NIMCET 2012 Question Paper with solutions

Given , $2^{\mathit{a}}$ = $3^{\mathit{b}}$ = $6^{-\mathit{c}}$ = K (say)
⇒ a = $\mathit{l}\mathit{o}{\mathit{g}}_2$ K , b = $\mathit{l}\mathit{o}{\mathit{g}}_3$ K , c = - $\mathit{l}\mathit{o}{\mathit{g}}_6$ K
⇒ a = $\frac{\mathit{l}\mathit{o}\mathit{g}\mathit{K}}{\mathit{l}\mathit{o}\mathit{g}2}$ , b = $\frac{\mathit{l}\mathit{o}\mathit{g}\mathit{K}}{\mathit{l}\mathit{o}\mathit{g}3}$ , c = - $\frac{\mathit{l}\mathit{o}\mathit{g}\mathit{K}}{\mathit{l}\mathit{o}\mathit{g}6}$
⇒ log 2 + log 3 = - $\frac{\mathit{l}\mathit{o}\mathit{g}\mathit{K}}{\mathit{c}}$ (Since log 6 = log 2 + log 3)
⇒ $\frac{\mathit{l}\mathit{o}\mathit{g}\mathit{K}}{\mathit{a}}$ + $\frac{\mathit{l}\mathit{o}\mathit{g}\mathit{K}}{\mathit{b}}$ = - $\frac{\mathit{l}\mathit{o}\mathit{g}\mathit{K}}{\mathit{c}}$
⇒ $\frac{1}{\mathit{a}}+\frac{1}{\mathit{b}}+\frac{1}{\mathit{c}}$ = 0 (Since log K ≠ 0)
⇒ $\frac{\mathit{b}\mathit{c}+\mathit{c}\mathit{a}+\mathit{a}\mathit{b}}{\mathit{a}\mathit{b}\mathit{c}}$ = 0 (Since abc ≠ 0)
⇒ ab + bc + ca = 0