NIMCET 2014 Question Paper with solutions

We have,
${\left(2\mathit{x}\right)}^{\mathit{l}\mathit{n}2}$ = ${\left(3\mathit{y}\right)}^{\mathit{l}\mathit{n}3}$ ... (i)
and $3^{\mathit{l}\mathit{o}\mathit{g}\mathit{x}}$ = $2^{\mathit{l}\mathit{o}\mathit{g}\mathit{y}}$ .... (ii)
If ${\mathit{x}}_0,{\mathit{y}}_0$are the solution of these equations
now , ${\left(2{\mathit{x}}_0\right)}^{\mathit{l}\mathit{o}\mathit{g}2}$ = ${\left(3{\mathit{y}}_0\right)}^{\mathit{l}\mathit{o}\mathit{g}3}$
Taking log on both sides,
log 2 log (2${\mathit{x}}_0$) = log 3 log (3${\mathit{y}}_0$)
⇒ $\frac{\mathit{l}\mathit{o}\mathit{g}2}{\mathit{l}\mathit{o}\mathit{g}3}$ log (2${\mathit{x}}_0$) = log $\left(3{\mathit{y}}_0\right)$
$\frac{\mathit{l}\mathit{o}\mathit{g}2}{\mathit{l}\mathit{o}\mathit{g}3}$ (log 2 + log ${\mathit{x}}_0$) = log 3 + log ${\mathit{y}}_0$ ... (iii)
Taking log on both sides of Eq. (ii)
log ${\mathit{x}}_0$ log 3 = log ${\mathit{y}}_0$ log 2
⇒ log ${\mathit{y}}_0$ = $\frac{\mathit{l}\mathit{o}\mathit{g}3}{\mathit{l}\mathit{o}\mathit{g}2}$ log ${\mathit{x}}_0$ ... (iv)
FromEqs. (iii) and (iv),
$\frac{\mathit{l}\mathit{o}\mathit{g}2}{\mathit{l}\mathit{o}\mathit{g}3}$ [log 2 + log ${\mathit{x}}_0$] = log 3 + $\frac{\mathit{l}\mathit{o}\mathit{g}3}{\mathit{l}\mathit{o}\mathit{g}2}$ log ${\mathit{x}}_0$
⇒ log ${\mathit{x}}_0\left[\frac{\mathit{l}\mathit{o}\mathit{g}2}{\mathit{l}\mathit{o}\mathit{g}3}-\frac{\mathit{l}\mathit{o}\mathit{g}3}{\mathit{l}\mathit{o}\mathit{g}2}\right]$ = log 3 - log 2 $\frac{\mathit{l}\mathit{o}\mathit{g}2}{\mathit{l}\mathit{o}\mathit{g}3}$
⇒ log ${\mathit{x}}_0\left[\frac{\mathit{l}\mathit{o}\mathit{g}2}{\mathit{l}\mathit{o}\mathit{g}3}-\frac{\mathit{l}\mathit{o}\mathit{g}3}{\mathit{l}\mathit{o}\mathit{g}2}\right]$ = log 2 $[\frac{\mathit{l}\mathit{o}\mathit{g}3}{\mathit{l}\mathit{o}\mathit{g}2}-\mathit{l}\mathit{o}\mathit{g}\frac{2}{\mathit{l}\mathit{o}\mathit{g}3}]$
⇒ log $\mathit{x}+0$ = - log 2
⇒ log ${\mathit{x}}_0$ = log $2^{-1}$
or ${\mathit{x}}_0$ = $\frac{1}{2}$