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Determinants — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDeterminants5 Marks
Question
Without expanding, show that the values of the following determinant are zero: $\begin{vmatrix}(2^{x}+2^{-x})^2&(2^{x}-2^{-x})^2&1\\(3^{x}+3^{-x})^2&(3^{x}-3^{-x})^2&1\\(4^{x}+4^{-x})^2&(4^{x}-4^{-x})^2&1\end{vmatrix}$
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Answer

$\begin{vmatrix}(2^{x}+2^{-x})^2&(2^{x}-2^{-x})^2&1\\(3^{x}+3^{-x})^2&(3^{x}-3^{-x})^2&1\\(4^{x}+4^{-x})^2&(4^{x}-4^{-x})^2&1\end{vmatrix}$
$=\begin{vmatrix}(2^{x}+2^{-x}+2)&(2^{x}-2^{-x}-2)&1\\(3^{x}+3^{-x}+2)&(3^{x}-3^{-x}-2)&1\\(4^{x}+4^{-x}+2)&(4^{x}-4^{-x}-2)&1\end{vmatrix}$
$=\begin{vmatrix}4&(2^{x}+2^{-x}-2)&1\\4&(3^{x}+3^{-x}-2)&1\\4&(4^{x}+4^{-x}-2)&1\end{vmatrix}$ [Applying C1 → C1 - C2] 
$=4\begin{vmatrix}1&(2^{x}+2^{-x}-2)&1\\1&(3^{x}+3^{-x}-2)&1\\1&(4^{x}+4^{-x}-2)&1\end{vmatrix}$
$=0$

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