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Determinants — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDeterminants5 Marks
Question
Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}(2^{\text{x}}+2^{-\text{x}})^2&(2^{\text{x}}-2^{-\text{x}})^2&1\$3^{\text{x}}+3^{-\text{x}})^2&(3^{\text{x}}-3^{-\text{x}})^2&1\$4^{\text{x}}+4^{-\text{x}})^2&(4^{\text{x}}-4^{-\text{x}})^2&1\end{vmatrix}$
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Answer

$\begin{vmatrix}(2^{\text{x}}+2^{-\text{x}})^2&(2^{\text{x}}-2^{-\text{x}})^2&1\$3^{\text{x}}+3^{-\text{x}})^2&(3^{\text{x}}-3^{-\text{x}})^2&1\$4^{\text{x}}+4^{-\text{x}})^2&(4^{\text{x}}-4^{-\text{x}})^2&1\end{vmatrix}$
$=\begin{vmatrix}(2^{\text{x}}+2^{-\text{x}}+2)&(2^{\text{x}}-2^{-\text{x}}-2)&1\$3^{\text{x}}+3^{-\text{x}}+2)&(3^{\text{x}}-3^{-\text{x}}-2)&1\$4^{\text{x}}+4^{-\text{x}}+2)&(4^{\text{x}}-4^{-\text{x}}-2)&1\end{vmatrix}$
$=\begin{vmatrix}4&(2^{\text{x}}+2^{-\text{x}}-2)&1\\4&(3^{\text{x}}+3^{-\text{x}}-2)&1\\4&(4^{\text{x}}+4^{-\text{x}}-2)&1\end{vmatrix} [$Applying $C_1 \rightarrow C_1 - C_2]$
$=4\begin{vmatrix}1&(2^{\text{x}}+2^{-\text{x}}-2)&1\\1&(3^{\text{x}}+3^{-\text{x}}-2)&1\\1&(4^{\text{x}}+4^{-\text{x}}-2)&1\end{vmatrix}$
$=0$

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