RD Sharma Class 12 Exercise 1.1 relations Solutions Maths - Download PDF Free Online

Access premium articles, webinars, resources to make the best decisions for career, course, exams, scholarships, study abroad and much more with

Plan, Prepare & Make the Best Career Choices

# RD Sharma Class 12 Exercise 1.1 relations Solutions Maths - Download PDF Free Online

Edited By Kuldeep Maurya | Updated on Jan 18, 2022 01:21 PM IST

Solutions that are accurate and beneficial for exam prepared Sharma is one of the most well-known books in the country. RD Sharma books are detailed, informative, and contain step-by-step solutions for their problems.

RD Sharma Class 12th Exercise 1.1 deals with also need to solve exercise problems. This is where Career360 comes to help. It contains expert-created solutions of the chapter 'Relations.' They have plenty of example problems which the students can practice to develop their skills.

## RD Sharma Class 12 Solutions Chapter 1 Relations - Other Exercise

Chapter 1 Relations Ex 1.2

## Relations Excercise: 1.1

Relations Exercise 1.1 Question 1 (i)

Hint :
If $R$ is reflexive $\Rightarrow(a, a) \in R \text { for all } a \in A$
If $R$ is symmetric $\Rightarrow(a, b) \in R \Rightarrow(b, a) \in R_{\text {for all }} a, b \in A$
If $R$ is transitive $\Rightarrow(a, b) \in R \text { and }(b, c) \in R \Rightarrow(a, c) \in R \text { for all } a, b, c \in A$
Given : $R=\{(x, y): x \text { and } y \text { work at same place }\}$
Solution : A relation $R$ on set A is said to be reflexive if every element of A is related to itself.
Thus, $R$ is reflexive $\Rightarrow(a, a) \in R \text { for all } a \in A$
A relation $R$ on set A is said to be symmetric relation if $(a, b) \in R \Rightarrow(b, a) \in R$ for all $a, b \in A$
i.e. $a R b \Rightarrow b R a \text { for all } a, b \in A$
A relation $R$ on set A is said to be transitive relation
If $(a, b) \in R \; {\text {and }}\; (b, a) \in R \Rightarrow(c, a) \in R\; {\text {for all }} \; a, b, c \in A$
i.e $a R b$ and $b R c \Rightarrow a R c$ for all $a, b, c \in A$
Now, let's get back to the actual problem
For Reflexive :
$x$ and $x$ work at same place
Similarly, $y$ and $y$ work at the same place.
So, $R$ is reflexive.
For Symmetric :
It is given that $x$ and $y$ work at the same place.
So, we can say that $y$ and $x$ work at the same place.
So, $R$ is symmetric.
For Transitive:
Let $z$ be a person such that $y$ and $z$ work at the same place.
And we know that $x$ and $y$ work at the same place.
So, we can say that $x$ and $z$ work at the same place.
So, $R$ is Transitive.

Relations Exercise 1.1 Question 1 (ii)

Hint :
If $R$ is reflexive $\Rightarrow(a, a) \in R \text { for all } a \in A$
If $R$ is symmetric $\Rightarrow(a, b) \in R \Rightarrow(b, a) \in R_{\text {for all }} a, b \in A$
If $R$ is transitive $\Rightarrow(a, b) \in R \text { and }(b, c) \in R \Rightarrow(a, c) \in R \text { for all } a, b, c \in A$
Given :
$\mathrm{R}=\{(x, y): x \text { and } y \text { live in same locality\} }$
Solution :
A relation $R$ on set A is said to be reflexive if every element of A is related to itself.
Thus, $R$ is reflexive $\Leftrightarrow(a, a) \in R \; {\text {for all }} a \in A$
A relation $R$ on set A is said to be symmetric relation if $(a, b) \in R \Rightarrow(b, a) \in R$for all$a, b \in A$
I.e. $a R b \Rightarrow b R a$ for all $a, b \in A$
A relation $R$ on set A is said to be transitive relation if $(a, b) \in R \; {\text {and }}(b, c) \in R \Rightarrow(c, a) \in R$ for all $a, b, c \in A$
i.e $a R b \text { and } b R c \Rightarrow a R c \text { for all } a, b, c \in A$
For Reflexive:
$x$ and $x$ live in the same locality.
Similarly, $y$ and $y$ live in same locality
So, $R$ is reflexive.
For Symmetric:
$x$ and $y$ live in same locality.
So, we can easily say that $y$ and $x$ live in same locality.
So, $R$ is symmetric.
For Transitive:
Let $z$ be a person; $z \in A$ and $z$ and $y$ live in same locality
And it is given that $x$ and $y$ live in same locality
So, we can say that $x$ and $z$ live in the same locality.
So, $R$ is Transitive.

Relations Exercise 1.1 Question 1 (iii)

Neither reflexive, nor symmetric, not transitive.
Hints :
If $R$ is reflexive $\Rightarrow(a, a) \in R \; {\text {for all }} a \in A$
If $R$ is symmetric $\Rightarrow(a, b) \in R \Rightarrow(b, a) \in R \text { for all } a, b \in A$
If $R$ is transitive $\Rightarrow(a, b) \in R,(b, c) \in R \Rightarrow(a, c) \in R \text { for all } a, b, c \in A$
Given :
$R=\left\{(x, y): x\; {\text {is wife of }} y\right\}$
Solution :
A relation $R$ on set $A$ is said to be reflexive if every element of $A$ is related to itself.
Thus, $R$ is reflexive $\Leftrightarrow(a, a) \in R\; {\text {for all }} a \in A$.
A relation $R$ on set $A$ is said to be symmetric relation if $(a, b) \in R \Rightarrow(b, a) \in R$for all
$a, b \in A$
i.e; $a R b \Rightarrow b R a \text { for all } a, b \in A$
A relation $R$ on set $A$ is said to be transitive relation if $(a, b) \in R$ and $(b, c) \in R \Rightarrow$ $(c, a) \in R$ for all $a, b, c \in A$
i.e; $a R b$ and $b R c \Rightarrow a R c$ for all $a, b, c \in A$
For Reflexive:
$x$ is not wife of $x$ and $y$ is not wife of $y$
So, $R$ is not reflexive.
For Symmetric:
$x$ is the wife of $y$ but $y$ is not the wife of $x$.
So, $R$ is not symmetric.
For Transitive:
Let $z$ be a person; $\mathrm{z} \in A$ such that $y$ is the wife of $z$ .
And it is given that $x$ is the wife of $y$ but this case is not possible. Also, here we can’t show x is the wife of z.
So, R is not transitive.

Relations Exercise 1.1 Question 1 (iv)

Neither reflexive, nor symmetric nor transitive.
Hints :
If $R$ is reflexive $\Rightarrow(a, a) \in R\; {\text {for all }} a \in A$
If $R$ is symmetric $\Rightarrow(a, b) \in R \Rightarrow(b, a) \in R\; {\text {for all }} a, b \in A$
If $R$ is transitive $\Rightarrow(a, b) \in R,(b, c) \in R \Rightarrow(a, c) \in R \text { for all } a, b, c \in A$
Given :
$R=\{(x, y): x \text { is father of } \mathrm{y}\}$
Solution :
A relation $R$ on set $A$ is said to be reflexive if every element of $A$ is related to itself.
Thus, $R$ is reflexive $\Leftrightarrow(a, a) \in R_{\text {for all }} a \in A$
A relation $R$ on set $A$ is said to be symmetric relation if $(a, b) \in R \Rightarrow(b, a) \in R$ for all $a, b \in A$
i.e $a R b \Rightarrow b R a$ for all $a, b \in A$
A relation $R$ on set $A$ is said to be transitive relation if $(a, b) \in R$ and $(b, c) \in R \Rightarrow$ $(c, a) \in R$ for all $a, b, c \in A$
i.e $a R b$ and $b R c \Rightarrow a R c$ for all $a, b, c \in A$
For Reflexive:
$x$ is not father of $x$ and $y$ is not father of $y$
So, $R$ is not reflexive
For Symmetric:
It is given that $x$ is the father of $y$.
But we can say that $y$ is not the father of $x$.
So $R$ is not symmetric.
For Transitive:
Let $z$ be a person; $z \in A$ such that $y$ is father of $z$ and it is given that x is a father of y.
Then $x$ is grandfather of $z$
So, $R$ is not Transitive.

Relations Exercise 1.1 Question 2

$R_{1}$ is reflexive and transitive but not symmetric
$R_{2}$ is reflexive, symmetric, and transitive.
$R_{3}$ is transitive but neither reflexive nor symmetric
$R_{4}$ is neither reflexive nor symmetric nor transitive
Hint :
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $(a, b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a, b)$and $(b,c)$ $\in\; R$, then $(a, c)$$\in\; R$ for every $a, b \in A$
Given :
Set $A=\{a, b, c\}$
\begin{aligned} &R_{1}=\{(a, a),(a, b),(a, c),(b, b),(b, c),(c, a),(c, b),(c, c)\} \\ &R_{2}=\{(a, a)\} \\ &R_{3}=\{(b, c)\} \\ &R_{4}=\{(a, b),(b, c),(c, a)\} \end{aligned}
Consider $R_{1}=\{(a, a),(a, b),(a, c),(b, b),(b, c),(c, a),(c, b),(c, c)\}$
Reflexive :
Given $(a, a),(b, b)$ and $(c, c) \in R_{1}$
So, $R_{1}$ is reflexive.
For Symmetric:
We see that the ordered pairs obtained by interchanging the components of $R_{1}$ are not in $R_{1}$.
For ex : $(a, b) \in R_{1} \text { but }(b, a) \notin R_{1}$
So, $R_{1}$ is not symmetric.
For Transitive:
Here, $(a, b) \in R_{1} \text { and }(b, c) \in R_{1}$ but $(a, c) \in R_{1}$
So, $R_{1}$ is transitive
(ii) Consider $R_{2}$
$R_{2}=\{(a, a)\}$
Reflexive:
clearly $(a, a) \in R_{2}$
So, $R_{2}$ is reflexive.
Symmetric:
Clearly $(a,a)\in \; R_{2}$
So, $R_{2}$ is symmetric.
Transitive:
$R_{2}$ is a transitive relation, since there is only one element in it.
(iii) Consider $R_{3}$
$R_{3}=\{(b, c)\}$
Reflexive:
Here neither $(b, b) \notin R_{3}$ nor $(c, c) \notin R_{3}$
So, $R_{3}$ is not reflexive
Symmetric:
Here neither $(b, c) \in R_{3}$ nor $(c, b) \notin R_{3}$
So, $R_{3}$ is not symmetric.
Transitive:
$R_{3}$ has only one element
Hence $R_{3}$ is transitive.
(iv) Consider $R_{4}=\{(a, b),(b, c),(c, a)\}$
Reflexive:
Here $(a, b) \in R_{4} \text { but }(b, a) \notin R_{4}$
So, $R_{4}$ is not reflexive
Symmetric:
Here $(a, b) \in R_{4} \text { but }(b, a) \notin R_{4}$
So,$R_{4}$ is not symmetric
Transitive:
Here $(a, b) \in R_{4},(b, c) \in R_{4}$ but $(a, c) \notin R_{4}$
Hence $R_{4}$ is not transitive.

Relations Exercise 1.1 Question 3 (i)

Answer: $R_{1}$is symmetric but neither reflexive nor transitive.
Hint :
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $(a,b)$ is true then $(b, a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a,b)$ and $(b, c) \in R$, then $(a, c) \in R$ for every $a, b, c \in A$
Given :
$R_{1} \text { on } \mathrm{Q}_{0}$ defined by $(a, b) \in R_{1} \Leftrightarrow a=\frac{1}{b}$
Solution :
Reflexivity:
Let $a$ be an arbitrary element of $R_{1}$
Then, $a \in R_{1}$
$a \neq \frac{1}{a} \text { for all } a \in Q_{0}$
So, $R_{1}$ is not reflexive
Symmetry:
Let $(a, b) \in R_{1}$
Therefore, we can write 'a' as $a=\frac{1}{b}$
$b=\frac{1}{a}$
Then $(b, a) \in R_{1}$
So, $R_{1}$ is symmetric.
For Transitive:
Let $(a, b) \in R_{1} \text { and }(b, c) \in R_{1}$
\begin{aligned} &a=\frac{1}{b} \text { and } b=\frac{1}{c} \\ &a=\frac{1}{\left(\frac{1}{c}\right)} \Rightarrow c \\ &a \neq \frac{1}{c} \\ &(a, c) \notin R_{1} \end{aligned}
So, $R_{1}$ is not transitive.

Relations Exercise 1.1 Question 3 (ii)

Answer:$R_{2}$ is reflexive and symmetric but not transitive.
Hint :
A relation R on set A is
Reflexive relation:

If $(a, a) \in R$ for every $a \in A$
Symmetric relation:
if $(a, b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a, b)$ and $(b, c) \in R$, then $(a, c) \in R$ for every $a, b, c \in A$
Given:
$R_{2} \text { on } Z$ defined by $(a, b) \in R_{2} \Leftrightarrow|a-b| \leq 5$
Solution :
Reflexivity:
Let a be an arbitrary element of $R_{2}$
Then, $a \in R_{2}$
On applying the given condition, we get
$|a-a|=0 \leq 5$
So, $R_{2}$ is reflexive
Symmetry :
Let $(a, b) \in R_{2} \quad|a-b| \leq 5$
[Since $|a-b|=|b-a|$ ]
Then $|b-a| \leq 5$
$(b, a) \in R_{2}$
So, $R_{2}$ is symmetric.
Transitivity :
Let $(1,3) \in R_{2} \text { and }(3,7) \in R_{2}$
$|1-3| \leq 5 \text { and }|3-7| \leq 5$
But, $|1-7| \leq 5$
$(1,7) \neq R_{2}$
So, $R_{2}$ is not transitive.

Relations Exercise 1.1 Question 3 (iii)

$R_{3}$ is reflexive but neither symmetric nor transitive
Hint :
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$ for every $a \in R$
Symmetric relation:
If $(a,b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
if $(a,b)$ and $(b, c) \in R$, then $(a, c) \in R$ for every $\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{A}$
Given : $R_{3}$ on R defined by $(a, b) \in R_{3} \Leftrightarrow a^{2}-4 a b+3 b^{2}=0$
Solution :
Reflexivity:
Let a be an arbitrary element of $R_{3}$
Then, $a \in R_{3}$
$a^{2}-4 a \times a+3 a^{2}=0$
So, $R_{3}$ is reflexive
Symmetry :
Let : $(a, b) \in R_{3}$
$a^{2}-4 a b+3 b^{2}=0$
But $b^{2}-4 b a+3 a^{2} \neq 0$ for all $a, b \in R$
So, $R_{3}$ is not symmetric.
Transitivity:
Let $(a, b) \in R_{3} \text { and }(b, c) \in R_{3}$
\begin{aligned} &a^{2}-4 a b+3 b^{2}=0\; \; \; \; \; \; ...(i)\\ &\text { and } b^{2}-4 b c+3 c^{2}=0 \; \; \; \; \; \; \; ....(ii) \end{aligned}
Adding (i) and (ii) we get,
\begin{aligned} &\qquad a^{2}-4 a b+3 b^{2}+b^{2}-4 b c+3 c^{2}=0 \\ &\Rightarrow a^{2}-4 a b+4 b^{2}-4 b c+3 c^{2}=0 \\ &\text { But } a^{2}-4 a c+3 c^{2}=-4 a c-4 a b+4 b^{2}-4 b c \neq 0 \\ &\Rightarrow a^{2}-4 a c+3 c^{2} \neq 0 \end{aligned}
$(a, c) \notin R_{3}$
So, $R_{3}$ is not transitive.

Relations Exercise 1.1 Question 4

Answer: $R_{1}$ is reflexive but neither symmetric nor transitive
$R_{2}$ is symmetric but neither reflexive nor transitive
$R_{3}$ is transitive but neither reflexive nor symmetric
Hint :
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$ for every $a \in R$
Symmetric relation:
If $(a,b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
if $(a,b)$ and $(b, c) \in R$, then $(a, c) \in R$ for every $\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{A}$
Given :
\begin{aligned} &R_{1}=\{(1,1),(1,3),(3,1),(2,2),(2,1),(3,3)\} \\ &R_{2}=\{(2,2),(3,1),(1,3)\} \\ &R_{3}=\{(1,3),(3,3)\} \end{aligned}
Solution :
Consider $R_{1}$
$R_{1}=\{(1,1),(1,3),(3,1),(2,2),(2,1),(3,3)\}$
Reflexivity:
Here $(1,1),(2,2),(3,3) \in R$
So, $R_{1}$ is Reflexive
Symmetric :
Here $(2,1) \in R_{1}$
But, $(1,2) \notin R_{1}$
So, $R_{1}$ is not symmetric
Transitivity:
\begin{aligned} &\text { Here }(2,1) \in R_{1} \text { and }(1,3) \in R_{1} \\ &\text { But }(2,3) \notin R_{1} \end{aligned}
So, $R_{1}$ is not transitive.
Now consider $R_{2}$
$R_{2}=\{(2,2),(3,1),(1,3)\}$
Reflexivity :
Clearly, $(1,1) \text { and }(3,3) \notin R_{2}$
So, $R_{2}$ is not reflexive.
Symmetric :
$\text { Here }(1,3) \in R_{2} \text { and }(3,1) \in R_{2}$
So, $R_{2}$ is symmetric.
Transitivity:
\begin{aligned} &\text { Here }\\ &(1,3) \in R_{2} \text { and }(3,1) \in R_{2}\\ &\text { But }(3,3) \notin R_{2} \end{aligned}
So, $R_{2}$ is not transitive.
Now consider $R_{3}$
$R_{3}=\{(1,3),(3,3)\}$
Reflexivity:
Clearly, $(1,1) \notin R_{3}$
So, $R_{3}$ is not reflexive.
Symmetry:
Here $(1,3) \in R_{3} \text { but }(3,1) \notin R_{3}$
So,$R_{3}$ is not symmetric.
Transitivity:
Here $(1,3) \in R_{3} \text { and }(3,3) \in R_{3}$
But $(1,3) \in R_{3}$
So, $R_{3}$ is transitive.

Relation Exercise 1.1 Question 5 (i)

Hint:
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $(a,b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a,b)$ and $(b, c) \in R$, then $(a, c) \in R$ for every $a, b, c \in A$
Given : $a R b \text { if } a-b>0$
Solution :
Reflexivity:
Let $a$ be an arbitrary element of $R$
Then,$a \in A$
But $a-a=0 \ngtr 0$
So, this relation is not reflexive.
Symmetry :
Let
\begin{aligned} &(a, b) \in R \\ &a-b>0 \\ &-(b-a)>0 \\ &b-a<0 \end{aligned}
So, the given relation is not symmetric.
Transitivity :
\begin{aligned} &\text { Let }(a, b) \in R \; {\text {and }}(b, c) \in R \\ &\text { Then, } a-b>0 \; \; \; \; \; \; \; ...(i)\\ &\; \; \; \; \qquad b-c>0 \; \; \; \; \; \; \;...(ii) \end{aligned}
Adding eq (i) & (ii), we get
\begin{aligned} &a-b+b-c>0 \\ &a-c>0 \\ &(a, c) \in R \end{aligned}
So, the given relation is transitive.

Relation Exercise 1.1 Question 5 (ii)

Reflexive and symmetric but not transitive.
Hint:
A relation R on set A is
Reflexive relation:
If $(a,a)\in \; R$ for every $a\in \; R$
Symmetric relation:
If $(a,b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a,b)$ and $(b, c) \in R$, then $(a, c) \in R$ for every $\mathrm{a}, \mathrm{b}, \mathrm{c} \in A$
Given :
$a R b \text { if } 1+a b>0$
Solution :
Reflexivity:
Let $a$ be an arbitrary element of $R$
Then, $a\in \; R$
\begin{aligned} &1+a \times a>0 \\ &1+a^{2}>0 \end{aligned}
Since, Square of any number is positive.
So, the given relation is reflexive.
Symmetry:
Let $(a, b) \in R$
\begin{aligned} &1+a b>0 \\ &1+b a>0 \\ &(b, a) \in R \end{aligned}
So, the given relation is symmetric.
Transitivity:
\begin{aligned} &\text { Let }(a, b) \in R \text { and }(b, c) \in R \\ &\text { Let } \mathrm{a}=-8, \mathrm{~b}=-2, \mathrm{c}=\frac{1}{4} \\ &\text { Then } 1+a b>0 \text { i.e; } 1+(-8)(-2)=17>0 \\ &\text { and } 1+b c>0 \text { i.e; } 1+(-2) \frac{1}{4}=\frac{1}{2}>0 \text { But } 1+a c \neq 0 \text { i.e; } 1+(-8) \frac{1}{4}=-1 \ngtr 0 \\ &(a, c) \epsilon R \end{aligned}

So, the given relation is not transitive.

Relation Exercise 1.1 Question 5 (iii)

Answer: Transitive neither reflexive nor symmetric.
Hint:
A relation R on set A is
Reflexive relation:
If $(a,a)\in \; R$ for every $a\in \; R$
Symmetric relation:
If $(a,b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a,b)$ and $(b, c) \in R$, then $(a, c) \in R$ for every $\mathrm{a}, \mathrm{b}, \mathrm{c} \in A$
Given : $a R b \text { if }|a| \leq b$
Solution :
Reflexivity:
Let -a be an arbitrary element of R
Then $-a \in R$
$\Rightarrow|-a| \neq-a$
So, $R$ is not reflexive
Symmetry:
Let $(a, b) \in R$
\begin{aligned} &|a| \leq b \\ &|b| \lessgtr a \text { for all } a, b \in R \\ &(b, a) \notin R \end{aligned}
So, $R$ is not symmetric.
Transitivity:
\begin{aligned} &\text { Let }(a, b) \in R_{\text {and }}(b, c) \in R \\ &|a| \leq b \text { and }|b| \leq c \text { for } a, b, c \in R \end{aligned}
Multiplying the corresponding sides, we get
\begin{aligned} &|a| \times|b| \leq b c \\ &|a| \leq c \\ &(a, c) \in R \end{aligned}
Thus, $R$ is transitive

Relation Exercise 1.1 Question 6

$R$ is neither reflexive nor symmetric nor transitive.
Hint:
A relation R on set A is
Reflexive relation:
If $(a,a)\in \; R$ for every $a\in \; R$
Symmetric relation:
If $(a,b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a,b)$ and $(b, c) \in R$, then $(a, c) \in R$ for every $\mathrm{a}, \mathrm{b}, \mathrm{c} \in A$
Given : $R=\{(a, b): b=a+1\}$
Solution :
Let $a$ be an arbitrary element of set A.
Then,
$a=a+1$ cannot be true for all $a\; \in\; A$
$(a, a) \notin R$
So, $R$ is not reflexive on $A$
Symmetry :
\begin{aligned} &\text { Let }(a, b) \in R\\ &b=a+1\\ &a=b-1\\ &-a=-b+1\\ &\text { Thus }\\ &(b, a) \notin R \end{aligned}
So, $R$ is not symmetric on $A$
Transitivity :
Let $(1,2) \text { and }(2,3) \in R$
\begin{aligned} &2=1+1 \text { and } \\ &3=2+1 \text { is true } \\ &\text { But } 3 \neq 1+1 \\ &(1,3) \notin R \end{aligned}
So, $R$ is not transitive on $A$

Relation Exercise 1.1 Question 7

$R$ is neither reflexive nor symmetric nor transitive.
Hint:
A relation R on set A is
Reflexive relation:
If $(a,a)\in \; R$ for every $a\in \; R$
Symmetric relation:
If $(a,b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a,b)$ and $(b, c) \in R$, then $(a, c) \in R$ for every $\mathrm{a}, \mathrm{b}, \mathrm{c} \in A$
Solution :
Reflexive :
It is observed that $(1 / 2,1 / 2) \text { is } R \text { as } 1 / 2>(1 / 2)^{3}=\frac{1}{8}$
$\therefore R \text { is not reflexive. }$
Symmetric :
Now $(1,2) \in R\left(\text { as } 1<2^{3}=8\right)$
But $(2,1) \notin R\left(\text { as } 2>1^{3}=1\right)$
$R$ is not symmetric
Transitive:
We have $\left(3, \frac{3}{2}\right),\left(\frac{3}{2}, \frac{6}{5}\right) \operatorname{in} R \text { as } 3<\left(\frac{3}{2}\right)^{3}$
and $\left(\frac{3}{2}\right)<\left(\frac{6}{5}\right)^{3}$
but $\left(3, \frac{6}{5}\right) \notin R \text { as } 3>\left(\frac{6}{5}\right)^{3}$
$R$ is not transitive
Hence, $R$ is neither reflexive, nor symmetric nor transitive

Relations Exercise 1.1 Question 8

Hence, prove every identity relation on a set is reflexive but the converse is not necessarily true.
Hint:
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$ for every$a \in A$
Symmetric relation:
If $\left ( a,b \right )$ is true then $\left ( b,a\right )$ is also true for every $a, b \in A$
Transitive relation:
$\text { If }(a, b) \text { and }(b, c) \in R, \text { then }(a, c) \in R \text { for every } \mathrm{a}, \mathrm{b}, \mathrm{c} \in A$
Given: Every Identity relation is reflexive.
Solution:
Let $A$ be a set $A=\{1,2,3\}$
Then, $I_{A}=\{(1,1),(2,2),(3,3)\}$
Identity relation is reflexive, since $(a, a) \in A \quad \forall a$
The converse of it need not be necessarily true.
Counter example:
Consider the set $A=\{1,2,3\}$
Here,
Relation$R=\{(1,1),(2,2),(3,3),(2,1),(1,3)\}$ is reflexive on $A$
However, $R$ is not an identity relation.

Relations Exercise 1.1 Question 9(i)

$R=\{(1,1),(2,2),(3,3),(4,4),(2,1)\}$
Hint:
A relation R on set A is
Reflexive relation:
If$(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $\left ( a,b \right )$is true then $\left ( b,a \right )$ is also true for every $a, b \in A$
Transitive relation:
$\text { If }(a, b) \text { and }(b, c) \in R, \text { then }(a, c) \in R \text { for every } \mathrm{a}, \mathrm{b}, \mathrm{c} \in A$
Given:
$A=\{1,2,3,4\}$
Solution:
The relation on $A$ having properties of being Reflexive, transitive but not symmetric is
$R=\{(1,1),(2,2),(3,3),(4,4),(2,1)\}$
Relation $R$ satisfies reflexivity and transitivity
$(1,1),(2,2),(3,3) \in R$
And $(2,2),(2,1) \in R \Rightarrow(2,1) \in R$
However,$(2,1) \in R_{\text {but }}(1,2) \notin R$

Relations Exercise 1.1 Question 9 (ii)

Answer: $R=\{(1,2),(2,1)\}$
Hint:
A relation R on set A is
Reflexive relation:
If$(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $\left ( a,b \right )$ is true then $\left ( b,a \right )$ is also true for every $a, b \in A$
Transitive relation:
$\text { If }(a, b) \text { and }(b, c) \in R, \text { then }(a, c) \in R \text { for every } \mathrm{a}, \mathrm{b}, \mathrm{c} \in A$
Given:
$A=\{1,2,3\}$
Solution:
The relation on A having properties of being symmetric but neither transitive nor reflexive.
Let $R=\{(1,2),(2,1)\}$
Now, $(1,2) \in R,(2,1) \in R$
So, it is symmetric.
Clearly $R$ is not transitive $(1,2) \in R,(2,1) \in R \text { but }(1,1) \notin R$

Relations Exercise 1.1 Question 9(iii)

$R=\{(1,1),(2,2),(3,3),(4,4),(1,2),(2,1)\}$
The relation $R$ is an equivalence relation on $A$
Hint:
A relation $R$ on set $A$ is
Reflexive relation:
If $(a, a) \in R$for every $a \in A$
Symmetric relation:
If $\left ( a,b \right )$ is true then $\left ( b,a \right )$ is also true for every $a, b \in A$
Transitive relation:
$\text { If }(a, b) \text { and }(b, c) \in R, \text { then }(a, c) \in R \text { for every } \mathrm{a}, \mathrm{b}, \mathrm{c} \in A$
Given:
$A=\{1,2,3,4\}$
Solution:
The relation on $A$ having properties of being
Symmetric, reflexive and transitive is
$R=\{(1,1),(2,2),(3,3),(4,4),(1,2),(2,1)\}$
The relation $R$ is an equivalence relation on $A$

Relations Exercise 1.1 Question 10

Domain of $R$ is x ∈ N where x∈$\left \{ 1,2,3............20 \right \}$and
range of $R$ is y ∈ N where $\left \{ 39,37,35,......3,1 \right \}$
The relation having properties of being neither symmetric nor transitive nor reflexive.
Hint:
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $\left ( a,b \right )$is true then $\left (b,a\right )$ is also true for every $a \in A$
Transitive relation:
$\text { If }(a, b) \text { and }(b, c) \in R, \text { then }(a, c) \in R \text { for every } \mathrm{a}, \mathrm{b}, \mathrm{c} \in A$
Given:
$R=\{(x, y): x, y \in N, 2 x+y=41\}$
Solution:
The domain of $R \text { is } x \in N, \text { such that } 2 x+y=41$
$x=(41-y) / 2$
Since $y \in N$, largest value that $x$ can take corresponds to the smallest value that $y$ can take.
$\therefore \quad x=\{1,2,3 \ldots .20\}$
Range $R$ of is $y \in N$such that
Since
$\begin{gathered} 2 x+y=41 \\ y=41-2 x \\ x=\{1,2,3 \ldots 20\} \end{gathered}$
$y=\{39,37,35, \ldots . .3,1\}$
Since $(2,2) \notin R, R$,is not reflexive.
Also, since $(1,39) \in R(39,1) \notin R, R$ is not symmetric.
Finally, since $(15,11) \in R \text { and }(11,19) \in R \text { but }(15,19) \notin R, R$ is not transitive.

Relations Exercise 1.1 Question 11

No, it is not true that every relation which is symmetric and transitive is also reflexive.
Hint:
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $(a, b)$ is true then$(b,a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a, b)$and $(b,a)$ $\in R$, then $(a, c) \in R$ for every $a, b \in A$
Given:
Answer that, whether every relation which is symmetric and transitive is also reflexive.
Solution:
No, it is not necessary that a relation that is symmetric and transitive is reflexive as well.
For example:
$R=\{(1,1),(1,2),(2,1)\} \text { on } A=\{1,2\}$ is symmetric and transitive but not reflexive.
Because $(2,2) \notin R$

Relations Exercise 1.1 Question 13

Hence proved, the relation “$\geq$” on the set $R$ of all real numbers is reflexive and transitive but not symmetric.
Hint:
A relation $R$ on set $A$ is
Reflexive relation:
If $(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $(a, b)$ is true then $(b, a)$is also true for every $a, b \in A$
Transitive relation:
If $(a, b) \text { and }(b, c) \in R$ then $(a, c) \in R$ for every $a, b, c \in A$
Given:
Relation is ”$\geq$”on the $R$ of all real numbers.
Solution:
Reflexivity:
Let $a \in R$
$a \geq a$
"$\geq$ "is reflexive.
Transitive:
Let $a, b, c \in R$
Such that $a \geq b \text { and } b \geq c$
Then$a \geq c$
$" \geq "$ is transitive.
Symmetric: Let $a, b \in R$
Such that $a \geq b \text { but } b \ngeqslant a$
$" \geq "$not symmetric

Relations Exercise 1.1 Question 14(ii)

Answer:$R=\left\{(a, b): a^{3} \geq b^{3}\right\}$
Hint:
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$for every $a \in A$
Symmetric relation:
If $\left ( a,b \right )$ is true then $\left (b,a\right )$ is also true for every $a, b \in A$
Transitive relation:
$\text { If }(a, b) \text { and }(b, c) \in R \text { , then }(a, c) \in R \text { for every } \mathrm{a}, \mathrm{b}, \mathrm{c} \in A$
Given:
We have to give the example of a relation which is reflexive and transitive but not symmetric.
Solution:
The relation having properties of being reflexive and transitive but not symmetric.
Define a relation $R$ in $R$ as:
$R=\left\{(a, b): a^{3} \geq b^{3}\right\}$
Clearly $(a, a) \in R \text { as } a^{3}=a^{3}$
Therefore $R$ is reflexive.
Now $(2,1) \in R\left(\text { as } 2^{3} \geq 1^{3}\right)$
But $(1,2) \notin R\left(\text { as } 1^{3}<2^{3}\right)$
Therefore $R$ is not symmetric.
Now let
\begin{aligned} &(a, b), \quad(b, c) \in R \\ &a^{3} \geq b^{3} \text{and }b^{3} \geq c^{3} \text { then } a^{3} \geq c^{3} \quad(a, c) \in R \end{aligned}

$R$ is transitive.
Hence, relation $R$ is reflexive and transitive but not symmetric.

Relations Exercise 1.1 Question 14(iii)

Answer: $R=\{(-5,-6),(-6,-5),(-5,-5)\}$
Hint:
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$for every $a \in A$
Symmetric relation:
If $\left ( a,b \right )$ is true then $\left (b,a\right )$ is also true for every $a, b \in A$
Transitive relation:
$\text { If }(a, b) \text { and }(b, c) \in R \text { , then }(a, c) \in R \text { for every } \mathrm{a}, \mathrm{b}, \mathrm{c} \in A$
Given:
We have to give the example of a relation which is symmetric and transitive but not reflexive.
Solution:
The relation having properties of being symmetric and transitive but not reflexive.
Let $A=\{-5,-6\}$
Define a relation $R$ on $A$ as
$R=\{(-5,-6),(-6,-5),(-5,-5)\}$
Relation $R$ is not reflexive as $(-6,-6) \notin R$
Relation $R$ is symmetric as $(-5,-6),(-6,-5) \in R$
It is seen that $(-5,-6),(-6,-5) \in R$
Also $(-5,-5) \in R$
The relation $R$ is transitive.
Hence relation $R$ is symmetric and transitive but not reflexive.

Relations Exercise 1.1 Question 14(iv)

Answer: $R=\{(5,6),(6,5)\}$
Hint:
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $\left ( a,b \right )$ is true then $\left ( b,a \right )$is also true for every $a, b \in A$
Transitive relation:
If $(a, b) \text { and }(b, c) \in R$, then $(a, c) \in R$ for every $a, b, c \in A$
Given:
$\text { Let } A=\{5,6,7\} \text { . }$
Solution:
$\\\text{Define a relation R on A as R}=\{(5,6),(6,5)\}. \\\text{Relation R is not reflexive as }(5,5),(6,6),(7,7) \notin \mathrm{R}.$
\begin{aligned} &\text { Now, as }(5,6) \in \mathrm{R} \text { and also }(6,5) \in \mathrm{R}, \mathrm{R} \text { is symmetric. }\\ &\Rightarrow(5,6),(6,5) \in \mathrm{R}, \text { but }(5,5) \notin \mathrm{R} \end{aligned}
$\\\text{Therefore, R is not transitive. }\\ \text{Hence, relation R is symmetric but not reflexive or transitive.}$

Relations Exercise 1.1 Question 14(v)

Answer:$R=\{(a, b): a
Hint:
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$for every $a \in A$
Symmetric relation:
If $\left ( a,b \right )$ is true then $\left ( b,a\right )$is also true for every $a, b \in A$
Transitive relation:
If $(a, b) \text { and }(b, c) \in R, \text { then }(a, c) \in R \text { for every } \mathrm{a}, \mathrm{b}, \mathrm{c} \in A$
Given:
We have to give the example of a relation which is transitive but neither symmetric nor reflexive.
Solution:
The relation having properties of being transitive but neither symmetric nor reflexive.
Consider a relation R in $R$ defined as:
$R=\{(a, b): a
For any $a \in R$ we have $(a, a) \notin R$, since $a$ can’t be strictly less than $a$ itself.
Infact $a=a$
∴ Relation $R$ is not reflexive.
Now, $(1,2) \in R(\text { as } 1<2)$
But 2 is not less than 1
$(2,1) \notin R$
$R$ is not symmetric
Now let $(a, b),(b, c) \in R$
\begin{aligned} &a
$R$ is transitive.

Relation Exercise 1.1 Question 15

Answer :$R=\{(1,2),(2,3),(1,1),(2,2),(3,3),(3,2),(2,1),(1,3),(3,1)\}$
Hint :
A relation R on set A is
Reflexive relation: $a, b, c \in A$
If $(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $(a,b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a,b)$ and , then $(b, c) \in R$ for every $(a, c) \in R$
Given :
\begin{aligned} &\text { Relation } R=\{(1,2),(2,3)\} \text { on the set }\\ &A=\{1,2,3\} \end{aligned}
Solution :
To make $R$ reflexive we will add $(1,1),(2,2),(3,3)$ to get $R^{\prime}=\{(1,2),(2,3),(1,1),(2,2),(3,3)\}$ is reflexive.
Again, to make $R$ symmetric we will add $(3,2)$ and $(2,1)$ $R^{\prime \prime}=\{(1,2),(2,3),(1,1),(2,2),(3,3),(3,2),(2,1)\}$is reflexive and symmetric .
To make $R$ transitive we will add $(1,3)$ and $(3,1)$ $R^{\prime \prime \prime}=\{(1,2),(2,3),(1,1),(2,2),(3,3),(3,2),(2,1),(1,3),(3,1)\}$ is reflexive and symmetric and transitive.

Relation Exercise 1.1 Question 16

only 1 ordered pair maybe added to $R$ so that it may become a transitive relation on $A$
Hint:
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $(a,b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a,b)$ and $(b, c) \in R$, then $(a, c) \in R$ for every $a, b, c \in A$
Given :$A=\{1,2,3\}$
Solution :
$R=\{(1,2),(1,1),(2,3)\}$ be a relation on $A$
To make $R$ transitive we shall add $(1,3)$ only $R^{\prime}=\{(1,2),(1,1),(2,3),(1,3)\}$
As we know,
Transitive relation
$\\\text {x=y} \; and \; \text {y=z}\\ \text {Then} \; \text {x=z}$

Note: for $R$ to be transitive $(a,c)$ must be in $R$ because $(a, b) \in R \text { and }(b, c) \in R$ So, $(a,c)$ must be in $R$

Relation Exercise 1.1 Question 17

At least 3 ordered pairs must be added for $R$ to be reflexive and transitive
Hint:
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $(a,b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a,b)$ and $(b, c) \in R$, then $(a, c) \in R$ for every $a, b, c \in A$
Given: Minimum number of order pair that makes R reflexive and transitive.
Solution:
A relation $R$ in $A$ is said to be reflexive if $aRa$ for all $a \in A$
$R$ is said to be transitive if $aRb$ and bRc then aRc for all $a, b, c \in A$
Hence for $R$ to be reflexive $(b,b)$ and $(c,c)$must be there in set $R$
Also for $R$ to be transitive $(a,c)$ must be in $R$ because $(a, b) \in R(b, c) \in R$
So, $(a,c)$ must be in $R$
So, at least 3 ordered pairs must be added for $R$ to be reflexive and transitive.

Relation Exercise 1.1 Question 18 (i)

Hint :
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $(a,b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a,b)$ and $(b, c) \in R$ then $(a, c) \in R$ for every $a, b, c \in A$
Given :$x>y, x, y \in N(x, y) \in\{(2,1),(3,1) \ldots \ldots(3,2),(4,2) \ldots\}$
Solution :
This is not reflexive as $(1,1),(2,2) \ldots \ldots$are absents.
This is not symmetric as $(2,1)$ is present but $(1,2)$ is absent
This is transitive as $(3,2) \in R \text { and }(2,1) \in R \text { also }(3,1) \in R$
Hence, this relation is not satisfying reflexivity and symmetricity.

Relation Exercise 1.1 Question 18 (ii)

Hint :
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $(a,b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a,b)$ and $(b, c) \in R$ then $(a, c) \in R$ for every $a, b, c \in A$
Given :
$x+y=10, x, y \in N(x, y) \in\{(9,1),(1,9),(2,8),(8,2),(3,7),(7,3),(4,6),(6,4),(5,5)\}$
Solution :
This is not reflexive as $(1,1),(2,2) \ldots \ldots$ are absent.
This only follows the condition of symmetry as $(1,9) \in R \operatorname{also}(9,1) \in R$
This is not transitive because $\{(1,9),(9,1)\} \in R \text { but }(1,1)$ is absent.
Hence, this relation is not satisfying reflexivity and transitivity.

Relation Exercise 1.1 Question 18 (iii)

Answer : Symmetric, Reflexive and transitive
Hint :
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $(a,b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a,b)$ and $(b, c) \in R$ then $(a, c) \in R$ for every $a, b, c \in A$
Given :
$x y \text { is square of an integer } \ x, y \in N$
$(x, y) \in\{(1,1),(2,2),(4,1),(1,4)(3,3),(9,1),(1,9),(4,4),(2,8),(8,2),(16,1),(1,16) \ldots\}$
Solution :
This is reflexive as $(1,1),(2,2) \ldots \ldots$are present.
This is also symmetric because $a R b \Leftrightarrow b R a \text { for all } a, b \in N$
This is transitive because if $a R b \text { and } b R c \Leftrightarrow a R c \ \ \ a, b, c \in N$
This relation is reflexive, symmetric, and transitive.

Relation Exercise 1.1 Question 18 (iv)

Answer : This relation is neither symmetric nor reflexive nor transitive.
Hint :
A relation R on set A is
Reflexive relation:
If $(a, a) \in R$ for every $a \in A$
Symmetric relation:
If $(a,b)$ is true then $(b,a)$ is also true for every $a, b \in A$
Transitive relation:
If $(a,b)$ and $(b, c) \in R$ then $(a, c) \in R$ for every $a, b, c \in A$
Given :
$\\x+4 y=10, x, y \in N \\ \\ (x, y) \in\{(6,1),(2,2)\}$ x

Solution :

This is not reflexive as $(1,1),(6,6)$ are absent.

This is not symmetric as $(6,1)$is present but $(1,6)$ is absent.

This is not transitive as there are only two elements in the set having no element in common.

This relation is neither symmetric nor reflexive nor transitive.

• Void relation

• Universal relation

• Identity relation

• Reflexive relation

• Symmetric relation

• Transitive relation

• Antisymmetric relation

• Equivalence relation

• Theorems based on relations

With such a vast number of topics, studying all of them at once gets pretty confusing. Therefore, the questions from RD Sharma Class 12th Exercise 1.1 are divided into two parts: Level 1 and Level 2. The best approach for this chapter it's to complete 20 Level 1 questions and 10 Level 2 questions per day to cover the entire chapter systematically. This will ensure that you cover a major part of the portion without the burden.

Why is it essential to study the RD Sharma Class 12 Chapter 1 Exercise-1.1 for the board exam?

• Class 12 RD Sharma Chapter 1 Exercise 1.1 Solution is designed to help students grasp the principles behind the numerous questions that will be asked on the test.

• The solutions pdf offers extensive and elaborate explanations to assist students in achieving a higher academic score.

• Students will study for the board exam and score well if they refer to these solutions.

• The solutions are given in plain English to assist students in their exam preparation.

• RD Sharma class 12th exercise 1.1 solutions are well-known, and hundreds of students have trusted them and received good results.

Our subject specialists have solved all of the problems from the textbook. Therefore, students can rely on RD Sharma Class 12th Exercise 1.1 for test preparation. Students will write precise and elaborate answers to textbook questions by practicing the solutions offered in the solutions PDF and following the latest CBSE recommendations.

RD Sharma Class 12 Solution Chapter 1 Exercise 1.1 Relations have the following features:

• Created by subject matter experts

• Easily accessible online

• Free of cost

• Based on the most recent CBSE syllabus pattern

• Chapter-by-chapter solutions

• A simple tool for preparing for an exam

All ideas and definitions have been described in-depth in a lucid manner. They have also been explained with relevant graphic patterns in RD Sharma Class 12 Solutions Relations Chapter 1 Ex 1.1.

For Maths, RD Sharma solutions are chosen because they are provided chapter-by-chapter materials. Because the RD Sharma Class 12th Exercise 1.1 Maths syllabus is large, RD Sharma Maths Solutions are organized by chapter.

With RD Sharma Class 12th Exercise 1.1 solutions, students can read each chapter in depth. Most CBSE Class 12 students prefer our RD Sharma Maths Solutions since they were created by experts with extensive expertise teaching Maths.

## Chapter-wise RD Sharma Class 12 Solutions

JEE Main Highest Scoring Chapters & Topics
Just Study 40% Syllabus and Score upto 100%

1. How can I get the RD Sharma chapter 1 exercise 1.1 solution for class 12?

It is quite simple to obtain the class 12 RD Sharma chapter 1 exercise 1.1 solution. All you have to do is go to Career360's website and get the book in PDF format for free. The book's pdf version is always available and requires no subscription costs.

2. Are the RD Sharma solutions up to current with the most recent syllabus?

In the NCERT book, RD Sharma aims to include all of the answers to the questions. As a result, they are revised every year to reflect the most recent CBSE class 12 syllabus. Therefore, you can rest assured that the book will contain all of the information you require. All you have to do now is select the appropriate file for your exam year.

3. Is it expensive to use RD Sharma's solutions?

The solutions provided by RD Sharma are not all expensive. They are, in fact, completely free and do not have any hidden costs. You can get a soft copy of the book from Careers360, including the most recent and updated editions.

4. How would the RD Sharma chapter 1 exercise 1.1 solution help me in Class 12?

For students who want to do well in their exams and have started studying ahead of time, the Class 12 RD Sharma chapter 1 exercise 1.1 solution book would be very helpful. It can also be used for self-evaluation, self-practice, and identifying your weak areas. Many students and teachers recommend the book, indicating that it can assist you in exam preparation.

5. Do the RD Sharma class 12 solutions Relations Ex 1.1 follow the most recent syllabus?

The NCERT book's RD Sharma 12 answers Relations Ex 1.1 is always updated with the latest syllabus. Likewise, the pdf of the RD Sharma 12 solutions chapter 1 Ex 1.1 is updated with each new edition of the NCERT book so that students may find answers to all of the questions in the book.

## Upcoming School Exams

#### All India Sainik Schools Entrance Examination

Application Date:06 November,2023 - 15 December,2023

#### National Institute of Open Schooling 12th Examination

Application Date:20 November,2023 - 19 December,2023

#### National Institute of Open Schooling 10th examination

Application Date:20 November,2023 - 19 December,2023

#### West Bengal Board 12th Examination

Exam Date:30 November,2023 - 14 December,2023

#### National Means Cum-Merit Scholarship

Application Date:03 December,2023 - 18 December,2023

Get answers from students and experts
##### Bio Medical Engineer

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

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
##### Database Architect

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
##### Ethical Hacker

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
##### Data Analyst

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
##### Geothermal Engineer

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
##### Geotechnical engineer

The role of geotechnical engineer starts with reviewing the projects needed to define the required material properties. The work responsibilities are followed by a site investigation of rock, soil, fault distribution and bedrock properties on and below an area of interest. The investigation is aimed to improve the ground engineering design and determine their engineering properties that include how they will interact with, on or in a proposed construction.

The role of geotechnical engineer in mining includes designing and determining the type of foundations, earthworks, and or pavement subgrades required for the intended man-made structures to be made. Geotechnical engineering jobs are involved in earthen and concrete dam construction projects, working under a range of normal and extreme loading conditions.

3 Jobs Available
##### Cartographer

How fascinating it is to represent the whole world on just a piece of paper or a sphere. With the help of maps, we are able to represent the real world on a much smaller scale. Individuals who opt for a career as a cartographer are those who make maps. But, cartography is not just limited to maps, it is about a mixture of art, science, and technology. As a cartographer, not only you will create maps but use various geodetic surveys and remote sensing systems to measure, analyse, and create different maps for political, cultural or educational purposes.

3 Jobs Available
##### Bank Probationary Officer (PO)

A career as Bank Probationary Officer (PO) is seen as a promising career opportunity and a white-collar career. Each year aspirants take the Bank PO exam. This career provides plenty of career development and opportunities for a successful banking future. If you have more questions about a career as  Bank Probationary Officer (PO), what is probationary officer or how to become a Bank Probationary Officer (PO) then you can read the article and clear all your doubts.

3 Jobs Available
##### Operations Manager

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
##### Data Analyst

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
##### Finance Executive

A career as a Finance Executive requires one to be responsible for monitoring an organisation's income, investments and expenses to create and evaluate financial reports. His or her role involves performing audits, invoices, and budget preparations. He or she manages accounting activities, bank reconciliations, and payable and receivable accounts.

3 Jobs Available
##### Investment Banker

An Investment Banking career involves the invention and generation of capital for other organizations, governments, and other entities. Individuals who opt for a career as Investment Bankers are the head of a team dedicated to raising capital by issuing bonds. Investment bankers are termed as the experts who have their fingers on the pulse of the current financial and investing climate. Students can pursue various Investment Banker courses, such as Banking and Insurance, and Economics to opt for an Investment Banking career path.

3 Jobs Available
##### Bank Branch Manager

Bank Branch Managers work in a specific section of banking related to the invention and generation of capital for other organisations, governments, and other entities. Bank Branch Managers work for the organisations and underwrite new debts and equity securities for all type of companies, aid in the sale of securities, as well as help to facilitate mergers and acquisitions, reorganisations, and broker trades for both institutions and private investors.

3 Jobs Available
##### Treasurer

Treasury analyst career path is often regarded as certified treasury specialist in some business situations, is a finance expert who specifically manages a company or organisation's long-term and short-term financial targets. Treasurer synonym could be a financial officer, which is one of the reputed positions in the corporate world. In a large company, the corporate treasury jobs hold power over the financial decision-making of the total investment and development strategy of the organisation.

3 Jobs Available
##### Product Manager

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
##### Transportation Planner

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
##### Construction Manager

Individuals who opt for a career as construction managers have a senior-level management role offered in construction firms. Responsibilities in the construction management career path are assigning tasks to workers, inspecting their work, and coordinating with other professionals including architects, subcontractors, and building services engineers.

2 Jobs Available
##### Carpenter

Carpenters are typically construction workers. They stay involved in performing many types of construction activities. It includes cutting, fitting and assembling wood.  Carpenters may help in building constructions, bridges, big ships and boats. Here, in the article, we will discuss carpenter career path, carpenter salary, how to become a carpenter, carpenter job outlook.

2 Jobs Available
##### Welder

An individual who opts for a career as a welder is a professional tradesman who is skilled in creating a fusion between two metal pieces to join it together with the use of a manual or fully automatic welding machine in their welder career path. It is joined by intense heat and gas released between the metal pieces through the welding machine to permanently fix it.

2 Jobs Available
##### Environmental Engineer

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
##### Naval Architect

A Naval Architect is a professional who designs, produces and repairs safe and sea-worthy surfaces or underwater structures. A Naval Architect stays involved in creating and designing ships, ferries, submarines and yachts with implementation of various principles such as gravity, ideal hull form, buoyancy and stability.

2 Jobs Available
##### Welding Engineer

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.

2 Jobs Available
##### Field Surveyor

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
##### Orthotist and Prosthetist

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
##### Veterinary Doctor

A veterinary doctor is a medical professional with a degree in veterinary science. The veterinary science qualification is the minimum requirement to become a veterinary doctor. There are numerous veterinary science courses offered by various institutes. He or she is employed at zoos to ensure they are provided with good health facilities and medical care to improve their life expectancy.

5 Jobs Available
##### Pathologist

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
##### Gynaecologist

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
##### Oncologist

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
##### Surgical Technologist

When it comes to an operation theatre, there are several tasks that are to be carried out before as well as after the operation or surgery has taken place. Such tasks are not possible without surgical tech and surgical tech tools. A single surgeon cannot do it all alone. It’s like for a footballer he needs his team’s support to score a goal the same goes for a surgeon. It is here, when a surgical technologist comes into the picture. It is the job of a surgical technologist to prepare the operation theatre with all the required equipment before the surgery. Not only that, once an operation is done it is the job of the surgical technologist to clean all the equipment. One has to fulfil the minimum requirements of surgical tech qualifications.

3 Jobs Available
##### Maxillofacial Surgeon

A Maxillofacial Surgeon is a medical professional who performs facial surgeries that include tooth implant, neck, head or other surgeries such as removal of tumours, cosmetic surgeries and treatment of injuries on the face.

2 Jobs Available
##### Surgical Assistant

Surgical assistants are professionals in the service of saving others’ lives. They perform various medical procedures. In a career as a surgical assistant, one works in a team and contributes to the success of operations. Surgical assistants learn new procedures and update their knowledge of new medical technology and equipment. Surgical assistants clean and sterilize the tools used in surgery. In a career as a surgical assistant, individuals perform all the basic duties that allow surgeons to keep their focus on essential technical functions.

2 Jobs Available
##### Actor

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
##### Acrobat

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
##### Video Game Designer

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
##### Talent Agent

The career as a Talent Agent is filled with responsibilities. A Talent Agent is someone who is involved in the pre-production process of the film. It is a very busy job for a Talent Agent but as and when an individual gains experience and progresses in the career he or she can have people assisting him or her in work. Depending on one’s responsibilities, number of clients and experience he or she may also have to lead a team and work with juniors under him or her in a talent agency. In order to know more about the job of a talent agent continue reading the article.

If you want to know more about talent agent meaning, how to become a Talent Agent, or Talent Agent job description then continue reading this article.

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
##### Videographer

Careers in videography are art that can be defined as a creative and interpretive process that culminates in the authorship of an original work of art rather than a simple recording of a simple event. It would be wrong to portrait it as a subcategory of photography, rather photography is one of the crafts used in videographer jobs in addition to technical skills like organization, management, interpretation, and image-manipulation techniques. Students pursue Visual Media, Film, Television, Digital Video Production to opt for a videographer career path. The visual impacts of a film are driven by the creative decisions taken in videography jobs. Individuals who opt for a career as a videographer are involved in the entire lifecycle of a film and production.

2 Jobs Available
##### Multimedia Specialist

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
##### Visual Communication Designer

Individuals who want to opt for a career as a Visual Communication Designer will work in the graphic design and arts industry. Every sector in the modern age is using visuals to connect with people, clients, or customers. This career involves art and technology and candidates who want to pursue their career as visual communication designer has a great scope of career opportunity.

2 Jobs Available
##### Copy Writer

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
##### Journalist

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
##### Publisher

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
##### Vlogger

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 cost got reduced the viewership for these types of content has increased on a large scale. Therefore, the career as vlogger has a lot to offer. If you want to know more about the career as vlogger, how to become a vlogger, so on and so forth then continue reading the article. Students can visit Jamia Millia Islamia, Asian College of Journalism, Indian Institute of Mass Communication to pursue journalism degrees.

3 Jobs Available
##### Editor

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

Advertising managers consult with the financial department to plan a marketing strategy schedule and cost estimates. We often see advertisements that attract us a lot, not every advertisement is just to promote a business but some of them provide a social message as well. There was an advertisement for a washing machine brand that implies a story that even a man can do household activities. And of course, how could we even forget those jingles which we often sing while working?

2 Jobs Available
##### Photographer

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
##### Social Media Manager

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
##### Product Manager

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
##### Quality Controller

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
##### Production Manager

Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.

3 Jobs Available

A Team Leader is a professional responsible for guiding, monitoring and leading the entire group. He or she is responsible for motivating team members by providing a pleasant work environment to them and inspiring positive communication. A Team Leader contributes to the achievement of the organisation’s goals. He or she improves the confidence, product knowledge and communication skills of the team members and empowers them.

2 Jobs Available
##### Quality Systems Manager

A Quality Systems Manager is a professional responsible for developing strategies, processes, policies, standards and systems concerning the company as well as operations of its supply chain. It includes auditing to ensure compliance. It could also be carried out by a third party.

2 Jobs Available
##### Merchandiser

A career as a merchandiser requires one to promote specific products and services of one or different brands, to increase the in-house sales of the store. Merchandising job focuses on enticing the customers to enter the store and hence increasing their chances of buying a product. Although the buyer is the one who selects the lines, it all depends on the merchandiser on how much money a buyer will spend, how many lines will be purchased, and what will be the quantity of those lines. In a career as merchandiser, one is required to closely work with the display staff in order to decide in what way a product would be displayed so that sales can be maximised. In small brands or local retail stores, a merchandiser is responsible for both merchandising and buying.

2 Jobs Available
##### Procurement Manager

The procurement Manager is also known as  Purchasing Manager. The role of the Procurement Manager is to source products and services for a company. A Procurement Manager is involved in developing a purchasing strategy, including the company's budget and the supplies as well as the vendors who can provide goods and services to the company. His or her ultimate goal is to bring the right products or services at the right time with cost-effectiveness.

2 Jobs Available
##### Production Planner

Individuals who opt for a career as a production planner are professionals who are responsible for ensuring goods manufactured by the employing company are cost-effective and meets quality specifications including ensuring the availability of ready to distribute stock in a timely fashion manner.

2 Jobs Available
##### Information Security Manager

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
##### Computer Programmer

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
##### Product Manager

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
##### ITSM Manager

ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.

3 Jobs Available
##### .NET Developer

.NET Developer Job Description: A .NET Developer is a professional responsible for producing code using .NET languages. He or she is a software developer who uses the .NET technologies platform to create various applications. Dot NET Developer job comes with the responsibility of  creating, designing and developing applications using .NET languages such as VB and C#.

2 Jobs Available
##### Corporate Executive

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
##### DevOps Architect

A DevOps Architect is responsible for defining a systematic solution that fits the best across technical, operational and and management standards. He or she generates an organised solution by examining a large system environment and selects appropriate application frameworks in order to deal with the system’s difficulties.

2 Jobs Available
##### Cloud Solution Architect

Individuals who are interested in working as a Cloud Administration should have the necessary technical skills to handle various tasks related to computing. These include the design and implementation of cloud computing services, as well as the maintenance of their own. Aside from being able to program multiple programming languages, such as Ruby, Python, and Java, individuals also need a degree in computer science.

2 Jobs Available