Question

In the set ${Q^ + }$ of all positive rational numbers, the operation $ * $ is defined by the formula $a * b = \dfrac{{ab}}{6}$ . Then the inverse of 9 with respect to $ * $ is
\[
  A.{\text{ }}4 \\
  B.{\text{ }}3 \\
  C.{\text{ }}\dfrac{1}{9} \\
  D.{\text{ }}\dfrac{1}{3} \\
 \]

Answer 1

Hint: In order to solve the given problem and find the inverse of the given number. First consider some identity element and with the help of the definition of the function given find the general inverse function in terms of the given variable and finally substitute the value to find the inverse of 9.

Complete step by step answer:
Given that:
${Q^ + }$ is a set of all positive numbers.
And also we have the operation $ * $ which is defined by the formula $a * b = \dfrac{{ab}}{6}$ .
And we have to find the inverse of 9 with respect to operation $ * $ .
Let us consider $x$ as the identity term.
According to the given operation we have:
$a * x = a$
Substituting the value of above term according to the given operation we have:
\[
  \because a * x = a \\
   \Rightarrow a * x = \dfrac{{ax}}{6} = a \\
   \Rightarrow x = \dfrac{{6a}}{a} \\
   \Rightarrow x = 6 \\
 \]
As we know that the operation of a variable term and its inverse is given by identity term.
$ \Rightarrow a * {a^{ - 1}} = x$
Let us substitute the value of the identity term as obtained above in this formula. So, we have:
$
   \Rightarrow a * {a^{ - 1}} = 6 \\
   \Rightarrow \dfrac{{a{a^{ - 1}}}}{6} = 6 \\
 $
From the above equation let us manipulate the term to find the value of the inverse variable.
$
  \because \dfrac{{a{a^{ - 1}}}}{6} = 6 \\
   \Rightarrow {a^{ - 1}} = \dfrac{{6 \times 6}}{a} \\
 $
Now let us substitute the value 9 in place of the inverse variable to find the inverse of 9 according to the given operation.
$
   \Rightarrow {9^{ - 1}} = \dfrac{{6 \times 6}}{9} \\
   \Rightarrow {9^{ - 1}} = \dfrac{{36}}{9} = 4 \\
 $
Hence, the inverse of 9 with respect to $ * $ is 4.
So, option A is the correct option.

Note: In order to solve the problems students must remember different identities related to function and the formulas for the identity term. Some of these identities are mentioned above. Sometimes it may feel awkward to get some inverse like in the above case we have got 9 inverses as 4. But students must understand that this inverse is for a particular defined function or operation.