Question

$\left|\begin{array}{ccc}{(a^x+a^{-x})}^2& {(a^x-a^{-x})}^2& 1\\ {(b^x+b^{-x})}^2& {(b^x-b^{-x})}^2& 1\\ {(c^x+c^{-x})}^2& {(c^x-c^{-x})}^2& 1\end{array}\right|$ is equal to

• A. $0$
• B. $2abc$
• C. $a^2{b}^2{c}^2$
• D. None of these

Answer 1

Answer: (A)

$0$

Let $\Delta =\left|\begin{array}{ccc}{(a^x+a^{-x})}^2& {(a^x-a^{-x})}^2& 1\\ {(b^x+b^{-x})}^2& {(b^x-b^{-x})}^2& 1\\ {(c^x+c^{-x})}^2& {(c^x-c^{-x})}^2& 1\end{array}\right|$

Put $x=0$ in the given determinant, we get

$\Delta =\left|\begin{array}{ccc}{(1+1)}^2& {(1-1)}^2& 1\\ {(1+1)}^2& {(1-1)}^2& 1\\ {(1+1)}^2& {(1-1)}^2& 1\end{array}\right|$

$=\left|\begin{array}{ccc}4& 0& 1\\ 4& 0& 1\\ 4& 0& 1\end{array}\right|$

$=0$