numbers between 10 and 20 }
b) A = { letters of the word ‘FOOTBALL’ }
c) F = {x : x is a A factor of 18 }.
d) M = {y : y is a multiple of 3, 5 < y < 10}.
8. Let's rewrite these sets in set builder method.
a) A= {1, 2, 3, 4, 5} b) B = { 2, 3, 5, 7}
c) C = {1, 4, 9, 16, 25} d) D = {1, 2, 4, 8}
9. Let's list the elements and write the cardinal numbers of these sets.
a) A = { composite numbers between 10 and 20}.
b) B = { all possible factors of 12 }.
c) Z = { x : x is an integer, –2 ≤ x ≤ 2}
d) W = { x : x is a whole number, x < 1 }
It's your time - Project work!
10. a) Let's write the whole numbers from 90 to 100. Select the appropriate
numbers to form the following sets. Then, write the types of sets.
(i) A={even numbers} (ii) B={odd numbers}
(iii) C={x : x is a prime number} (iv) D={y : y is a composite number}
(v) E={z : z is a square number} (vi) F={cube numbers}
(vii) G={multiples of 7} (viii) H={x : x is divisible by 11}
b) Let's observe around the kitchen of your house and select any five objects
as the members of a set. Then, express the set in description, roster, and
rule methods.
1.6 Relationships between sets
According to the types and number of elements contained by two or more sets, there
are various types of relationships between the sets, such as equal sets, equivalent
sets, disjoint sets, and overlapping sets.
9 Vedanta Excel in Mathematics - Book 7
Set
(i) Equal sets
Let’s take any two sets, A = {s, v, u, 3, ª} and B = {ª, u, s, 3, v}.
Here, n (A) = 5 and n (B) = 5. Thus, they have the equal cardinal number and
both the sets have exactly the same elements. Therefore, sets A and B are said
to be the equal sets and written as A = B.
Thus, two or more than two sets are said to be equal if they have exactly the
same elements and equal cardinal number.
(ii) Equivalent sets
Let’s take any two sets, A = { c, o, w } and B = { g, o, d}. Here,
n (A) = 3 and n (B) = 3. They have the equal cardinal number.
However, the elements c, w of set A are not contained by the set B and the
elements g, d of the set B are not contained by the set A. So, they are not equal
and they are said to be equivalent sets.
We write it as A ~ B.
Thus, two or more than two sets are said to be equivalent if they have the equal
cardinal number but they do not have exactly the same elements.
(iii) Overlapping sets
Let’s take any two sets: A = {1, 2, 3, 6} and B = { 1, 2, 4, 8}.
In these two sets, the elements 1 and 2 are common to both the sets. Therefore,
sets A and B are overlapping sets.
AB
Thus, two or more than two sets are said to be overlapping 3 14
if they contain at least one element common. The common
elements of overlapping sets are shown in the shaded 62 8
region of the two intersecting diagrams.
(iv) Disjoint sets
Let’s take any two sets : A = { 3, 6, 9, 12} and B = { 5, 10, 15, 20 }
In these two sets, there is no any common element. Therefore, sets A and B are
disjoint sets. Of course, non overlapping sets are the disjoint sets.
Thus, two or more than two sets are said to be A B
disjoint if they do not have any element common. 36 5 10
The elements of disjoint sets are shown in 9 12 15 20
non-intersecting diagrams.
1.7 Universal set and subset
Let’s take a set of natural numbers less than 15.
N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}
Now, let's select certain elements from this set and make a few other sets.
Vedanta Excel in Mathematics - Book 7 10
Set
A = {even numbers less than 15 = {2, 4, 6, 8, 10, 12, 14}
B = {prime numbers less than 15} = {2, 3, 5, 7, 11, 13}
C = {square number less than 15} = {1, 4, 9}
D = {possible factors of 12} = {1, 2, 3, 4, 6, 12}
NA N B 14 N C 14 13 N D
2 1 12 5 1 14
12 4 14 1 2 3 12
12 13 5 7 10 34 11 2 3 4 13
3 6 11 11 13 5 9 10 7 6
58 10 4 68 8 12
9 6 78 9 10 11
79
Here, the set of natural numbers less than 15 is known as the universal set.
Furthermore, every element of the sets A, B, C, and D is also an element of the set
N. In such a case, sets A, B, C and D are called the subsets of the set N. We use the
symbol ‘’ to represent a set as a subset of another set. For example:
‘A is a subset of N’ is written as A N .
‘B is a subset of N’ is written as B N, and so on.
Remember that, every set is a subset of itself and an empty set (I) is a subset of very
set.
On the other hand, if a set is not the subset of a given set, we denote it by the symbol ''.
Thus, a set under the consideration from which many other subsets can be formed is
called a universal set. The set of teachers of a school is a universal set from which the
subsets like set of Maths teachers, set of Science teachers, set of English, etc. can be
formed. We usually denote a universal set by the capital letter 'U'.
Super set
If the set A is a subset of N, N is called the super set of A. It is denoted as N A and
read as ‘N is a super set of A’.
Proper subset
Let's take any two sets, A = {p, e, n, c, i, l} and B = {p, e, n}.
Here, B is a subset of A and B is not equal to A. In such a case, set B is said to be a
proper subset of A.
Thus, the set B is said to be a proper subset of the set A if it contains at least one
element less than A.
We use a symbol ‘’ to represent a set as a proper subset of another set. For example:
B A and we read it as ‘B is a proper subset of A’.
Improper subset
Let's take any two set, A = {g], kf, n} and B = {kf, n, g}]
Here, B is a subset of A and B is equal to A. In such a case, the set B is said to be an
improper subset of A.
11 Vedanta Excel in Mathematics - Book 7
Set
Thus, the set B is said to be an improper subset of A, if B is equal to A, i.e., B = A. We
use the symbol to represent a set as an improper subset of another set. For example,
B A and we read it as 'B is an improper subset of A'.
Number of subsets of a given set
Let's study the following table and draw the conclusion about the number of subsets
of a given set.
Set Subsets No. of elements No. of subsets
A={} n(A) = 0 1 m 2°
B = {a} {} n(B) = 1 2 m 21
C = {a, b} n(C) = 2 4 m 22
D = {a, b, c} {a}, I
n(D) = 3 8 m 23
{a}, {b}, {a, b}, I
{a}, {b}, {c}, {a, b},
{b, c}, {a, c} {a, b, c} I
From the above table, we conclude that the number of subsets of a set is given by the
formula 2n, where n is the cardinal number of the given set.
EXERCISE 1.2
General section - Classwork
1. Let's say whether these pairs of sets are equal or equivalent sets and write
A = B or A ~ B in the blank spaces.
(a) A = {1, 2, 4, 8 } and B = { 1, 4, 9, 16}, .......................................
(b) A = { 4, 6, 8, 9, 10} and B = { 6, 9, 4, 8, 10}, ......................................
2. Let's say and write whether these pairs of sets are overlapping or disjoint sets.
(a) A = { 1, 3, 5, 15 } and B = {5, 10, 15, 20}, A and B are .........................
(b) P = { 1, 3, 5, 7, 9 } and Q = { 4, 8, 12 }, P and Q are ........................
3. Let's say and fill in the blanks with appropriate words.
(a) If each element of set A is also an element of B, A is said to be
a ............................... of B and B is said to be a ............................... of A.
(b) Every set is a ............................... of itself.
(c) The empty set is a ............................... of every set.
(d) The number of possible subsets of a set containing 'n' number of elements
is given by the formula ...............................
4. Let's say and write the possible number of subsets of these sets.
(a) In {1, 2}, number of subsets = .................................................................
(b) In {g, o, d }, number of subsets = .................................................................
Vedanta Excel in Mathematics - Book 7 12
Set
5. Let's say and write the possible number of proper subsets of these sets.
(a) In {k} number of proper subsets = .....................................................
(b) In { 1, 4, 9 } number of proper subsets = .....................................................
6. Let's say and write the possible subsets of these sets.
(a) {P} ...............................................................................................................
(b) { 2, 7} ..............................................................................................................
7. Let's say and write which one is the universal set and its subset in the given
pairs of sets below.
a) A = { students of class 7 } and B = { boys of class 7 }
Universal set is .................. and subset is ..................
b) X = { even numbers less than 20 } and W = {whole numbers less than 20}.
Universal set is ....................... and subset is.......................
Creative section
8. a) Define the following types of sets with examples.
(i) Equal and equivalent sets (ii) Overlapping and disjoints sets
b) Define universal set and subset with examples.
c) Define proper and improper subsets with examples.
9. Let's list the elements of the following sets and write with reasons whether
they are equal or equivalent sets:
a) A = {days of the week starting with the letter S} and
B = {days of the week starting with the letter T}
b) M = {letters of the word 'FOLLOW'} and
N = {letters of the word 'WOLF'}
10. Let's list the elements of the following sets and write with reasons whether
they are overlapping or disjoint sets:
a) O = {odd numbers less than 10} and
S = {perfect square numbers less than 10}
b) X = {x : x is a prime number which exactly divides 30}
Y = {y : y is a prime number which exactly divides 77}
11. Let's write the possible subsets of the following sets and list the proper subsets
separately:
(a) F = {apple} (b) A = {1, 2} (c) E = {a, b, c} (d) T = {1, 2, 3, 4}
13 Vedanta Excel in Mathematics - Book 7
Set
12. At first, let's list the elements of each of these sets from the diagrams. Then,
select the common elements and list them in separate sets.
a) A B b) P Q
3 14 m t i
a h
62 8 n
k
13. Let's list the elements and common elements of these pairs of sets and show
them in diagrams.
(a) A = { x : x is a whole number less than 10 }and B = { factors of 24 }
(b) P = { x : x is a multiple of 4, x ≤ 20 } and
Q = { x : x is a multiple of 5, x ≤ 20 }
It's your time - Project work!
14. a) Let's take a universal set under your consideration. Then, write as many
subsets as possible from your universal set.
b) Let's conduct a survey inside your classroom among your friends. Then,
list the name of your friends and make separate sets in the following cases:
(i) Sets of friends who like tea, coffee or milk.
(ii) Sets of friends who like Mo:Mo, chowmein or Thukpa.
(iii) How many overlapping sets are formed? Show them in diagrams.
(iv) How many disjoint sets are formed? Show them in diagrams.
(v) How many equal sets are formed?
(vi) How many equivalent sets are formed?
c) Is there any possibility to form overlapping sets of the sets of teachers
who are teaching different subjects in your school? If so, make these sets
and show in diagrams.
15. Let's make groups of 3 students and play game!
Each student of each group should make two disjoint sets with maximum
5 members taking the natural numbers from 1 to 20. Next day, from the sets
of numbers made by the students of each group, form as many number of
overlapping sets as possible. The group which has the maximum number of
overlapping sets is the winner!
1.8 Set Operations
Sets can be combined in a number of different ways to make another set. It is known
as set operations. There are four basis set operations. They are:
(i) Union of sets (ii ) Intersection of sets
(iii) Difference of sets (iv) Complement of a set
(i) Union of sets
Let's make a sport committee P with 3 members Ram, Sita, Laxmi and a
cultural committee Q with 4 members Sita, Laxman, Ajay, and Sudip. When
the committees P and Q have a joint meeting and a new committee R is
Vedanta Excel in Mathematics - Book 7 14
Set
formed; then, clearly the new committee R (the union of committees P and Q)
has 6 members: Ram, Sita, Laxmi, Laxman, Ajay, and Sudip. It does not have
7 members.
When the elements of two or more sets are combined and listed together in a
single set, it is called the union of these sets. Now, let's take another example.
If A = {1, 3, 5, 7, 9} and B = {2, 3, 5, 7},
the union of sets A and B = {1, 2, 3, 5, 7, 9},
it is denoted as A B = {1, 2, 3, 5, 7, 9}.
Thus, the union of two sets A and B is the set consisting of all elements that
belong to A or B (or both), and it is denoted by A B (read as ‘A union B’).
The symbol ‘’ (cup) denotes the union of sets.
It is noted that while making the union of sets, the common elements should
be listed only once.
(ii) Intersection of sets
In the above example, set P = {Ram, Sita, Laxmi} and set Q = {Sita, Laxman,
Ajay, Sudip}. Here, Sita is the member common to the sets P and Q. So, the
intersection of these two sets is {Sita}.
When the common elements of two or more sets are listed in a separate set, it
is called intersection of sets. Now, let's take another example.
If A = {1, 2, 3, 4, 5, 6} and B = {1, 3, 5, 7, 9, 11},
the intersection of sets A and B = {1, 3, 5}
It is denoted as A B = {1, 3, 5}
Thus, the intersection of sets A and B is the set consisting of all elements that
belong to A and belong to B. It is denoted by A B (read as ‘A inter section B’).
The symbol ‘’ (cap) denotes the intersection of sets.
(iii) Difference of sets
The difference of two sets A and B denoted by A – B is the set of all elements
contained only by A but not by B. For example:
If A = {2, 4, 6, 8, 10} and B = {1, 2, 4, 8},
the difference of A and B = {6, 10}, which is only A.
It is denoted as A – B = {6, 10}
Similarly, the difference of B and A denoted by B – A is the set of all elements
contained only by B but not by A.
So, B – A = {1}, which is only B.
(iv) Complement of a set
When your teacher asks students of odd roll numbers to raise their hands, the
students of even roll numbers who do not raise hands are the complement
of odd roll numbers. Thus, if a set A is the subset of a universal set U, the
complement of A denoted by A or A' or Ac is the set which is formed due to
the difference of U and A, i.e. U – A.
For example, if U = {1, 2, 3, … 8} and A = {1, 2, 4, 8}, the complement of
A = U – A = {3, 5, 6, 7} and it is denoted as A = {3, 5, 6, 7}.
15 Vedanta Excel in Mathematics - Book 7
Set
1.9 Venn-diagrams
We can show the sets and set operations by using diagrams. The concept was at
first introduced by Euler, the Swiss Mathematician and it was further developed by
the British Mathematician John Venn. So, the diagrams are famous as Venn Euler
diagrams or Venn diagrams.
In Venn diagrams, a universal set is represented by a rectangle and its subsets are
represented by circles or ovals.
Let’s learn to represent sets and set operations by using Venn diagrams.
U A U U
A BA B
U U U
A BA BA B
U U U
A BA BA B
The shaded region The shaded region The shaded region
represents A – B represents A – B represents B – A
U U U
AB AB
A
U A B
AB A U
U
AB
AB
The shaded region The shaded region The shaded region
represent A B. represent A – B. represent B – A .
Vedanta Excel in Mathematics - Book 7 16
Set
Worked-out examples
Example 1: If M = {m, a, t, h, s} and N = {m, a, g, i, c}, find
(a) M N (b) M N (c) M – N, (d) N – M. Also, represent them in
Venn diagrams.
Solution:
Here, M = {m, a, t, h, s} and N = {m, a, g, i, c} MN
Now,
t mg
a) M N = {m, a, t, h, s} {m, a, g, i, c} = {m, a, t, h, s, g, i, c}
h a i
s c
The shaded region represents M N. MN
b) M N = {m, a, t, h, s} {m, a, g, i, c} = {m, a}
t mg
The shaded region represents M N.
c) M – N = {m, a, t, h, s} – {m, a, g, i, c} = {t, h, s} h a i
s c
The shaded region represents M – N.
d) N – M = {m, a, g, i, c} – {m, a, t, h, s} = {g, i, c} MN
The shaded region represents N – M. t mg
h a i
s c
MN
t mg
h a i
s c
Example 2: If U = {0, 1, 2, …, 10}, A = {1, 2, 3, 4, 5}, and B = {1, 3, 5, 7, 9},
complete the following set operations and represent them in
Venn-diagrams.
a) A B b) A B c) A – B d) B – A
e) A B f) A B g) A – B h) B – A
Solution:
Here, U = {0, 1, 2, …, 10}, A = {1, 2, 3, 4, 5} and B = {1, 3, 5, 7, 9} U
Now, A B
a) A B = {1, 2, 3, 4, 5} {1, 3, 5, 7, 9} and 02
1 7
B = {1, 2, 3, 5, 7, 9} 4 3
6
59
8
10
The shaded region represents A B. U
b) A B = {1, 2, 3, 4, 5} {1, 3, 5, 7, 9} = {1, 3, 5} A B
The shaded region represents A B. 02
1 7
4 3
6
59
8
10
17 Vedanta Excel in Mathematics - Book 7
Set
U
c) A – B = {1, 2, 3, 4, 5} – {1, 3, 5, 7, 9} = {2, 4} A B
The shaded region represents A – B. 02
1 7
d) B – A = {1, 3, 5, 7, 9} – {1, 2, 3, 4, 5} = {7, 9} 4 3
The shaded region represents the elements of B – A. 6
59
e) A B = U – (A B) 8
= {0, 1, 2, …, 10} – {1, 2, 3, 4, 5, 7, 9} 10
= {0, 6, 8, 10} A
02 U
The shaded region represents A B .
4 B
f) A B = U – (A B) 6
= {0, 1, 2, …, 10} – {1, 3, 5} 1 7
= {0, 2, 4, 6, 7, 8, 9, 10} 8 3
The shaded region represents A B . A 59
02
g) A – B = U – (A – B) 10
= {0, 1, 2, …, 10} – {2, 4} 4
= {0, 1, 3, 5, 6, 7, 8, 9, 10} 6 U
The shaded region represents A – B .
8 B
h) B – A = U – (B – A)
= {0, 1, 2, …, 10} – {7, 9} A 1 7
= {0, 1, 2, 3, 4, 5, 6, 8, 10} 02 3
The shaded region represent B – A 4 59
6
10
8
U
A
02 B
4 1 7
6 3
8 59
A 10
02
U
4
6 B
8 1 7
3
59
10
U
B
1 7
3
59
10
Example 3: From the Venn-diagram given alongside, list the P U
elements of the following sets. 6 Q
a) P Q b) P Q c) P – Q d) Q – P 10
e) P Q f) P Q g) P – Q h) Q – P 2
3 41
Solution: 8
Here, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 5 79
a) P Q = {1, 2, 4, 6, 8, 10} b) P Q = {2, 4 ,8}
c) P – Q = {6, 10} d) Q – P = {1}
e) P Q = {3, 5, 7, 9} f) P Q = {1, 3, 5, 6, 7, 9, 10}
g) P – Q = {1, 2, 3, 4, 5, 7, 8, 9} h) Q – P = {2, 3, 4, 5, 6, 7, 8, 9, 10}
Vedanta Excel in Mathematics - Book 7 18
Set
Example 4: If U = {1, 2, 3, … 15}, A = {2, 4, 6, 8, 10, 12, 14}, B = {3, 6, 9, 12, 15}, and
C = {2, 3, 5, 7, 11, 13} , list the elements of the following sets.
a) A B C b) A B C c) (A B) C d) (A B) C
e) (A B) – C f) A – (B C) g) A B C h) A B C
Solution:
Here, U = {1, 2, 3, …15}, A = {2, 4, 6, 8, 10, 12, 14}, B = {3, 6, 9, 12, 15}
C = {2, 3, 5, 7, 11, 13}
a) A B = {2, 4, 6, 8, 10, 12, 14} {3, 6, 9, 12, 15}
= {2, 3, 4, 6, 8, 9, 10, 12, 14, 15}
? A B C = (A B) C
= {2, 3, 4, 6, 8, 9, 10, 12, 14, 15} {2, 3, 5, 7, 11, 13}
= {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
b) A B = {2, 4, 6, 8, 10, 12, 14} {3, 6, 9, 12, 15}
? ABC = {2, 4, 6, 8, 10, 12, 14} {3, 6, 9, 12, 15} = {6, 12}
= (A B) C
= {6, 12} {2, 3, 5, 7, 11, 13} = I
c) A B = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15}
? (A B) C = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15} {2, 3, 5, 7, 11, 13}
= {2, 3}
d) A B = {6, 12}
? (A B) C = {6, 12} {2, 3, 5, 7, 11, 13}
= {2, 3, 5, 6, 7, 11, 12, 13}
e) (A B) = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15}
? (A B) – C = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15} – {2, 3, 5, 7, 11, 13}
= {4, 6, 8, 9, 10, 12, 14, 15}
f) B C = {3, 6, 9, 12, 15} {2, 3, 5, 7, 11, 13} = {3}
? A – (B C) = {2, 4, 6, 8, 10, 12, 14} – {3}
= {2, 4, 6, 8, 10, 12, 14}
g) A B C = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 ,15}
? A B C = U – (A B C)
= {1, 2, 3, …, 15} – {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
= {1}
h) A B C =I
? A B C = U – {A B C} = {1, 2, 3, …15} – I
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
19 Vedanta Excel in Mathematics - Book 7
Set
EXERCISE 1.3
General Section - Classwork
1. Let's say and write the members of the following set operations.
U = {a, b, c, d, e} A = {a, b, c}, and B = { b, c, d}
(i) A B = ............................. (ii) A B = .............................
(iii) A – B = .............................. (iv) B – A = ..............................
(v) A = .................................. (vi) B = ....................................
(vii) A B = ............................. (viii) A B = .............................
2. Let's say and write the required set operations represented by the shaded
regions of Venn-diagrams.
a) b) c) d) U
U U U
A B P QX YA B
.......................... .......................... .......................... ..........................
3. Let’s identify and shade the region to show the given operations.
(i) (ii) (iii) (iv)
A U U U U
BP QM NX Y
AB P–Q MN Y–X
Creative Section - A
4. S is the set of students in sport-club and C is the set of students in
cultural club of a school. If S = {Ram, Sita, Krishna, Shiva, Abdul} and
C = {Sita, Laxmi, Shova, Hari, Joseph}, answer the following questions:
a) Make a separate set of students who are in both the clubs.
b) Make a separate set of students at the time of their joint meeting and
write the number of members in the joint meeting.
c) Make a set of students who are only the member of Sport-club.
d) Make a set of students who are only the member of Cultural-club.
Vedanta Excel in Mathematics - Book 7 20
Set
5. a) If A = {a, e, i, o, u} and B = {a, b, c, d, e}, find: (i) A B (ii) A B
(iii) A – B, (iv) B – A and show these operations in Venn-diagrams.
b) If P ={1, 3, 5, 7, 9} and Q = {1, 3, 6, 9, 18}, find: (i) P Q (ii) P Q
(iii) P – Q (iv) Q – P and show these operations in Venn-diagrams.
c) If M = {1, 2, 3, 6} and N = {factor of 8}, find: (i) M N (ii) M N
(iii) M – N (iv) N – M and show these operations in Venn-diagrams.
d) If A = {x : x ≤ 5, x N} and B = {2, 4}, find: (i) A B (ii) A B
(iii) A – B (iv) B – A and show these operations in Venn-diagrams.
6. a) If U = {1, 2, 3, 4, … 15}, A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8, 10, 12},
find: (i) A (ii) B and show them in Venn - diagrams.
b) If U = {1, 2, 3, ... 10}, P = {1, 4, 9} and Q = {2, 3, 5, 7}, find:
(i) P (ii) Q and show these operations in Venn-diagrams.
7. a) If U = {1, 2, 3, ... 10}, A = {2, 4, 6, 8}, and B = {2, 3, 5}, find:
(i) A B (ii) A B and show these operations in Venn-diagrams.
b) If U = {1, 2, 3, ... 12}, A = {3, 6, 9, 12}, and B = {2, 4, 6, 8}, find:
(i) A B (ii) A B and show these operations in Venn-diagrams.
c) If U = {1, 2, 3, ... 10}, A = {1, 2, 4, 8}, and B = {2, 4, 8, 10}, find:
(i) A – B (ii) A –B and show these operations in Venn-diagrams.
d) If U = {1, 2, 3, ... 10}, A = {1, 2, 4, 8}, and B = {2, 4, 8, 10}, find:
(i) B – A (ii) B – A and show these operations in Venn-diagrams.
8. From the given Venn – diagram, list the elements of X U
Y
the following sets.
2 1
4 3
a) U b) X Y and X Y c) X Y and X Y 5 9
d) X – Y and X Y e) Y – X and Y X 10 68 7 11 15
12 13 14
Creative Section - B
9 . If U = { 1, 2, 3, ... 12}, A = { 3, 4, 5, 6, 7}, B = { 2, 4, 6, 8, 10 } and
C = { 2, 3, 4, 6, 12}, list the elements of the following set operations.
a) A B C and A B C b) A B C and A B C
c) (A B) C d) (A B) C
21 Vedanta Excel in Mathematics - Book 7
Set
10. If U = {x : x ≤ 15, x N}, A = {even numbers less than 15}, and
B = {factors of 12}, list the elements of the following set operation:
a) A (A B) b) A (A B) c) A – (A B) d) B – (A B)
e) A B f) A B g) A B h) A B
11. If A = { 1, 2, 3, 4, 5}, B = { 1, 3, 5, 7}, and C = { 2, 3, 5, 7 }, show that:
a) A (A B) = A B b) B (A B) = B A
c) A (B C) = (A B ) C d) (A B ) C = A ( B C)
12. a) If F6 and F8 denote the sets of all possible factors of 6 and 8 respectively,
list the elements of F6 and F8 , then find F6 F8 .
b) F12 and F18 are the sets of all possible factors of 12 and 18 respectively.
List the elements of F12 F18
13. a) If M6 and M8 denote the sets of the first six multiples of 6 and 8
respectively, list the elements of M6 and M8 and find M6 M8.
b) M5 and M10 are the sets of the first five multiples of 5 and 10 respectively.
List the elements of M5 M10
It's your time - Project work!
14. Let's make groups of 10 students and conduct a survey in your class to find how
many students like apple or orange in each group. Then make two separate sets
of students who like apple or orange and answer the following questions.
a) Make a set of students of each group who like apple or orange.
b) Make a set of students of each group who like apple as well as orange.
c) Make a set of students of each group who like only apple.
d) Make a set of students of each group who like only orange.
e) Are there any students in your group who do not like apple or orange? Make
a set to show these students.
15. Write a universal set 'U' of the natural numbers from 1 to 15. Write any two
overlapping subsets 'A' and 'B' of this universal set U. Then, draw Venn-diagrams
to show the members of the following set operations by shading to show the
members of the following set operations by shading.
a) A B b) A B c) A – B d) B – A e) A B
f) A B g) A – B h) B – A i) A j) B
Vedanta Excel in Mathematics - Book 7 22
Unit Number Systems in Different Bases
2
2.1 Whole numbers - Looking back
Classroom - Exercise
1. Let's tick () the correct numeral of the number name.
a) Seven lakh nine thousand: (i) 79000 (ii) 700900 (iii) 709000
b) Three crore six hundred: (i) 30006000 (ii) 30000600 (iii) 30060000
c) Eight million ninety-nine thousand: (i) 8099000 (ii) 899000 (iii) 8990000
d) If x is the digit at tens and y is at ones place, the number is:
(i) xy (ii) x + y (iii) 10x + y
e) The place name of five in 950038120 is:
(i) Ten-lakhs (ii) Crores (iii) Ten-crores
f) The place value of 9 in 20900470850 is:
(i) 90000000 (ii) 9000000 (iii) 900000000
2. Let's say and write the answers as quickly as possible.
a) The place name of 6 in 2460178953 in Nepali system is ................................
and in International system is ...........................................
b) Rewrite 5639240000 using commas and express in words in Nepali system.
............................................................................................................................
c) Rewrite 8719480000 using commas and express in words in International
system. ..............................................................................................................
Now, let’s study about the following sets of whole number and natural numbers.
W = {0, 1, 2, 3, 4, 5, ...} is the set of whole numbers. The set of natural numbers,
i.e. N = {1, 2, 3, 4, 5, ...} is a subset of the set of whole numbers.
Zero (0) is the least whole number, whereas 1 is the least natural number. The
greatest whole number or natural number is infinite.
2.2 Decimal or Denary number system
Hindu-Arabic number system is based on decimal or denary number system. In this
system, we use ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Any whole number can be
written using a combination of these ten digits. The system is based on grouping of
tens. So, it is also known as the Base Ten or Decimal system.
23 Vedanta Excel in Mathematics - Book 7
Number Systems in Different Bases
Let’s take 17 blocks of cubes and regroup them into the group of 10 blocks.
17 = 10 + 7 = 1 × 101 + 7 × 10°
Let’s take 39 pencils and regroup them into the group of 10 blocks.
39 = 30 + 9 = 3 × 101 + 9 × 10°
Similarly, 594 = 500 + 90 + 4 = 5 × 102 + 9 × 101 + 4 × 100.
In this way, whole numbers can be regrouped into the base of 10 with some power
of 10. It is called the decimal numeration system or denary system.
Each digit of a numeral has its own place and its place value is obtained multiplying
the digit by its place. For example, let's take a numeral 7425.
7425
It is at ones place and place value is 5 × 1 =5
It is at tens place and place value is 2 × 10 = 20
It is at hundreds place and place value is 4 × 100 = 400
It is at thousands place and place value is 7 × 1000 = 7000
Now, we can write the numeral 7425 in the expanded form in the following way.
7425 = 7 × 1000 + 4 × 100 + 2 × 10 + 5 × 1
= 7 × 103 + 4 × 102 + 2 × 101 + 5 × 10°
In this way, if x, y and z are the digits at hundreds, tens and ones place respectively
in a number, then the number can be expressed as 100x + 10y + z.
2.3 Periods and place
The tables given below show the periods and places in Nepali system and
International system of numerations.
9 16 7 24 3 8 5 6 30
9 16 7 24 38 5 6 30
Vedanta Excel in Mathematics - Book 7
24
Number Systems in Different Bases
From the table, the number name of 916724385630 in Nepali system is:
Nine kharab sixteen arab seventy-two crore forty-three lakh eighty-five thousand
six hundred thirty.
In International system the number name is:
Nine hundred sixteen billion seven hundred twenty-four million three hundred
eighty-five thousand six hundred thirty.
Comparison between Nepali and International numeration system.
100 thousand = 1 lakh , 1 million = 10 lakhs, 10 million = 1 crore
1 billion = 1 arab, 100 billions = 1 kharab
Placement of commas
We can read and write the larger numbers more easily and comfortably when the
periods of the digits are separated by using commas. Let's take a numeral 507490680
and rewrite it using commas in both Nepali and International system.
Nepali system International system
50,74,90,680 507,490,680
Separating unit period Separating unit period
Separating thousands period
Separating lakhs period Separating thousands period
The number name is fifty crore The number name is five hundred
seventy-four lakh ninety thousand seven million four hundred ninety
six hundred eighty thousand six hundred eighty.
EXERCISE 2.1
General Section - Classwork
1. Let's say and write the number names in Nepali system and in International
system.
Numerals Number name in Nepali Number name in
system
a) 100000 International system
b) 4000000
c) 70000000
d) 200000000
e) 5000000000
f) 90000000000
g) 800000000000
25 Vedanta Excel in Mathematics - Book 7
Number Systems in Different Bases
2. Let's write the numerals which have the following expanded forms.
a) 2 × 103 + 4 × 102 + 7 × 101 + 5 × 100 = ..........................................
b) 7 × 104 + 5 × 102 + 9 × 101 + 8 × 100 = ..........................................
c) 4 × 105 + 6 × 104 + 1 × 101 + 7 × 100 = ..........................................
d) 9 × 106 +3 × 102 + 5 × 100 = ..........................................
3. a) If x is the digit at tens place and y is the digit at ones place, the two-digit
number formed by these digits is ..............................................
b) If y is the digit at tens place and x is the digit at ones place, the two-digit
number formed by these digits is ..............................................
c) If a is digit at hundreds place, b is at tens place and c is at ones place, the
three-digit number formed by these digit is ..............................................
4. Rewrite these numerals using commas in Nepali system and in International
system.
Numerals Nepali system International system
a) 18576390
b) 420198675
c) 99999999999
Creative Section A
5. a) The estimated cost of construction of a hydro-power project is
Rs 2562880000. Rewrite this cost in words in Nepali as well as International
numeration system.
b) The world population estimated by US Census Bureau in June 2019 was
7577130400. Rewrite this population in words according to Nepali numeration
system.
c) The annual budget of Government of Nepal was Rs 1315161700000 in the
fiscal year 2075/076. Rewrite it in words according to Nepali and International
system.
6. Rewrite these number names in numerals using commas both in Nepali system
and International system.
a) Six kharab eighty-four arab ninety-one crore forty-seven lakh five thousand
three hundred forty.
b) Thirty-seven billion eight hundred fifteen million six hundred sixty-eight
thousand five hundred twenty-one.
7. Let's write the following numbers in the expanded forms:
a) 52063709 b) 400801530 c) 7502600048 d) 23900068407
Vedanta Excel in Mathematics - Book 7 26
Number Systems in Different Bases
8. 2.5 arab = 2500000000 = Two arab fifty crore = Two billion five hundred million.
7.38 billion = 7380000000 = Seven billion three hundred eighty million
= Seven arab thirty-eight crore
Now, let's write the values of the following numbers and rewrite the number
names in Nepali or in International systems.
a) 1.6 crore b) 4.75 crore c) 3.4 arab d) 5.13 arab
e) 2.7 million f) 6.99 million g) 7.2 billion h) 8.36 billion
Creative Section B
9. a) Find the difference between one million and two hundred fifty thousand.
b) Express the difference in words in International and Nepali Systems.
Find the difference between two billion and eighty-five million. Express the
difference in words in International and Nepali System.
c) Find the difference between seven crore and seventy-five lakh. Express the
difference in words in Nepali and International Systems.
d) Find the difference between three arab and forty-eight crore. Express the
difference in words in Nepali and International Systems.
10. a) By how much is Rs 3.6 million more than Rs 3.6 lakh? Express in Nepali
b) System of numeration.
By how much is Rs 7.5 crore more than Rs 7.5 million? Express in
International System of numeration.
It's your time - Project work!
11. Using the digits from 0 to 9, let's write the greatest and the least ten-digit
numerals.
a) Express these numerals in words in Nepali and International Systems.
b) Express the difference between these numerals in Nepali and International
Systems.
12. Let's visit the available and reliable website. Then, search and find the
following facts and figures:
a) Today's live population of the three most populated countries in the world.
Express the population in Nepali and International Systems of numeration.
b) Find the total population of these three countries and express in words in
Nepali and International System of numeration.
c) The distance between the Sun and its four nearest planets. Express the
distance in words in Nepali and International System of numeration.
27 Vedanta Excel in Mathematics - Book 7
Number Systems in Different Bases
2.4 Binary number system
In denary number system, we use ten digits 0 to 9 to write any number. However, in
binary number system, we use only two digits 0 to 1 to express any number. Binary
number system has its broad applications in digital electronics.
Computers and hand held calculators actually use the binary system for their
internal calculations since the system consists of only two symbols, 0 and 1. All
numbers can then be represented by electronic “switches”, of one kind or another,
where “on” indicates 1 and “off” indicates 0.
2.5 Conversion of binary numbers to decimal numbers
To convert a binary number into decimal, it is expanded in the power of 2. Then, by
simplifying the expanded form of the binary number, we obtain a decimal number.
For example:
(i) 110112 = 1 u 24 + 1 u 23 + 0 u 22 + 1 u 21 + 1 u 10
? 110112 = 1 u 16 + 1 u 8 + 0 u 4 + 1 u 2 + 1 u 1
= 16 + 8 + 0 + 2 + 1 = 27
= 27
(ii) 10110012 = 1 u 26 + 0 u 25 + 1 u 24 + 1 u 23 + 0 u 22 + 0 u 21+ 1 u 20
? 10110012 = 1 u 64 + 0 u 32 + 1 u 16 + 1 u 8 + 0 u 4 + 0 u 2 + 1 u 1
= 64 + 0 + 16 + 8 + 0 + 0 + 1 = 89
= 89
2.6 Conversion of decimal numbers to binary numbers
Again, let’s take 15 blocks of cubes and regroup them into the group of 2 blocks.
7 pairs blocks of cube and 1 cube
Now, let’s arrange the groups of 2 blocks into the base of 2 with the maximum
possible powers.
8 4 21
23 22 21 20
So, 15 = 1 × 23 + 1 × 22 + 1 × 21 + 1 × 2° = 11112
In this way, the denary number 15 can be expressed in binary number as 11112.
We can convert a decimal number into a binary number by using the place value
table of the binary system. For example:
Vedanta Excel in Mathematics - Book 7 28
Number Systems in Different Bases
Convert 19 into binary system.
Power of base 2 26 24 23 22 21 2°
Decimal Equivalent 32 16 8 4 2 1
Binary number 1 0011
There is one 16 in 19. So There is no 4 in 3. So,
insert 1. insert 0.
Remainder = 19– 16 = 3 Remainder is again 3.
There is no 8 in 3. So, There is one 2 in 3. So
insert 0. insert 1.
Remainder is still 3. Remainder =3 – 2 = 1.
There is one in 1. So
insert 1.
From the table, 19 = 1 × 16 + 0 × 8 + 0 × 4 + 1 × 2 + 1 × 1
= 1 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 1 × 2°
= 100112
Convert 25 into binary system
Power of base 2 25 24 23 22 21 20
Decimal equivalent 32 16 8 4 2 1
Binary number 1 1001
From the table, 25 = 1 u 16 + 1 u 8 + 0 u 4 + 0 u 2 + 1 u 1
= 1 u 24 + 1 u 23 + 0 u 22 + 0 u 21 + 1 u 2q = 110012
? 25 = 110012
Alternative method
We can convert a decimal number into binary system also by another method. In
this method, we should divide the given number successively by 2 until the quotient
is zero. The remainder obtained in each successive division is listed in a separate
column. For example:
2 19 RemainderArranging the
29 1 remainders in
24 1 reverse order:
22 0 100112
21 0
1
0
? 19 = 100112
29 Vedanta Excel in Mathematics - Book 7
Number Systems in Different Bases
2.7 Quinary number system
We have already discussed on denary (or decimal) number system as the base
ten system and binary number system as the base two system. Likewise, quinary
(or pental) number system is known as the base five system. In this system, we
use only five digits: 0, 1, 2, 3, and 4. The numbers in the quinary system can be
expressed in the power of 5. For example:
14 = 1 u 51 + 4 u 5q, 321 = 3 u 52 + 2 u 51 + 1 u 5q, and so on.
For the proper identification of quinary numbers, they are written with their base 5
in the suffix. Some examples are 345, 1025, 312405, etc.
2.8 Conversion of quinary numbers to decimal numbers
To covert a quinary number into decimal number, it is expanded in the power of 5.
Then, by simplifying the expanded form of the quinary number, we get a decimal
number. For example:
(i) 325 = 3 u 51 + 2 u 5q (ii) 12345 = 1 u 53 + 2 u 52 + 3 u 51 + 4 u 5q
=3u5+2u1 = 1 u 125 + 2 u 25 + 3 u 5 + 4 u 1
= 15 + 2 = 17 = 125 + 50 + 15 + 4 = 194
2.9 Conversion of decimal numbers to quinary numbers
We can convert a decimal number into quinary number by using the place value
table of the quinary system. For example:
Convert 134 into quinary system.
Power of base : 5 54 53 52 51 5°
Decimal Equivalent 625 125 25 5 1
178 1×125 2 × 25 0×5 3×1
Quinary number 2 0 3
1
There is no 5 in 3.
There is one 125 in 178. So, insert 0.
So insert 1. Remainder = 3 – 0 = 3
Remainder = 178 – 125 = 53
There is two 25 in 53. There are three 1 in 3.
So insert 2. So, insert 3.
Remainder = 53 – 50 = 3.
From table, 178 = 1 u 125 + 2 u 25 + 0 u 5 + 3 u 1
= 1 × 53 + 2 × 52 + 0 × 51 + 3 × 50 = 12035
Alternative method
In this method, to convert a decimal number into quinary number, we should divide
the given number successively by 5 until the quotient is zero. The remainders of
each successive division are then arranged in the reverse order to get the required
quinary number. For example:
Vedanta Excel in Mathematics - Book 7 30
Number Systems in Different Bases
5 178 Remainders Arranging the
5 35 3 remainders in
57 0 reverse order:
51 2 12035
1
0
? 178 = 12035
EXERCISE 2.2
General Section - Classwork
1. The glowing lamp indicates 1 and the lamp which is glowing out indicates
0 of the binary system. Let's say and write the binary numbers indicated by
these lamps.
a) b) c)
........................... ........................... ...........................
d) e) f)
........................... ........................... ...........................
2. Let’s study the given examples carefully and write the correct numbers in
the blank space as quickly as possible after shading the appropriate blank
circles.
Decimal Number 1×23 1×22 1×21 1×2° Binary numbers
8 4 2 1
1 = 12
2 = 102
3 = 112
4 = 1002
5 = 1012
6 =?
7 =?
8 =?
9 = 10012
31 Vedanta Excel in Mathematics - Book 7
Number Systems in Different Bases
10 = ?
11 = ?
12 = ?
13 = 11012
14 = ?
15 = 11112
3. a) Let's say and write the binary values of these powers of twos.
2° = 1, 21 = 10, 22 = ................, 23 = 1000, 24 = ..................
b) Let's say and write the quinary values of these powers of fives.
5° = 1, 51 = 10, 52 = ................., 53 = .................. , 54 = .................
Creative section - A
4. Let's convert each of the following binary numbers into decimal number.
a) 112 b) 1012 c) 1102 d) 10102 e) 11002
f) 100102 g) 111112 h) 1110102 i) 1110012 j) 11001012
5. Let's convert each of the following decimal numbers into binary numbers .
a) 7 b) 18 c) 25 d) 72 e) 99
f) 124 g) 145 h) 216 i) 308 j) 417
6. Let's convert each of the following quinary numbers into decimal numbers.
a) 135 b) 245 c) 325 d) 1205 e) 2035
f) 3405 g) 12035 h) 21135 i) 12345 j) 44445
7. Let's convert each of the following decimal numbers into quinary numbers.
a) 9 b) 13 c) 27 d) 55 e) 126
f) 212 g) 512 h) 670 i) 1126 j) 4025
Creative section - B
8. Let's convert each of the binary numbers into quinary numbers and quinary
number into binary number.
a) 11102 b) 110112 c) 345 d) 425
9. Lets change each pair of numbers in denary numbers and compare them.
a) 1112 and 125 b) 101012 and 325
It's your time - Project work!
10. Let's learn with fun to convert a decimal number in to binary system.
Process 1: If the given decimal number is even, write zero (0) just below it. If it is
odd, write 1 just below it.
Process 2: Now divide the given number into half and write to the left of the
number.
Vedanta Excel in Mathematics - Book 7 32
Number Systems in Different Bases
Process 3: For each half write 0, if it is even and write 1 if it is odd.
For example 1: Let’s convert 14 and 67 into binary number.
? 14 = 11102 a) 12 ? 67 = 10000112
Now, let's try for the numbers: b) 43 c) 216
11. a) Let's take a few number of matchsticks and represent the following binary
numbers by pairing the required number of matchsticks.
(i) 102 (ii) 112 (iii) 1012 (iv) 1112
(v) 1102 (vi) 11102 (vii) 11112
b) Let's draw the required number of pairs of dots in the table to convert the
given denary numbers into binary numbers.
Denary numbers 24 23 22 21 20
3
11
17
c) Let's draw the required number of groups of 5 dots in the table to convert
given denary numbers into quinary numbers.
Denary numbers 52 51 50
7
8
10
14
28
33 Vedanta Excel in Mathematics - Book 7
Unit Algebraic Expressions
9
9.1 Algebraic terms and expressions – Looking back
Classroom - Exercise
1. Let's tell and write the answers as quickly as possible.
a) How many terms are there in each of the following expressions? Also write
the terms.
(i) In 2xy, number of terms: .................., terms are .......................................
(ii) In 2x + y, number of terms: .................., terms are ...................................
(iii) In 2+x+y, number of terms: .................., terms are ..................................
b) In 5x3 , coefficient is ............... base is .............. and power is ......................
c) If l = 3 and b = 2 then (i) 2 (l + b) = .................... (ii) l × b = ....................
2. a) The mathematical expression for the sum of 2x and 3y is .............................
b) The mathematical expression for the difference of 7ab and 3bc is ................
c) The mathematical expression for the product of 5p and 2q is .......................
3. Let's tell and write the sum, difference, or product.
a) 7x 2x = ......... 7x 2x = ......... 7x u 2x = .........
b) 5p2 2p2 = ......... 5p2 2p2 = ......... 5p2 × 2p2 = .........
x, 2x, 3ab, p2q, etc. are algebraic terms. An algebraic expressions is a collection of
one or more terms. Which are separated to each other by either addition (+) or
subtraction (–) sign.
For example: 3xyz, 7x – 2y, x + y – z, etc are algebraic expressions. We can represent
the terms and factors of the terms of an expression by a tree diagram.
For example: Expression (5x yz) (2xy 6)
Terms 5x yz 2xy 6
Factors 5x yz 2 x y 2 3
141 Vedanta Excel in Mathematics - Book 7
Algebraic Expressions
9.2 Types of algebraic expressions
Algebraic expressions are categorized according to the number of terms contained
by the expressions. The table given below shows the types of expressions.
Monomial An algebraic expression with only one term is called a
Binomial monomial . For example : 5xy, – 8m, 9x2y, 11 etc are monomials.
Trinomial
An algebraic expression with two unlike terms is called a
binomial. For example : x + y, x – 4, 2xy + 3x, a2 – b2, pq – r,
x y
a b etc are binomials.
An algebraic expression with three unlike terms is called a trinomial.
For example : 2x +y – 1, a2 + ab + b2, xy + x + y etc are trinomial.
9.3 Polynomial
An algebraic expression with one or more terms and powers of the variables being
whole numbers in each term is called polynomial.
For example : x2y3z, x2 – 4, x + y + 7 etc. are polynomials.
However, x2 + 1 , x2/3 – y2/3 are not polynomials. Because the powers of the variables
x2
in these expressions are not whole numbers.
9.4 Degree of polynomials
Let's study the illustrations given in the table and learn about the degree of
polynomials.
Polynomials Degree of polynomials
2x Power of the variable x is 1. So, its degree is 1.
3y2 Power of the variable y is 2. So, its degree is 2.
x2yz The sum of the powers of the variables x, y, and z
= 2 + 1 + 1 = 4. So, it's degree is 4.
2p3 – 3p2 + 5 The highest power of the variable p is 3. So, its degree is 3.
a2b2 + 2a2b – 4ab2 The highest sum of the powers of ab = 2 + 2 = 4. So, its
degree is 4.
9.5 Evaluation of algebraic expressions
Let's take an algebraic expression 2x – 3y and evaluate it when x = 2 and y = 1.
Here, if x = 2 and y = 1, then 2x – 3y = 2 × 2 – 3 × 1 = 4 – 3 = 1
In this way, the process of finding the value of an algebraic expression by replacing
the variables with numbers is called evaluation.
Vedanta Excel in Mathematics - Book 7 142
Algebraic Expressions
Worked-out examples
Example 1 : Which of the following expressions are polynomials? Write with
reason. 3
x p
a) 2 + 5 b) 3p2 – c) √ 5 y2 + 3 d) 2√ x – 1
Solution:
x
a) 2 + 5 is a polynomial because the power of is 1, which is a whole number.
b) 3p2 – 3 is not a polynomial because the power of the term 3 is –1, which is not
p p
a whole number.
c) √ 5 y2 + 3 is a polynomial because the power of y is 2, which is a whole number.
d) 2√ x – 1 is not a polynomial because the power of x is 1 , which is not a whole
number. 2
Example 2 : Find the degree of a) 4x2 b) 3x2y c) 7x5y2 9x2y3 4xy5
Solution: d) (xy)2 + x2 – y2
a) The degree of 4x2 is 2
b) The degree of 3x2y is 2 + 1 = 3
c) In 7x5y2 , the sum of powers of variables = 5 2 = 7
In 9x2y3, the sum of powers of variables = 2 3 = 5
In 4xy5, the sum of powers of variables = 1 5 = 6
Since the highest sum of powers of variables is 7,
the degree of 7x5y2 –9x2y3 + 4xy5 is 7.
d) Here, (xy)2 + x2 – y2 = x2y2 + x2 + y2
The highest sum of the powers of variables is 2 +2 = 4. So, its degree is 4.
Example 3 : If l = 5 and b = 3, evaluate 2(l + b).
Solution:
Here, when l = 5 and b = 3, then 2(l + b) = 2(5 + 3) = 2 × 8 = 16
EXERCISE 9.1
General Section - Classwork
1. Let's tick ( ) the correct answer.
a) The terms of expression 5x2 – 3xy are
(i) 5x2 and 3xy (ii) x2 and xy (iii) 5x2 and – xy (iv) 5x2 and –3xy
b) The number of terms in the expression x2 + y2 + z2 is
7
(i) 2 (ii) 3 (iii) 4 (iv) 7
143 Vedanta Excel in Mathematics - Book 7
Algebraic Expressions
c) An algebraic expression with three unlike terms is called a
(i) Monomial ii) binomial (iii) trinomial (iv) all of these
d) Which of following expressions is a binomial?
(i) 3m2n – mn2 + m2n (ii) pq + qr + pr
(iii) √ 3 xy + z – 7 (iv) x2y –3x – x2y
e) Which one of the following expressions is not a polynomial? 4
x3 y3 x3 x2
(i) √ 5 x2y + z (ii) 2x3 – xy + y2 (iii) 2 3 (iv) 2
f) The degree of polynomial 2x4yz3 is
(i) 2 (ii) 8 (iii) 4 (iv) 3
g) The degree of polynomial 5x2y – xy + y2 is
(i) 2 (ii) 3 (iii) 5 (iv) 7
2. Let's tell and write the value of the expressions quickly.
a) If x = 3, y = 2, then (i) x + y = ........... (ii) x – y = ........... (iii) xy = ...........
b) If a = 2, b = 3, then (i) 2(a + b) = ........ (ii) a2 = ............ (iii) b2 = ............
Creative Section
3. a) Define algebraic expressions with examples.
b) What are monomial, binomial, and trinomial expressions? Write with
examples.
c) Is x – y + 2x a trinomial expression? Why?
d) What is a polynomial? Give an example of a polynomial.
1
e) Why is x2 a polynomial, but x2 is not a polynomial?
4. Let's identify and then classify the given expressions as monomial, binomial
or trinomial.
a) x2y + xy2 b) 9 – x2 c) XYZ d) pq + p + q
2
e) x2 + y2 f) a2 + a + 1 g) 3x2 + 7xy + 6y2 h) 3x + xy – 8y2
(i) x2 + x (j) – 6x2 k) 1 + x + xy l) 64
y2
5. Let's state with reason whether the given expressions are polynomials.
a) x3 + x2 b) 3 c) ab a b d) x2 + x –2
x2 3
e) √ 3 x2 – xy f) x1/2 + y1/2 g) √ x + 2x + 1 h) 5x3 – 4x2 + 6xy – 7
6. Let's find the degree of the following polynomials.
a) 3x2 b) –2xy c) 4x2yz
d) x2 + 5x + 6 e) 3y3 – 2y2 + 5y – 6 f) x2yz + xyz – 6
g) 2x2y2 + x2y – xy2 – 3xy + 4 h) x – x2y3 + (xy)3 i) (xy)2 + (xy)3 + (xy)4
Vedanta Excel in Mathematics - Book 7 144
Algebraic Expressions
7. a) If l = 6, b = 4 and h = 2, evaluate the following expressions.
(i) l × b (ii) l × b × h (iii) 2(l + b)
(iv) 2h(l + b) (v) l2 (vi) 6l2
8. If x = 2 and y = 3, show that: a) (x + y)2 = x2 + 2xy + y2
b) (x – y)2 = x2 – 2xy + y2 c) x2 – y2 = (x + y) (x – y)
9.6 Addition and subtraction of algebraic expressions
x, 2x, 5x, etc. are the like terms.
The sum of x and 2x = x + 2x = 3x (1 + 2)x = 3x
The sum of 2x and 5x = 2x + 5x = 7x (2 + 5)x = 7x
The sum of x, 2x and 5x = x + 2x + 5x = 8x (1 + 2 + 5)x = 8x
The difference of 5x and 2x = 5x – 2x = 3x (5 – 2)x = 3x
The difference of 2x and x = 2x – x = x (2 – 1)x = x
Thus, when we add or subtract like terms, we should add or subtract the coefficients
of the like terms.
On the other hand, x, x2, y, 2y2 are unlike terms.
Sum of x and x2 = x + x2, difference of x and x2 = x – x2
Sum of x and y = x + y, difference of x and y = x – y
Thus, we do not add or subtract the coefficient of unlike terms.
Worked-out examples
Example 1: Add (i) 2a, 3a2, 4a and a2 (ii) 3x2y, 2x2y, 3xy2 and – 2xy2.
Solution:
(i) 2a + 3a2 + 4a + a2 = 2a + 4a + 3a2 + a2
= 6a + 4a2
(ii) 3x2y + 2x2y + 3xy2 + (– 2xy2) = 5x2y + 3xy2 – 2xy2
= 5x2y – xy2
Example 2: Add 4ab + 7bc – 5, 3bc – 8ab + 6 and 9ab – bc – 2.
Solution:
Addition by vertical arrangement Addition by horizontal arrangement
4ab + 7bc – 5 (4ab + 7bc – 5) + (3bc – 8ab + 6) + (9ab – bc – 2)
– 8ab + 3bc + 6 = 4ab + 7bc – 5 + 3bc – 8ab + 6 + 9ab – bc – 2
9ab – bc – 2 = 4ab 9ab 8ab 7bc 3bc – bc – 5 6 – 2
5ab + 9bc 1 = 5ab + 9bc 1
= 5ab + 9bc 1
145 Vedanta Excel in Mathematics - Book 7
Algebraic Expressions
Example 3: Subtract 2a2 + 5ab – b2 from 5a2 – ab + 3b2.
Solution:
Subtraction by vertical arrangement Subtraction by horizontal arrangement
5a2 ab 3b2
± 2a2 ± 5ab b2 (5a2 – ab + 3b2) (2a2 + 5ab – b2)
= 5a2 – ab + 3b2 2a2 5ab b2)
3a2 – 6ab + 4b2 = 5a2 2a2 – ab 5ab + 3b2 b2
= 3a2 6ab + 4b2
Example 4: What should be added to 3a + 4x to get 7a – 2x ?
Solution:
Here, the required expression to be added is Let’s think, what should be
(7a 2x) (3a 4x) added to 3 to get 7.
= 7a – 2x – 3a – 4x It’s 4 and it is 7 – 3. It’s my
= 7a – 3a – 2x – 4x investigation to work out
= 4a 6x such problems.
Example 5: What should be subtracted from 8a – 5b + 2 to get 2a + 3b – 9?
Solution:
Here, the required expression to be subtracted is
(8a – 5b + 2) – (2a + 3b – 9) Let’s think, what should be
= 8a – 5b + 2 – 2a – 3b + 9 subtracted from to 8 to get 5.
= 8a – 2a 5b 3b 2 9 It’s 3 and it is 8 – 5. It’s my rule
= 6a 8b 11 to work out such problems.
EXERCISE 9.2
General Section - Classwork
1. Let's tell and write the sums or differences as quickly as possible.
a) 4x + 3x = ............... (b) 3xy + 6xy = ............. (c) 8a2 + 5a2 = .............
(d) 8p – 3p = ............. (e) 9ab2 – 6ab2 = ............. (f) 10x3 – 3x3 = .............
2. a) What is the sum of a2 and a ? ...............................
b) What is the difference of 2x and 3y? ...............................
c) What should be added to 3x2 to get 7x2? .............................
d) What should be subtracted from 11a3 to get 6a3? ...............................
Creative Section - A
3. a) What are like and unlike algebraic terms? Write with examples.
b) How do we add or subtract like terms? Write with examples.
c) Can we add or subtract the coefficients of unlike terms? Write with examples.
d) In what way the sum of x + x and the product of x × x different with each
other?
Vedanta Excel in Mathematics - Book 7 146
Algebraic Expressions
4. Let's add.
a) 3x, 5y, 4x and 6y b) 6ab, 8bc, (–2ab) and (–3bc)
c) 7at2, (–2at), (–4at2) and at d) (–5p3q2), (–9p2q3), 6p3q2 and 10p2q3
e) 3t – 2tz + 4 and 5tz + 2t – 10 f) 2x + 3y – 6 and 5x – 4y + 1
g) 7a + 4b – 8 and 3b – 5a + 9 h) 5x2 + 3x + 4 and x2 + 2x – 7
i) 8a2 + 3ab – 2b2, 3a2 – ab + 5b2 and 5ab – 7a2 – b2
j) a2 + 3ab – bc, b2 + 3bc – ca and c2 + 3ca – ab
5. Let's subtract.
a) 5pq from 8pq b) 2x2y from 9x2y
c) –3ab from 5ab d) 2a2b3 from – 3a2b3
e) 4m + 5n from 6m + 9n f) 7p2 – 6q2 from 5p2 + 2q2
g) 2x2 – 3x + 6 from 4x2 + 5x – 3
h) y3 – 5y2 + y – 11 from 4y3 – 3y2 – y – 6
i) a3b3 – 2a2b2 + 3ab – 4 from –5a3b3 – a2b2 – 4ab – 7
j) 2.6x4 – 3.8x3 – 1.2x2 + 4.6x – 5.4 from 6.2x4 + 8.3x3 –2.1x2 + 6.4x – 4.5
6. Let's simplify. b) 9a – 2b – 4a + 7b
a) 3x + 2y + 5x – 9y d) 9a2 5a – 6a2 3a – 2
c) 4x2 – 2y2 + 2x2 – 5y2 f) 5x2 – (2x2 – y2) – 4y2
e) 7p2 + p + p2 – 6p + 3 h) 7p – 5q – (2p – 8q)
g) 10a + 4b – (3a + 2b) j) 13a2 – (3b2 – 4c2) + a2 – (8a2 – 5b2 + 7c2)
i) 12x – (5x + 4y) – (2y + 3x)
7. a) What should be added to 5xy to get 9xy ?
b) What should be added to ab bc ca to get ab bc ca ?
c) What should be subtracted from 9x2y to get 4x2y?
d) What should be subtracted from 3p2 + 2p – 1 to get p2 – 3p 4 ?
8. a) To what expression must 5a2 – 4a + 3 be added to make the sum zero?
b) From what expression must x2 + 5x – 7 be subtracted to make the difference
unity?
9. a) If a = x + y and b = x – y, show that (i) ( a + b)2 = 4x2 (ii) (a –b)2 = 4y2
b) If x = p + 2 and y = p – 3, show that (i) x + y + 1 = 2p (ii) x y – 5 = 0
It's your time – Project work
10. a) Let's write any three different pairs of like terms. Then, find the sum and
difference of each pair.
b) Write any three binomial algebraic expressions and denote them by A, B and C
respectively.
Then, find (i) A +B – C (ii) A – (B – C) (iii) (A + B) – (A – C)
147 Vedanta Excel in Mathematics - Book 7
Laws of Indices Laws of Indices
Unit
10
10.1 Laws of indices (or exponents)
Let’s consider an algebraic term 2x3.
Here, 2 is called the coefficient, x is the base and 3 is the exponent of the base. The
exponent is also called the index of the base. The plural form of index is indices.
An index of a base shows the number of times the base is multiplied. For example:
x×x o x is multiplied two times = x2 (x squared)
x × x × x o x is multiplied three times = x3 (x cubed)
x × x × x × x o x is multiplied four times = x4 (x raise to the power 4)
While performing the operations of multiplication and division of algebraic
expressions we need to work out indices of the same bases under the certain rules.
These rules are also called laws of indices.
1. Product law of indices
Study the following illustrations and investigate the idea of the product law of
indices.
1 2 = 4 unit squares 123
3 4 2 = 21 × 21 = 21 +1 = 9 unit squares
22 32 4 5 6 3 = 31 × 31 = 31 +1
2 789
3
Similarly,
x2 x = x1 × x1 y2 y = y1 × y1
= x1 +1 = y1
+1
x y
Again,
3 = 8 unit cubes
2 = 27 unit cubes
14 = 31 × 31 × 31
2 = 21 × 21 × 21
23 7 8 = 21 + 1 + 1 3
56 2 33 = 31 + 1 + 1
2
3 3
x3 x = x1 × x1 × x1 y3 y = y1 × y1 × y1
= x1 + 1 + 1 = y1 + 1 + 1
xx y
y
Vedanta Excel in Mathematics - Book 7 148
Laws of Indices
Furthermore, 23 = 21 2 I understood!
21 u 22 = 2 u (2 u 2) 25 = 22 3 When the same bases are
27 = 23 4 multiplied, we should
22 u 23 = (2 u u (2 u 2 u 2) a5 = a2 3 add their indices!!
23 u 24 = (2 u u 2) u (2 u 2 u 2 u 2)
Similarly, a2 u a3 = (a u a) u (a u a u a)
Thus, if am and an are any two terms with the same base a and the powers m and n
respectively, then, am u an = am + n
2. Quotient law of indices
Let's study the following illustrations and try to investigate the idea of quotient law
of indices.
22 ÷ 2 = 22 = 2×2 = 2 = 22 – 1 I also understood!
2 2 When a base is divided
by another same base,
23 ÷ 2 = 23 = 2 × 2 × 2 = 22 = 23 – 1 powers should be
2 2 subtracted.
25 ÷ 22 = 25 = 2×2×2×2×2 = 23 = 25– 2
22 2×2
24 ÷ 26 = 24 = 2×2×2×2 = 1 = 1
26 2×2×2×2×2×2 22 26 – 4
Similarly, a5 y a2 = a5 = a×a×a×a×a = a3 =a5-2
a2 a×a
Thus, if am and an are any two terms with the same base a and the powers m and n
1
respectively, then, am ÷ an = am – n if m > n and am ÷ an = an – m if n > m
3. Power law of indices
(i) Let's study the following illustrations and try to investigate the idea of power
law of indices.
(22)3 = 22 u 22 u 22 = 22 2 2 = 26 = 22 u 3 When a base with some
(22)4 = 22 u 22 u 22 u 22 = 22 2 2 2 = 22 u 4 = 28 power has another power,
Similarly, the powers are multiplied.
(a3)2 = a3 × a3 = a3 3 = a2 u 3 = a6
Thus, if am is any term with the base a and the index m, then, (am) n = am u n
(ii) Let's study the following illustrations.
(2 u 3)2 = 22 u 32, (4 u 5)3 = 43 u 53
Similarly, (a u b)3 = a3 u b3
Thus, if a and b are any two terms, then, (a × b)m = am u bm Also, am bm = (ab)m
(iii) Let's study the following illustrations.
2 2 22 , 4 3 43 , Similarly, a 4 a4
3 32 5 53 b b4
= = =
Thus, if a and b are any two terms, then a m = am
b bm
149 Vedanta Excel in Mathematics - Book 7
Laws of Indices
4. Law of zero index
Let's study the following illustrations and investigate the result when a base has
zero index. It’s interesting!
The value of a base with
20 = 21– 1 = 21 ÷ 21 = 21 =1 power 0, is always 1!
21
30 = 31 – 1 = 31 ÷ 31 = 31 =1
31
Thus, if a0 is any term with base a and power 0, then, a0 = 1 where a z 0.
Worked-out examples
Example 1: Which one is greater 25 or 52 ?
Solution:
Here, 25 = 2 × 2 × 2 × 2 × 2 = 32 and 52 = 5 × 5 = 25
Since 32 > 25, 25 is greater than 52.
Example 2: Express the numbers as a product of prime factor in exponential
form. a) 432 b) 675
Solution:
a) 2 432 b) 3 675
2 216 3 225
2 108 3 75
2 54 5 25
3 27 5
39
3 ? 675 = 3 × 3 × 3 × 5 × 5
?432 = 2 × 2 × 2 × 2 ×3 × 3 × 3 = 33 × 52
= 24 × 33
Example 3: Factorize the number and express in exponential form.
a) 216 b) 2744
Solution: b) 2 2744
a) 2 216
2 108 2 1372
2 54 2 686
3 27 7 343
3 9 7 49
37
? 216 = 2 × 2 × 2 × 3 × 3 × 3 ? 2744 = 2 × 2 × 2 × 7 × 7 × 7
= 23 u 33 = (2 u 3)3 = 63 = 23 u 73 = (2 u 7)3 = 143
Vedanta Excel in Mathematics - Book 7 150
Laws of Indices
Example 4: Express the numbers as the products of powers of 10.
a) 600 b) 900000 c) 11000000
Solution: = 6 u 100 = 6 u 10 u 10 = 6 u 102
a) 600
b) 900000 = 9 u 100000 = 9 u 10 u 10 u 10 u 10 u 10 = 9 u 105
c) 11000000 = 11 u 1000000 = 11 u 10 u 10 u 10 u 10 u 10 u 10 = 11 u 106
Example 5: Find the product in their exponential forms.
a) 52 × 54 × 56 b) 3 3 × 43 4 × 34 5 c) 34 × 93 × 272
4
Solution:
a) 52 × 54 × 56 = 52+4+6 = 512
b) 3 3 × 43 2 × 43 5 = 3 3 – 2 5 = 3 6 = 36
4 4 4 4
c) 34 × 93 × 272 = 34 × (32)3 × (33)2 = 34 × 36 × 36 = 3 4+6+6 = 316
Example 6: Find the quotients in their exponential forms.
a) 35 ÷ 32 b) (2x)7 ÷ (2x) –2 c) 85 ÷ 44 d) 34 ÷ 93
Solution:
a) 35 ÷ 32 = 35–2 = 33
b) (2x)7 ÷ (2x) –2 = (2x)7–(–2) = (2x) 7+2 = (2x)9
c) 85 ÷ 44 = (23)5 ÷ (22)4 = 215 ÷ 28 = 215–8 = 27
1
d) 34 ÷ 93 = 34 ÷ (32)3 = 34 ÷ 36 = 1 = 32
36 – 4
Example 7: Find the value of a) 4 1 b) 8 2
2 3
Solution: 9 27
a) 4 1 22 1 = 2 2×12 = 21 = 2
9 32 2 3 3 3
2=
b) 8 2 = 23 2 = 2 3× 2 = 2 2 = 22 = 4
3 3 3 3 3 32 9
27 33
Example 8: Simplify a) 25 3 b) 33 × 35 × 37 c) 253 × 52
Solution: 23 32 × 95 54 × 1252
a) 25 3 = (25 – 3)3 = (22)3 = 22 u 3 = 26 = 64
23
b) 33 × 35 × 37 = 33 + 5 + 7 = 315 = 315 = 315 – 12 = 33 = 27
32 × 95 32 × (32)5 32 × 310 312
c) 253 × 52 = (52)3 × 52 = 56 × 52 = 56 + 2 = 58 = 1 = 1 = 1
54 × 1252 54 × (53)2 54 × 56 54 + 6 510 510 – 8 52 25
xa + b × xb + c × xc + a
Example 9: Simplify a) xb – c u xc – a u xa – b b) x2a × x2b × x2c
Solution:
151 Vedanta Excel in Mathematics - Book 7
Laws of Indices
a) xb – c u xc – a u xa – b = xb – c + c – a + a – b = xq = 1.
b) xa + b × xb + c × xc + a a+b+b+c+c+a
x2a × x2b × x2c
= x x2a + 2b + 2c
= x2a + 2b + 2c = x2a + 2b + 2c – 2a –2b – 2c = x° = 1
x2a + 2b + 2c
EXERCISE 10.1
General Section - Classwork
1. Let's tell and write the answers as quickly as possible.
a) a × a2 = ......... b) a2 × a3 = ......... c) a5 × a4 = .........
d) am × an = ......... e) x2 ÷x = ......... f) x5 ÷ x2 = .........
g) x10 ÷ x7 = ......... h) xm ÷ xn = ......... i) (y2)3 = .........
j) (y3)4 = ......... k) (y6)3 = ......... l) (ya)b = .........
m) x° = ......... n) (2x)° = ......... o) p) xm–m = .........
2. Let's tell and write the products in their exponential forms.
a) 7 × 7 × 7 = ......... b) (–2)×(–2)×(–2)×(–2) = .........
c) (5p) × (5p) × (5p) × (5p) × (5p) = ......... d) 23 × 23 × 23 = .........
Creative Section - A
3. a) Define coefficient, base and index of an algebraic term with an example.
b) Define product law of indices taking two terms xa and xb.
c) Define quotient law of indices taking two terms px and py.
d) Define power law of indices taking (xa)b.
e) Write the law of zero index with an example.
4. Let's evaluate and identify which one is greater?
a) 23 and 32 b) 43 and 34 c) 25 and 52
d) 210 and 102 e) 54 and 45 f) 53 and 35
5. Let's factorize the following numbers and express in exponential form.
a) 16 b) 27 c) 128 d) 343 e) 625 f) 400 g) 864 h) 1125
6. Let's find the prime factors and express in exponential forms.
a) 36 b) 100 c) 216 d) 225 e) 196 f) 441 g) 1000 h) 1296
7. Let's express the following numbers as the products of power of 10.
a) 200, 2000, 20000, 200000 b) 5000, 50000, 500000, 5000000
8. Let's find the products in their exponential forms.
a) 2 ×22 × 23 b) 22 × 42 × 82 c) 273 × 92 × 3 45
d) 54 × 252 × 125 e) (3a)4 ×(3a)3 × (3a)–2 2 3 7
f) 4 × 4 ×
7 7
Vedanta Excel in Mathematics - Book 7 152
Laws of Indices
9. Let's find the quotients in their exponential forms.
a) 27 ÷ 23 b) 39 ÷ 34 c) (2x)8 ÷ (2x)2 d) (3a)4 ÷ (3a)–3
e) 95 ÷ 36 f) 163 ÷ 82 g) 4 8 ÷ 43 h) 4 5 26
5 5 9 3
÷
10. Let's find the quotients in their exponential forms.
a) 34 ÷ 39 b) 53 ÷ 510 c) (5a) 4 ÷ (5a)7 d) (4p) –3 y (4p)2
e) 22 ÷ 43 f) 35 ÷ 94 g) 54 ÷ 253 h) 7 ÷ 492
11. Let's evaluate. 1 2 3 3
2 3 4 5
a) 4 1 b) 25 1 c) 4 d) 8 e) 16 f) 32
2 2 9 27 81 243
12. Let's simplify.
a) a5 u a7 b) p6 u p4 c) x7 ux 2 d) y10 u y 4 e) 4x5 u 3x
a9 p3 x3 y4 6x2
13. Let's simplify.
a) 22 3 b) 34 3 c) 57 2 d) 92 3 e) 43 4 f) 82 5 g) 272 2
54 32 23 29 310
Creative Section - B
14. Let's simplify.
a) 23 × 43 × 84 b) 32 × 93 × 274 c) 22 × 34 × 123
22 × 45 × 162 33 × 94 × 812 25 × 63
d) 64 × 92 × 253 e) 43 2 34 2 92 2 f) 53 2 u 82 3 54
32 × 42 × 156 25 32 272 252 42 2
u u u
15. Let's simplify.
a) xa b × xb – a b) xa – b × xb – c × xc – a
c) (xa)b – c × (xb)c – a × (xc)a – b d) (xp q)r × (xq r)p × (xr p)q
e) xa + b × xb + c × xc + a f) (x2)a + b × (x2)b + c × (x2)c + a
x2a × x2b × x2c (xa × xb × xc)4
16. a) If a = 1 and b = 2, find the value of (i) ab (ii) ba (iii) (a + b)a + b
b) If x = 5 and y = 3, find the value of (i) xy (ii) yx (iii) (x – y)x – y
c) If p = 10 and q = 1, find the value of p2 2pq q2
p q
d) If m = 15 and n = 5, find the value of = m2 n2 .
m n
17. a) If x = 5a and y = 5b, show that xy = 5a + b
b) If x = 7a – b and y = 7b a, show that xy = 1.
c) If x y = x, show that xy = 1.
d) If a = 2 and b = 1, show that (a + b)2 = a2 + 2ab + b2
e) If a = 5 and b = 3, show that (a – b)2 = a2 – 2ab + b2
153 Vedanta Excel in Mathematics - Book 7
Laws of Indices
It's your time - Project work 14 = .........
18. a) Let's fill in the blanks with correct numbers. 19 = .........
1° = ......... 11 = ......... 12 = ......... 13 = ......... (–1)5 = ......,
15 = ......... 16 = ......... 17 = ......... 18 = ......... (–1)10 = ......,
i) What idea did you investigate ?
ii) What would be the value of: 150 , 11000 and 1x ?
b) Lets find the value of :
(–1)1 = ......, (–1)2 = ......, (–1)3 = ......, (–1)4 = ......,
(–1)6 = ......, (–1)7 = ......, (–1)8 = ......, (–1)9 = ......,
i) What idea did you investigate?
ii) What would be the value of (–1)20 , (–1)99 and (–1)500?
c) Write a short report about your investigation on a) and b), then present in
the class.
10.2 Multiplication of algebraic expressions
While multiplying algebraic expressions, the coefficients of the terms are multiplied
and the power of the same bases are added. For example:
Example 1: Multiply: 4x2 by 3x
Solution: 4 × 3 = 12 (Coefficients are multiplied.)
x2 × x = x2 + 1 = x3 (Power of the same bases are added.)
Here, 4x2 u 3x = 12x3
(i) Multiplication of polynomials by monomials
While multiplying a polynomial by a monomial, we multiply each term of a
polynomial separately by the monomial. For example:
Example 2: Multiply: (x + y) by a xy
Solution: ax ay a
Here, a u (x + y) = ax + ay x+y
Example 3: Multiply (3m2 – 2n2) by 5mn
Solution:
Here, 5mn u (3m2 – 2n2) = 5mn u 3m2 – 5mn u 2n2 Each term of 3m2 – 2n2 is separately
= 15m3n – 10mn3 multiplied by 5mn.
(ii) Multiplication of polynomials
While multiplying two polynomials, each term of a polynomial is separately
multiplied by each term of another polynomial. Then, the product is simplified.
For example: ab
Example 4: Multiply (a + b) by (x + y) x ax bx x+y
Solution: y ay by
Here, (x + y) u (a + b) = x (a + b) + y (a + b) a+b
= ax + bx + ay + by
Vedanta Excel in Mathematics - Book 7 154
Laws of Indices
Example 5: Multiply (3x2 + 2x – 4) by (2x – 3) By vertical arrangement
Solution: 3x2 + 2x – 4
By horizontal arrangement
(2x – 3) (3x2 + 2x – 4) u 2x – 3
= 2x (3x2 + 2x – 4) – 3 (3x2 + 2x – 4) x3 4x2 – 8x
= 6x3 + 4x2 – 8x – 9x2 – 6x + 12
= 6x3 – 5x2 – 14x + 12 – 9x2 – 6x + 12
6x3 – 5x2 – 14x +12
EXERCISE 10.2
General Section - Classwork
Let's tell and write the products as quickly as possible.
1. a) 2x × 3x × 4x = ............. b) 4y × (– 5y3) = .............
c) ( 3ab × ( 4a2b) = ............. d) (– 2xy) × y × (– 3x2yz) = .............
e) p (p 5) = ............. f) 2x (3x 7) = .............
g) 5x2 (3x 2y) = ............. h) ab (a2 + b2) = .............
i) x(xy x y) = ............. j) ab (ab a 1) = .............
2. a) If a = x and b = 2x2, then 2ab = ..................
b) If x = 2p2 and y = 3p3, then 3xy = ..................
3. Let's investigate the tricky process of multiplication shown below.
1+ 2
x×x 1× 2
(x + 1)(x + 2)= x2... (x + 1)(x + 2) = x2 + 3x... (x + 1)(x + 2) = x2 + 3x + 2
a× a 2–3 2× –3
(a + 2 )(a–3) = a2 ... (a + 2)(a – 3) = a2 – 1.a ... (a + 2)(a – 3) = a2 – a – 6
Now, apply the tricky process shown above. Then, tell and write the products as
quickly as possible.
(a) (x + 2) (x + 1) = ......................... (b) (x + 2) (x + 3)= .........................
(c) (a + 3) (a + 4) = ......................... (d) (a + 3) (a – 2) = .........................
(e) (x – 3) (x + 5) = ......................... (f) (x 2) (x – 5) = .........................
(g) (x – 3) (x 2) = ......................... (h) (a 6) (a – 3) = .........................
Creative Section - A b) (–3x2) × 4x × (–x)
4. Let's simplify: d) (–3pq) × (–5qr) × pqr
f) (– 4x) × (–x2yz) × (–3y) × (–2z)
a) x × 2x × 3x2
c) (–2ab) × (–5a) × (–b)
e) mn × (–m2) × (–3n2)
155 Vedanta Excel in Mathematics - Book 7
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