Q84 / 200 · Manipal (MET) 2014 · Mathematics

The value of the determinant $\left|\begin{array}{ccc}\left(a^x+a^{-x}\right)^2 & \left(a^x-a^{-x}\right)^2 & 1 \\ \left(b^x+b^{-x}\right)^2 & \left(b^x-b^{-x}\right)^2 & 1 \\ \left(c^x+c^{-x}\right)^2 & \left(c^x-c^{-x}\right)^2 & 1\end{array}\right|$ is

  • Correct answer
    A
    0
  • B
    $2 a b c$
  • C
    $a^2 b^2 c^2$
  • D
    None of these

Solution

Given determinant is $\Delta=\left|\begin{array}{ccc} \left(a^x+a^{-x}\right)^2 & \left(a^x-a^{-x}\right)^2 & 1 \\ \left(b^x+b^{-x}\right)^2 & \left(b^x-b^{-x}\right)^2 & 1 \\ \left(c^x+c^{-x}\right)^2 & \left(c^x-c^{-x}\right)^2 & 1 \end{array}\right|$
Applying $C_1 \rightarrow C_1-C_2$, we get $\begin{aligned} \Delta & =\left|\begin{array}{ccc} 4 & \left(a^x-a^{-x}\right)^2 & 1 \\ 4 & \left(b^x-b^{-x}\right)^2 & 1 \\ 4 & \left(c^x-c^{-x}\right)^2 & 1 \end{array}\right|=4\left|\begin{array}{ccc} 1 & \left(a^x-a^{-x}\right)^2 & 1 \\ 1 & \left(b^x-b^{-x}\right)^2 & 1 \\ 1 & \left(c^x-c^{-x}\right)^2 & 1 \end{array}\right| \\ & =0 \quad(\because \text { two columns are identical) } \end{aligned}$

Hence, the correct answer is option: 'A'.

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