Question

The value of $\left|\begin{array}{ccc}1& 1& 1\\ {(2^x+2^{-x})}^2& {(3^x+3^{-x})}^2& {(5^x+5^{-x})}^2\\ {(2^x-2^{-x})}^2& {(3^x-3^{-x})}^2& {(5^x-5^{-x})}^2\end{array}\right|$

• A. 0
• B. ${30}^x$
• C. ${30}^{-x}$
• D. None of these

Answer 1

The correct option is A 0

$\left|\begin{array}{ccc}1& 1& 1\\ (2^x+2^{-x}{)}^2& (3^x+3^{-x}{)}^2& (5^x+5^{-x}{)}^2\\ (2^x-2^{-x}{)}^2& (3^x-3^{-x})& (5^x-5^{-x}{)}^2\end{array}\right|$
$=\left|\begin{array}{ccc}1& 1& 1\\ 2+2^{2x}+2^{-2x}& 2+3^{2x}+3^{-2x}& 2+5^{2x}+5^{-2x}\\ -2+2^{2x}+2^{-2x}& -2+3^{2x}+3^{-2x}& -2+5^{2x}+5^{-2x}\end{array}\right|$
$\Rightarrow C_1\to C_1-C_3$ & $C_2\to C_2-C_3$
$=\left|\begin{array}{ccc}0& 0& 1\\ 2^{2x}+2^{-2x}-(5^{2x}+5^{-2x})& 3^{2x}+3^{-2x}-(5^{2x}+5^{-2x})& 5^{2x}+5^{-2x}+2\\ 2^{2x}+2^{-2x}-(5^{2x}+5^{-2x})& 3^{2x}+3^{-2x}-(5^{2x}+5^{-2x})& 5^{2x}+5^{-2x}-2\end{array}\right|$
$\Rightarrow R_2\to R_2-R_3$
$=\left|\begin{array}{ccc}0& 0& 1\\ 0& 0& 4\\ 2^{2x}+2^{-2x}-(5^{2x}+5^{-2x})& 3^{2x}+3^{-2x}-(5^{2x}+5^{-2x})& 5^{2x}+5^{-2x}-2\end{array}\right|$

$=0$