(i) Given a*b = 3a + b²
\(\Rightarrow 4*3=3(4)+(3)^{ 2 }\)
= 12 + 9
4*3 = 21
(ii) Given a*b = 3a + b²
\(\Rightarrow 5*2=3(5)+2^{ 2 }\)
\(\Rightarrow 5*2=15\)
\(\therefore 5*2=19\)

We have 2*(3*x) = 33
\(\Rightarrow 2*(3+x+3x)=33\)
\(\Rightarrow 2*(3+4x)=33\)
\(\Rightarrow 2+(3+4x)+2(3+4x)=33\)
\(\Rightarrow 2+3+4x+6+8x=33\)
\(\Rightarrow 12x+11=33\)
\(\Rightarrow 12x=22\Rightarrow x=\frac { 11 }{ 6 } \)

for a*b = a^b is a unique natural number * is a binary operation on N
(ii) Let a = 0, b = 0 , a*b = 0*0 = 0⁰ \(\in Z^{ ' }\) 
* is a binary operation on Z

\(f(x)=x^{ 2 },x\in Ra\)
\(f(1.1)=(1.1)^{ 2 }z\)
\(=1.21\)
\(f(1)=(1)^{ 2 }=1\)
\(\frac { f(1.1)-f(1) }{ 1.1-1 } =\frac { 1.21 }{ 1.1-1 } =\frac { 0.21 }{ 0.1 } \)
\(=2.1\)

A function \(f:A\rightarrow B\) is said to be one-one if
\(a\neq b\)
\(\Rightarrow f(a)\neq f(b)for\quad all\quad a,b,\in A\)
or f(a) = f(b)
a = b for all a,b,A
In other words, is one-one, if no two elements of A have same image, i.e., no two elements of A are mapped to same elements of B.

Let the relation \(\bot \) on the set \(\angle \) be denoted by R
For symmetry : \(l_{ 1 }Rl_{ 2 }\Rightarrow l_{ 1 }\bot l_{ 1 }\Rightarrow l_{ 2 }\bot l_{ 1 }Rl_{ 1 }\)
The relation R is symmetric
For reflective \(l_{ 1 }Rl_{ 2 }\Rightarrow l_{ 1 }\bot l\)
No line can be perpendicular to it self
For transitive : \(l_{ 1 }Rl_{ 2 }\Rightarrow l_{ 1 }\) for \(I\ \in \ \angle \)
It is not true
Relation is not transitive

A relation R on a non-empty set A is called a transitive relation if (a, b), (b, c) \(\in R\) then (a, c) \(\in R\) , i.e., aRb, bRc implies aRc.
Thus a relation R on a non empty set A is said to be transitive if there exist a, b, c \(\in A\) such that (a, b)(b, c) \(\in R\)  implies (a, c) . \(\in R\)
Example
Let A = (1, 2, 3, 6)
R = (3, 6) (6, 1) (3, 1)
3 R 6 and 6 R 1 \(\Rightarrow \)(3, 1)  \(\in R\)
\(\therefore\) A is transitive.

Symmetric Relation : A relation R on a set A is called symmetric relation if aRb implies bRa, for every a,b \(\in a\) i.,e if (a,b)  \(\in R\) \(\Rightarrow \) (b,a)  \(\in R\)For every a,b \(\in A\)
Example 
A = (1,2,3)
A x A =(1,2) (2,1) (1,1) (2,2) (3,3) (1,3) (2,3)  (3,1) (3,2 )  \(\in R\) 
since (a,b)  \(\in R\)  (b,a)  \(\in R\) for every a, b  \(\in A\) 
Relation is said to be symmetric 

Reflexive Relation : A relation R on a set A is called reflexive relation if aRa for every \(a\in A\) ; if (a, a) \(\in \) R, for every \(a\in A\)
Example let
A = [1, 2, 3]
A x A =(1, 1) (1, 2)(1, 3) (2, 1) (2, 2) (2, 3) (3, 1)(3, 2) (3, 3)  \(\in R\) 
Since (a, a) \(\in R\) for every \(a\in A\)