Consider a binary operation $*$ on N defined as a $*$ b $= a^3 + b^3$. Choose the correct answer.
(A) Is $*$ both associative and commutative?
(B) Is $*$ commutative but not associative?
(C) Is $*$ associative but not commutative?
(D) Is $*$ neither commutative nor associative?
(B) For commutative
a $*$ b $= a^3 + b^3 = b^3 + a^3 =$ b $*$ a.
$\therefore$ $*$ is a commutative operation.
For associative :
a $*$ ( b $*$ c ) $=$ a $* (b^3 + c^3 ) = a^3 + (b^3 + c^3)^3$
and (a $*$ b) $*$ c $= (a^3 + b^3 ) *$ c $= (a^3 + b^3 )^3 + c^3$
.
$\Rightarrow$ a $*$ ( b $*$ c ) $\ne$ (a $*$ b ) $*$ c. $*$ is not associative.
NCERT & Exemplar solution for CBSE Class 12 Mathematics, Relations and Functions. Curated by Sachin Sharma. Free for all students.