ACE booklet

Question

R be a set of real # and ( R, *) is a group with respect to operation a*b= a+b+2, then inverse of 2 is......

(a) -2        (b) -4         (c) -6          (d) -8

Answer 1

Answer : Option C

Before proceeding to find Inverse of an element, First we need to find the Identity element of the given Group.

Identity element $e$ of any group is defined as : $x * e = x = e*x$

here, $x * e = x + e + 2$, So, $x + e + 2 = x$ .....$\Rightarrow$ $e = -2$

Hence, The identity element of the given Group is $-2$.

Now, Let's find Inverse of an element $x$.

Inverse $y$ of an element $x$ is such an element in the Group for which :

$x * y = e = y*x$

So, $x + y+ 2 = e$   $\Rightarrow$   $y = e -2 -x$   $\Rightarrow$     $y = -2-2-x$  $\Rightarrow$   $y = -4 -x$

So, For any element $x$, the Inverse will be $-4-x$

Hence, the Inverse of $2$ will be $-6$.