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NCERT Solutions for Exercise 1.1 Class 12 Maths Chapter 1 Relations and Functions are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the latest syllabus and pattern of CBSE 2023-24. Class 12 Maths ex 1.1 deals with questions related to various concepts of Relations and Functions which includes types of relations, functions, binary operations etc. Exercise 1.1 Class 12 Maths will help students to grasp the basic concepts of sets and relations. It is highly recommended to students to practise the NCERT Solutions for Class 12 Maths chapter 1 exercise 1.1 to score well in CBSE class 12 board exam. In competitive exams also like JEE main ,some questions can be asked from Class 12 Maths chapter 1 exercise 1.1. Concepts related to functions discussed in Class 12th Maths chapter 1 exercise 1.1 are important for Board examination also.
12th class Maths exercise 1.1 answers are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise together using the link provided below.
Question1(i) . Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) Relation in the set defined as
Answer:
Since, so is not reflexive.
Since, but so is not symmetric.
Since, but so is not transitive.
Hence, is neither reflexive nor symmetric and nor transitive.
Question 1(ii) . Determine whether each of the following relations are reflexive, symmetric and
transitive:
(ii) Relation R in the set N of natural numbers defined as
Answer:
Since,
so is not reflexive.
Since, but
so is not symmetric.
Since there is no pair in such that so this is not transitive.
Hence, is neither reflexive nor symmetric and
nor transitive.
Question1(iii) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(iii) Relation R in the set as
Answer:
Any number is divisible by itself and .So it is reflexive.
but .Hence,it is not symmetric.
and 4 is divisible by 2 and 4 is divisible by 4.
Hence, it is transitive.
Hence, it is reflexive and transitive but not symmetric.
Question.1(iv) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(iv). Relation R in the set Z of all integers defined as
Answer:
For , as which is an integer.
So,it is reflexive.
For , and because are both integers.
So, it is symmetric.
For , as are both integers.
Now, is also an integer.
So, and hence it is transitive.
Hence, it is reflexive, symmetric and transitive.
Question:1(v) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
(a)
Answer:
,so it is reflexive
means .
i.e. so it is symmetric.
means also .It states that i.e. .So, it is transitive.
Hence, it is reflexive, symmetric and transitive.
Question:1(v) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
(b)
Answer:
as and is same human being.So, it is reflexive.
means .
It is same as i.e. .
So,it is symmetric.
means and .
It implies that i.e. .
Hence, it is reflexive, symmetric and
transitive.
Question:1(v) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
(c)
Answer:
means but i.e. .So, it is not reflexive.
means but i.e .So, it is not symmetric.
means and .
i.e. .
Hence, it is not reflexive,not symmetric and
not transitive.
Question:1(v) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(v). Relation R in the set A of human beings in a town at a particular time given by
(d)
Answer:
means but i.e. .
So, it is not reflexive.
means but i.e. .
So, it is not symmetric.
Let, means and .
This case is not possible so it is not transitive.
Hence, it is not reflexive, symmetric and
transitive.
Question:1(v) Determine whether each of the following relations are reflexive, symmetric and
transitive:
(v) Relation R in the set A of human beings in a town at a particular time given by
(e)
Answer:
means than i.e. .So, it is not reflexive..
means than i.e. .So, it is not symmetric.
Let, means and than i.e. .
So, it is not transitive.
Hence, it is neither reflexive nor symmetric and nor transitive.
Answer:
Taking
and
So, R is not reflexive.
Now,
because .
But, i.e. 4 is not less than 1
So,
Hence, it is not symmetric.
as
Since because
Hence, it is not transitive.
Thus, we can conclude that it is neither reflexive, nor symmetric, nor transitive.
Question:3 Check whether the relation R defined in the set as
is reflexive, symmetric or transitive.
Answer:
R defined in the set
Since, so it is not reflexive.
but
So, it is not symmetric
but
So, it is not transitive.
Hence, it is neither reflexive, nor symmetric, nor transitive.
Question:4 Show that the relation R in R defined as , is reflexive and
Answer:
As so it is reflexive.
Now we take an example
as
But because .
So,it is not symmetric.
Now if we take,
Than, because
So, it is transitive.
Hence, we can say that it is reflexive and transitive but not symmetric.
Question:5 Check whether the relation R in R defined by is reflexive,
symmetric or transitive.
Answer:
because
So, it is not symmetric
Now, because
but because
It is not symmetric
as .
But, because
So it is not transitive
Thus, it is neither reflexive, nor symmetric, nor transitive.
Question:6 Show that the relation R in the set given by is
symmetric but neither reflexive nor transitive.
Answer:
Let A=
We can see so it is not reflexive.
As so it is symmetric.
But so it is not transitive.
Hence, R is symmetric but neither reflexive nor transitive.
Answer:
A = all the books in a library of a college
because x and x have the same number of pages so it is reflexive.
Let means x and y have same number of pages.
Since y and x have the same number of pages so .
Hence, it is symmetric.
Let means x and y have the same number of pages.
and means y and z have the same number of pages.
This states,x and z also have the same number of pages i.e.
Hence, it is transitive.
Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence
relation.?
Answer:
Let there be then as which is even number. Hence, it is reflexive
Let where then as
Hence, it is symmetric
Now, let
are even number i.e. are even
then, is even (sum of even integer is even)
So, . Hence, it is transitive.
Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence relation.
The elements of are related to each other because the difference of odd numbers gives even number and in this set all numbers are odd.
The elements of are related to each other because the difference of even number is even number and in this set, all numbers are even.
The element of is not related to because a difference of odd and even number is not even.
Question:9(i) Show that each of the relation R in the set , given by
(i) is an equivalence relation. Find the set of all elements related to 1 in each case.
Answer:
For , as which is multiple of 4.
Henec, it is reflexive.
Let, i.e. is multiple of 4.
then is also multiple of 4 because = i.e.
Hence, it is symmetric.
Let, i.e. is multiple of 4 and i.e. is multiple of 4 .
is multiple of 4 and is multiple of 4
is multiple of 4
is multiple of 4 i.e.
Hence, it is transitive.
Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence relation.
The set of all elements related to 1 is
is multiple of 4.
is multiple of 4.
is multiple of 4.
Question:9(ii) Show that each of the relation R in the set , given by
(ii) is an equivalence relation. Find the set of all elements related to 1 in each case.
Answer:
For , as
Henec, it is reflexive.
Let, i.e.
i.e.
Hence, it is symmetric.
Let, i.e. and i.e.
i.e.
Hence, it is transitive.
Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence relation.
The set of all elements related to 1 is {1}
Question:10(i) Give an example of a relation.
(i) Which is Symmetric but neither reflexive nor transitive.
Answer:
Let
so it is not reflexive.
and so it is symmetric.
but so it is not transitive.
Hence, symmetric but neither reflexive nor transitive.
Question:10(ii) Give an example of a relation.
(ii) Which is transitive but neither reflexive nor symmetric.
Answer:
Let
Now for , so it is not reflexive.
Let i.e.
Then is not possible i.e. . So it is not symmetric.
Let i.e. and i.e.
we can write this as
Hence, i.e. . So it is transitive.
Hence, it is transitive but neither reflexive nor symmetric.
Question:10(iii) Give an example of a relation.
(iii) Which is Reflexive and symmetric but not transitive.
Answer:
Let
Define a relation R on A as
If , i.e. . So it is reflexive.
If , and i.e. . So it is symmetric.
and i.e. . and
But So it is not transitive.
Hence, it is Reflexive and symmetric but not transitive.
Question:10(iv) Give an example of a relation.
(iv) Which is Reflexive and transitive but not symmetric.
Answer:
Let there be a relation R in R
because
Let i.e.
But i.e.
So it is not symmetric.
Let i.e. and i.e.
This can be written as i.e. implies
Hence, it is transitive.
Thus, it is Reflexive and transitive but not symmetric.
Question:10(v) Give an example of a relation.
(v) Which is Symmetric and transitive but not reflexive.
Answer:
Let there be a relation A in R
So R is not reflexive.
We can see and
So it is symmetric.
Let and
Also
Hence, it is transitive.
Thus, it Symmetric and transitive but not reflexive.
Answer:
The distance of point P from the origin is always the same as the distance of same point P from origin i.e.
R is reflexive.
Let i.e. the distance of the point P from the origin is the same as the distance of the point Q from the origin.
this is the same as distance of the point Q from the origin is the same as the distance of the point P from the origin i.e.
R is symmetric.
Let and
i.e. the distance of point P from the origin is the same as the distance of point Q from the origin, and also the distance of point Q from the origin is the same as the distance of the point S from the origin.
We can say that the distance of point P, Q, S from the origin is the same. Means distance of point P from the origin is the same as the distance of point S from origin i.e.
R is transitive.
Hence, R is an equivalence relation.
The set of all points related to a point are points whose distance from the origin is the same as the distance of point P from the origin.
In other words, we can say there be a point O(0,0) as origin and distance between point O and point P be k=OP then set of all points related to P is at distance k from the origin.
Hence, these sets of points form a circle with the centre as the origin and this circle passes through the point.
Answer:
All triangles are similar to itself, so it is reflexive.
Let,
i.e.T _{ 1 } is similar to T2
T _{ 1 } is similar to T2 is the same asT2 is similar to T _{ 1 } i.e.
Hence, it is symmetric.
Let,
and i.e. T _{ 1 } is similar to T2 and T2 is similar toT _{ 3 } .
T _{ 1 } is similar toT _{ 3 } i.e.
Hence, it is transitive,
Thus, , is equivalence relation.
Now, we see the ratio of sides of triangle T _{ 1 } andT _{ 3 } are as shown
i.e. ratios of sides of T _{ 1 } and T _{ 3 } are equal.Hence, T _{ 1 } and T _{ 3 } are related.
Answer:
The same polygon has the same number of sides with itself,i.e. , so it is reflexive.
Let,
i.e.P _{ 1 } have same number of sides as P _{ 2 }
P _{ 1 } have the same number of sides as P _{ 2 } is the same as P _{ 2 } have same number of sides as P _{ 1 } i.e.
Hence,it is symmetric.
Let,
and i.e. P _{ 1 } have the same number of sides as P _{ 2 } and P _{ 2 } have same number of sides as P _{ 3 }
P _{ 1 } have same number of sides as P _{ 3 } i.e.
Hence, it is transitive,
Thus, , is an equivalence relation.
The elements in A related to the right angle triangle T with sides 3, 4 and 5 are those polygons which have 3 sides.
Hence, the set of all elements in A related to the right angle triangle T is set of all triangles.
Answer:
All lines are parallel to itself, so it is reflexive.
Let,
i.e.L _{ 1 } is parallel to L _{ 2 } .
_{ L1 } is parallel to L _{ 2 } is same as L _{ 2 } is parallel to L _{ 1 } i.e.
Hence, it is symmetric.
Let,
and i.e. _{ L1 } is parallel to L _{ 2 } and L _{ 2 } is parallel to L _{ 3 } .
L _{ 1 } is parallel to L _{ 3 } i.e.
Hence, it is transitive,
Thus, , is equivalence relation.
The set of all lines related to the line are lines parallel to
Here, Slope = m = 2 and constant = c = 4
It is known that the slope of parallel lines are equal.
Lines parallel to this ( ) line are ,
Hence, set of all parallel lines to are .
Question:15 Let R be the relation in the set A= {1,2,3,4}
given by . Choose the correct answer.
(A) R is reflexive and symmetric but not transitive.
(B) R is reflexive and transitive but not symmetric.
(C) R is symmetric and transitive but not reflexive.
(D) R is an equivalence relation.
Answer:
A = {1,2,3,4}
For every there is .
R is reflexive.
Given, but
R is not symmetric.
For there are
R is transitive.
Hence, R is reflexive and transitive but not symmetric.
The correct answer is option B.
Question:16 Let R be the relation in the set N given by . Choose the correct answer.
(A)
(B)
(C)
(D)
Answer:
(A) Since, so
(B) Since, so
(C) Since, and so
(d) Since, so
The correct answer is option C.
The NCERT Class 12 maths chapter Relations and Functions has a total of 5 exercises including miscellaneous. Exercise 1.1 Class 12 Maths covers solutions to 16 main questions and their sub-questions. The initial 10 questions are based on concepts like symmetric, reflexive and transitive relation and subsequent questions upto 15 are based in equivalence relation etc. NCERT Solutions for Class 12 Maths chapter 1 exercise 1.1 is good source to learn concepts related to symmetric relations, equivalence of a relation etc. Students can get access of NCERT Notes For Class 12 Mathematics Chapter 1 which can be used for quick revision of important concepts of this chapter.
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Concepts related to symmetric, reflexive and transitive, equivalence relations etc, are discussed in the Exercise 1.1 Class 12 Maths
Topics like
Two chapters 'relation and function' and 'inverse trigonometry' combined has 10 % weightage in the CBSE final board exam.
From the analysis of previous year questions of Board exams, it is clear that direct questions are asked from the NCERT questions. Also Some of the questions are repeated year after year. Hence it is said that NCERT solutions are low hanging fruits. Every serious student must practice NCERT questions to score well in the exam.
In maths, relation defines the relationship between sets of values of ordered pairs
symmetric, reflexive and transitive, equivalence relations etc, are discussed in the Exercise 1.1 Class 12 Maths
There are 15 questions in Exercise 1.1 Class 12 Maths
In NCERT class 12 maths chapter 1 relations and function, there are a total of 5 exercises which includes a miscellaneous exercise also.
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If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.
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If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.
You can get the Previous Year Questions (PYQs) on the official website of the respective board.
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Hello student,
If you are planning to appear again for class 12th board exam with PCMB as a private candidate here is the right information you need:
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