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Mathematics Question on Determinants

Which of the following values of α\alpha satisfy the equation. ∣(1+α)2(1+2α)2(1+3α)2 (2+α)2(2+2α)2(2+3α)2 (3+α)2(3+2α)2(3+3α)2∣=−648α?\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

-9

Explanation

Solution

∣121 441 961∣∣111 α2α3α α24α29α2∣=−648 α\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
⇒ (−4)(2a3)=−648 a\Rightarrow\,\left(-4\right) \left(2a^3\right) = - 648 \,a
? a2=81?\, a^{2} = 81
? a=±9? \,a = \pm9