Question

$\text{If}x,y,z\in R$, then the value of determinant
$$\left|\begin{array}{ccc}(5^x+5^{-x}{)}^2& (5^x-5^{-x}{)}^2& 1\\ (6^x+6^{-x}{)}^2& (6^x-6^{-x}{)}^2& 1\\ (7^x+7^{-x}{)}^2& (7^x-7^{-x}{)}^2& 1\end{array}\right|$$ is

• A. $10$
• B. $12$
• C. $1$
• D. $0$

Answer 1

The correct option is D $0$

$△=\left|\begin{array}{ccc}(5^x+5^{-x}{)}^2& (5^x-5^{-x}{)}^2& 1\\ (6^x+6^{-x}{)}^2& (6^x-6^{-x}{)}^2& 1\\ (7^x+7^{-x}{)}^2& (7^x-7^{-x}{)}^2& 1\end{array}\right|$

Applying $C_1\to C_1-C_2$, we get
$△=\left|\begin{array}{ccc}4& (5^x-5^{-x}{)}^2& 1\\ 4& (6^x-6^{-x}{)}^2& 1\\ 4& (7^x-7^{-x}{)}^2& 1\end{array}\right|$

$\Rightarrow △=4\left|\begin{array}{ccc}1& (5^x-5^{-x}{)}^2& 1\\ 1& (6^x-6^{-x}{)}^2& 1\\ 1& (7^x-7^{-x}{)}^2& 1\end{array}\right|$

$\Rightarrow △=4\times 0=0$