Q. If $x^{a}y^{b} = e^{m},\ x^{c}y^{d} = e^{n},\ \mathrm{\Delta}_{1} = \left| \begin{matrix} m & b \\ n & d \end{matrix} \right|,\ \mathrm{\Delta}_{2} = \left| \begin{matrix} a & m \\ c & n \end{matrix} \right|$ and $\mathrm{\Delta}_{3} = \left| \begin{matrix} a & b \\ c & d \end{matrix} \right|$, then the values of $x$ and $y$ are respectively
A) $\frac{\mathrm{\Delta}_{1}}{\mathrm{\Delta}_{3}}$ and $\frac{\mathrm{\Delta}_{2}}{\mathrm{\Delta}_{3}}$
B) $\frac{\mathrm{\Delta}_{2}}{\mathrm{\Delta}_{1}}$ and $\frac{\mathrm{\Delta}_{3}}{\mathrm{\Delta}_{1}}$
C) log $(\ \frac{\mathrm{\Delta}_{1}}{\mathrm{\Delta}_{3}}$ ) and log $\left( \ \frac{\mathrm{\Delta}_{2}}{\mathrm{\Delta}_{3}} \right)$
D) $e^{\mathrm{\Delta}_{1}/\mathrm{\Delta}_{3}}$ and $e^{\mathrm{\Delta}_{2}/\mathrm{\Delta}_{3}}$
Explanation
(d)
Given that, $x^{a}y^{b} = e^{m},\ x^{c}y^{d} = e^{n}$
and $\mathrm{\Delta}_{1} = \left| \begin{matrix}
m & b \\
n & d
\end{matrix} \right|,\ \mathrm{\Delta}_{2} = \left| \begin{matrix}
a & m \\
c & n
\end{matrix} \right|,$ $\mathrm{\Delta}_{3} = \left| \begin{matrix}
a & b \\
c & d
\end{matrix} \right|$
$\Rightarrow a\log{x + b\log{y = m}}$
$\Rightarrow c\log{x + d\log{y = n}}$
By Cramer’s rule
$\log{x =}\frac{\mathrm{\Delta}_{1}}{\mathrm{\Delta}_{3}}\ and\log{y = \frac{\mathrm{\Delta}_{2}}{\mathrm{\Delta}_{3}}}\$
$\Rightarrow x = e^{\mathrm{\Delta}_{1}/\mathrm{\Delta}_{3}}\ and\ y = e^{\mathrm{\Delta}_{2}/\mathrm{\Delta}_{3}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.