If ${x^a}{y^b} = {e^m},{x^c}{y^d} = {e^n},{\Delta _1} = \left| {\begin{array}{*{20}{c}} m&b\\ n&d \end{array}} \right|,{\Delta _2} = \left| {\begin{array}{*{20}{c}} a&m\\ c&n \end{array}} \right|$ and ${\Delta _3} = \left| {\begin{array}{*{20}{c}} a&b\\ c&d \end{array}} \right|$, then the values of $x$ and $y$ are respectively

$\frac{{{\Delta _1}}}{{{\Delta _3}}}$ and $\frac{{{\Delta _2}}}{{{\Delta _3}}}$

$\frac{{{\Delta _2}}}{{{\Delta _1}}}$ and $\frac{{{\Delta _2}}}{{{\Delta _1}}}$

$\log \left( {\frac{{{\Delta _1}}}{{{\Delta _2}}}} \right)$ and $\log \left( {\frac{{{\Delta _2}}}{{{\Delta _3}}}} \right)$

JEE MATHEMATICS MCQ: MATRICES

1 Medium

1 of 1

Application +4 | -1

{x^a}{y^b} = {e^m},{x^c}{y^d} = {e^n},{\Delta _1} = \left| {\begin{array}{cccccccccccccccccccc} m&b\\ n&d \end{array}} \right|,{\Delta _2} = \left| {\begin{array}{cccccccccccccccccccc} a&m\\ c&n \end{array}} \right|

and

{\Delta _3} = \left| {\begin{array}{cccccccccccccccccccc} a&b\\ c&d \end{array}} \right|

, then the values of

x

and

y

are respectively

View answer and explanation

Correct option: D

{e^{{\Delta _1}/{\Delta _3}}}

and

{e^{{\Delta _2}/{\Delta _3}}}

Option D (

{e^{{\Delta _1}/{\Delta _3}}}

and

{e^{{\Delta _2}/{\Delta _3}}}

) is correct because using Cramer's rule after taking logarithms yields these expressions for x and y.