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Edited By Ramraj Saini | Updated on Sep 12, 2023 09:06 PM IST | #CBSE Class 12th

Relations and Functions class 12 solutions are provided here. These NCERT solutions are created by expert team at careers360 keeping in mind of latest syllabus of CBSE 2023-24. This is the first chapter of Class 12 math. NCERT solutions class 12 maths chapter 1 Relations and Functions contains the answer and step-by-step solution to each question asked in the exercise of NCERT Class 12 maths book. NCERT Class 12 maths solutions Chapter 1 will help you to understand the concepts and score well in CBSE 12th board exam. Here you will find all NCERT solutions of chapter 1 maths class 12 at a single place which will be helpful when you are not able to solve the NCERT questions.

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- Relations And Functions Class 12 Questions And Answers
- Relations and Functions Class 12 Questions And Answers PDF Free Download
- Relations and Functions Class 12 Solutions - Important Formulae
- Relations and Functions Class 12 NCERT Solutions (Intext Questions and Exercise)
- NCERT Exemplar Class 12th Maths Solutions
- Class 12 Maths chapter 1 ncert solutions - Topics
- Class 12 maths chapter 1 NCERT Solutions - Chapter wise
- Class 12 maths chapter 1 NCERT solutions - Subject Wise
- NCERT Solutions Class wise
- NCERT Books and NCERT Syllabus

In Relations and Functions class 12 maths chapter 1 question answer, there are four exercises with 55 questions and one miscellaneous exercise with 19 questions. relations and functions class 12 solutions are very important for students because they comprise quality practice questions. In this article, you will find the detailed NCERT solutions for class 12 maths chapter 1. Here you will get NCERT solutions for class 12 also.

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**Also Read:**

- Class 12 Maths Chapter 1 Relations And Functions Notes
- NCERT Exemplar Solutions For Class 12 Maths Chapter 1 Relations And Functions

** >> Relations:** A relation R is a subset of the cartesian product of A × B, where A and B are non-empty sets.

R-1, the inverse of relation R, is defined as R-1 = {(b, a) : (a, b) ∈ R}

Domain of R = Range of R^{-1}

Range of R = Domain of R^{-1}

** >> Functions:** A relation f from set A to set B is a function if every element in A has one and only one image in B.

A × B = {(a, b): a ϵ A, b ϵ B}

If (a, b) = (x, y), then a = x and b = y

n(A × B) = n(A) * n(B), where n(A) is the cardinality of set A.

A × ϕ = ϕ (where ϕ is the empty set)

A function f: A → B is denoted as f(x) = y.

*Algebra of functions:*

(f + g)(x) = f(x) + g(x)

(f - g)(x) = f(x) - g(x)

(f * g)(x) = f(x) * g(x)

(kf)(x) = k * f(x), where k is a real number

{f/g}(x) = f(x)/g(x), where g(x) ≠ 0

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Download EBookFree download **NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions **for CBSE Exam.

**NCERT Solutions for Class 12 relations and functions NCERT solutions: Exercise 1.1** ** **

**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

** 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

** 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

** 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

** 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

** 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 .

(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.

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.

(A) Since, so

(B) Since, so

(C) Since, and so

(d) Since, so

The correct answer is option C.

** Relations and Functions Class 12 NCERT Solutions: Exercise 1.2 **

** Answer: **

Given, is defined by .

One - One :

f is one-one.

Onto:

We have , then there exists ( Here ) such that

.

Hence, the function is one-one and onto.

If the domain R _{ ∗ } is replaced by N with co-domain being same as R _{ ∗ i.e. } defined by

g is one-one.

For ,

but there does not exists any x in N.

Hence, function g is one-one but not onto.

** Question:2(i) ** Check the injectivity and surjectivity of the following functions:

** Answer: **

One- one:

then

f is one- one i.e. injective.

For there is no x in N such that

f is not onto i.e. not surjective.

Hence, f is injective but not surjective.

** Question:2(ii) ** Check the injectivity and surjectivity of the following functions:

Answer:

One- one:

For then

but

f is not one- one i.e. not injective.

For there is no x in Z such that

f is not onto i.e. not surjective.

Hence, f is neither injective nor surjective.

** Question:2(iii) ** Check the injectivity and surjectivity of the following functions:

Answer:

One- one:

For then

but

f is not one- one i.e. not injective.

For there is no x in R such that

f is not onto i.e. not surjective.

Hence, f is not injective and not surjective.

** Question:2(iv) ** Check the injectivity and surjectivity of the following functions:

** Answer: **

One- one:

then

f is one- one i.e. injective.

For there is no x in N such that

f is not onto i.e. not surjective.

Hence, f is injective but not surjective.

** Question:2(v) ** Check the injectivity and surjectivity of the following functions:

** Answer: **

One- one:

For then

f is one- one i.e. injective.

For there is no x in Z such that

f is not onto i.e. not surjective.

Hence, f is injective but not surjective.

** Answer: **

One- one:

For then and

but

f is not one- one i.e. not injective.

For there is no x in R such that

f is not onto i.e. not surjective.

Hence, f is not injective but not surjective.

** Answer: **

One- one:

For then

f is not one- one i.e. not injective.

For ,

We know is always positive there is no x in R such that

f is not onto i.e. not surjective.

Hence, , is neither one-one nor onto.

** Question:5 ** Show that the Signum Function , given by

** Answer: **

is given by

As we can see , but

So it is not one-one.

Now, f(x) takes only 3 values (1,0,-1) for the element -3 in codomain ,there does not exists x in domain such that .

So it is not onto.

Hence, signum function is neither one-one nor onto.

** Question:6 ** Let , and let be a function from A to B. Show that f is one-one.

** Answer: **

Every element of A has a distant value in f.

Hence, it is one-one.

** Question:7(i) ** In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

** Answer: **

Let there be such that

f is one-one.

Let there be ,

Puting value of x,

f is onto.

f is both one-one and onto hence, f is bijective.

** Question:7(ii) ** In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

** Answer: **

Let there be such that

For and

f is not one-one.

Let there be (-2 in codomain of R)

There does not exists any x in domain R such that

f is not onto.

Hence, f is neither one-one nor onto.

** Question:8 ** Let A and B be sets. Show that such that is

bijective function.

** Answer: **

Let

such that

and

f is one- one

Let,

then there exists such that

f is onto.

Hence, it is bijective.

** Question:9 ** Let be defined by for all . State whether the function f is bijective. Justify your answer.

** Answer: **

,

Here we can observe,

and

As we can see but

f is not one-one.

Let, (N=co-domain)

case1 n be even

For ,

then there is such that

case2 n be odd

For ,

then there is such that

f is onto.

f is not one-one but onto

hence, the function f is not bijective.

** Question:10 ** Let and . Consider the function defined by . Is f one-one and onto? Justify your answer.

** Answer: **

Let such that

f is one-one.

Let, then

such that

For any there exists such that

f is onto

Hence, the function is one-one and onto.

** Question:11 ** Let be defined as . Choose the correct answer.

(D) f is neither one-one nor onto.

** Answer: **

One- one:

For then

does not imply that

example: and

f is not one- one

For there is no x in R such that

f is not onto.

Hence, f is neither one-one nor onto.

Option D is correct.

** Question:12 ** Let be defined as . Choose the correct answer.

(D) f is neither one-one nor onto.

** Answer: **

One - One :

Let

f is one-one.

Onto:

We have , then there exists such that

.

Hence, the function is one-one and onto.

The correct answer is A .

**Relation and Function Class 12 maths chapter 1 question answer: Exercise 1.3**

** Question:1 ** Let and be given by and . Write down .

** Answer: **

Given : and

and

Hence, =

** Question:4 ** If show that , for all . What is the inverse of ?

** Answer: **

, for all

Hence,the given function is invertible and the inverse of is itself.

** Question:5(i) ** State with reason whether following functions have inverse

(i)

with

** Answer: **

(i) with

From the given definition,we have:

f is not one-one.

Hence, f do not have an inverse function.

** Question:5(ii) ** State with reason whether following functions have inverse

** Answer: **

(ii) with

From the definition, we can conclude :

g is not one-one.

Hence, function g does not have inverse function.

** Question:5(iii) ** State with reason whether following functions have inverse

** Answer: **

(iii) with

From the definition, we can see the set have distant values under h.

h is one-one.

For every element y of set ,there exists an element x in such that

h is onto

Thus, h is one-one and onto so h has an inverse function.

** Question:6 ** Show that , given by is one-one. Find the inverse of the function

** Answer: **

One -one:

f is one-one.

It is clear that is onto.

Thus,f is one-one and onto so inverse of f exists.

Let g be inverse function of f in

let y be an arbitrary element of range f

Since, is onto, so

for

,

** Question:7 ** Consider given by . Show that f is invertible. Find the inverse of .

** Answer: **

is given by

One-one :

Let

f is one-one function.

Onto:

So, for there is ,such that

f is onto.

Thus, f is one-one and onto so exists.

Let, by

Now,

and

Hence, function f is invertible and inverse of f is .

** Answer: **

It is given that

, and

Now, Let f(x) = f(y)

⇒ x ^{ 2 } + 4 = y ^{ 2 } + 4

⇒ x ^{ 2 } = y ^{ 2 }

⇒ x = y

⇒ f is one-one function.

Now, for y [4, ∞), let y = x ^{ 2 } + 4.

⇒ x ^{ 2 } = y -4 ≥ 0

⇒ for any y R, there exists x = R such that

= y -4 + 4 = y.

⇒ f is onto function.

Therefore, f is one–one and onto function, so f-1 exists.

Now, let us define g: [4, ∞) → R+ by,

g(y) =

Now, gof(x) = g(f(x)) = g(x ^{ 2 } + 4) =

And, fog(y) = f(g(y)) = =

Therefore, gof = gof = I _{ R } .

Therefore, f is invertible and the inverse of f is given by

f-1(y) = g(y) =

** Question:9 ** Consider given by . Show that is invertible with

** Answer: **

One- one:

Let

Since, x and y are positive.

f is one-one.

Onto:

Let for ,

f is onto and range is .

Since f is one-one and onto so it is invertible.

Let by

Hence, is invertible with the inverse of given by

** Answer: **

Let be an invertible function

Also, suppose f has two inverse

For , we have

[f is invertible implies f is one - one]

[g is one-one]

Thus,f has a unique inverse.

** Question:11 ** Consider given by , and . Find and show that .

** Answer: **

It is given that

Now,, lets define a function g :

such that

Now,

Similarly,

And

Hence, and , where and

Therefore, the inverse of f exists and

Now,

is given by

Now, we need to find the inverse of ,

Therefore, lets define such that

Now,

Similarly,

Hence, and , where and

Therefore, inverse of exists and

Therefore,

** Hence proved **

** Question:12 ** Let be an invertible function. Show that the inverse of is , i.e.,

** Answer: **

To prove:

Let be a invertible function.

Then there is such that and

Also,

and

and

Hence, is invertible function and f is inverse of .

i.e.

** Question:14 ** Let be a function defined as . The inverse of is the map given by

** Answer: **

Let f inverse

Let y be the element of range f.

Then there is such that

Now , define as

Hence, g is inverse of f and

The inverse of f is given by .

The correct option is B.

** Class 12 maths chapter 1 NCERT solutions: Exercise 1.4 **

** Answer: **

(i) On , define ∗ by

It is not a binary operation as the image of under * is .

** Answer: **

(ii) On , define ∗ by

We can observe that for ,there is a unique element ab in .

This means * carries each pair to a unique element in .

Therefore,* is a binary operation.

** Answer: **

(iii) On , define ∗ by

We can observe that for ,there is a unique element in .

This means * carries each pair to a unique element in .

Therefore,* is a binary operation.

** Answer: **

(iv) On , define ∗ by

We can observe that for ,there is a unique element in .

This means * carries each pair to a unique element in .

Therefore,* is a binary operation.

** Answer: **

(v) On , define ∗ by

* carries each pair to a unique element in .

Therefore,* is a binary operation.

** Question:2(i) ** For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

** Answer: **

a*b=a-b

b*a=b-a

so * is not commutative

(a*b)*c=(a-b)-c

a*(b*c)=a-(b-c)=a-b+c

(a*b)*c not equal to a*(b*c), so * is not associative

** Question:2(ii) ** For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

** Answer: **

(ii) On , define

ab = ba for all

ab+1 = ba + 1 for all

for

where

operation * is not associative.

** Question:2(iii) ** For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

** Answer: **

(iii) On , define

ab = ba for all

for all

for

operation * is commutative.

operation * is associative.

** Question:2(iv) ** For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

** Answer: **

(iv) On , define

ab = ba for all

2ab = 2ba for all

for

the operation is commutative.

where

operation * is not associative.

** Question:2(v) ** For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

** Answer: **

(v) On , define

and

for

the operation is not commutative.

where

operation * is not associative.

** Question:2(vi) ** For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

** Answer: **

(iv) On , define

and

for

the operation is not commutative.

where

operation * is not associative.

** Question:3 **

Consider the binary operation Λ on the set {1, 2, 3, 4, 5} defined by

a Λ b = min {a, b}. Write the operation table of the operation Λ .

** Answer: **

for

The operation table of the operation is given by :

| 1 | 2 | 3 | 4 | 5 |

1 | 1 | 1 | 1 | 1 | 1 |

2 | 1 | 2 | 2 | 2 | 2 |

3 | 1 | 2 | 3 | 3 | 3 |

4 | 1 | 2 | 3 | 4 | 4 |

5 | 1 | 2 | 3 | 4 | 5 |

** Question:4(i) ** Consider a binary operation ∗ on the set given by the following multiplication table (Table 1.2).

(Hint: use the following table)

** Answer: **

(i)

** Question:4(ii) ** Consider a binary operation ∗ on the set given by the following multiplication table (Table 1.2).

(Hint: use the following table)

** Answer: **

(ii)

For every , we have . Hence it is commutative.

** Question:4(iii) ** Consider a binary operation ∗ on the set { given by the following multiplication table (Table 1.2).

(iii) Compute (2 ∗ 3) ∗ (4 ∗ 5).

(Hint: use the following table)

** Answer: **

(iii) (2 ∗ 3) ∗ (4 ∗ 5).

from the above table

** Answer: **

for

The operation table is as shown below:

| 1 | 2 | 3 | 4 | 5 |

1 | 1 | 1 | 1 | 1 | 1 |

2 | 1 | 2 | 1 | 2 | 1 |

3 | 1 | 1 | 3 | 1 | 1 |

4 | 1 | 2 | 1 | 4 | 1 |

5 | 1 | 1 | 1 | 1 | 5 |

The operation ∗′ same as the operation ∗ defined in Exercise 4 above.

** Question:6(i) ** let ∗ be the binary operation on N given by . Find

** Answer: **

a*b=LCM of a and b

(i) 5 ∗ 7, 20 ∗ 16

** Question:6(ii) ** Let ∗ be the binary operation on N given by . Find

** Answer: **

(ii) for all

Hence, it is commutative.

** Question:6(iii) ** Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find

** Answer: **

a b = L.C.M. of a and b

(iii)

Hence, the operation is associative.

** Question:6(iv) ** Let ∗ be the binary operation on N given by . Find

** Answer: **

(iv) the identity of ∗ in N

We know that

for

Hence, 1 is the identity of ∗ in N.

** Question 6(v) ** Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find

(v) Which elements of N are invertible for the operation ∗?

** Answer: **

An element a is invertible in N

if

Here a is inverse of b.

a*b=1=b*a

a*b=L.C.M. od a and b

a=b=1

So 1 is the only invertible element of N

** Question:7 ** Is ∗ defined on the set by a binary operation? Justify your answer.

** Answer: **

A =

Operation table is as shown below:

| 1 | 2 | 3 | 4 | 5 |

1 | 1 | 2 | 3 | 4 | 5 |

2 | 2 | 2 | 6 | 4 | 10 |

3 | 3 | 6 | 3 | 12 | 15 |

4 | 4 | 4 | 12 | 4 | 20 |

5 | 5 | 10 | 15 | 20 | 5 |

From the table, we can observe that

Hence, the operation is not a binary operation.

Answer:

a ∗ b = H.C.F. of a and b for all

H.C.F. of a and b = H.C.F of b and a for all

Hence, operation ∗ is commutative.

For ,

Hence, ∗ is associative.

An element will be identity for operation * if for .

Hence, the operation * does not have any identity in N.

** Question:9(i) ** Let ∗ be a binary operation on the set Q of rational numbers as follows:

(i) Find which of the binary operations are commutative and which are associative.

** Answer: **

On the set Q ,the operation * is defined as .It is observed that:

here

Hence, the * operation is not commutative.

It can be observed that

for all

The operation * is not associative.

** Question:9(ii) ** Let ∗ be a binary operation on the set Q of rational numbers as follows:

(ii) Find which of the binary operations are commutative and which are associative.

** Answer: **

On the set Q ,the operation * is defines as .It is observed that:

For

Hence, the * operation is commutative.

It can be observed that

for all

The operation * is not associative.

** Question:9(iii) ** Let ∗ be a binary operation on the set Q of rational numbers as follows:

(iii) Find which of the binary operations are commutative and which are associative.

** Answer: **

On the set Q ,the operation * is defines as .It is observed that:

For

for

Hence, the * operation is not commutative.

It can be observed that

for all

The operation * is not associative.

** Question:9(iv) ** Let ∗ be a binary operation on the set Q of rational numbers as follows:

(iv) Find which of the binary operations are commutative and which are associative.

** Answer: **

On the set Q ,the operation * is defined as .It is observed that:

For

for

Hence, the * operation is commutative.

It can be observed that

for all

The operation * is not associative.

** Question:9(v) ** Let ∗ be a binary operation on the set Q of rational numbers as follows:

(v) Find which of the binary operations are commutative and which are associative.

** Answer: **

On the set Q ,the operation * is defines as .It is observed that:

For

for

Hence, the * operation is commutative.

It can be observed that

for all

The operation * is associative.

** Question:9(vi) ** Let ∗ be a binary operation on the set Q of rational numbers as follows:

(vi) Find which of the binary operations are commutative and which are associative.

** Answer: **

On the set Q ,the operation * is defines as .It is observed that:

For

for

Hence, the * operation is not commutative.

It can be observed that

for all

The operation * is not associative.

** Question:10 ** Find which of the operations given above has identity.

** Answer: **

An element will be identity element for operation *

if for all

when .

Hence, has identity as 4.

However, there is no such element which satisfies above condition for all rest five operations.

Hence, only (v) operations have identity.

** Question:11 ** Let and ∗ be the binary operation on A defined by Show that ∗ is commutative and associative. Find the identity element for ∗ on A, if any.

** Answer: **

and ∗ be the binary operation on A defined by

Let

Then,

We have

Thus it is commutative.

Let

Then,

Thus, it is associative.

Let will be a element for operation * if for all .

i.e.

This is not possible for any element in A .

Hence, it does not have any identity.

** Question:12(i) ** State whether the following statements are true or false. Justify.

(i) For an arbitrary binary operation ∗ on a set N,

** Answer: **

(i) For an arbitrary binary operation ∗ on a set N,

An operation * on a set N as

Then , for b=a=2

Hence, statement (i) is false.

** Question:12(ii) ** State whether the following statements are true or false. Justify.

(ii) If ∗ is a commutative binary operation on N, then

** Answer: **

(ii) If ∗ is a commutative binary operation on N, then

R.H.S

(* is commutative)

( as * is commutative)

= L.H.S

Hence, statement (ii) is true.

** Question:13 ** Consider a binary operation ∗ on N defined as . Choose the correct answer.

** Answer: **

A binary operation ∗ on N defined as .

For

Thus, it is commutative.

where

Hence, it is not associative.

Hence, B is the correct option.

**NCERT solutions for class 12 maths chapter 1 Relations and Functions: Miscellaneous Exercise **

** Question:1 ** Let be defined as . Find the function such that .

** Answer: **

and

For one-one:

Thus, f is one-one.

For onto:

For ,

Thus, for , there exists such that

Thus, f is onto.

Hence, f is one-one and onto i.e. it is invertible.

Let as

and

Hence, defined as

** Answer: **

if n is odd

if n is even.

For one-one:

Taking x as odd number and y as even number.

Now, Taking y as odd number and x as even number.

This is also impossible.

** If both x and y are odd ** :

** If both x and y are even : **

f is one-one.

Onto:

Any odd number 2r+1 in codomain of N is an image of 2r in domain N and any even number 2r in codomain N is the image of 2r+1 in domain N.

Thus, f is onto.

Hence, f is one-one and onto i.e. it is invertible.

Sice, f is invertible.

Let as if m is even and if m is odd.

When x is odd.

When x is even

Similarly, m is odd

m is even ,

and

Hence, f is invertible and the inverse of f is g i.e. , which is the same as f.

Hence, inverse of f is f itself.

** Question:3 ** If f : R → R is defined by f(x) = x ^{ 2 } – 3x + 2, find f (f (x)).

** Answer: **

This can be solved as following

f : R → R

** Question:4 ** Show that the function defined by is one one and onto function.

** Answer: **

The function defined by

,

One- one:

Let ,

It is observed that if x is positive and y is negative.

Since x is positive and y is negative.

but 2xy is negative.

Thus, the case of x is positive and y is negative is removed.

Same happens in the case of y is positive and x is negative so this case is also removed.

** When x and y both are positive: **

** When x and y both are negative ** :

f is one-one.

Onto:

Let such that

If y is negative, then

If y is positive, then

Thus, f is onto.

Hence, f is one-one and onto.

** Question:5 ** Show that the function given by is injective.

** Answer: **

One-one:

Let

We need to prove .So,

Let then there cubes will not be equal i.e. .

It will contradict given condition of cubes being equal.

Hence, and it is one -one which means it is injective

** Question:6 ** Give examples of two functions and such that is injective but g is not injective. (Hint : Consider and ).

** Answer: **

One - one:

Since

As we can see but so is not one-one.

Thus , g(x) is not injective.

Let

Since, so x and y are both positive.

Hence, gof is injective. ** **

** Question:7 ** Give examples of two functions and such that is onto but is not onto.

** Answer: **

and

and

Onto :

Consider element in codomain N . It is clear that this element is not an image of any of element in domain N .

f is not onto.

Now, it is clear that , there exists such that .

Hence, is onto.

** Answer: **

Given a non empty set X, consider P(X) which is the set of all subsets of X.

Since, every set is subset of itself , ARA for all

R is reflexive.

Let

This is not same as

If and

then we cannot say that B is related to A.

R is not symmetric.

If

this implies

R is transitive.

Thus, R is not an equivalence relation because it is not symmetric.

** Answer: **

Given is defined as .

As we know that

Hence, X is the identity element of binary operation *.

Now, an element is invertible if there exists a ,

such that (X is identity element)

i.e.

This is possible only if .

Hence, X is only invertible element in with respect to operation *

** Question:10 ** Find the number of all onto functions from the set to itself.

** Answer: **

The number of all onto functions from the set to itself is permutations on n symbols 1,2,3,4,5...............n.

Hence, permutations on n symbols 1,2,3,4,5...............n = n

Thus, total number of all onto maps from the set to itself is same as permutations on n symbols 1,2,3,4,5...............n which is n. ** **

** Question:11(i) ** Let and . Find of the following functions F from S to T, if it exists.

** Answer: **

is defined as

is given by

** **

** Question:11(ii) ** Let and . Find of the following functions F from S to T, if it exists.

** Answer: **

is defined as

, F is not one-one.

So inverse of F does not exists.

Hence, F is not invertible i.e. does not exists. ** **

** Answer: **

Given and is defined as

and

For , we have

the operation is commutative.

where

the operation is not associative

Let . Then we have :

Hence,

Now,

for

Hence, operation o does not distribute over operation *. ** **

** Question:13 ** ** ** Given a non-empty set X, let ∗ : P(X) × P(X) → P(X) be defined as A * B = (A – B) ∪ (B – A), ∀ A, B ∈ P(X). Show that the empty set φ is the identity for the operation ∗ and all the elements A of P(X) are invertible with A–1 = A. (Hint : (A – φ) ∪ (φ – A) = A and (A – A) ∪ (A – A) = A ∗ A = φ).

** Answer: **

It is given that

be defined as

Now, let .

Then,

And

Therefore,

Therefore, we can say that is the identity element for the given operation *.

Now, an element A P(X) will be invertible if there exists B P(X) such that

Now, We can see that

such that

Therefore, by this we can say that all the element A of P(X) are invertible with

** Question:14 ** Define a binary operation ∗ on the set as Show that zero is the identity for this operation and each element of the set is invertible with being the inverse of .

** Answer: **

X = as

An element is identity element for operation *, if

For ,

Hence, 0 is identity element of operation *.

An element is invertible if there exists ,

such that i.e.

means or

But since we have X = and . Then .

is inverse of a for .

Hence, inverse of element , is 6-a i.e. ,

** Question:15 ** Let , and be functions defined by and . Are and equal? Justify your answer. (Hint: One may note that two functions and such that , are called equal functions).

** Answer: **

Given :

,

are defined by and .

It can be observed that

Hence, f and g are equal functions.

** Question:16 ** Let . Then number of relations containing and which are reflexive and symmetric but not transitive is

** Answer: **

The smallest relations containing and which are

reflexive and symmetric but not transitive is given by

, so relation R is reflexive.

and , so relation R is symmetric.

but , so realation R is not transitive.

Now, if we add any two pairs and to relation R, then relation R will become transitive.

Hence, the total number of the desired relation is one.

Thus, option A is correct.

** Question:17 ** Let . Then number of equivalence relations containing is

(A) 1

(B) 2

(C) 3

(D) 4

** Answer: **

The number of equivalence relations containing is given by

We are left with four pairs , , .

, so relation R is reflexive.

and , so relation R is not symmetric.

, so realation R is not transitive.

Hence , equivalence relation is bigger than R is universal relation.

Thus the total number of equivalence relations cotaining is two.

Thus, option B is correct.

** Question:18 ** Let be the Signum Function defined as and be the Greatest Integer Function given by , where is greatest integer less than or equal to . Then, does and coincide in ?

** Answer: **

is defined as

is defined as

Let

Then we have , if x=1 and

Hence,for , and .

Hence , gof and fog do not coincide with . ** **

** Question:19 ** Number of binary operations on the set are(A) 10(B) 16(C) 20(D ) 8

** Answer: **

Binary operations on the set are is a function from

i.e. * is a function from

Hence, the total number of binary operations on set is

Hence, option B is correct

If you are looking for relation and function class 12 ncert solutions of exercises then these are listed below.

Relation And Function Class 12 Ncert Solutions Exercise 1.1

Relation And Function Class 12 Ncert Solutions Exercise 1.2

Relation And Function Class 12 Ncert Solutions Exercise 1.3

Relation And Function Class 12 Ncert Solutions Exercise 1.4

Relation And Function Class 12 Ncert Solutions Miscellaneous Exercise

We have relations like father, mother, brother, sister, husband, wife. Relation becomes a function when there is only one output for every input. In NCERT class 11 maths solutions you have already learnt in brief about relations and functions, range, domain and co-domain with different types of specific real-valued functions and their graphs.

In Class 12 maths chapter 1 question answer, you will learn about different types of relations and functions, invertible functions, the composition of functions, and binary operations. Also you can find ncert solutions for class 12 chapter 1 by careers360 expert team.

Concepts of this chapter are very useful in various other topics of calculus and are also very important from the exam point of view. Unit "Relation and Function" of NCERT class 12th maths syllabus includes two chapters i.e. relation and function, and inverse trigonometry which together has 10 % weightage in the CBSE class 12th final examination. So, you should study class 12 maths ch 1 question answer carefully, and solve every question on your own including solved examples.

NCERT Exemplar Solutions Chapter 1 - Relations and Functions

NCERT Exemplar Solutions Chapter 2 - Inverse Trigonometric Functions

NCERT Exemplar Solutions Chapter 3 - Matrices

NCERT Exemplar Solutions Chapter 4 - Determinants

NCERT Exemplar Solutions Chapter 5 - Continuity and Differentiability

NCERT Exemplar Solutions Chapter 6 - Application of Derivatives

NCERT Exemplar Solutions Chapter 7 - Integrals

NCERT Exemplar Solutions Chapter 8 - Application of Integrals

NCERT Exemplar Solutions Chapter 9 - Differential Equations

NCERT Exemplar Solutions Chapter 10 - Vector Algebra

NCERT Exemplar Solutions Chapter 11 - Three Dimensional Geometry

NCERT Exemplar Solutions Chapter 12 - Linear Programming

NCERT Exemplar Solutions Chapter 13 - Probability

These Relations and Functions class 12 NCERT solutions are explained in a step-by-step method, so it will be very easy to understand the concepts. Still if you are in a doubt anywhere, you can contact our subject matter experts who are available to help you out and make learning easier for you.

The meaning of the term ‘relation’ in mathematics is the same as the meaning of ' relation' in the English language. Relation means two quantities or objects are related if there is a link between them. In other words, we can say that it is a connection between or among things.

Let's understand with an example - let A is the set of students of class XII of a school and B is the set of students of class XI of the same school. Then some of the examples of relations from A to B are-

(i) {(a, b) ∈A × B: a is a brother of b},

(ii) {(a, b) ∈A × B: a is a sister of b},

(iii) {(a, b) ∈A × B: age of a is less than the age of b}.

If (a, b) ∈ R, we can say that ‘a’ is related to ‘b’ under the relation ‘R’ and we write as ‘a R b’. To understand the topic in-depth, after every concept, some topic wise questions are given in the textbook of CBSE class 12. In this article, you will find solutions of NCERT for class 12 maths chapter 1 Relations and Functions for such type of questions also.

The main topics covered in chapter 1 maths class 12 are:

- Type of Relation

In this ch 1 maths class 12 topics discuss different types of relations namely reflexive, symmetric, and transitive. we also study the concept of empty relation, universal relation, trivial relation, and equivalence relation in the chapter relations and functions. there are good quality questions in functions and relations class 12 solutions.

- type of functions

This ch 1 maths class 12 concerns different types of functions like constant function, polynomial function, identity function, rational function, modulus function, signum function, etc. this chapter also contains the concept of one-one (or injective), onto (or subjective), one-one and onto (or bijective) functions. The concept of addition, subtraction, multiplication, and division of two functions have also been discussed. to get command on these concepts you can refer to NCERT solutions for class 12 maths chapter 1.

- composition of function and Invertible function

we understand the concept of composition of a function in this chapter of class 12 NCERT. also we get a good hold of invertible functions concepts in this chapter. for questions, you can browse class 12 NCERT solutions.

- Binary operations

this ch 1 maths class 12 also includes concepts of binary operations. terms like commutative, associative invertible, inverse, identity are also discussed in class 12 NCERT. you can refer to class 12 NCERT solutions for questions about these concepts.

Topics enumerated in class 12 NCERT are very important and students are advised to go through all the concepts discussed in the topics. Questions related to all the above topics are covered in the class 12 maths ch 1 question answer.

**Also read, **

- NCERT Exemplar Class 12 Chemistry Solutions
- NCERT Exemplar Class 12 Mathematics Solutions
- NCERT Exemplar Class 12 Biology Solutions
- NCERT Exemplar Class 12 Physics Solutions

Chapter 1 | Relations And Functions |

Chapter 2 | |

Chapter 3 | |

Chapter 4 | |

Chapter 5 | |

Chapter 6 | |

Chapter 7 | |

Chapter 8 | |

Chapter 9 | |

Chapter 10 | |

Chapter 11 | |

Chapter 12 | |

Chapter 13 |

- NCERT solutions for class 12 mathematics
- NCERT solutions class 12 chemistry
- NCERT solutions for class 12 physics
- NCERT solutions for class 12 biology

Class 12 relations and functions NCERT solutions is helpful for the students who wish to perform well in the CBSE 12 board examination. Some guidelines to follow to make the best use of NCERT solutions:

- Before solving an exercise, first, go through the examples that are given in the NCERT class 12 maths textbook.
- Also, try to solve every exercise including miscellaneous exercise, NCERT chapter examples, miscellaneous examples on your own, if you are not able to do it, then you can take help of NCERT solutions for class 12 maths chapter 1 Relations and functions.
- Reading the solutions is not enough, you have to solve it on your own, even after reading the solutions
- Stick to the syllabus that is provided by NCERT and solve it completely including all examples and all exercises
- If you have solved all NCERT questions, then you can solve previous years papers of CBSE board to get familiar with the pattern of the exam.
- NCERT solutions for class 12 maths chapter 1 pdf download will also be made available soon.

Happy learning !!!

1. How the NCERT solutions are helpful in the board exam ?

As CBSE board exam paper is designed entirelly based on NCERT textbooks and most of the questions in CBSE board exam are directly asked from NCERT textbook, students must know the relations and functions class 12 questions and answers very well to perform well in the exam. NCERT solutions are not only important when you stuck while solving the problems but students will get how to answer in the board exam in order to get good marks in the board exam.

2. What are the important topics in chapter relations and functions?

Definitions of relations and functions, types of relations, types of functions, composition of functions, invertible function and binary operations are the important topics in this maths chapter 1 class 12. these topics are important because concepts are used in calculus and other topics as well as exams therefore students are recommended ncert solutions and ncert exercise to get command on the concepts.

3. What concepts are covered in the class 12 maths chapter 1 NCERT solutions?

The NCERT Solutions for maths chapter 1 class 12 provide in-depth explanations of several important concepts, including types of relations, different types of functions, composition of functions, invertible functions, and binary operations. These solutions for class 12 maths ch 1 are created by a team of highly qualified and experienced teachers and their primary goal is to assist students in achieving a high score on their Class 12 Maths board exams.

4. What are the key attributes of chapter 1 maths class 12 ncert solutions?

Following are some key attributes of relation and function class 12 solutions.

The maths chapter 1 class 12 are created by experienced subject matter experts who possess a deep understanding of the key concepts.

The solutions for relation function class 12 are presented in a clear and straightforward language to make it easy for students to grasp the methods for solving complex problems.

The step-by-step solutions for class 12 relation and function are designed based on the marks weightage assigned by the CBSE exam, ensuring that students can maximize their scores on the exam.

5. Is relying solely on NCERT Solutions for Class 12 Maths Chapter 1 sufficient for preparing for the CBSE exams?

The Maths class 12 relations and functions ncert solutions are updated with the latest CBSE guidelines, ensuring that all the important topics are covered. The chapter contains four exercises, offering students a variety of problems to solve on their own. The class 12 maths ch 1 ncert solutions are structured to build confidence in students ahead of the CBSE exams. For ease, students can study relations and functions class 12 ncert pdf online and offline in both modes.

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5 minHave a question related to CBSE Class 12th ?

Hello aspirant,

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Thank you

Hope this information helps you.

hello,

Yes you can appear for the compartment paper again since CBSE gives three chances to a candidate to clear his/her exams so you still have two more attempts. However, you can appear for your improvement paper for all subjects but you cannot appear for the ones in which you have failed.

I hope this was helpful!

Good Luck

Hello dear,

If you was not able to clear 1st compartment and now you giving second compartment so YES, you can go for your improvement exam next year but if a student receives an improvement, they are given the opportunity to retake the boards as a private candidate the following year, but there are some requirements. First, the student must pass all of their subjects; if they received a compartment in any subject, they must then pass the compartment exam before being eligible for the improvement.

As you can registered yourself as private candidate for giving your improvement exam of 12 standard CBSE(Central Board of Secondary Education).For that you have to wait for a whole year which is bit difficult for you.

Positive side of waiting for whole year is you have a whole year to preparing yourself for your examination. You have no distraction or something which may causes your failure in the exams. In whole year you have to stay focused on your 12 standard examination for doing well in it. By this you get a highest marks as a comparison of others.

Believe in Yourself! You can make anything happen

All the very best.

Hello Student,

I appreciate your Interest in education. See the improvement is not restricted to one subject or multiple subjects and we cannot say if improvement in one subject in one year leads to improvement in more subjects in coming year.

You just need to have a revision of all subjects what you have completed in the school. have a revision and practice of subjects and concepts helps you better.

All the best.

Hi,

You just need to give the exams for the concerned two subjects in which you have got RT. There is no need to give exam for all of your subjects, you can just fill the form for the two subjects only.

Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.

4 Jobs Available

The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.

4 Jobs Available

A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.

3 Jobs Available

GIS officer work on various GIS software to conduct a study and gather spatial and non-spatial information. GIS experts update the GIS data and maintain it. The databases include aerial or satellite imagery, latitudinal and longitudinal coordinates, and manually digitized images of maps. In a career as GIS expert, one is responsible for creating online and mobile maps.

3 Jobs Available

The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.

Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

3 Jobs Available

Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.

3 Jobs Available

If you are intrigued by the programming world and are interested in developing communications networks then a career as database architect may be a good option for you. Data architect roles and responsibilities include building design models for data communication networks. Wide Area Networks (WANs), local area networks (LANs), and intranets are included in the database networks. It is expected that database architects will have in-depth knowledge of a company's business to develop a network to fulfil the requirements of the organisation. Stay tuned as we look at the larger picture and give you more information on what is db architecture, why you should pursue database architecture, what to expect from such a degree and what your job opportunities will be after graduation. Here, we will be discussing how to become a data architect. Students can visit NIT Trichy, IIT Kharagpur, JMI New Delhi.

3 Jobs Available

Individuals who opt for a career as a remote sensing technician possess unique personalities. Remote sensing analysts seem to be rational human beings, they are strong, independent, persistent, sincere, realistic and resourceful. Some of them are analytical as well, which means they are intelligent, introspective and inquisitive.

Remote sensing scientists use remote sensing technology to support scientists in fields such as community planning, flight planning or the management of natural resources. Analysing data collected from aircraft, satellites or ground-based platforms using statistical analysis software, image analysis software or Geographic Information Systems (GIS) is a significant part of their work. Do you want to learn how to become remote sensing technician? There's no need to be concerned; we've devised a simple remote sensing technician career path for you. Scroll through the pages and read.

3 Jobs Available

Budget analysis, in a nutshell, entails thoroughly analyzing the details of a financial budget. The budget analysis aims to better understand and manage revenue. Budget analysts assist in the achievement of financial targets, the preservation of profitability, and the pursuit of long-term growth for a business. Budget analysts generally have a bachelor's degree in accounting, finance, economics, or a closely related field. Knowledge of Financial Management is of prime importance in this career.

4 Jobs Available

The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.

Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

3 Jobs Available

An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.

3 Jobs Available

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available

Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.

3 Jobs Available

Individuals who opt for a career as a stock analyst examine the company's investments makes decisions and keep track of financial securities. The nature of such investments will differ from one business to the next. Individuals in the stock analyst career use data mining to forecast a company's profits and revenues, advise clients on whether to buy or sell, participate in seminars, and discussing financial matters with executives and evaluate annual reports.

2 Jobs Available

A Researcher is a professional who is responsible for collecting data and information by reviewing the literature and conducting experiments and surveys. He or she uses various methodological processes to provide accurate data and information that is utilised by academicians and other industry professionals. Here, we will discuss what is a researcher, the researcher's salary, types of researchers.

2 Jobs Available

Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.

5 Jobs Available

A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.

3 Jobs Available

Individuals who opt for a career as an environmental engineer are construction professionals who utilise the skills and knowledge of biology, soil science, chemistry and the concept of engineering to design and develop projects that serve as solutions to various environmental problems.

2 Jobs Available

A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.

2 Jobs Available

A Conservation Architect is a professional responsible for conserving and restoring buildings or monuments having a historic value. He or she applies techniques to document and stabilise the object’s state without any further damage. A Conservation Architect restores the monuments and heritage buildings to bring them back to their original state.

2 Jobs Available

A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.

2 Jobs Available

**Highway Engineer Job Description: **A Highway Engineer is a civil engineer who specialises in planning and building thousands of miles of roads that support connectivity and allow transportation across the country. He or she ensures that traffic management schemes are effectively planned concerning economic sustainability and successful implementation.

2 Jobs Available

Are you searching for a Field Surveyor Job Description? A Field Surveyor is a professional responsible for conducting field surveys for various places or geographical conditions. He or she collects the required data and information as per the instructions given by senior officials.

2 Jobs Available

Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.

6 Jobs Available

A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.

5 Jobs Available

Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.

4 Jobs Available

The audiologist career involves audiology professionals who are responsible to treat hearing loss and proactively preventing the relevant damage. Individuals who opt for a career as an audiologist use various testing strategies with the aim to determine if someone has a normal sensitivity to sounds or not. After the identification of hearing loss, a hearing doctor is required to determine which sections of the hearing are affected, to what extent they are affected, and where the wound causing the hearing loss is found. As soon as the hearing loss is identified, the patients are provided with recommendations for interventions and rehabilitation such as hearing aids, cochlear implants, and appropriate medical referrals. While audiology is a branch of science that studies and researches hearing, balance, and related disorders.

3 Jobs Available

An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.

3 Jobs Available

Are you searching for an ‘Anatomist job description’? An Anatomist is a research professional who applies the laws of biological science to determine the ability of bodies of various living organisms including animals and humans to regenerate the damaged or destroyed organs. If you want to know what does an anatomist do, then read the entire article, where we will answer all your questions.

2 Jobs Available

For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.

4 Jobs Available

Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.

3 Jobs Available

Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.

Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.

3 Jobs Available

Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.

A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.

3 Jobs Available

The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.

2 Jobs Available

A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.

2 Jobs Available

Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.

2 Jobs Available

An individual who is pursuing a career as a producer is responsible for managing the business aspects of production. They are involved in each aspect of production from its inception to deception. Famous movie producers review the script, recommend changes and visualise the story.

They are responsible for overseeing the finance involved in the project and distributing the film for broadcasting on various platforms. A career as a producer is quite fulfilling as well as exhaustive in terms of playing different roles in order for a production to be successful. Famous movie producers are responsible for hiring creative and technical personnel on contract basis.

2 Jobs Available

In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.

5 Jobs Available

In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion.

Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article.

3 Jobs Available

For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

3 Jobs Available

Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

3 Jobs Available

Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

3 Jobs Available

Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.

2 Jobs Available

Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

2 Jobs Available

A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.

2 Jobs Available

5 Jobs Available

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available

3 Jobs Available

A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.

2 Jobs Available

2 Jobs Available

The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.

2 Jobs Available

An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.

4 Jobs Available

An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.

4 Jobs Available

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available

3 Jobs Available

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack

3 Jobs Available

An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.

2 Jobs Available

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