The study of Algebra requires a deep and intrinsic understanding of concepts, terms and formulas. Hence, to ease this task, we present “Std. IX: Algebra”, ...

Written as per the revised syllabus prescribed by the Maharashtra State Board of Secondary and Higher Secondary Education, Pune.

STD. IX

Algebra

Fifth Edition: March 2016

Salient Features • Written as per the new textbook.

• Exhaustive coverage of entire syllabus.

• Topic-wise distribution of all textual questions and practice problems at the beginning of every chapter.

• Covers answers to all textual exercises and problem set.

• Includes additional problems for practice.

• Multiple choice questions for effective preparation.

• Comprehensive solution to Question Bank.

Printed at: India Printing Works, Mumbai

No part of this book may be reproduced or transmitted in any form or by any means, C.D. ROM/Audio Video Cassettes or electronic, mechanical including photocopying; recording or by any information storage and retrieval system without permission in writing from the Publisher.

P.O. No. 15194

10040_10510_JUP

PREFACE Algebra is the branch of mathematics which deals with the study of rules of operations and relations, and the concepts arising from them. It has wide applications in different fields of science and technology. It deals with concepts like linear equations, quadratic equations etc. Its application in statistics deals with measures of central tendency, representation of statistical data etc. The study of Algebra requires a deep and intrinsic understanding of concepts, terms and formulae. Hence, to ease this task, we present “Std. IX: Algebra”, a complete and thorough guide, extensively drafted to boost the students confidence. The question answer format of this book helps the student to understand and grasp each and every concept thoroughly. The book is based on the new text book and covers the entire syllabus. At the beginning of every chapter, topic-wise distribution of all textual questions and practice problems has been provided for simpler understanding of different types of questions. It contains answers to textual exercises, problems sets and Question bank. It also includes additional questions and multiple choice questions for practice. Graphs are drawn with proper scale. Another feature of the book is its layout which is attractive and inspires the student to read. Lastly, I would like to thank all those who have helped me in preparing this book. There is always room for improvement and hence I welcome all suggestions and regret any errors that may have occurred in the making of this book. A book affects eternity; one can never tell where its influence stops.

Best of luck to all the aspirants! Yours faithfully, Publisher

No.

Topic Name

Page No.

1

Sets

1

2

Real Numbers

19

3

Algebraic Expressions

62

4

Linear Equations in Two Variables

97

5

Graphs

133

6

Ratio and Proportion

194

7

Statistics

224

8

Question Bank

259

01 Sets Type of Problems Definition of Sets

Chapter 01: Sets

Exercise 1.1

Q.1

Problem set-1

Q.1

1.1 Method of Writing Sets

Types of Sets

Practice Problems (Based on Exercise 1.1) 1.2

Q.1, 2, 3, 4

Practice Problems (Based on Exercise 1.2)

Practice Problems (Based on Exercise 1.3)

Q.1, 2, 4, 5 Q.1, 3, 4

1.4

Q.1, 2, 3, 4, 5

Practice Problems (Based on Exercise 1.4)

Practice Problems (Based on Exercise 1.5) Problem set-1 1.5 Practice Problems (Based on Exercise 1.5) Problem set-1 1.3 Practice Problems

Draw a Venn Diagram

Q.4, 5, 9

Q.11, 12, 22

1.5

Word Problems on Sets

Q.1, 2

Problem set-1

Problem set-1

Number of elements in a the Set

Q.1, 2 Q.2, 3, 10

1.3

Operations on Sets and their Properties

Q.2, 3, 4

Problem set-1

Problem set-1

Subset and Universal Set

Q. Nos.

(Based on Exercise 1.3) Practice Problems

Q.1, 2, 3, 4 Q.6, 7, 8, 13, 14, 21, 23 Q.1, 2, 5 Q.2 Q.15, 18 Q. 3, 4 Q.3, 4 Q.16, 17, 19 Q.3 Q.2 Q.1, 5

(Based on Exercise 1.5) Problem set-1

Q.20 1

Std. IX : Algebra Introduction Consider the following examples: i. Collection of books in a library. ii. Collection of cloths in a shop. Objects in each of these examples can be seen clearly. Such collections are well defined collections. Consider the following examples: i. Brilliant students in a class. ii. Happy people in the city. The term “brilliant” and “happy” are relative terms. A person may be brilliant or happy according to one person but he may not be so according to the other person. It is important to determine whether a given collection is well defined or not. Well defined collections or groups are termed as “Sets”. George Cantor, (1845-1918) a German Mathematician is a creator of “Set theory” which has become a fundamental theory in Mathematics. 1.1

Definition of Sets

Set: A well defined collection of objects is called a “set”. Example: i. Collection of odd natural numbers. ii. Collection of whole numbers. Each object in the set is called as an “element” or a “member” of the set. Example: i. For a set containing odd natural numbers, elements are 1, 3, 5, 7, … ii. For a set of whole numbers, elements are 0, 1, 2, 3, … Collection of elements which are not well defined, do not form a set. Such sets usually contain relative terms like easy, good, favourite, etc. Example: The collection of good books in a library. Here, ‘good’ is a relative term whose meaning will vary from person to person. Important Points to Remember: 1.

Sets are denoted by capital alphabets. e.g. A, B, C, X, Y, Z, etc. But the elements of a set are denoted by small alphabets e.g. a, b, p, q, r, etc. If ‘r’ is an element of set P, then it is written as r P and is read as: i. ‘r’ belongs to set P or ii. ‘r’ is a member of set P or iii. ‘r’ is an element of set P.

2.

2

3.

4.

1.2

Symbol ‘’ stands for ‘belongs to’, ‘is a member of’ or ‘is an element of’. If ‘r’ is not an element of set P, then it is written as r P and it is read as: i. ‘r’ does not belong to set P or ii. ‘r’ is not a member of set P or iii. ‘r’ is not an element of set P . The symbol stands for ‘does not belong to’ or ‘not a member of’ or ‘not an element of’. The set of Natural numbers, Whole numbers, Integers, Rational numbers, Real numbers are denoted by N, W, I, Q, R respectively. Methods of Writing Sets

There are two methods of writing a set: a. Listing method or Roster form b. Rule method or Set builder form a. Listing method or Roster form In this method: i. Elements of the set are enclosed within curly brackets. ii. Each element is written only once. iii. Elements are separated by commas. iv. The order of writing the elements in a set is not important. Example: A = {a, b, c, d, e} or A = {b, d, a, c, e} are same or equal sets that represent first five letters of the English alphabet. Few examples of writing a set by listing method are: i. L is a set of letters of the word “fatal”. L = {f, a, t, l} ii. M is a set of integers less than 5. M = {… , 3, 2, 1, 0, 1, 2, 3, 4} iii. O is a set of even natural numbers from 1 to 100. O = {2, 4, 6, 8, … , 100} b.

Rule method or Set builder form In this method, elements of the set are described by specifying the property or rule that uniquely determines the elements of a set. Example: i. Y = {x|x is a vowel in the English alphabet} In the above notation, curly brackets denotes ‘set of’, vertical line (|) denotes ‘such that’. Set Y is read as: “Y is a set of all ‘x’ such that ‘x’ is a vowel in the English alphabet”.

Chapter 01: Sets ii.

B = {x|x W, x < 10} Set B is read as: “B is a set of all ‘x’ such that ‘x’ is a whole number less than 10”.

Solution: i. F = {x|x = 5n, n N, n 4} ii.

G = {x|x = n2, n N, 3 n < 10}

iii.

H = {x|x = 5n, n N, n 4}

iv.

X = {x| square of x is 64} or X = {x|x is a square root of 64}

Which of the following collections are sets? i. The collection of prime numbers. ii. The collection of easy sub topics in this chapter. iii. The collection of good teachers in your school. iv. The collection of girls in your class. v. The collection of odd natural numbers. Solution: i. It is a set. ii. Meaning of ‘easy sub topics’ may vary from person to person, as it is a relative term. Therefore, it is not a set. iii. Choice of good teachers varies from student to student as ‘good’ is a relative term. Therefore, it is not a set. iv. It is a set. v. It is a set.

v.

Y = {x|x =

2.

Examples: A

Note: Instead of ‘|’ sometimes two vertical dots ‘:’ are also used. Exercise 1.1 1.

Write the following sets in the roster form: i. A = {x|x is a month of the Gregarian year not having 30 days} ii. B = {y|y is a colour in the rainbow} iii. C = {x|x is an integer and 4 < x < 4} iv. D = {x|x I, 3 < x 3} v. E = {x|x = (n 1)3, n < 3, n W} Solution: i. A = {January, February, March, May, July, August, October, December} ii. B = {violet, indigo, blue, green, yellow, orange, red} iii. C = {3, 2, 1, 0, 1, 2, 3} iv. D = {2, 1, 0, 1, 2, 3} v. Putting n = 0, 1, 2, we have, E = {1, 0, 1} 3.

Write the following sets in the set builder form: i. F = {5, 10, 15, 20} ii. G = {9, 16, 25, 36, … , 81} iii. H = {5, 52, 53, 54} iv. X = {8, 8} v.

1 1 1 1 Y = 1, , , , 8 27 64 125

1 n3

, n N, n 5}

4.

Write the set of first five positive integers whose square is odd. Solution: P = {1, 3, 5, 7, 9} 1.3

Venn Diagrams

L. Euler, a great Mathematician, introduced the idea of diagrammatic representation of sets. Later, British logician, John-Venn (1834-1923) used and developed the idea of the above concept to study sets. Such representations are called Venn Diagrams. A set is represented by a ‘closed’ figure in a Venn Diagram, where the elements of the set are represented by points in the closed figure. Some of the closed figures used to represent Venn Diagrams are: rectangle, circle, triangle, etc.

.a

.e .i .o .u C

. 0

B

.

2

.

1

.a

.

.

.b .c

2

4

.d

. 8

.

.e

D

6

1.4

Types of Sets

i.

Singleton set: A set containing exactly one element is called as a singleton set. Example: a. A = {5} b. B = {x|x + 3 = 0} Set B having only one element i.e., 3

ii.

Empty set: A set which does not contain any element is called as an empty or a null set. It is represented as {} or (phi). 3

Std. IX : Algebra

iii.

iv.

Example: a. A = {a|a is a natural number, 5 < a < 6} A = { } or A =

iii.

p3 = 8

p3 = (2)3

b.

p = 2

C = {2}

It is a singleton set.

iv.

(q 4)2 = 0

q4=0

q=4

D = {4}

It is a singleton set.

v.

1 + 2x = 3x

1 = 3x 2x

1=x

x=1

E = {1}

It is a singleton set.

B = {x|x is a natural number, x < 1} B=

Finite set: If counting of elements in a set terminates at a certain stage, the set is called as finite set. Example: A = {1, 2, 3, 4, 5, 6, 7} B = {x|x is days in a week} The above sets A and B have finite elements. Set A and set B are finite sets. Infinite set: If counting of elements in a set does not terminate at any stage, the set is called as infinite set. Example: P = {1, 2, 3, 4, 5, 6, …} W = {x|x is a whole number} The above sets P and W have elements that cannot be counted. They are sets that do not terminate at any stage. Therefore, P and W are infinite sets.

Note: i. X = {0} is not a null set as ‘0’ is an element of set X. ii. An empty set is a finite set. iii. Sets of Natural numbers, Whole numbers, Integers, Rational numbers and Real numbers are all infinite sets. Exercise 1.2 1.

State which of the following sets are singleton sets:

i.

A= x

x 16

ii. B = {y|y2 = 36} iii. C = {p|p I, p3 = 8} iv. D = {q|(q 4)2 = 0} v. E = {x|1 + 2x = 3x, x W} Solution: x = 16 i. x = 256 A = {256} It is a singleton set. y2 = 36 y = 6 B = {–6, +6} It is not a singleton set.

ii. 4

2.

Which of the following sets are empty? i. A set of all even prime numbers ii. B = {x|x is a capital of India} iii. F = {y|y is a point of intersection of two parallel lines} iv. G = {z|z N, 3 < z < 4} v. H = {t|t is a triangle having four sides} Solution: i. A = {2} It is not an empty set. ii.

B = {Delhi} It is not an empty set.

iii.

Parallel lines do not intersect each other. F={} It is an empty set.

iv.

z is a natural number. There is no natural number between 3 and 4. G={} It is an empty set.

v.

A triangle is a three-sided figure. H={} It is an empty set.

Chapter 01: Sets 3.

Classify the following sets into finite or infinite: i. A = {1, 3, 5, 7, …} ii. B = {101, 102, 103, … , 1000} iii. C = {x|x Q, 3 < x < 5} iv. D = {y|y = 3n, n N} Solution: i. Here, counting of elements do not terminate at any stage. A is an infinite set. ii.

Here, counting of elements terminate at 1000. B is a finite set.

iii.

There is infinite number of rational elements between 3 and 5. C is an infinite set.

iv.

Here, counting of elements do not terminate at any stage. D is an infinite set.

4.

Let G = {x|x is a boy of your class} and H = {y|y is a girl of your class}. What type of sets G and H are? Solution: Set G and set H are finite sets. 1.5

Subset

If every element of set Y is an element of set X, then Y is said to be subset of set X. Symbolically, it is represented as Y X If we have say ‘a’, an element which belongs to set Y, we can say that, it (‘a’) also belongs to set X. But if a Y and a X then it is said that set Y is not a subset of X or Y X. Example: If Y = {b, z} and X = {b, l, z} then we say that Y X. If Y is a subset of X and set X contains atleast one element which is not in set Y, then set Y is the proper subset of set X. It is denoted as Y X. Set X is said to be the ‘super set’ of set Y and is denoted as X Y. If X = {a, b} and Y = {b, a}, then set X is a subset of set Y and Y is also subset of set X. In this case set X is the improper subset of the set Y. It is denoted as X Y and it is read as “X is an improper subset of Y.” Also set Y is the improper subset of the set X.

It is denoted as Y X and it is read as “Y is an improper subset of X.” Note: i. Every set is a subset of itself i.e. Y Y. ii. Empty set is a subset of every set i.e., X. 1.6

Universal Set

A suitably chosen non-empty set of which all the sets under consideration are the subsets of that set is called the Universal set. It is denoted by ‘U’. Example: A = {x|x is Physics laboratory in your school} B = {y|y is Chemistry laboratory in your school} C = {z|z is Biology laboratory in your school} U = {l|l is laboratories in your school} It can be seen that A U, B U, C U. Set U is the universal set of sets A, B and C. Note: Universal set is a set that cannot be changed once fixed for a particular solution. In Venn diagram, generally universal set is represented by a rectangle. Exercise 1.3 1.

Observe the following sets and answer the questions given below: A = The set of all residents in Mumbai B = The set of all residents in Bhopal C = The set of all residents in Maharashtra D = The set of all residents in India E = The set of all residents in Madhya Pradesh i. Write the subset relation between the sets A and C. ii. Write the subset relation between the sets E and D. iii. Which set can be chosen suitably as the universal set? Solution: i. All residents of Mumbai are residents of Maharashtra. AC ii. iii.

All residents of Madhya Pradesh are residents of India. ED Mumbai, Maharashtra, Bhopal, Madhya Pradesh are parts of India. Set D can be chosen as the universal set. 5

Std. IX : Algebra 2.

Let A = {a, b, c}, B = {a}, C = {a, b}, then i. Which sets given above are the proper subsets of the set A? ii. Which set is the super set of set C? Solution: i. Elements of set B and set C are the elements of set A. Also, there exists an element viz. c which is not an element of set B and set C but is in set A. Set B and set C are the proper subsets of set A. ii. Set A is the super set of set C i.e. A C. 3.

Draw a Venn diagram, showing sub set relations of the following sets: A = {2, 4} B = {x|x = 2n, n < 5, n N} C = {x|x is an even natural number 16} Solution: A = {2, 4} B = {2, 4, 8, 16} C = {2, 4, 6, 8, 10, 12, 14, 16} ABC C B A .12 .10 .8 .2 .16

.4 .6

.14

Prove that, if A B and B C, then A C. (Hint: Start with an arbitrary element x A and show that x C) Solution: Let us assume that x A ….(i) But, A B xB BC 4.

xC From (i) and (ii), AC

….(ii)

5. If X = {1, 2, 3}, write all possible subsets of X. Solution: All possible subsets of X are as follows: i. { } or .…[a null set is a subset of every set] ii. {1} iii. {2} iv. {3} 6

v. vi. vii. viii.

{1, 2} {1, 3} {2, 3} {1, 2, 3}

1.7

Operations on Sets

.…[every set is a subset of itself]

a.

Equality: If A is a subset of B and B is a subset of A, then A and B are said to be equal sets and are denoted by A = B. Both the sets A and B contain exactly the same elements. If the elements of A and B are not same, then we write A B. Note: To prove that sets A and B are equal, it is always necessary to prove that A B and B A. i. Let A = {x|x = 2n, n N and x < 10} and B = {2, 4, 6, 8} A = {2, 4, 6, 8} A B and B A A=B ii. Let P = {x|x is an odd natural number, x < 8} and Q = {y|y is an even natural number, y 0} Solution: i. A = {1, 2, 4, 5, 7}, B = {2, 3, 4, 8} A B = {2, 4}

Let A = {a|a is a letter in the word ‘college’} and B = {b|b is a letter in the word ‘luggage’} and U = {a, b, c, d, e, f, g, l, o, u}. Verify: i. (A B) = A B ii. (A B) = A B Proof: i. In roster form, set A and set B can be written as: A = {c, o, l, e, g} B = {l, u, g, a, e} U = {a, b, c, d, e, f, g, l, o, u} A = {a, b, d, f, u} B = {b, c, d, f, o} A B = {a, c, e, g, l, u, o} L.H.S. = (A B) = {b, d, f} .… (i) R.H.S. = A B = {b, d, f} .… (ii) From (i) and (ii), we get L.H.S. = R.H.S. (A B) = A B

ii.

ii.

A B = {l, g, e} L.H.S. = (A B) = {a, b, c, d, f, o, u} .… (iii) R.H.S. = A B = {a, b, c, d, f, o, u} …. (iv) From (iii) and (iv), we get L.H.S. = R.H.S. (A B) = A B

1.8

Number of Elements in the Set

ii.

C = {a, e, i, o, u}, D = {a, b, c, d} C D = {a, b, c, d, e, i, o, u}

iii.

The Roster form of set E and set F is as follows: E = {1, 2, 3, 4, 6, 12} F = {1, 2, 3, 6, 9, 18} E F = {1, 2, 3, 4, 6, 9, 12, 18}

3.

The Roster form of set C and set D is as follows: C = {6, 7, 8, 9, 10} D = {5, 6, 7, 8, 9} C D = {6, 7, 8, 9}

iii.

The Roster form of set E and set F is as follows: E = {… , 4, 3, 2, 1} F = {1, 2, 3, 4, …} E F = { } or

Let U = {x|x = 2n, n W, n < 8} be the universal set. A = {y|y = 4n, n W, n < 4}; B = {z|z = 8n, n W, n 2}. Then find: i. A ii. B iii. (A B) iv. (A B) Solution: The Roster form of set U, set A and set B is as follows: U = {20, 21, 22, 23, 24, 25, 26, 27} = {1, 2, 4, 8, 16, 32, 64, 128}

4.

8

If A is any set then the number of elements in set A is denoted by n (A). Illustrations: i. Let A = {x|x N, 7 < x 12} A = {8, 9, 10, 11, 12} n(A) = 5 ii. For an empty set, n() = 0 iii. n(A B) = n(A) + n(B) n(A B) To verify this identity, let us consider the following example, A = {2, 3, 4} and B = {3, 4, 5, 6} A B = {2, 3, 4, 5, 6} and A B = {3, 4}

Chapter 01: Sets

n(A) = 3, n(B) = 4, n(A B) = 5 and n(A B) = 2 L.H.S. = n(A B) = 5 .... (i) R.H.S. = n(A) + n(B) n(A B) =3+42 =5 .... (ii) n(A B) = n(A) + n(B) n(A B) .... [From (i) and (ii)]

n(T C) = 80 There are 80 students who either drink tea or coffee or both. But there are 100 students in the hostel. Number of students who neither drink tea nor coffee = n(U) n(T C) = 100 80 = 20 Students who do not drink tea or coffee is 20.

4. Exercise 1.5 1.

Let A = {1, 3, 5, 6, 7}, B = {4, 6, 7, 9}, then verify the following: n (A B) = n(A) + n(B) n(A B) Proof: A = {1, 3, 5, 6, 7} and B = {4, 6, 7, 9} A B = {1, 3, 4, 5, 6, 7, 9} and A B = {6, 7} n(A) = 5, n(B) = 4, n(A B) = 7 and n(A B) = 2 L.H.S. = n(A B) = 7 ....(i) R.H.S. = n(A) + n(B) n(A B) =5+42=7 ....(ii) L.H.S. = R.H.S. ....[From (i) and (ii)] n(A B) = n(A) + n(B) n(A B)

2.

Let A and B be two sets such that n(A) = 5, n(A B) = 9, n(A B) = 2. Find n(B). Solution: Given, n(A) = 5, n(A B) = 9, n(A B) = 2 n(B) = ? By using identity, n(A B) = n(A) + n(B) n(A B) 9 = 5 + n(B) 2 9 – 5 + 2 = n(B) n(B) = 6

3.

In a school hostel, there are 100 students, out of which 60 drink tea, 50 drink coffee and 30 drink both tea and coffee. Find the number of students who do not drink tea or coffee. Solution: Let U be the universal set of students in hostel, T be the set of students who drink tea and C be the set of students who drink coffee. n(U) = 100, n(T) = 60, n(C) = 50, n(T C) = 30 By using the identity, n(T C) = n(T) + n(C) n(T C) = 60 + 50 30 = 110 30

110 children choose their favourite colour from blue and pink. Every student has to choose at least one of the colours. 60 children choose blue colour, while 70 children choose pink colour. How many children choose both the colours as their favourite colour? Solution: Let the number of children who choose blue colour be n(B) and number of children who choose pink colour be n(P). n(B) = 60 and n(P) = 70 Number of children who choose their favourite colour from blue or pink. n(B P) = 110 By using the identity, n(B P) = n(B) + n(P) n(B P) 110 = 60 + 70 n(B P) n(B P) = 60 + 70 110 n(B P) = 20 The number of students who choose both the colours as their favourite colours is 20. 5.

Observe the figure and verify the following equation: A

.1 .5

.4

.2 .3 .9

B

.8 .7 .6 C

n(A B C) = n(A) + n(B) + n(C) n(A B) n(B C) n(C A) + n(A B C) Proof: L.H.S. = n(A B C) A B C = {1, 2, 3, 4, 5, 6, 7, 8, 9} n(A B C) = 9 …. (i) Now, A = {1, 2, 3, 4, 5} n(A) = 5 B = {2, 3, 6, 7, 8} n(B) = 5 C = {3, 4, 6, 9} n(C) = 4 A B = {2, 3} n(A B) = 2 9

Std. IX : Algebra

B C = {3, 6} n(B C) = 2 C A = {3, 4} n(C A) = 2 A B C = {3} n(A B C) = 1 R.H.S. = n(A) + n(B) + n(C) n(A B) n(B C) n(C A) + n(A B C) =5+5+4222+1 =9 …. (ii) L.H.S. = R.H.S. ....[From (i) and (ii)] n(A B C) = n(A) + n(B) + n(C) n(A B) n(B C) n(C A) + n(A B C)

Solution: i. A = {… , 3, 2, 1} ii. B = {25, 45, 52, 54, 56, 58, 65, 85} iii. C = {2, 3, 5} iv. D = {3, 2, 1, 0, 1, 2, 3} v. Since, 2 n 4 n = 2, 3, 4 n 2 2 for n = 2, 2 = = 2 n 1 (2) 1 3

for n = 3,

n 3 3 = = 2 n 1 (3) 1 8

for n = 4,

n 4 4 = = 2 n 1 (4) 1 15

2

2

Problem Set - 1 1.

Which of the following collections are sets? i. The collection of rich people in your district. ii. The collection of natural numbers less than 50. iii. The collection of most talented persons of India. iv. The collection of first ten prime integers. v. The collection of all days in a week starting with the letter ‘T’. vi. The collection of some months in a year. vii. The collection of all books in your school library. viii. The collection of smart boys in your class. ix. The collection of multiples of 7. x. The collection of students in your class who got a lot of marks in the first unit test. Solution: From the given collections (ii), (iv), (v), (vii) and (ix) are sets. Remaining collections are not considered as sets as they have relative terms and their meaning may vary from person to person. 2.

10

E= , ,

3.

Write the following sets in the set builder form: i. F = {I, N, D, A} ii. G = {1, 1} iii. H = {3, 9, 27, 81, 243} iv. J = {15, 24, 33, 42, 51, 60}

E = x x

, 2 n 4,n N n 1 n

2

1 2 3 4 5 , , 2 5 10 17 26

v.

K = , ,

Solution: i. F = {x|x is a letter in the word ‘INDIA’} ii. G = {y| square of y is 1} or G = {y|y is a square root of 1} iii. H = {a|a = 3n, n N, n 5} iv. J = {b|b is a two digit number whose sum of digits is 6}

v.

Write the following sets in roster form: i. A = {x|x I, x W} ii. B = {x|x is two digit number such that the product of its digits is a multiple of ten} iii. C = {x|x is a prime divisor of 120} iv. D = {x|x I and x2 < 10} v.

2 3 4 3 8 15

When n = 1, c =

1 (1) 1 2

When n = 2, c =

2 2 2 (2) 1 5 3 (3) 1 10

When n = 4, c =

4 4 2 (4) 1 17

n

n 2 +1

K = c c =

3

When n = 3, c =

When n = 5, c =

1

2

2

5

5 (5) 1 26 2

, n N, n 5

Chapter 01: Sets 4.

Classify the following sets as ‘singleton’ or ‘empty’: i. A = {x|x is a negative natural number} ii. B = {y|y is an odd prime number < 4} iii. C = {z|z is a natural number, 5 < z < 7} iv. D = {d|d N, d2 0}

Solution: i. Each natural number is positive. A={} It is an empty set.

ii.

B = {3} It is a singleton set.

iii.

C = {6} It is a singleton set.

iv.

There is no natural number whose square is less than or equal to zero. D={} It is an empty set.

5.

Classify the following sets as ‘finite’ or ‘infinite’: i. A = {x|x is a multiple of 3} ii. B = {y|y is a factor of 13} iii. C = {…, 3, 2, 1, 0} iv. D = {x|x = 2n, n N} Solution: i. A = {3, 6, 9, 12, …} It is an infinite set.

ii.

B = {1, 13} It is a finite set.

iii.

C is an infinite set.

iv.

D = {20, 21, 22, 23, 24,...} = {2, 4, 8, 16, 32, …} D is an infinite set.

6.

State which of the following sets are equal. i. N = {1, 2, 3, 4, …} ii. W = {0, 1, 2, 3, …} iii. A = {x|x = 2n, n W} iv. B = W {0} Solution: i. N ={1, 2, 3, 4, …} ii. W = {0, 1, 2, 3, …} iii. A = {20, 21, 22, 23, …} = {1, 2, 4, 8, …} iv. B = W {0} = {1, 2, 3, 4, …} Here, set N and set B are subset of each other. In set N and set B, the elements are the same. N=B

7.

Let A = {7, 5, 2} and B =

3

125, 4, 49 .

Are the sets A and B equal? Justify your answer. Solution: A = {7, 5, 2}, B = 3 125, 4, 49

B = {5, 2, 2, 7, 7} Here, A is a subset of B, but B is not a subset of A. Elements of set A and set B are not equal. AB

8.

If A = {1, 2, 3, 4}, B = {2, 4, 6, 8}, C = {3, 4, 5, 6} and U = {x|x N, x < 10}. Verify the following properties: i. A (B C) = (A B) C ii. A (B C) = (A B) (A C) iii. A (C) = (A B) (A C) iv. (A B) = A B v. (A B) = A B vi. (A) = A Solution: Roster form of set U is as follows: U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {1, 2, 3, 4} B = {2, 4, 6, 8} C = {3, 4, 5, 6} i. B C = {2, 3, 4, 5, 6, 8} A B = {1, 2, 3, 4, 6, 8} L.H.S. = A (B C) = {1, 2, 3, 4, 5, 6, 8} …. (i) R.H.S. = (A B) C = {1, 2, 3, 4, 5, 6, 8} …. (ii) From (i) and (ii), we get L.H.S. = R.H.S. A (B C) = (A B) C

ii.

iii.

B C = {4, 6} A B = {1, 2, 3, 4, 6, 8} A C = {1, 2, 3, 4, 5, 6} L.H.S. = A (B C) = {1, 2, 3, 4, 6} …. (i) R.H.S. = (A B) (A C) = {1, 2, 3, 4, 6} …. (ii) From (i) and (ii), we get L.H.S. = R.H.S. A (B C) = (A B) (A C) B C = {2, 3, 4, 5, 6, 8} L.H.S. = A (B C) = {2, 3, 4}

…. (i) 11

Std. IX : Algebra

iv.

v.

vi.

9.

A B = {2, 4} A C = {3, 4} R.H.S. = (A B) (A C) = {2, 3, 4} .… (ii) From (i) and (ii), we get L.H.S. = R.H.S. A (C) = (A B) (A C) A B = {1, 2, 3, 4, 6, 8} (A B) = {5, 7, 9} A = {1, 2, 3, 4} A = {5, 6, 7, 8, 9} B = {2, 4, 6, 8} B = {1, 3, 5, 7, 9} L.H.S. = (A B) = {5, 7, 9} …. (i) R.H.S. = A B = {5, 7, 9} …. (ii) From (i) and (ii), we get L.H.S. = R.H.S. (A B) = A B A B = {2, 4} (A B) = {1, 3, 5, 6, 7, 8, 9} A = {5, 6, 7, 8, 9} B = {1, 3, 5, 7, 9} L.H.S. = (A B) = {1, 3, 5, 6, 7, 8, 9} R.H.S. = A B = {1, 3, 5, 6, 7, 8, 9} From (i) and (ii), we get L.H.S. = R.H.S. (A B) = A B A = {5, 6, 7, 8, 9} L.H.S. = (A) = {1, 2, 3, 4} R.H.S. = A = {1, 2, 3, 4} From (i) and (ii), we get L.H.S. = R.H.S. (A) = A

A = {0} Set A is not a null set.

ii.

Here, 2b is an even number and 1 is an odd number. Since, addition of even and odd number always give odd number, value of 2b + 1, for any value of b N is an odd number. Set B is a null set.

iii.

Square of an odd number is always an odd number. c2 cannot be even. Set C is a null set.

10.

Give an example of the set which can be written in set builder form but cannot be written in roster form. Solution: Consider the set of rational numbers ‘Q’. In set builder form, it is a b

Q = a I, b I and b 0 But same set Q cannot be written in roster form. …. (i) …. (ii)

…. (i)

11.

Write down all possible subsets of each of the following sets: i. ii. A = {1} iii. B = {1, 2} iv. C = {a, b, c, d} Solution: i. Subset of a null set is only one i.e.

ii.

Subsets of set A are empty set { } and set A itself. i.e. and {1}

iii.

All possible subsets of set B are , {1}, {2}, {1, 2}.

iv.

All possible subsets of set C are {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c, d}

…. (ii)

For each of the following sets, state with reasons, whether it is a null set or not: i. A = {x|x I, x2 is not positive} ii. B = {b|b N, 2b + 1 is even} iii. C = {c|c N, c is odd and c2 is even} Solution: i. Whether a number is positive or negative its square is always a positive. Square of an integer cannot be negative, except zero, whose square is neither positive nor negative. 12

12.

Write proper subsets of the following sets: i. A = {a, b} ii. B = {a, b, c} Solution: i. Proper subsets of A are {a} and {b}. ii. Proper subsets of B are {a}, {b}, {c}, {a, b}, {b, c}, {c, a}.

Chapter 01: Sets 13.

Write the sets A and B such that A is finite, B is finite, A and B are disjoint sets. Solution: Let B = {1, 3, 5, 7}, A = {2, 4, 6, 8} be the finite sets. But their intersection is a null set, i.e. A B = A and B are disjoint sets.

14.

Let A = {a, b, c, d}, B = {a, b, c}, C = {b, d, e}, then find the sets D and E satisfying the following conditions: i. D A, D B ii. C E, B E = Solution: i. Since, D A and D B D = {a, b}/{a, c}/{a, d}/{a, b, d}/{b, c, d} /{a, c, d} ii. Since, C E and B E = E must not contain any element of set B. C is superset of set E. E = {d}/{e}/{d, e} Let U = {x|x N, x < 10}, A = {a|a is even, a U}, B = {b|b is a factor of 6, b U}. Verify that: n(A) + n(B) = n(A B) + n(A B). Solution: Roster form of set U and set A is as follows: U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {2, 4, 6, 8} ….(Even numbers within the universal set) n(A) = 4 Roster form of set B is as follows: B = {1, 2, 3, 6} ….( b U) 15.

n(B) = 4 A B = {1, 2, 3, 4, 6, 8} n(A B) = 6 A B = {2, 6} n(A B) = 2 L.H.S. = n(A) + n(B) =4+4 =8 ....(i) R.H.S. = n(A B) + n(A B) =6+2 =8 ....(ii) From (i) and (ii), we get L.H.S. = R.H.S. n(A) + n(B) = n(A B) + n(A B)

16.

In a group of students, 50 students passed in English, 60 students passed in Mathematics and 40 students passed in both. Find the number of students who passed either in English or in Mathematics. Solution: Let the number of students who passed in English be denoted by set E. n(E) = 50 Let the number of students who passed in Mathematics be denoted by set M. n(M) = 60 The number of students who passed in English and Mathematics = n(M E) = 40 By using identity, n(E M) = n(E) + n(M) n(E M) = 50 + 60 – 40 = 110 40 = 70 The number of students who passed either in English or in Mathematics is 70. 17.

A T.V. survey says 136 students watch only programme P1, 107 watch only programme P2, 27 watch only programme P3. 25 students watch P1 and P2 but not P3. 37 watch P2 and P3 but not P1. 53 students watch P1 and P3 but not P2. 40 students watch all three programmes and 80 students do not watch any programme. Find, with the help of Venn diagram. i. Number of P1 viewers. ii. Number of P2 or P3 viewers. iii. Total number of viewers surveyed. Solution: Given data represented by Venn diagram is as follows: U P2 P1 136

107

25

53

40

37

27 80

P3 i.

From the Venn diagram, Number of P1 viewers = n(P1) = 136 + 25 + 53 + 40 = 254 Number of P1 viewers is 254. 13

Std. IX : Algebra

ii. iii.

Number of P2 or P3 viewers = n(P2 P3) = 107 + 25 + 40 + 37 + 53 + 27 = 289 Number of P2 or P3 viewers is 289. Total number of viewers surveyed = Number of only P1 viewers + Number of P2 or P3 viewers + 80 = 136 + 289 + 80 = 505 Total number of viewers surveyed is 505.

18.

Show that, it is impossible to have sets A and B such that set A has 32 elements, set B has 42 elements, A B has 12 elements and A B has 64 elements. Solution: Set A has 32 elements. n(A) = 32 Set B has 42 elements. n(B) = 42 Given that n(A B) = 12 and n(A B) = 64 By using the identity, n(A B) = n(A) + n(B) n(A B), L.H.S. = n(A B) = 64 ....(i) R.H.S. = n(A) + n(B) n(A B) = 32 + 42 12 = 62 ....(ii) From (i) and (ii), we get L.H.S. R.H.S. Sets A and B are impossible. 19.

Let the universal set U be a set of all students of your school. A is the set of boys, B is the set of girls and C is the set of students participating in sports. Describe the following sets in words and represent them by a Venn diagram: i. BC ii. A (B C) Solution: i. B C represents the set of girls participating in sports. A B C

Represent sets A, B, C such that A B, A C = and B C by Venn diagram and shade the portion representing A (B C). Solution: i. AB set A is proper subset of set B. i.e. set A is inside set B. ii. A C = set A and set C do not intersect. iii. B C set B and set C intersect. B C B C A 20.

A (B C) Let A, B, C be sets such that A B , B C and A C . Do you claim that A B C ? Justify your answer. Solution: There are two possibilities: i. If A = {a, b} B = {b, c} C = {c, a} then, it is observed that A B , B C , C A , but, A B C = ii. If A = {a, b} B = {a, c} C = {a, b, c} then, it is observed that A B , B C , C A , but, A B C = {a} We cannot claim that A B C . 21.

22.

ii.

BC A (B C) represents the set of all boys or set of girls that participate in sports. BC A C B

A (B C) 14

With the help of suitable example, verify the following statements: If A B, B C, then A C. Solution: Let A = {x, y, z}, B = {a, x, y}, C = {y, w} Since, each element of set A does not exist in set B. A B Each element of set B does not exist in set C. B C Each element of set A does not exist in set C. A C

Chapter 01: Sets 23.

If A and B are any two sets, then prove that i. (A B) = A B ii. (A B) = A B [Hint: Show (A B) A B and vice versa] Solution: Let U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 3, 4, 5} B = {3, 4, 5, 6, 7} i. A B = {1, 2, 3, 4, 5, 6, 7} L.H.S. = (A B) = {8} …. (i) A = {6, 7, 8} B = {1, 2, 8} R.H.S. = A B = {8} …. (ii) From (i) and (ii), we get L.H.S. = R.H.S. (A B) = A B

ii.

Now, A B = {3, 4, 5} L.H.S. = (A B) = {1, 2, 6, 7, 8} R.H.S. = A B = {1, 2, 6, 7, 8} From (i) and (ii), we get L.H.S. = R.H.S. (A B) = A B

…. (i) …. (ii)

One-Mark Questions 1.

Write the following set in set builder form. A = {2, 3, 5, 7, 11, 13, 17} Solution: The set builder r form of set B is: A = {x|x is a prime number, x < 18} 2.

Write the following set in roster form. B = {x|x is a natural number and 4 x < 10 Solution: The Roster form of set B is: B = {4, 5, 6, 7, 8, 9} 3.

If A = {1, 2, 3, 4, 5, 6} and B = {1, 3, 5, 7} then draw Venn diagram for A B. Solution: U B A 2 4 6

1 3 5 AB

7

4.

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} is the universal set and C = {5, 6, 7, 8} then find C. Solution: Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and C = {5, 6, 7, 8} C = {1, 2, 3, 4, 9} 5.

If A = {9, 11, 13, 15} and B = {1, 3, 5, 7} then find A B. Solution: Given, A = {9, 11, 13, 15} and B = {1, 3, 5, 7} A B = { } or 6.

If A and B are two sets such that n(B) = 8, n(A B) = 11, n(A B) = 6, find n(A). Solution: By using identify, n(A B) = n(A) + n(B) n(A B) 11 = n(A) + 8 6 n(A) = 11 2 n(A) = 9

7.

State which of the following sets are equal: A = {x|x W, x < 6} B = {1, 2, 3, 4, 5, 6} C = {0, 1, 2, 3, 4, 5} Solution: Here, A = {0, 1, 2, 3, 4, 5} and C = {0, 1, 2, 3, 4, 5} A=C 8.

Classify the following sets as singleton or empty. i. A = {x|x is a natural number, x 5 and x 7} ii. B = {x|x is an even prime number} Solution: i. There is no common number for x < 5 and x > 7 A = {} Set A is empty set. ii B = {2} Set B is singleton set. 9.

If A = {2, 3, 4, 5} and B = {1, 2, 5, 6}, then find A B. Solution: Given, A = {2, 3, 4, 5} and B = {1, 2, 5, 6} A B = {1, 2, 3, 4, 5, 6} 15

Std. IX : Algebra 10. If A = {3}, write all possible subsets of set A. Solution: and {3} 11. If U = {1, 2, 3, 4} and X = {2, 4}, then find X. Solution: Given, U = {1, 2, 3, 4} and X = {2, 4} X = {1, 3}

2.

Draw a Venn diagram showing subset relations of the following sets: A = {2 , 8} B = {x|x = 2n, n 4 and n N} C = {x|x is an even natural number 20}

3.

If A = {x, y}, write all possible subsets of A.

4.

State true or false: i. is a subset of itself. ii. If A B and B A, then A = B. iii. The empty set is a subset of all sets.

Additional Problems for Practice

Based on Exercise 1.1 1.

2.

Write the following sets in the roster form: i. A = {x|x is a prime number which is a divisor of 30} ii. B = {x|x is an even natural number} iii. C = {x|x is an integer and x2 < 5} iv. F = {x|x is a letter in the word ‘LITTLE’} v. E = {x|x W, x N} vi. D = {x|x is a square root of 81}

Based on Exercise 1.4 1.

Find the union of each of the following pairs of sets: i. A = {5, 15, 25}, B = {10, 20, 30} ii. H = {3, 6, 9, 12, 15} , F = {3, 4, 5, 6} iii. M = {x|x N and x is a divisor of 12} N = {x|x N and x is a prime divisor of 12}

Write the following sets in the set builder form: i. A = {2, 4, 6, 8, 10, 12, 14} ii. B = {5, 10, 15, 20, ….} iii. C = {7, 72, 73, 74} iv. D = {51, 53, 55, 57, 59} v. E = {2, 3, 5, 7, 11, 13, 17, 19}

2.

Find the Intersection of the following pairs of sets: i. A = {5, 6, 7}, B = {8, 9, 10} ii. M = {10, 20, 30, 40, 50}, N = {20, 40, 60} iii. N is a set of natural numbers and W is a set of whole numbers. iv. P = {a, b, p, d, q}, R = {q, r, s, p}

3.

If U = {x|x is a natural number less than 15} is a universal set A = {1, 3, 4, 5, 9}, B = {3, 5, 7, 9, 12} Verify that (A B) = A B

4.

U = {x|x I and 3 x 3}, A = {2, 0, 2}, B = {0, 1, 2, 3} Find i. A ii. B iii. (A B) iv. A B

Based on Exercise 1.2 1.

2.

State which of the following sets are singleton or empty sets: i. A = {x|x 5 = 0} ii. B = {y|y is an even prime number greater than 2} iii. D = {x|x N and 3x 1 = 0} iv. E = {x|x I, x is neither a positive nor a negative number} v. C = {x|x N and x < 7 and x > 11} Classify the following sets into finite or infinite: i. A = {x|x is a multiple of 1} ii. C = {x|x is a point on a line} iii. D = {1, 2, 3, 4, …., 100} iv. E = {x|x N and x is an odd number}

Based on Exercise 1.3 1.

16

Write the subset relations among the following sets: P = set of all residents in Nagpur X = set of all residents in Vadodara Y = set of all residents in Maharashtra T = set of all residents in Gujarat

Based on Exercise 1.5 1.

With the help of following figure, write the following sets: U B A 1 4

3

i. iv.

9

5 10 11

A AB

7 2 8

12 6

ii. v.

B AB

iii.

U

Chapter 01: Sets

2.

Let A and B be two sets such that n(A) = 17, n(B) = 23, n(A B) = 38. Find n(A B)

7.

If A P, B P, then (A B) ________. (A) A (B) B (C) P (D)

3.

240 students in a school were interviewed and their hobbies were noted. 150 students were interested in stamp collection. 80 took delight in reading books, 40 of them do not like either. What is the number of students who liked both stamp collection and reading books?

8.

If A = {3, 4, 7, 8, 9} and B = {7, 8, 10, 11} then A B = ? (A) {3, 4} (B) {7, 8} (C) {10, 11} (D) {10}

9.

For any two sets A and B, A B = ? (A) {x|x B or x A} (B) {x|x A or x B} (C) {x|x A and x B} (D) {x|x A and x B}

10.

If U = {1, 2, 3, 4, ….} and A = {2, 4, 6, 8, ….} then A = ? (A) {2, 4, 6, ….} (B) {1, 3, 5, 7, ….} (C) {0, 1, 3, 5, ….} (D) {0, 2, 4, 6, 8, ….}

11.

If U = {4, 5, 6, 7, 8, 9}, P = {5, 6, 7, 8}, Q = {4, 6, 8, 9} then, P Q = ? (A) {4, 5, 7, 8, 9} (B) {4, 5, 7, 9} (C) {6, 7, 8} (D) {4, 6, 7, 8, 9}

12.

In the following Venn n(P Q) = 70, then x = ?

4.

5.

In a class of 50 students, 35 like Physics, 30 like Mathematics and 3 like neither. How many like both the subjects and how many like Physics only? U

A 3 6 15 12 9

B 18 24

From the above diagram find: i. AB ii. n (A B) iii. (A B) iv. n (A B) v. A B Multiple Choice Questions

1.

If B = {x|x is a vowel in English alphabet}, then Roster form of B is (A) {a, e, i, u} (B) {a, e, p, o} (C) {a, e, c, d} (D) {a, e, i, o, u}

2.

Which of the following is not an Infinite set (A) N (B) W (C) I (D) None of these

3.

A = {z|z + 6 = 0} is a _______. (A) empty set (B) singleton set (C) infinite set (D) finite set

4.

Empty set is a _______ of every set. (A) subset (B) proper subset (C) super set (D) universal set

5.

If A = {x|x is worker in department I of your company} B = {y|y is worker in department II of your company} and C = {z|z is worker of your company) then (A) C A (B) A B (C) A C (D) C B

6.

If A B and B A, then set A and B are _______ sets. (A) equal (B) disjoint (C) super (D) universal

P

13.

5

If

Q 40 x

(A)

diagram.

(B)

3

x

35 x

(C)

6

(D)

8

If n(A) = 10, n(B) = 25 and n(A B) = 15, then n(A B) = ? (A) 20 (B) 0 (C) 10 (D) 5

Answers to additional problems for practice

Based on Exercise 1.1 1.

i. ii. iii. iv. v. vi.

A = {2, 3, 5} B = {2, 4, 6, 8, ….} C = {2, 1, 0, 1, 2} F = {L, I, T, E} E = {0} D = {9, 9}

2.

i. ii. iii. iv.

A = {x|x = 2n, n N and n < 8} B = {x|x = 5n and n N} C = {x|x = 7n, 1 n 4} D = {x|x N, x is an odd integer and 50 < x < 60} E = {x|x is a prime number and 1 < x < 20}

v.

17

Std. IX : Algebra

5.

Based on Exercise 1.2 1.

Singleton sets are A, E Empty sets are B, D, C

2.

Finite set is D. Infinite sets are A, C, E.

1.

P Y, X T

2.

A = {2, 8}, B = {2, 4, 8, 16} C = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} C B 6

A

12 4

2 8

16 18

20

14

3.

, {x}, {y}, {x, y}

4.

i.

True

ii.

True

iii.

True

Based on Exercise 1.4 1.

i. ii. iii.

A B = {5, 10, 15, 20, 25, 30} H F = {3, 4, 5, 6, 9, 12, 15} M N = {1, 2, 3, 4, 6, 12}

2.

i. ii. iii. iv.

AB= M N = {20, 40} N W = {1, 2, 3, ….} P R = {p, q}

4.

i. ii. iii. iv.

A = {3, 1, 1, 3} B = {3, 2, 1} (A B) = {3, 1} A B = {3, 1}

Based on Exercise 1.5 1.

i. ii. iii. iv. v.

2.

2

3.

30

4.

18 students like both subjects and 17 students like only Physics.

18

A B = {6, 12} n(A B) = 7 (A B) = {3, 9, 15, 18, 24} n(A B) = 2 A B = {3, 9, 15, 18, 24}

Answers to Multiple Choice Questions

Based on Exercise 1.3

10

i. ii. iii. iv. v.

A = {2, 3, 6, 7, 8, 11, 12} B = {1, 3, 4, 6, 9, 11, 12} U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} A B = {1, 2, 4, 5, 7, 8, 9, 10} A B = {5, 10}

1. 5. 9. 13.

(D) (C) (B) (A)

2. (D) 6. (A) 10. (B)

3. (B) 7. (C) 11. (B)

4. (A) 8. (B) 12. (A)

STD. IX

Algebra

Fifth Edition: March 2016

Salient Features • Written as per the new textbook.

• Exhaustive coverage of entire syllabus.

• Topic-wise distribution of all textual questions and practice problems at the beginning of every chapter.

• Covers answers to all textual exercises and problem set.

• Includes additional problems for practice.

• Multiple choice questions for effective preparation.

• Comprehensive solution to Question Bank.

Printed at: India Printing Works, Mumbai

No part of this book may be reproduced or transmitted in any form or by any means, C.D. ROM/Audio Video Cassettes or electronic, mechanical including photocopying; recording or by any information storage and retrieval system without permission in writing from the Publisher.

P.O. No. 15194

10040_10510_JUP

PREFACE Algebra is the branch of mathematics which deals with the study of rules of operations and relations, and the concepts arising from them. It has wide applications in different fields of science and technology. It deals with concepts like linear equations, quadratic equations etc. Its application in statistics deals with measures of central tendency, representation of statistical data etc. The study of Algebra requires a deep and intrinsic understanding of concepts, terms and formulae. Hence, to ease this task, we present “Std. IX: Algebra”, a complete and thorough guide, extensively drafted to boost the students confidence. The question answer format of this book helps the student to understand and grasp each and every concept thoroughly. The book is based on the new text book and covers the entire syllabus. At the beginning of every chapter, topic-wise distribution of all textual questions and practice problems has been provided for simpler understanding of different types of questions. It contains answers to textual exercises, problems sets and Question bank. It also includes additional questions and multiple choice questions for practice. Graphs are drawn with proper scale. Another feature of the book is its layout which is attractive and inspires the student to read. Lastly, I would like to thank all those who have helped me in preparing this book. There is always room for improvement and hence I welcome all suggestions and regret any errors that may have occurred in the making of this book. A book affects eternity; one can never tell where its influence stops.

Best of luck to all the aspirants! Yours faithfully, Publisher

No.

Topic Name

Page No.

1

Sets

1

2

Real Numbers

19

3

Algebraic Expressions

62

4

Linear Equations in Two Variables

97

5

Graphs

133

6

Ratio and Proportion

194

7

Statistics

224

8

Question Bank

259

01 Sets Type of Problems Definition of Sets

Chapter 01: Sets

Exercise 1.1

Q.1

Problem set-1

Q.1

1.1 Method of Writing Sets

Types of Sets

Practice Problems (Based on Exercise 1.1) 1.2

Q.1, 2, 3, 4

Practice Problems (Based on Exercise 1.2)

Practice Problems (Based on Exercise 1.3)

Q.1, 2, 4, 5 Q.1, 3, 4

1.4

Q.1, 2, 3, 4, 5

Practice Problems (Based on Exercise 1.4)

Practice Problems (Based on Exercise 1.5) Problem set-1 1.5 Practice Problems (Based on Exercise 1.5) Problem set-1 1.3 Practice Problems

Draw a Venn Diagram

Q.4, 5, 9

Q.11, 12, 22

1.5

Word Problems on Sets

Q.1, 2

Problem set-1

Problem set-1

Number of elements in a the Set

Q.1, 2 Q.2, 3, 10

1.3

Operations on Sets and their Properties

Q.2, 3, 4

Problem set-1

Problem set-1

Subset and Universal Set

Q. Nos.

(Based on Exercise 1.3) Practice Problems

Q.1, 2, 3, 4 Q.6, 7, 8, 13, 14, 21, 23 Q.1, 2, 5 Q.2 Q.15, 18 Q. 3, 4 Q.3, 4 Q.16, 17, 19 Q.3 Q.2 Q.1, 5

(Based on Exercise 1.5) Problem set-1

Q.20 1

Std. IX : Algebra Introduction Consider the following examples: i. Collection of books in a library. ii. Collection of cloths in a shop. Objects in each of these examples can be seen clearly. Such collections are well defined collections. Consider the following examples: i. Brilliant students in a class. ii. Happy people in the city. The term “brilliant” and “happy” are relative terms. A person may be brilliant or happy according to one person but he may not be so according to the other person. It is important to determine whether a given collection is well defined or not. Well defined collections or groups are termed as “Sets”. George Cantor, (1845-1918) a German Mathematician is a creator of “Set theory” which has become a fundamental theory in Mathematics. 1.1

Definition of Sets

Set: A well defined collection of objects is called a “set”. Example: i. Collection of odd natural numbers. ii. Collection of whole numbers. Each object in the set is called as an “element” or a “member” of the set. Example: i. For a set containing odd natural numbers, elements are 1, 3, 5, 7, … ii. For a set of whole numbers, elements are 0, 1, 2, 3, … Collection of elements which are not well defined, do not form a set. Such sets usually contain relative terms like easy, good, favourite, etc. Example: The collection of good books in a library. Here, ‘good’ is a relative term whose meaning will vary from person to person. Important Points to Remember: 1.

Sets are denoted by capital alphabets. e.g. A, B, C, X, Y, Z, etc. But the elements of a set are denoted by small alphabets e.g. a, b, p, q, r, etc. If ‘r’ is an element of set P, then it is written as r P and is read as: i. ‘r’ belongs to set P or ii. ‘r’ is a member of set P or iii. ‘r’ is an element of set P.

2.

2

3.

4.

1.2

Symbol ‘’ stands for ‘belongs to’, ‘is a member of’ or ‘is an element of’. If ‘r’ is not an element of set P, then it is written as r P and it is read as: i. ‘r’ does not belong to set P or ii. ‘r’ is not a member of set P or iii. ‘r’ is not an element of set P . The symbol stands for ‘does not belong to’ or ‘not a member of’ or ‘not an element of’. The set of Natural numbers, Whole numbers, Integers, Rational numbers, Real numbers are denoted by N, W, I, Q, R respectively. Methods of Writing Sets

There are two methods of writing a set: a. Listing method or Roster form b. Rule method or Set builder form a. Listing method or Roster form In this method: i. Elements of the set are enclosed within curly brackets. ii. Each element is written only once. iii. Elements are separated by commas. iv. The order of writing the elements in a set is not important. Example: A = {a, b, c, d, e} or A = {b, d, a, c, e} are same or equal sets that represent first five letters of the English alphabet. Few examples of writing a set by listing method are: i. L is a set of letters of the word “fatal”. L = {f, a, t, l} ii. M is a set of integers less than 5. M = {… , 3, 2, 1, 0, 1, 2, 3, 4} iii. O is a set of even natural numbers from 1 to 100. O = {2, 4, 6, 8, … , 100} b.

Rule method or Set builder form In this method, elements of the set are described by specifying the property or rule that uniquely determines the elements of a set. Example: i. Y = {x|x is a vowel in the English alphabet} In the above notation, curly brackets denotes ‘set of’, vertical line (|) denotes ‘such that’. Set Y is read as: “Y is a set of all ‘x’ such that ‘x’ is a vowel in the English alphabet”.

Chapter 01: Sets ii.

B = {x|x W, x < 10} Set B is read as: “B is a set of all ‘x’ such that ‘x’ is a whole number less than 10”.

Solution: i. F = {x|x = 5n, n N, n 4} ii.

G = {x|x = n2, n N, 3 n < 10}

iii.

H = {x|x = 5n, n N, n 4}

iv.

X = {x| square of x is 64} or X = {x|x is a square root of 64}

Which of the following collections are sets? i. The collection of prime numbers. ii. The collection of easy sub topics in this chapter. iii. The collection of good teachers in your school. iv. The collection of girls in your class. v. The collection of odd natural numbers. Solution: i. It is a set. ii. Meaning of ‘easy sub topics’ may vary from person to person, as it is a relative term. Therefore, it is not a set. iii. Choice of good teachers varies from student to student as ‘good’ is a relative term. Therefore, it is not a set. iv. It is a set. v. It is a set.

v.

Y = {x|x =

2.

Examples: A

Note: Instead of ‘|’ sometimes two vertical dots ‘:’ are also used. Exercise 1.1 1.

Write the following sets in the roster form: i. A = {x|x is a month of the Gregarian year not having 30 days} ii. B = {y|y is a colour in the rainbow} iii. C = {x|x is an integer and 4 < x < 4} iv. D = {x|x I, 3 < x 3} v. E = {x|x = (n 1)3, n < 3, n W} Solution: i. A = {January, February, March, May, July, August, October, December} ii. B = {violet, indigo, blue, green, yellow, orange, red} iii. C = {3, 2, 1, 0, 1, 2, 3} iv. D = {2, 1, 0, 1, 2, 3} v. Putting n = 0, 1, 2, we have, E = {1, 0, 1} 3.

Write the following sets in the set builder form: i. F = {5, 10, 15, 20} ii. G = {9, 16, 25, 36, … , 81} iii. H = {5, 52, 53, 54} iv. X = {8, 8} v.

1 1 1 1 Y = 1, , , , 8 27 64 125

1 n3

, n N, n 5}

4.

Write the set of first five positive integers whose square is odd. Solution: P = {1, 3, 5, 7, 9} 1.3

Venn Diagrams

L. Euler, a great Mathematician, introduced the idea of diagrammatic representation of sets. Later, British logician, John-Venn (1834-1923) used and developed the idea of the above concept to study sets. Such representations are called Venn Diagrams. A set is represented by a ‘closed’ figure in a Venn Diagram, where the elements of the set are represented by points in the closed figure. Some of the closed figures used to represent Venn Diagrams are: rectangle, circle, triangle, etc.

.a

.e .i .o .u C

. 0

B

.

2

.

1

.a

.

.

.b .c

2

4

.d

. 8

.

.e

D

6

1.4

Types of Sets

i.

Singleton set: A set containing exactly one element is called as a singleton set. Example: a. A = {5} b. B = {x|x + 3 = 0} Set B having only one element i.e., 3

ii.

Empty set: A set which does not contain any element is called as an empty or a null set. It is represented as {} or (phi). 3

Std. IX : Algebra

iii.

iv.

Example: a. A = {a|a is a natural number, 5 < a < 6} A = { } or A =

iii.

p3 = 8

p3 = (2)3

b.

p = 2

C = {2}

It is a singleton set.

iv.

(q 4)2 = 0

q4=0

q=4

D = {4}

It is a singleton set.

v.

1 + 2x = 3x

1 = 3x 2x

1=x

x=1

E = {1}

It is a singleton set.

B = {x|x is a natural number, x < 1} B=

Finite set: If counting of elements in a set terminates at a certain stage, the set is called as finite set. Example: A = {1, 2, 3, 4, 5, 6, 7} B = {x|x is days in a week} The above sets A and B have finite elements. Set A and set B are finite sets. Infinite set: If counting of elements in a set does not terminate at any stage, the set is called as infinite set. Example: P = {1, 2, 3, 4, 5, 6, …} W = {x|x is a whole number} The above sets P and W have elements that cannot be counted. They are sets that do not terminate at any stage. Therefore, P and W are infinite sets.

Note: i. X = {0} is not a null set as ‘0’ is an element of set X. ii. An empty set is a finite set. iii. Sets of Natural numbers, Whole numbers, Integers, Rational numbers and Real numbers are all infinite sets. Exercise 1.2 1.

State which of the following sets are singleton sets:

i.

A= x

x 16

ii. B = {y|y2 = 36} iii. C = {p|p I, p3 = 8} iv. D = {q|(q 4)2 = 0} v. E = {x|1 + 2x = 3x, x W} Solution: x = 16 i. x = 256 A = {256} It is a singleton set. y2 = 36 y = 6 B = {–6, +6} It is not a singleton set.

ii. 4

2.

Which of the following sets are empty? i. A set of all even prime numbers ii. B = {x|x is a capital of India} iii. F = {y|y is a point of intersection of two parallel lines} iv. G = {z|z N, 3 < z < 4} v. H = {t|t is a triangle having four sides} Solution: i. A = {2} It is not an empty set. ii.

B = {Delhi} It is not an empty set.

iii.

Parallel lines do not intersect each other. F={} It is an empty set.

iv.

z is a natural number. There is no natural number between 3 and 4. G={} It is an empty set.

v.

A triangle is a three-sided figure. H={} It is an empty set.

Chapter 01: Sets 3.

Classify the following sets into finite or infinite: i. A = {1, 3, 5, 7, …} ii. B = {101, 102, 103, … , 1000} iii. C = {x|x Q, 3 < x < 5} iv. D = {y|y = 3n, n N} Solution: i. Here, counting of elements do not terminate at any stage. A is an infinite set. ii.

Here, counting of elements terminate at 1000. B is a finite set.

iii.

There is infinite number of rational elements between 3 and 5. C is an infinite set.

iv.

Here, counting of elements do not terminate at any stage. D is an infinite set.

4.

Let G = {x|x is a boy of your class} and H = {y|y is a girl of your class}. What type of sets G and H are? Solution: Set G and set H are finite sets. 1.5

Subset

If every element of set Y is an element of set X, then Y is said to be subset of set X. Symbolically, it is represented as Y X If we have say ‘a’, an element which belongs to set Y, we can say that, it (‘a’) also belongs to set X. But if a Y and a X then it is said that set Y is not a subset of X or Y X. Example: If Y = {b, z} and X = {b, l, z} then we say that Y X. If Y is a subset of X and set X contains atleast one element which is not in set Y, then set Y is the proper subset of set X. It is denoted as Y X. Set X is said to be the ‘super set’ of set Y and is denoted as X Y. If X = {a, b} and Y = {b, a}, then set X is a subset of set Y and Y is also subset of set X. In this case set X is the improper subset of the set Y. It is denoted as X Y and it is read as “X is an improper subset of Y.” Also set Y is the improper subset of the set X.

It is denoted as Y X and it is read as “Y is an improper subset of X.” Note: i. Every set is a subset of itself i.e. Y Y. ii. Empty set is a subset of every set i.e., X. 1.6

Universal Set

A suitably chosen non-empty set of which all the sets under consideration are the subsets of that set is called the Universal set. It is denoted by ‘U’. Example: A = {x|x is Physics laboratory in your school} B = {y|y is Chemistry laboratory in your school} C = {z|z is Biology laboratory in your school} U = {l|l is laboratories in your school} It can be seen that A U, B U, C U. Set U is the universal set of sets A, B and C. Note: Universal set is a set that cannot be changed once fixed for a particular solution. In Venn diagram, generally universal set is represented by a rectangle. Exercise 1.3 1.

Observe the following sets and answer the questions given below: A = The set of all residents in Mumbai B = The set of all residents in Bhopal C = The set of all residents in Maharashtra D = The set of all residents in India E = The set of all residents in Madhya Pradesh i. Write the subset relation between the sets A and C. ii. Write the subset relation between the sets E and D. iii. Which set can be chosen suitably as the universal set? Solution: i. All residents of Mumbai are residents of Maharashtra. AC ii. iii.

All residents of Madhya Pradesh are residents of India. ED Mumbai, Maharashtra, Bhopal, Madhya Pradesh are parts of India. Set D can be chosen as the universal set. 5

Std. IX : Algebra 2.

Let A = {a, b, c}, B = {a}, C = {a, b}, then i. Which sets given above are the proper subsets of the set A? ii. Which set is the super set of set C? Solution: i. Elements of set B and set C are the elements of set A. Also, there exists an element viz. c which is not an element of set B and set C but is in set A. Set B and set C are the proper subsets of set A. ii. Set A is the super set of set C i.e. A C. 3.

Draw a Venn diagram, showing sub set relations of the following sets: A = {2, 4} B = {x|x = 2n, n < 5, n N} C = {x|x is an even natural number 16} Solution: A = {2, 4} B = {2, 4, 8, 16} C = {2, 4, 6, 8, 10, 12, 14, 16} ABC C B A .12 .10 .8 .2 .16

.4 .6

.14

Prove that, if A B and B C, then A C. (Hint: Start with an arbitrary element x A and show that x C) Solution: Let us assume that x A ….(i) But, A B xB BC 4.

xC From (i) and (ii), AC

….(ii)

5. If X = {1, 2, 3}, write all possible subsets of X. Solution: All possible subsets of X are as follows: i. { } or .…[a null set is a subset of every set] ii. {1} iii. {2} iv. {3} 6

v. vi. vii. viii.

{1, 2} {1, 3} {2, 3} {1, 2, 3}

1.7

Operations on Sets

.…[every set is a subset of itself]

a.

Equality: If A is a subset of B and B is a subset of A, then A and B are said to be equal sets and are denoted by A = B. Both the sets A and B contain exactly the same elements. If the elements of A and B are not same, then we write A B. Note: To prove that sets A and B are equal, it is always necessary to prove that A B and B A. i. Let A = {x|x = 2n, n N and x < 10} and B = {2, 4, 6, 8} A = {2, 4, 6, 8} A B and B A A=B ii. Let P = {x|x is an odd natural number, x < 8} and Q = {y|y is an even natural number, y 0} Solution: i. A = {1, 2, 4, 5, 7}, B = {2, 3, 4, 8} A B = {2, 4}

Let A = {a|a is a letter in the word ‘college’} and B = {b|b is a letter in the word ‘luggage’} and U = {a, b, c, d, e, f, g, l, o, u}. Verify: i. (A B) = A B ii. (A B) = A B Proof: i. In roster form, set A and set B can be written as: A = {c, o, l, e, g} B = {l, u, g, a, e} U = {a, b, c, d, e, f, g, l, o, u} A = {a, b, d, f, u} B = {b, c, d, f, o} A B = {a, c, e, g, l, u, o} L.H.S. = (A B) = {b, d, f} .… (i) R.H.S. = A B = {b, d, f} .… (ii) From (i) and (ii), we get L.H.S. = R.H.S. (A B) = A B

ii.

ii.

A B = {l, g, e} L.H.S. = (A B) = {a, b, c, d, f, o, u} .… (iii) R.H.S. = A B = {a, b, c, d, f, o, u} …. (iv) From (iii) and (iv), we get L.H.S. = R.H.S. (A B) = A B

1.8

Number of Elements in the Set

ii.

C = {a, e, i, o, u}, D = {a, b, c, d} C D = {a, b, c, d, e, i, o, u}

iii.

The Roster form of set E and set F is as follows: E = {1, 2, 3, 4, 6, 12} F = {1, 2, 3, 6, 9, 18} E F = {1, 2, 3, 4, 6, 9, 12, 18}

3.

The Roster form of set C and set D is as follows: C = {6, 7, 8, 9, 10} D = {5, 6, 7, 8, 9} C D = {6, 7, 8, 9}

iii.

The Roster form of set E and set F is as follows: E = {… , 4, 3, 2, 1} F = {1, 2, 3, 4, …} E F = { } or

Let U = {x|x = 2n, n W, n < 8} be the universal set. A = {y|y = 4n, n W, n < 4}; B = {z|z = 8n, n W, n 2}. Then find: i. A ii. B iii. (A B) iv. (A B) Solution: The Roster form of set U, set A and set B is as follows: U = {20, 21, 22, 23, 24, 25, 26, 27} = {1, 2, 4, 8, 16, 32, 64, 128}

4.

8

If A is any set then the number of elements in set A is denoted by n (A). Illustrations: i. Let A = {x|x N, 7 < x 12} A = {8, 9, 10, 11, 12} n(A) = 5 ii. For an empty set, n() = 0 iii. n(A B) = n(A) + n(B) n(A B) To verify this identity, let us consider the following example, A = {2, 3, 4} and B = {3, 4, 5, 6} A B = {2, 3, 4, 5, 6} and A B = {3, 4}

Chapter 01: Sets

n(A) = 3, n(B) = 4, n(A B) = 5 and n(A B) = 2 L.H.S. = n(A B) = 5 .... (i) R.H.S. = n(A) + n(B) n(A B) =3+42 =5 .... (ii) n(A B) = n(A) + n(B) n(A B) .... [From (i) and (ii)]

n(T C) = 80 There are 80 students who either drink tea or coffee or both. But there are 100 students in the hostel. Number of students who neither drink tea nor coffee = n(U) n(T C) = 100 80 = 20 Students who do not drink tea or coffee is 20.

4. Exercise 1.5 1.

Let A = {1, 3, 5, 6, 7}, B = {4, 6, 7, 9}, then verify the following: n (A B) = n(A) + n(B) n(A B) Proof: A = {1, 3, 5, 6, 7} and B = {4, 6, 7, 9} A B = {1, 3, 4, 5, 6, 7, 9} and A B = {6, 7} n(A) = 5, n(B) = 4, n(A B) = 7 and n(A B) = 2 L.H.S. = n(A B) = 7 ....(i) R.H.S. = n(A) + n(B) n(A B) =5+42=7 ....(ii) L.H.S. = R.H.S. ....[From (i) and (ii)] n(A B) = n(A) + n(B) n(A B)

2.

Let A and B be two sets such that n(A) = 5, n(A B) = 9, n(A B) = 2. Find n(B). Solution: Given, n(A) = 5, n(A B) = 9, n(A B) = 2 n(B) = ? By using identity, n(A B) = n(A) + n(B) n(A B) 9 = 5 + n(B) 2 9 – 5 + 2 = n(B) n(B) = 6

3.

In a school hostel, there are 100 students, out of which 60 drink tea, 50 drink coffee and 30 drink both tea and coffee. Find the number of students who do not drink tea or coffee. Solution: Let U be the universal set of students in hostel, T be the set of students who drink tea and C be the set of students who drink coffee. n(U) = 100, n(T) = 60, n(C) = 50, n(T C) = 30 By using the identity, n(T C) = n(T) + n(C) n(T C) = 60 + 50 30 = 110 30

110 children choose their favourite colour from blue and pink. Every student has to choose at least one of the colours. 60 children choose blue colour, while 70 children choose pink colour. How many children choose both the colours as their favourite colour? Solution: Let the number of children who choose blue colour be n(B) and number of children who choose pink colour be n(P). n(B) = 60 and n(P) = 70 Number of children who choose their favourite colour from blue or pink. n(B P) = 110 By using the identity, n(B P) = n(B) + n(P) n(B P) 110 = 60 + 70 n(B P) n(B P) = 60 + 70 110 n(B P) = 20 The number of students who choose both the colours as their favourite colours is 20. 5.

Observe the figure and verify the following equation: A

.1 .5

.4

.2 .3 .9

B

.8 .7 .6 C

n(A B C) = n(A) + n(B) + n(C) n(A B) n(B C) n(C A) + n(A B C) Proof: L.H.S. = n(A B C) A B C = {1, 2, 3, 4, 5, 6, 7, 8, 9} n(A B C) = 9 …. (i) Now, A = {1, 2, 3, 4, 5} n(A) = 5 B = {2, 3, 6, 7, 8} n(B) = 5 C = {3, 4, 6, 9} n(C) = 4 A B = {2, 3} n(A B) = 2 9

Std. IX : Algebra

B C = {3, 6} n(B C) = 2 C A = {3, 4} n(C A) = 2 A B C = {3} n(A B C) = 1 R.H.S. = n(A) + n(B) + n(C) n(A B) n(B C) n(C A) + n(A B C) =5+5+4222+1 =9 …. (ii) L.H.S. = R.H.S. ....[From (i) and (ii)] n(A B C) = n(A) + n(B) + n(C) n(A B) n(B C) n(C A) + n(A B C)

Solution: i. A = {… , 3, 2, 1} ii. B = {25, 45, 52, 54, 56, 58, 65, 85} iii. C = {2, 3, 5} iv. D = {3, 2, 1, 0, 1, 2, 3} v. Since, 2 n 4 n = 2, 3, 4 n 2 2 for n = 2, 2 = = 2 n 1 (2) 1 3

for n = 3,

n 3 3 = = 2 n 1 (3) 1 8

for n = 4,

n 4 4 = = 2 n 1 (4) 1 15

2

2

Problem Set - 1 1.

Which of the following collections are sets? i. The collection of rich people in your district. ii. The collection of natural numbers less than 50. iii. The collection of most talented persons of India. iv. The collection of first ten prime integers. v. The collection of all days in a week starting with the letter ‘T’. vi. The collection of some months in a year. vii. The collection of all books in your school library. viii. The collection of smart boys in your class. ix. The collection of multiples of 7. x. The collection of students in your class who got a lot of marks in the first unit test. Solution: From the given collections (ii), (iv), (v), (vii) and (ix) are sets. Remaining collections are not considered as sets as they have relative terms and their meaning may vary from person to person. 2.

10

E= , ,

3.

Write the following sets in the set builder form: i. F = {I, N, D, A} ii. G = {1, 1} iii. H = {3, 9, 27, 81, 243} iv. J = {15, 24, 33, 42, 51, 60}

E = x x

, 2 n 4,n N n 1 n

2

1 2 3 4 5 , , 2 5 10 17 26

v.

K = , ,

Solution: i. F = {x|x is a letter in the word ‘INDIA’} ii. G = {y| square of y is 1} or G = {y|y is a square root of 1} iii. H = {a|a = 3n, n N, n 5} iv. J = {b|b is a two digit number whose sum of digits is 6}

v.

Write the following sets in roster form: i. A = {x|x I, x W} ii. B = {x|x is two digit number such that the product of its digits is a multiple of ten} iii. C = {x|x is a prime divisor of 120} iv. D = {x|x I and x2 < 10} v.

2 3 4 3 8 15

When n = 1, c =

1 (1) 1 2

When n = 2, c =

2 2 2 (2) 1 5 3 (3) 1 10

When n = 4, c =

4 4 2 (4) 1 17

n

n 2 +1

K = c c =

3

When n = 3, c =

When n = 5, c =

1

2

2

5

5 (5) 1 26 2

, n N, n 5

Chapter 01: Sets 4.

Classify the following sets as ‘singleton’ or ‘empty’: i. A = {x|x is a negative natural number} ii. B = {y|y is an odd prime number < 4} iii. C = {z|z is a natural number, 5 < z < 7} iv. D = {d|d N, d2 0}

Solution: i. Each natural number is positive. A={} It is an empty set.

ii.

B = {3} It is a singleton set.

iii.

C = {6} It is a singleton set.

iv.

There is no natural number whose square is less than or equal to zero. D={} It is an empty set.

5.

Classify the following sets as ‘finite’ or ‘infinite’: i. A = {x|x is a multiple of 3} ii. B = {y|y is a factor of 13} iii. C = {…, 3, 2, 1, 0} iv. D = {x|x = 2n, n N} Solution: i. A = {3, 6, 9, 12, …} It is an infinite set.

ii.

B = {1, 13} It is a finite set.

iii.

C is an infinite set.

iv.

D = {20, 21, 22, 23, 24,...} = {2, 4, 8, 16, 32, …} D is an infinite set.

6.

State which of the following sets are equal. i. N = {1, 2, 3, 4, …} ii. W = {0, 1, 2, 3, …} iii. A = {x|x = 2n, n W} iv. B = W {0} Solution: i. N ={1, 2, 3, 4, …} ii. W = {0, 1, 2, 3, …} iii. A = {20, 21, 22, 23, …} = {1, 2, 4, 8, …} iv. B = W {0} = {1, 2, 3, 4, …} Here, set N and set B are subset of each other. In set N and set B, the elements are the same. N=B

7.

Let A = {7, 5, 2} and B =

3

125, 4, 49 .

Are the sets A and B equal? Justify your answer. Solution: A = {7, 5, 2}, B = 3 125, 4, 49

B = {5, 2, 2, 7, 7} Here, A is a subset of B, but B is not a subset of A. Elements of set A and set B are not equal. AB

8.

If A = {1, 2, 3, 4}, B = {2, 4, 6, 8}, C = {3, 4, 5, 6} and U = {x|x N, x < 10}. Verify the following properties: i. A (B C) = (A B) C ii. A (B C) = (A B) (A C) iii. A (C) = (A B) (A C) iv. (A B) = A B v. (A B) = A B vi. (A) = A Solution: Roster form of set U is as follows: U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {1, 2, 3, 4} B = {2, 4, 6, 8} C = {3, 4, 5, 6} i. B C = {2, 3, 4, 5, 6, 8} A B = {1, 2, 3, 4, 6, 8} L.H.S. = A (B C) = {1, 2, 3, 4, 5, 6, 8} …. (i) R.H.S. = (A B) C = {1, 2, 3, 4, 5, 6, 8} …. (ii) From (i) and (ii), we get L.H.S. = R.H.S. A (B C) = (A B) C

ii.

iii.

B C = {4, 6} A B = {1, 2, 3, 4, 6, 8} A C = {1, 2, 3, 4, 5, 6} L.H.S. = A (B C) = {1, 2, 3, 4, 6} …. (i) R.H.S. = (A B) (A C) = {1, 2, 3, 4, 6} …. (ii) From (i) and (ii), we get L.H.S. = R.H.S. A (B C) = (A B) (A C) B C = {2, 3, 4, 5, 6, 8} L.H.S. = A (B C) = {2, 3, 4}

…. (i) 11

Std. IX : Algebra

iv.

v.

vi.

9.

A B = {2, 4} A C = {3, 4} R.H.S. = (A B) (A C) = {2, 3, 4} .… (ii) From (i) and (ii), we get L.H.S. = R.H.S. A (C) = (A B) (A C) A B = {1, 2, 3, 4, 6, 8} (A B) = {5, 7, 9} A = {1, 2, 3, 4} A = {5, 6, 7, 8, 9} B = {2, 4, 6, 8} B = {1, 3, 5, 7, 9} L.H.S. = (A B) = {5, 7, 9} …. (i) R.H.S. = A B = {5, 7, 9} …. (ii) From (i) and (ii), we get L.H.S. = R.H.S. (A B) = A B A B = {2, 4} (A B) = {1, 3, 5, 6, 7, 8, 9} A = {5, 6, 7, 8, 9} B = {1, 3, 5, 7, 9} L.H.S. = (A B) = {1, 3, 5, 6, 7, 8, 9} R.H.S. = A B = {1, 3, 5, 6, 7, 8, 9} From (i) and (ii), we get L.H.S. = R.H.S. (A B) = A B A = {5, 6, 7, 8, 9} L.H.S. = (A) = {1, 2, 3, 4} R.H.S. = A = {1, 2, 3, 4} From (i) and (ii), we get L.H.S. = R.H.S. (A) = A

A = {0} Set A is not a null set.

ii.

Here, 2b is an even number and 1 is an odd number. Since, addition of even and odd number always give odd number, value of 2b + 1, for any value of b N is an odd number. Set B is a null set.

iii.

Square of an odd number is always an odd number. c2 cannot be even. Set C is a null set.

10.

Give an example of the set which can be written in set builder form but cannot be written in roster form. Solution: Consider the set of rational numbers ‘Q’. In set builder form, it is a b

Q = a I, b I and b 0 But same set Q cannot be written in roster form. …. (i) …. (ii)

…. (i)

11.

Write down all possible subsets of each of the following sets: i. ii. A = {1} iii. B = {1, 2} iv. C = {a, b, c, d} Solution: i. Subset of a null set is only one i.e.

ii.

Subsets of set A are empty set { } and set A itself. i.e. and {1}

iii.

All possible subsets of set B are , {1}, {2}, {1, 2}.

iv.

All possible subsets of set C are {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c, d}

…. (ii)

For each of the following sets, state with reasons, whether it is a null set or not: i. A = {x|x I, x2 is not positive} ii. B = {b|b N, 2b + 1 is even} iii. C = {c|c N, c is odd and c2 is even} Solution: i. Whether a number is positive or negative its square is always a positive. Square of an integer cannot be negative, except zero, whose square is neither positive nor negative. 12

12.

Write proper subsets of the following sets: i. A = {a, b} ii. B = {a, b, c} Solution: i. Proper subsets of A are {a} and {b}. ii. Proper subsets of B are {a}, {b}, {c}, {a, b}, {b, c}, {c, a}.

Chapter 01: Sets 13.

Write the sets A and B such that A is finite, B is finite, A and B are disjoint sets. Solution: Let B = {1, 3, 5, 7}, A = {2, 4, 6, 8} be the finite sets. But their intersection is a null set, i.e. A B = A and B are disjoint sets.

14.

Let A = {a, b, c, d}, B = {a, b, c}, C = {b, d, e}, then find the sets D and E satisfying the following conditions: i. D A, D B ii. C E, B E = Solution: i. Since, D A and D B D = {a, b}/{a, c}/{a, d}/{a, b, d}/{b, c, d} /{a, c, d} ii. Since, C E and B E = E must not contain any element of set B. C is superset of set E. E = {d}/{e}/{d, e} Let U = {x|x N, x < 10}, A = {a|a is even, a U}, B = {b|b is a factor of 6, b U}. Verify that: n(A) + n(B) = n(A B) + n(A B). Solution: Roster form of set U and set A is as follows: U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {2, 4, 6, 8} ….(Even numbers within the universal set) n(A) = 4 Roster form of set B is as follows: B = {1, 2, 3, 6} ….( b U) 15.

n(B) = 4 A B = {1, 2, 3, 4, 6, 8} n(A B) = 6 A B = {2, 6} n(A B) = 2 L.H.S. = n(A) + n(B) =4+4 =8 ....(i) R.H.S. = n(A B) + n(A B) =6+2 =8 ....(ii) From (i) and (ii), we get L.H.S. = R.H.S. n(A) + n(B) = n(A B) + n(A B)

16.

In a group of students, 50 students passed in English, 60 students passed in Mathematics and 40 students passed in both. Find the number of students who passed either in English or in Mathematics. Solution: Let the number of students who passed in English be denoted by set E. n(E) = 50 Let the number of students who passed in Mathematics be denoted by set M. n(M) = 60 The number of students who passed in English and Mathematics = n(M E) = 40 By using identity, n(E M) = n(E) + n(M) n(E M) = 50 + 60 – 40 = 110 40 = 70 The number of students who passed either in English or in Mathematics is 70. 17.

A T.V. survey says 136 students watch only programme P1, 107 watch only programme P2, 27 watch only programme P3. 25 students watch P1 and P2 but not P3. 37 watch P2 and P3 but not P1. 53 students watch P1 and P3 but not P2. 40 students watch all three programmes and 80 students do not watch any programme. Find, with the help of Venn diagram. i. Number of P1 viewers. ii. Number of P2 or P3 viewers. iii. Total number of viewers surveyed. Solution: Given data represented by Venn diagram is as follows: U P2 P1 136

107

25

53

40

37

27 80

P3 i.

From the Venn diagram, Number of P1 viewers = n(P1) = 136 + 25 + 53 + 40 = 254 Number of P1 viewers is 254. 13

Std. IX : Algebra

ii. iii.

Number of P2 or P3 viewers = n(P2 P3) = 107 + 25 + 40 + 37 + 53 + 27 = 289 Number of P2 or P3 viewers is 289. Total number of viewers surveyed = Number of only P1 viewers + Number of P2 or P3 viewers + 80 = 136 + 289 + 80 = 505 Total number of viewers surveyed is 505.

18.

Show that, it is impossible to have sets A and B such that set A has 32 elements, set B has 42 elements, A B has 12 elements and A B has 64 elements. Solution: Set A has 32 elements. n(A) = 32 Set B has 42 elements. n(B) = 42 Given that n(A B) = 12 and n(A B) = 64 By using the identity, n(A B) = n(A) + n(B) n(A B), L.H.S. = n(A B) = 64 ....(i) R.H.S. = n(A) + n(B) n(A B) = 32 + 42 12 = 62 ....(ii) From (i) and (ii), we get L.H.S. R.H.S. Sets A and B are impossible. 19.

Let the universal set U be a set of all students of your school. A is the set of boys, B is the set of girls and C is the set of students participating in sports. Describe the following sets in words and represent them by a Venn diagram: i. BC ii. A (B C) Solution: i. B C represents the set of girls participating in sports. A B C

Represent sets A, B, C such that A B, A C = and B C by Venn diagram and shade the portion representing A (B C). Solution: i. AB set A is proper subset of set B. i.e. set A is inside set B. ii. A C = set A and set C do not intersect. iii. B C set B and set C intersect. B C B C A 20.

A (B C) Let A, B, C be sets such that A B , B C and A C . Do you claim that A B C ? Justify your answer. Solution: There are two possibilities: i. If A = {a, b} B = {b, c} C = {c, a} then, it is observed that A B , B C , C A , but, A B C = ii. If A = {a, b} B = {a, c} C = {a, b, c} then, it is observed that A B , B C , C A , but, A B C = {a} We cannot claim that A B C . 21.

22.

ii.

BC A (B C) represents the set of all boys or set of girls that participate in sports. BC A C B

A (B C) 14

With the help of suitable example, verify the following statements: If A B, B C, then A C. Solution: Let A = {x, y, z}, B = {a, x, y}, C = {y, w} Since, each element of set A does not exist in set B. A B Each element of set B does not exist in set C. B C Each element of set A does not exist in set C. A C

Chapter 01: Sets 23.

If A and B are any two sets, then prove that i. (A B) = A B ii. (A B) = A B [Hint: Show (A B) A B and vice versa] Solution: Let U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 3, 4, 5} B = {3, 4, 5, 6, 7} i. A B = {1, 2, 3, 4, 5, 6, 7} L.H.S. = (A B) = {8} …. (i) A = {6, 7, 8} B = {1, 2, 8} R.H.S. = A B = {8} …. (ii) From (i) and (ii), we get L.H.S. = R.H.S. (A B) = A B

ii.

Now, A B = {3, 4, 5} L.H.S. = (A B) = {1, 2, 6, 7, 8} R.H.S. = A B = {1, 2, 6, 7, 8} From (i) and (ii), we get L.H.S. = R.H.S. (A B) = A B

…. (i) …. (ii)

One-Mark Questions 1.

Write the following set in set builder form. A = {2, 3, 5, 7, 11, 13, 17} Solution: The set builder r form of set B is: A = {x|x is a prime number, x < 18} 2.

Write the following set in roster form. B = {x|x is a natural number and 4 x < 10 Solution: The Roster form of set B is: B = {4, 5, 6, 7, 8, 9} 3.

If A = {1, 2, 3, 4, 5, 6} and B = {1, 3, 5, 7} then draw Venn diagram for A B. Solution: U B A 2 4 6

1 3 5 AB

7

4.

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} is the universal set and C = {5, 6, 7, 8} then find C. Solution: Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and C = {5, 6, 7, 8} C = {1, 2, 3, 4, 9} 5.

If A = {9, 11, 13, 15} and B = {1, 3, 5, 7} then find A B. Solution: Given, A = {9, 11, 13, 15} and B = {1, 3, 5, 7} A B = { } or 6.

If A and B are two sets such that n(B) = 8, n(A B) = 11, n(A B) = 6, find n(A). Solution: By using identify, n(A B) = n(A) + n(B) n(A B) 11 = n(A) + 8 6 n(A) = 11 2 n(A) = 9

7.

State which of the following sets are equal: A = {x|x W, x < 6} B = {1, 2, 3, 4, 5, 6} C = {0, 1, 2, 3, 4, 5} Solution: Here, A = {0, 1, 2, 3, 4, 5} and C = {0, 1, 2, 3, 4, 5} A=C 8.

Classify the following sets as singleton or empty. i. A = {x|x is a natural number, x 5 and x 7} ii. B = {x|x is an even prime number} Solution: i. There is no common number for x < 5 and x > 7 A = {} Set A is empty set. ii B = {2} Set B is singleton set. 9.

If A = {2, 3, 4, 5} and B = {1, 2, 5, 6}, then find A B. Solution: Given, A = {2, 3, 4, 5} and B = {1, 2, 5, 6} A B = {1, 2, 3, 4, 5, 6} 15

Std. IX : Algebra 10. If A = {3}, write all possible subsets of set A. Solution: and {3} 11. If U = {1, 2, 3, 4} and X = {2, 4}, then find X. Solution: Given, U = {1, 2, 3, 4} and X = {2, 4} X = {1, 3}

2.

Draw a Venn diagram showing subset relations of the following sets: A = {2 , 8} B = {x|x = 2n, n 4 and n N} C = {x|x is an even natural number 20}

3.

If A = {x, y}, write all possible subsets of A.

4.

State true or false: i. is a subset of itself. ii. If A B and B A, then A = B. iii. The empty set is a subset of all sets.

Additional Problems for Practice

Based on Exercise 1.1 1.

2.

Write the following sets in the roster form: i. A = {x|x is a prime number which is a divisor of 30} ii. B = {x|x is an even natural number} iii. C = {x|x is an integer and x2 < 5} iv. F = {x|x is a letter in the word ‘LITTLE’} v. E = {x|x W, x N} vi. D = {x|x is a square root of 81}

Based on Exercise 1.4 1.

Find the union of each of the following pairs of sets: i. A = {5, 15, 25}, B = {10, 20, 30} ii. H = {3, 6, 9, 12, 15} , F = {3, 4, 5, 6} iii. M = {x|x N and x is a divisor of 12} N = {x|x N and x is a prime divisor of 12}

Write the following sets in the set builder form: i. A = {2, 4, 6, 8, 10, 12, 14} ii. B = {5, 10, 15, 20, ….} iii. C = {7, 72, 73, 74} iv. D = {51, 53, 55, 57, 59} v. E = {2, 3, 5, 7, 11, 13, 17, 19}

2.

Find the Intersection of the following pairs of sets: i. A = {5, 6, 7}, B = {8, 9, 10} ii. M = {10, 20, 30, 40, 50}, N = {20, 40, 60} iii. N is a set of natural numbers and W is a set of whole numbers. iv. P = {a, b, p, d, q}, R = {q, r, s, p}

3.

If U = {x|x is a natural number less than 15} is a universal set A = {1, 3, 4, 5, 9}, B = {3, 5, 7, 9, 12} Verify that (A B) = A B

4.

U = {x|x I and 3 x 3}, A = {2, 0, 2}, B = {0, 1, 2, 3} Find i. A ii. B iii. (A B) iv. A B

Based on Exercise 1.2 1.

2.

State which of the following sets are singleton or empty sets: i. A = {x|x 5 = 0} ii. B = {y|y is an even prime number greater than 2} iii. D = {x|x N and 3x 1 = 0} iv. E = {x|x I, x is neither a positive nor a negative number} v. C = {x|x N and x < 7 and x > 11} Classify the following sets into finite or infinite: i. A = {x|x is a multiple of 1} ii. C = {x|x is a point on a line} iii. D = {1, 2, 3, 4, …., 100} iv. E = {x|x N and x is an odd number}

Based on Exercise 1.3 1.

16

Write the subset relations among the following sets: P = set of all residents in Nagpur X = set of all residents in Vadodara Y = set of all residents in Maharashtra T = set of all residents in Gujarat

Based on Exercise 1.5 1.

With the help of following figure, write the following sets: U B A 1 4

3

i. iv.

9

5 10 11

A AB

7 2 8

12 6

ii. v.

B AB

iii.

U

Chapter 01: Sets

2.

Let A and B be two sets such that n(A) = 17, n(B) = 23, n(A B) = 38. Find n(A B)

7.

If A P, B P, then (A B) ________. (A) A (B) B (C) P (D)

3.

240 students in a school were interviewed and their hobbies were noted. 150 students were interested in stamp collection. 80 took delight in reading books, 40 of them do not like either. What is the number of students who liked both stamp collection and reading books?

8.

If A = {3, 4, 7, 8, 9} and B = {7, 8, 10, 11} then A B = ? (A) {3, 4} (B) {7, 8} (C) {10, 11} (D) {10}

9.

For any two sets A and B, A B = ? (A) {x|x B or x A} (B) {x|x A or x B} (C) {x|x A and x B} (D) {x|x A and x B}

10.

If U = {1, 2, 3, 4, ….} and A = {2, 4, 6, 8, ….} then A = ? (A) {2, 4, 6, ….} (B) {1, 3, 5, 7, ….} (C) {0, 1, 3, 5, ….} (D) {0, 2, 4, 6, 8, ….}

11.

If U = {4, 5, 6, 7, 8, 9}, P = {5, 6, 7, 8}, Q = {4, 6, 8, 9} then, P Q = ? (A) {4, 5, 7, 8, 9} (B) {4, 5, 7, 9} (C) {6, 7, 8} (D) {4, 6, 7, 8, 9}

12.

In the following Venn n(P Q) = 70, then x = ?

4.

5.

In a class of 50 students, 35 like Physics, 30 like Mathematics and 3 like neither. How many like both the subjects and how many like Physics only? U

A 3 6 15 12 9

B 18 24

From the above diagram find: i. AB ii. n (A B) iii. (A B) iv. n (A B) v. A B Multiple Choice Questions

1.

If B = {x|x is a vowel in English alphabet}, then Roster form of B is (A) {a, e, i, u} (B) {a, e, p, o} (C) {a, e, c, d} (D) {a, e, i, o, u}

2.

Which of the following is not an Infinite set (A) N (B) W (C) I (D) None of these

3.

A = {z|z + 6 = 0} is a _______. (A) empty set (B) singleton set (C) infinite set (D) finite set

4.

Empty set is a _______ of every set. (A) subset (B) proper subset (C) super set (D) universal set

5.

If A = {x|x is worker in department I of your company} B = {y|y is worker in department II of your company} and C = {z|z is worker of your company) then (A) C A (B) A B (C) A C (D) C B

6.

If A B and B A, then set A and B are _______ sets. (A) equal (B) disjoint (C) super (D) universal

P

13.

5

If

Q 40 x

(A)

diagram.

(B)

3

x

35 x

(C)

6

(D)

8

If n(A) = 10, n(B) = 25 and n(A B) = 15, then n(A B) = ? (A) 20 (B) 0 (C) 10 (D) 5

Answers to additional problems for practice

Based on Exercise 1.1 1.

i. ii. iii. iv. v. vi.

A = {2, 3, 5} B = {2, 4, 6, 8, ….} C = {2, 1, 0, 1, 2} F = {L, I, T, E} E = {0} D = {9, 9}

2.

i. ii. iii. iv.

A = {x|x = 2n, n N and n < 8} B = {x|x = 5n and n N} C = {x|x = 7n, 1 n 4} D = {x|x N, x is an odd integer and 50 < x < 60} E = {x|x is a prime number and 1 < x < 20}

v.

17

Std. IX : Algebra

5.

Based on Exercise 1.2 1.

Singleton sets are A, E Empty sets are B, D, C

2.

Finite set is D. Infinite sets are A, C, E.

1.

P Y, X T

2.

A = {2, 8}, B = {2, 4, 8, 16} C = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} C B 6

A

12 4

2 8

16 18

20

14

3.

, {x}, {y}, {x, y}

4.

i.

True

ii.

True

iii.

True

Based on Exercise 1.4 1.

i. ii. iii.

A B = {5, 10, 15, 20, 25, 30} H F = {3, 4, 5, 6, 9, 12, 15} M N = {1, 2, 3, 4, 6, 12}

2.

i. ii. iii. iv.

AB= M N = {20, 40} N W = {1, 2, 3, ….} P R = {p, q}

4.

i. ii. iii. iv.

A = {3, 1, 1, 3} B = {3, 2, 1} (A B) = {3, 1} A B = {3, 1}

Based on Exercise 1.5 1.

i. ii. iii. iv. v.

2.

2

3.

30

4.

18 students like both subjects and 17 students like only Physics.

18

A B = {6, 12} n(A B) = 7 (A B) = {3, 9, 15, 18, 24} n(A B) = 2 A B = {3, 9, 15, 18, 24}

Answers to Multiple Choice Questions

Based on Exercise 1.3

10

i. ii. iii. iv. v.

A = {2, 3, 6, 7, 8, 11, 12} B = {1, 3, 4, 6, 9, 11, 12} U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} A B = {1, 2, 4, 5, 7, 8, 9, 10} A B = {5, 10}

1. 5. 9. 13.

(D) (C) (B) (A)

2. (D) 6. (A) 10. (B)

3. (B) 7. (C) 11. (B)

4. (A) 8. (B) 12. (A)