Question

Which of the following values of $\alpha$ satisfy the equation.
$\begin{vmatrix}\left(1+\alpha\right)^{2}&\left(1+2\alpha \right)^{2}&\left(1+3\alpha \right)^{2}\\ \left(2+\alpha \right)^{2}&\left(2+2\alpha \right)^{2}&\left(2+3\alpha \right)^{2}\\ \left(3+\alpha \right)^{2}&\left(3+2\alpha \right)^{2}&\left(3+3\alpha \right)^{2}\end{vmatrix} = -648\alpha ?$

• A. - 4
• B. 9
• C. - 9
• D. 4

Answer 1

Answer: (B), (C)

$\begin{vmatrix}1&2&1\\ 4&4&1\\ 9&6&1\end{vmatrix}\begin{vmatrix}1&1&1\\ \alpha&2\alpha&3\alpha\\ \alpha^{2}&4\alpha^{2}&9\alpha^{2}\end{vmatrix} = -648\,\alpha$
$\Rightarrow \,\left(-4\right) \left(2α^3\right) = - 648 \,α$
$⇒\, α^{2} = 81$
$⇒ \,α = \pm9$