Kc Sinha Class 12 PDF Free Download | Kc Sinha 12th Math Solution PDF Download Part 1 Chapter

Let us begin by learning the unit and chapter names for Kc Sinha Mathematics Class 12 Part 1. The following are the unit and chapter names for Kc Sinha Mathematics Class 12 Part 1.

Aug 6, 2023 - 19:16 Updated: Aug 6, 2023 - 14:48

Kc Sinha 12th Math Solution PDF Download

KC Sinha Class 12 PDF Free Download | KC Sinha 12th Math Solution PDF Download Part 1 Chapter. Hello everyone. It is now time to discover Kc Sinha Mathematics Class 12 Solutions Pdf Download. In general, mathematics is a difficult topic for many students. Dr. Kc Sinha creates the Kc Sinha Mathematics Book for students in grades 10th, 11th, and 12. Kc Sinha's Class 12th Mathematics book is the most popular among pupils. The finest books for Class 12th students are two books (NCERT Book and Kc Sinha Book).

Dr. Kc Sinha's Class 12 Mathematics Book is divided into two volumes or portions. You can find the Kc Sinha Mathematics Class 12 Solutions Pdf Download Part 1 of Book Volume 1 on this page. According to the most recent edition of Kc Sinha Mathematics Class 12, the first volume of the Kc Sinha Book contains 18 chapters. Because all of the Chapters are under six units, it contains six units.

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KC Sinha 12th Math Solution PDF Download Part 1

Let us begin by learning the unit and chapter names for Kc Sinha Mathematics Class 12 Part 1. The following are the unit and chapter names for Kc Sinha Mathematics Class 12 Part 1.

Relations and Functions Unit

Relations and Function is the first unit of kc Sinha mathematics class 12 pdf. This unit of 12th class math topic contains four chapters, some of which are significant for your board test. Our master teacher briefly describes each chapter.

Algebra Unit

The second unit of kc Sinha mathematics class 12 pdf is algebra. There are four chapters in this unit of class 12th math topic, and some of them are significant for your board test. Our master teacher provides a brief description of each chapter.

Calculus Unit

The third unit of kc Sinha maths class 12 pdf is calculus. There are 10 chapters in this unit of class 12th math topic, and some of them are significant for your board test. In brief, the master teacher prepares and designs all of the chapters.

Let us begin by asking, "What will you learn in the chapters of Kc Sinha Class 12 Mathematics Book in brief?" All Class 12 Chapter subjects and subtopics are listed here.

Kc Sinha Class 12 PDF Free Download | Kc Sinha 12th Math Solution PDF Download Part 1 Chapter

Index of Kc Sinha Mathematics Class 12 Solutions Pdf

Kc Sinha Mathematics Class 11 Book Pdf is now available on our website in a nutshell.

Chapter Names of 12th class math	Unit Names
Relation	Relation and Function
Function	Relation and Function
Binary Operation	Relation and Function
Inverse Trigonometric Functions	Relation and Function
Matrices	Algebra
Determinant	Algebra
Adjoint and Inverse of a Square Matrix	Algebra
Solution of System of Linear Equations	Algebra
Continuity	Calculus
Differentiability	Calculus
Differentiation	“
Second-Order Derivative	“
Rolle’s Theorem & Lagrange’s Mean Value Theorem	“
Application of Derivatives	“
Increasing and Decreasing Functions	“
Tangents and Normals	“
Approximation	“
Maxima and Minima	“

Chapter 1: Relations

Kc Sinha 12th Math Solution PDF Download Part 1 Chapter

KC Sinha Math Chapter 1 Ex 1.1

We will learn Types of relations: reflexive, symmetric, transitive, and equivalence relations in brief in the Relation and Function Chapter or Binary Operation Chapter meaning 1st and 2nd or 3rd chapter of Kc Sinha Mathematics Class 12 Solutions Pdf Download. Binary operations include one-to-one and onto functions, composite functions, and the inverse of a function. In the fourth chapter of Kc Sinha maths, we will learn about Definition, range, domain, and principal value branches in brief. In summary, graphs of inverse trigonometric functions. In summary, the basic features of inverse trigonometric functions.

Matrices and Determinants Chapter

We will learn about matrices in the Matrices Chapter, which is the fifth chapter of Kc Sinha Mathematics Class 12 Solutions Pdf Download. In a nutshell, concepts, notation, order, equality, types of matrices, zero matrices, matrix transpose, symmetric and skew-symmetric matrices. Matrix addition, multiplication, and scalar multiplication, in brief addition, multiplication, and scalar multiplication. In brief, non-commutativity of matrix multiplication and the existence of non-zero matrices whose product is the zero matrices (restrict to square matrices of order 2). The concept of basic row and column operations. Invertible matrices and proof of the uniqueness of the inverse, if it exists, in brief; (In this case, all matrices will have real entries) in brief.

We will learn the Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, cofactors, and applications of determinants in computing the area of a triangle in brief in the Determinants Chapter, which is the sixth chapter of kc Sinha maths class 12 pdf. In a nutshell, a square matrix's adjoint and inverse. In brief, consistency, inconsistency, and a number of solutions of a system of linear equations are demonstrated, as is solving a system of linear equations in two or three variables (with a unique solution) using the inverse of a matrix.

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Calculus Unit’s Chapter

Continuity, Differentiability, Differentiation, Second-order Derivative, Rolle's Theorem, and Lagrange's Mean Value Theorem are all examples of theorems. In brief, we will learn continuity and differentiability, a derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, and the derivative of an implicit function in the 9th, 10th, 11th, 12th, 13th, and 14th chapters of kc Sinha maths class 12 pdf. In a nutshell, exponential and logarithmic functions, as well as their derivative logarithmic differentiation. In a nutshell, derivatives of functions are expressed in parametric forms. In a nutshell, second-order derivatives. In summary, the Mean Value Theorems of Rolle and Lagrange (without proof) and their geometric interpretations.

Kc Sinha Mathematics Class 12 Pdf Chapter

Derivatives, increasing/decreasing functions, tangents and normals, approximation, maxima and minima In brief, we will learn Applications of derivatives: rate of change, increasing/decreasing functions, tangents & normals, approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool) in the 15th, 16th, 17th, and 18th chapters of Kc Sinha maths. Simple problems (that demonstrate fundamental ideas and grasp of the subject as well as real-life scenarios).

Kc Sinha Class 12 PDF Free Download | Kc Sinha 12th Math Solution PDF Download Part 1 Chapter

KC Sinha Mathematics Solution Class 12 Chapter 1 संबंध (Relation) Exercise 1.1

Question 1

(i) माना कि A={1,9} , B={5,13} तथा R={(a,b):a∈A,b∈B तथा a-b}से विभाज्य है । दिखाएँ कि R,A से B मे सार्वत्रिक सम्बन्ध है ।
Sol :
1-5=-4, 4 से विभाजय है ।
1-13=-12, 4 से विभाजय है ।
9-5=4 , 4 से विभाजय है ।
9-13=-4 , 4 से विभाजय है ।

R={(1,5),(1,13),(9,5),(9,13)}

R=A×B

∴ R, A से B मे universal relation हैं ।

(ii) माना कि A={1,5}, B={3,7} तथा R={(a,b):a∈A,b∈B तथा a-b,4 का अपवर्त्य है ।} दिखाएँ कि ,R,A से B मे रिक्त सम्बन्ध है ।
Sol :
1-3=-2, 4 का अपवर्त्य नहीं है।

1-7=-6, 4 का अपवर्त्य नहीं है।

5-3=2, 4 का अपवर्त्य नहीं है।
5-7=-2, 4 का अपवर्त्य नहीं है।

R=ϕ

∴ R, A से B मे empty relation हैं ।

Question 2

माना कि लड़को के किसी विद्यालय के सभी लड़को का समुच्चय A है । दिखाएँ कि समुच्चय A पर निम्न प्रकार परिभाषित सम्बन्ध R
(i) R={(a,b):a,b की बहन है } A मे एक रिक्त सम्बन्ध है ।
(ii) R'={(a,b):a औऱ b के ऊँचाइयो का अन्तर 3 मीटर से कम है }A पर एक सार्वत्रिक सम्बन्ध है ।
Sol :
(i) A=सभी लड़को का समुच्चय

माना a,b की बहन है ।

a∉A

R=ϕ

∴ R is a empty relation

(ii)
∴R'=A×A

∴R' is a universal relation

Question 3

Let A={1,2,3} and R be a relation on A defined by aRb⇔a=b.Show that R is an indentity relation on A
Sol :
R={(a,b): a,b∈A and a=b}

R={(1,1),(2,2),(3,3)}

∴ एक तत्समक सम्बन्ध है ।

Question 4

Let R be a relation from Q into Q defined by
R={(a,b):a,b∈Q and a-b∈Z}. Show that
(i) (a,a)∈R for all a∈Q
Sol :
(a,a)∈R⇒a-a=0∈R,∀a∈Q

(ii) (a,b)∈R⇒(b,a)∈R
Sol :
(a,b)∈R⇒
a-b∈Z
Then b-a∈Z
(b,a)∈R

i.e.
(43,13)∈R⇒43−13=1∈Z

then 13−43=−1∈R

(11,43)∈R

(iii) (a,b)∈R,(b,c)∈R⇒(a,c)∈R
Sol :
(a,b)∈R⇒a-b∈Z

and (b,c)∈R⇒b-c∈Z

Then (a,c)⇒a-c∈Z

i.e.
(73,43)∈R⇒73−43=1∈Z

and (43,13)∈R⇒43−13=1∈Z

(73,13)∈R⇒73−13=2∈Z

Question 5

Let A={1,2,3} and R1={(1,2),(2,2),(1,3),(3,2)}

R2={(3,3),(2,1),(1,2)}

R3={(1,2)},R4={(1,1)}

Which of these relations are reflexive , symmetric and transitive ?
Sol :
(i) R1={(1,2),(2,2),(1,3),(3,2)}

(a) For reflexive
(1,1)∉R1

∴R1 reflexive नहीं है ।

(b) For symmetric
(1,2)∈R1⇒(2,1)∉R1

∴R1 symmetric नहीं है ।

(c) For transitive

R1 is transitive.

Kc Sinha Class 12 PDF Free Download | Kc Sinha 12th Math Solution PDF Download Part 1 Chapter

Question 7

Determine whether each of the following relations are reflexive, symmetric and transitive.

(i)
Sol :
(a) For reflexive
x-x=0∈Z
(x,x)∈R ,∀x∈Z

∴ R reflexive है ।

(b) For symmetric
(x,y)∈R⇒x-y∈Z,∀x,y∈Z

then (y,x)∈R⇒y-x∈Z,∀x,y∈Z

∴ R symmetric है ।

i.e.
(2,1)∈R
2-1=1∈Z

(1,2)∈R
1-2=-1∈Z

(c) For transitive
(x,y)∈R and (y,z)∈R,∀,x,y,z∈Z

x-y∈Z and y-z∈Z⇒x-z∈Z ,(x,z∈R)

∴ R transitive है ।

i.e.
2-3=-1∈Z
3-4=-1∈Z
2-4=-2∈Z

(ii)
Sol :
R={(a,b) b=a+1 ; a,b∈A}

b=a+1; a=1⇒b=1+1=2
a=2⇒b=2+1=3
a=3⇒b=3+1=4
a=4⇒b=4+1=5
a=5⇒b=5+1=6

R={(1,2),(2,3),(3,4),(4,5),(5,6)}
(i)(1,1)∉R
(ii)(1,2)∈R⇒(2,1)∉R
(iii)(1,2)∈R and (2,3)∈R⇒(1,3)∉R

(iii)
Sol :
(i) For reflexive
(x,x)∈R,x÷x=1
i.e.
x,x से भाजय है ।

∴ R reflexive है ।

(ii) For symmetric
(x,y)∈R⇒y,x से भाजय है ,
(y,x)∉R⇒x,y से भाजय नहीं है

i.e.
4,2 से भाजय है
2,4 से भाजय नहीं है

(iii) For transitive
(x,y)∈R and (y,z)∈R ,∀x,y,z∈A

∴z,x से भाजय होगा ।
(x,z)∈R

∴R is transitive

(iv)
Sol :
x,y∈A :
2x-y=10
2x-10=y

x=6⇒y=2(6)-10=2
x=7⇒y=2(7)-10=4
x=8⇒y=2(8)-10=6
x=9⇒y=2(9)-10=8
x=10⇒y=2(10)-10=10

R={(6,2),(7,4),(8,6),(9,8),(10,10)}

(i) For reflexive
(1,1)∉R
∴R is not reflexive

(ii) For symmetric
(6,2)∈R⇒(2,1)∉R
∴R is not symmetric

(iii) For transitive
(9,8)∈R and (8,1)∈R
(9,6)∉R
∴R is not transitive

(vi)
Sol :
x,y∈N and y=x+5, x<4
x=1,2,3

x=1⇒y=1+5=6
x=2⇒y=2+5=7
x=3⇒y=3+5=8

R={(1,6),(2,7),(3,8)}

Question 8

Determine whether each of the following relations on the set A of all human beings in a town at a particular time are reflexive, symmetric and transitive :

(i) R1={(x,y):x is wife of y}
Sol :
(i) For reflexive
(x,x)∉R1⇒x,x की पत्नी है ।
∴R1 is not reflexive

(ii) For symmetric
(x,y)∈R1⇒x,y की पत्नी है ।
⇒y,x की पत्नी है ।
⇒(y,x)≠R1
∴R1 is not symmetric

(iii) For transitive :
(x,y)∈R1 and (y,z)∈R1
⇒x,y की पत्नी है और y,z की पत्नी है
⇒x,z की पत्नी है
⇒(x,z)∈R1

∴R1 is transitive

(ii) R2={(x,y):x is father of y}
Sol :
माना x,y,z तीन ही नगर के वासी हो ।
x,y,z∈A

(i) For reflexive
(x,x)∉R2⇒x,x का पिता नहीं हो सकता है ।

∴R2 is not reflexive

(ii) For symmetric
(x,y)∈R2⇒x,y के पिता है ।
⇒y,x के पिता है
⇒(y,x)∉$R_{2}$

∴R2 is not symmetric

(iii) For transitive
(x,y)∈R2 and (y,z)∈R2
⇒x,z के पिता है और y,z के पिता है ।
⇒x,z के दादा है ।
⇒(x,z)∉R2

∴R2 is not transitive

(iii) R3={(x,y):x and y live in the same locality}
Sol :
माना x,y,z तीन एक ही नगर के है ।
x,y,z∈A

(i) For reflexive
(x,x)∈R3⇒x तथा x एक ही मुहल्ले में रहते है ।
∴ R3 is reflexive

(ii) For symmetric
(x,y)∈R3⇒x तथा y एक ही मुहल्ले में रहते है ।
⇒(y,x)∈R3
∴ R3 is symmetric

(iii) For transitive
(x,y)∈R3 and (y,z)∈R3
⇒x तथा y एक ही मुहल्ले मे रहते है और y तथा z एक ही मुहल्ले मे रहते है
⇒x तथा z एक भी ही मुहल्ले मे रहते है
⇒(x,z)∈R3
R3 is transitive

(v) R5={(x,y):x is 7cm taller than y}
Sol :
(i) For reflexive
(x,x)∉R5⇒x,x से 7cm लंबा नहीं हो सकता है ।

∴R5 is not reflexive

(ii) For symmetric:
(x,y)∈Rs⇒x,y से 7cm लंबा है ।
⇒y,x से 7cm लंबा नहीं हो सकता है ।
⇒(y,x)∉R5
∴R5 is not symmetric

(iii) For transitive
(x,y)∈R5 and (y,z)∈R5
⇒x,y से 7cm लंबा है तथा y,z से 7cm लंबा है ।
⇒x,y से 14cm लंबा है ।
⇒(x,z)∉R5
∴R5 is not transitive

Question 9

(i)
Sol :
(i) For reflexive
(a,a)∈R1⇒a≤a is always true
∴R1 is reflexive

(ii) For symmetric
(a,b)∈R1⇒a≤b
⇒b≰a
⇒(b,a)∉R
∴R1 is not symmetric

(iii) For transitive:
(a,b)∈R₁ and (b,c)∈R₁⇒a≤b and b≤c
⇒a≤c
⇒(a,c)∈R₁
∴R₁ is transitive
(ii) Show that the relation R₂ in the set of all real numbers R defined as
R={(a,b):a≤b²} is neither reflexive nor symmetric nor transitive
Sol :
माना a,b,c तीन वास्तविक संख्याएँ है।
a,b,c∈R
(i) For not reflexive
⇒(a,a)∉R₂⟺a≰a²
∴R₂ is not symmetric

i.e.
12≤(12)2
12≤14

(ii) For not symmetric
(a, b)∉R₂⇒a≤b²
⇒b²≰a
⇒(b,a)∉R₂
∴R₂ is not symmetric

(iii) For not transitive

(a,b)∈R₂ and (b,c)∈R₂

⇒a≤b² and b≤c²

⇒a≰c²

⇒(a,c)∉R₂

∴R₂ is not transitive

Kc Sinha Class 12 PDF Free Download | Kc Sinha 12th Math Solution PDF Download Part 1 Chapter

Question 10

Show that the relation R in the set {1,2,3} given by R={(1,2),(2,1)}
is symmetric but neither reflexive nor transitive.
Sol :
R={(1,2),(2,1)}

(i) For symmetric
(1,2)∈R⇒(2,1)∈R
∴ R is symmetric

(ii) For not reflexive
(1,1)∉R,(2,2)∉R
∴ R is not reflexive

(iii) For not transitive
(1,2)∈R and (2,1)∈R
(1,1)∉R
∴R is not transitive

(ii) Show that the relation R in the set {1,2,3} given by
R={(1,1),(2,2),(3,3),(1,2),(2,3)}
is reflexive but neither symmetric nor transitive
Sol :
R={(1,1),(2,2),(3,3),(1,2),(2,3)}

(i) For reflexive
(1,1)∈R,(2,2)∈R,(3,3)∈R
∴R is reflexive

(ii) For not symmetric
(1,1)∈R⇒(2,1)∉R
∴R is not symmetric

(iii) For not transitive
(1,2)∈R,(2,3)∈R
⇒(1,3)∉R
∴R is not transitive

Question 11

Let S be the set of all points in a plane and R be a relation in S defined as
R={(a, b): distance between points a and b is less than 2units. Show that R is reflexive and symmetric but not transitive.
Sol :
माना a,b तथा c तीन बिंदु एक एक तल मे स्थित है ।
a,b,c∈S
(i) For reflexive

(a,a)∈R⇒a तथा a के तीन की दूरी 2 unit से कम है ।

∴R is reflexive

(ii) For symmetric
(a,b)∈R⇒a तथा b के बीच की दूरी 2 unit से कम है ।
⇒b तथा a के बीच की दूरी 2 unit से कम है ।
⇒(b,a)∈R
∴R is not transitive

Question 12

Write True or False for each of the following statements:
(i) The relation "is greater than" in the set of integers is reflexive.
Sol :
असत्य

(ii) The relation 'is a factor of ' in the set of positive integers is symmetric
Sol :
असत्य

(iii) The relation 'is similar to' in the set of triangles is transitive.
Sol :
सत्य

(iv) The relation 'is perpendicular to' in the set of lines is transitive.
Sol :
असत्य

(v) Identity relation on a nonempty set A is reflexive
Sol :
सत्य

(vi) Every reflexive relation on a nonempty set A is identity relation on A.
Sol :
असत्य

(vii) Identity relation on a non empty set A is symmetric
Sol :
सत्य

Question 13

Given Example of a relation which is
(i) Symmetric and transitive but not reflexive.
(ii) Symmetric but neither reflexive nor transitive.
(iii) Transitive but neither symmetric nor reflexive
(iv) Reflexive and symmetric but not transitive
Sol :
माना A={1,2,3}
(i) R₁={(1,1),(2,2)}

(ii) R₂={(1,3),(3,1)}

(iii) R₃={(1,2)}

(iv) R₄={(1,1),(2,2),(3,3),(2,3),(3,2),(3,1),(1,3)}

(v) R₅={(2,2),(2,3),(3,1)}

Question 14

Write True or False for each of the following statement
(i) An identity relation on a non empty set A is an equivalence relation
Sol :
True (सत्य)

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

(ii) Universal relation on a non empty set A is an equivalent relation
Sol :
R=A×A={(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}

Question 15

Let a relation R be defined on set Z of integers by
x R y⇔x=y∀x,y∈Z

माना x,y,z∈Z

(i) For reflexive:
(x,y)∈R⇒x=x is always true
∴R is reflexive

(ii) For symmetric
(z,y)∈R⇒x=y
⇒y=x
⇒(y,x)∈R
∴R is symmetric

(iii) For transitive
(x,y)∈R and (y,z)∈R
⇒x=y and y=z
⇒x=z
⇒(x,z)∈R
∴R is transitive

Question 16

Let relation R be defined on set Z of integers by
xRy⇔x-y is an even integer. Is R an equivalence relation?
Sol :
R={(x,y): x-y एक समपूर्णांक है तथा x,y∈z}

माना x,y,z∈z

(i) For reflexive

(x,x)∈R⇒x-x=0 एक समपूर्णांक है

∴R is reflexive

(ii) For Symmetric
(x,y)∈R⇒x-y एक समपूर्णांक है
⇒-(x-y) एक समपूर्णांक है
⇒y-x एक समपूर्णांक है
⇒(y,x)∈R
∴R is symmetric

(iii) For transitive
(x,y)∈R and (y,z)∈R
⇒x-y एक समपूर्णांक है y-z एक समपूर्णांक है
⇒x-z एक समपूर्णांक है
⇒(x,z)∈R
∴R is transitive

Hence , R is equivalence relation

Question 17

Show that the relation "is congruent to", on the set of all triangles in a plane is an equivalence relation.
Sol :
R={(Δ₁,Δ₂): Δ₁≅Δ₂ ,जहाँ Δ₁ तथा Δ₂ एक तल में स्थित हैं}

माना Δ₁,Δ₂ तथा Δ₃ एक तल में स्थित हैं ।
(i) For reflexive
(Δ₁,Δ₂)∈R⇒Δ₁≅Δ₂ is always true
∴R is reflexive

(ii) For symmetric
(Δ₁,Δ₂)∈R⇒Δ₁≅Δ₂
⇒Δ₁≅Δ₂
⇒(Δ₁≅Δ₂)∈R
∴R is symmetric

(iii) For transitive
(Δ₁,Δ₂)∈R and (Δ₂,Δ₃)∈R
⇒(Δ₁≅Δ₂) तथा (Δ₂≅Δ₃)
⇒Δ₁≅Δ₃
⇒(Δ₁≅Δ₃)∈R
∴R is transitive

Hence , R is an equivalence relation

Question 18

Let A be the set of all books in a library and R be a relation on A defined as
R={(x,y):x and y have same number of pages}.Show that R is an equivalence relation
Sol :
R={(x,y)x और y मे पुस्तको की संख्या समान हैं}

माना x,y,z तीन पुस्तके हैं और x,y,z∈A

(i) For reflexive
(x,x)∈R⇒x और x मे पुस्तको की संख्या समान हैं
∴R is reflexive

(ii) For symmetric
(x,y)∈R⇒x और y मे पुस्तको की संख्या समान हैं
⇒y और x मे पुस्तको की संख्या समान हैं
⇒(y,x)∈R
∴R is symmetric

(iii) For transitive
(x,y)∈R and (y,z)∈R
⇒x और y मे पुस्तको की संख्या समान हैं और y तथा z मे पुस्तको की संख्या समान हैं
⇒x और z मे पुस्तको की संख्या समान हैं
⇒(x,z)∈R
∴R is transitive

Hence, R is an equivalence relation

Question 19

Prove that the relation of 'congruence modulo m' in the set of integers Z is an equivalence relation
Sol :
माना , R={(a,b):a≅b(mod m)}∀a,b∈Z

माना a,b तथा c तीन पूर्णांक है और a,b,c∈Z

(i) For reflexive
(a,a)∈R⇒a≅a(mod m)⇒a-a=0 ,m से विभाजय है
∴R is reflexive

(ii) For symmetric
(a,b)∈R⇒a≅b(mod m)⇒a-b=0 ,m से विभाजय है ।
⇒b-a, m से विभाजय है ।
⇒b≅a(mod m)
⇒(b,a)∈R
∴R is symmetric

(iii) For transitive
(a,b)∈R and (b,c)∈R
⇒a≅b(mod m) and b≅c(mod m)
⇒a-b, m से विभाजय है और b-c,m से विभाजय है ।
⇒a≅c(mod m)
⇒(a,c)∈R
∴R is transitive
Hence,R is an equivalence relation

Kc Sinha Class 12 PDF Free Download | Kc Sinha 12th Math Solution PDF Download Part 1 Chapter

Question 21

Let A be the set of all the points in a given plane. A relation R is defined on A by PRQ⇔P and Q are equidistant from the origin i.e.
R={(P,Q): P and Q are equidistant from the origin}.Show that R is an equivalence relation .Further show that the set of all points related to a point P≠O(0,0) is the circle passing through P with origin as centre.
Sol :
R={(P,Q):P तथा Q मूल बिन्दु से समान दूरी पर है}

माना p,Q तथा S तीन बिंदु किसी तल में स्थित है और P,Q,S∈A
(i) For reflexive
(P,P)∈R⇒P तथा P मूल बिन्दु से समान दूरी पर है
∴R is reflexive

(ii) For symmetric
(P,Q)∈R⇒P तथा Q मूल बिन्दु से समान दूरी पर है
⇒Q तथा P मूल बिन्दु से समान दूरी पर है
⇒(Q,P)∈R
∴R is symmetric

(iii) For transitive
(P,Q)∈R and (Q,S)∈R
⇒P तथा Q मूल बिन्दु से समान दूरी पर है और Q तथा S मूल बिन्दु से समान दूरी पर है
⇒P तथा S मूल बिन्दु से समान दूरी पर है
⇒(P,S)∈R
∴R is transitive
Hence,R is an equivalence relation

∵हम जानते है , कि वृत्त पर स्थित बिंदु से केनद्र समान दूरी पर होता है

OP=OQ=OS=r

Diagram

Question 22

Is the relation R defined on the set Q* of non-zero rational number, by
xRy⇔xy=1∀x,y∈Q*; an equivalence relation ?
Sol :
R={(x,y): xy=1∀x,y∈Q}

For reflexive
(x,x)∉R
⇒x.x=x²≠1,∀x∈Q

i.e.
(12,12)∉R
⇒12×12=14≠1
∴R is not reflexive

Question 23

Let N be the set of natural numbers. Let a relation R be defined on N×N by
(a,a)R(c,d)⇔ad=bc, prove that R is an equivalence relation
Sol :
माना a,b,c,d,e,f प्रकृत संख्याएँ है ,
(a,b),(c,d).(e,f)∈R

(i) For reflexive
(a,b)R(a,b)
⇒ab=ba
∴(a,b)R(a,b)∀(a,b)∈R
∴R is reflexive

(ii) For symmetric
(a,b)R(c,d)⇒ad=bc
⇒bc=ad
⇒cb=da
⇒(c,d)R(a,b)
∴R is symmetric

(iii) For transitive
(a,b)R(c,d) and (c,d)R(e,f)
⇒ad=bc and cf=de
⇒adcf=bcde
⇒(af)(cd)=(be)(cd)
⇒af=be
⇒(a,b)R(e,f)
∴R is transitive
Hence, R is an equivalence relation

Question 24

(i) Is the relation '>' defined on N an equivalence relation?
Sol :
Let R={(a,b): a>b,∀a,b}∈N

(ii) Show that in the set of real number the relation '>' is transitive but not reflexive
Sol :
R={(a,b): a>b,∀a,b∈R}

Question 25

Let a relation R' in the set of real numbers be defined by
xR'y⇔1+xy>0. Show that R' is reflexive and symmetric but not transitive
Sol :
R={(x,y): 1+xy>0,x,y∈R}

माना x,y,z तीन वास्तविक संख्याएँ है ।
x,y,z∈R
(i) For reflexive
(x,x)∈R
⇒1+x.x=1+x²>0
∴R is reflexive

(ii) For symmetric
(x,y)∈R⇒1+xy>0
⇒1+yx>0
⇒(y,x)∈R
∴R is symmetric

(iii) For transitive
(x,y)∈R and (y,z)∈R
⇒1+xy>0 and 1+yz>0
⇒1+xz≰0

i.e.
(12,−13),∈R and (−13,−6)<R
⇒1+12×(−13)>0 and 1+(−13)(−6)>0
⇒1+12(−6)=1-3
=-2<0
⇒(12,−6)∉R

Question 26

Let a relation R in the set of natural numbers N be defined by
mRn⇔(m-n)(m-3n)=0. Is R an equivalence relation ?
Sol :
R={(m,n):(m-n)(m-3n)=0,m,n∈N}

माना m,n तथा p तीन प्राकृत संख्याए है ।
m,n,p∈N

(i) For reflexive
(m,m)∈R⇒(m-n)(m-3n)=0
∴R is reflexive

(ii) For symmetric
(m,n)∈R⇒(m-n)(m-3n)=0
⇒(n-m)(m-3n)≠0
⇒(m,m)∉R
∴R is not symmetric

i.e.
m=9,n=3
(9,3)∈R
⇒(9-3)(9-3×3)=0
⇒(3-9)(3-3×9)≠0
(3,9)∉R

Hence, R is not equivalence relation

Question 27

Let f:X→Y be a function. Define a relation R in X as
R={(a,b):f(a)=f(b)}
Examine , if R is an equivalence relation
Sol :
R={(a,b):f(a)=f(b),∀a,b∈X}

(i) For reflexive
(a,a)∈R⇒f(a)=f(a) is always true ∀a∈X
∴R is reflexive

(ii) For symmetric
(a,b)∈R⇒f(a)=f(b)
⇒f(b)=f(a)
⇒(b,a)∈R,∀a,b∈X
∴R is symmetric

(iii) For transitive
(a,b)∈R and (b,c)∈R,∀a,b,c∈X
⇒f(a)=f(b) and f(b)=f(c)

⇒f(a)=f(c)

⇒(a,c)∈R,∀a,c∈X

∴R is transitive

Question 28

Sol :
R={(a,b).|a-b| सम है}
|1-3|=|-2|=2 सम है⇒(1,3)∈R
|1-5|=|-4|=2 सम है⇒(1,5)∈R
|3-1|=|2|=2 सम है⇒(3,1)∈R
|5-1|=|4|=4 सम है⇒(5,1)∈R
|3-5|=|-2|=2 सम है⇒(3,5)∈R
|5-3|=|2|=2 सम है⇒(5,3)∈R
|2-4|=|-2|=2 सम है⇒(2,4)∈R
|4-2|=|2|=2 सम है⇒(4,2)∈R
|1-1|=0 सम है⇒(1,1)∈R
|2-2|=0 सम है⇒(2,2)∈R
|3-3|=0 सम है⇒(3,3)∈R
|4-4|=0 सम है⇒(4,4)∈R
|5-5|=0 सम है⇒(5,5)∈R

R={(1,3),(1,5),(3,1),(5,1),(3,5),(5,3),(2,4),(4,2),(1,1),(2,2),(3,3),(4,4),(5,5)}

Kc Sinha Class 12 PDF Free Download | Kc Sinha 12th Math Solution PDF Download Part 1 Chapter

Question 30

Show that the relation R in the set
A={x∈Z:0≤x≤12 given by
R={a,b}:a=b is an equivalence relation. Find set of all elements related to 1
Sol :
A={0,1,2,3...12}

R={(a,b):a=b}

R={(0,0),(1,1),(2,2),(3,3)...(12,12)}

(i) For reflexive
(a,a)∈R,∀a∈A
∴R is reflexive

(ii) For symmetric
(a,b)∈R⇒a=b
⇒b=a
⇒(b,a)∈R,∀a,b∈A
∴R is symmetric

(iii) For transitive:
(a,b)∈R and (b,c)∉R
∴R is transitive
Hence, R is an equivalence relation

Question 31

Let A={1,2,3}. Then show that the number of equivalence relations on A containing (2,3) and (3,2) is 2
Sol :
A={1,2,3}

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

(2,3) तथा (3,2) को शामिल करने वाले न्यूनतम वाला equivalence relation

R₁={(1,1),(2,2),(3,3),(2,3),(3,2)}

अब A×A में 4 (1,2),(1,3),(2,1),(3,1)

यदि R₁में (1,2) शामिल करे तो सममित होने के लिए (2,1) भी

R₂={(1,1),(2,2),(3,3),(2,3),(3,2),(2,1),(1,2),(1,3),(3,1)}

(2,3) तथा (3,2) को शामिल करते हुए equivalence relation की संख्या 2 है ।

Question 32

Let A={a,b,c}. Then show that the number of relation on A containing (b,c) and (c,a) which are reflexive and transitive but not symmetric is 4
Sol :
A×A={(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),(c,c)}

(b,c) तथा (c,a) को शामिल करने वाले reflexive, transitive and not symmetric होने वाले संबंध जिलके अवयवो की संख्या न्यूनतम है ।

R₁={(a,a),(b,b),(c,c),(b,c),(c,a),(b,a)}

R₁ मे (c,b) को शामिल करने पर

R₂={(a,a),(b,b),(c,c),(b,c),(c,a),(b,a),(a,b)}

R₁मे (a,c) तथा (c,b) शामिल करने पर

R₄={(a,a),(b,b),(c,c),(b,c),(c,a),(b,a),(a,b),(c,b)}

(b,c) तथा (c,a) को शामिल करते हुए reflexive , transitive but not symmetric संबंधो की संख्या 4 है ।

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