A) \[{{\Delta }_{1}}=3{{({{\Delta }_{2}})}^{2}}\]
B) \[\left( \frac{d}{dx} \right)({{\Delta }_{1}})=3{{\Delta }_{2}}\]
C) \[\left( \frac{d}{dx} \right)({{\Delta }_{1}})=3{{({{\Delta }_{2}})}^{2}}\]
D) \[{{\Delta }_{1}}\,=3{{({{\Delta }_{2}})}^{3/2}}\]
E) \[\left( \frac{d}{dx} \right)({{\Delta }_{1}})={{\Delta }_{2}}\]
Correct Answer: B
Solution :
Given, \[{{\Delta }_{1}}=\left| \begin{matrix} x & a & b \\ b & x & a \\ a & b & x \\ \end{matrix} \right|,\]\[{{\Delta }_{2}}=\left| \begin{matrix} x & b \\ a & x \\ \end{matrix} \right|\] \[\frac{d}{dx}({{\Delta }_{1}})=\left| \begin{matrix} 1 & 0 & 0 \\ b & x & a \\ a & b & x \\ \end{matrix} \right|+\left| \begin{matrix} x & a & b \\ 0 & 1 & 0 \\ a & b & x \\ \end{matrix} \right|\] \[+\left| \begin{matrix} x & a & b \\ b & x & a \\ 0 & 0 & 1 \\ \end{matrix} \right|\] \[=\left| \begin{matrix} x & a \\ b & x \\ \end{matrix} \right|+\left| \begin{matrix} x & b \\ a & x \\ \end{matrix} \right|+\left| \begin{matrix} x & a \\ b & x \\ \end{matrix} \right|=3{{\Delta }_{2}}\] \[\Rightarrow \] \[\frac{d}{dx}({{\Delta }_{1}})=3{{\Delta }_{2}}\]
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