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CET Karnataka Engineering CET - Karnataka Engineering Solved Paper-2005

  • question_answer
    In the group \[({{Q}^{+}},*)\] of positive rational numbers w.r.t. the binary operation * defined by \[a*b=\frac{ab}{3}\,\,\forall \,\,a,b\in {{Q}^{+}}\] the solution of the equation \[5*x={{4}^{-1}}\] in \[{{Q}^{+}}\] is :

    A)  \[\frac{27}{20}\]                                             

    B)  \[\frac{20}{27}\]

    C)  \[\frac{1}{20}\]                               

    D)  20

    Correct Answer: A

    Solution :

    Let e is the identity element of \[{{Q}^{+}}\] then, \[e*a=a,\,\forall \,a\in {{Q}^{+}}\]                 \[\frac{ea}{3}=a\]                            \[[\,\because \,a*b=\frac{ab}{3}]\] \[\therefore \]  \[e=3\] Again, let b be die inverse of a then, \[a*b=e\]                 \[\frac{ab}{3}=3\,\,\,[\]                \[[\because \,\,e=4]\]                 \[b=\frac{9}{a}\] \[\therefore \] Inverse of 4 is \[\frac{9}{4}\]. From the given condition,                 \[5*x={{4}^{-1}}\]                 \[\frac{5x}{3}=\frac{9}{4}\] \[\Rightarrow \]               \[x=\frac{27}{20}\]


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