Which of the following values of α 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?

We have ( 1 + α ) 2 ( 1 + 2 α ) 2 ( 1 + 3 α ) 2 ( 2 + α ) 2 ( 2 + 2 α ) 2 ( 2 + 3 α ) 2 ( 3 + α ) 2 ( 3 + 2 α ) 2 ( 3 + 3 α ) 2 = − 648 α C 3 → C 3 − C 2 , C 2 → C 2 − C 1 ⇒ α ( 1 + α ) 2 2 + 3 α 2 + 5 α ( 2 + α ) 2 4 + 3 α 4 + 5 α ( 3 + α ) 2 6 + 3 α 6 + 5 α C 3 → C 3 − C 2 ⇒ α ( 1 + α ) 2 2 + 3 α 2 α ( 2 + α ) 2 4 + 3 α 2 α ( 3 + α ) 2 6 + 3 α 2 α ⇒ 2 α 2 ( 1 + α ) 2 2 + 3 α 1 ( 2 + α ) 2 4 + 3 α 1 ( 3 + α ) 2 6 + 3 α 1 Applying R 1 → R 1 − R 2 and R 2 → R 2 − R 3 ⇒ 2 α 2 ( 1 + α ) 2 2 + 3 α 0 ( 2 + α ) 2 4 + 3 α 0 ( 3 + α ) 2 6 + 3 α 1 = − 648 α - 8 α 3 = - 648 α α = ± 9

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Which of the following values of α satisfy the equation $|\begin{array}{lll} (1 + α)^{2} (1 + 2 α)^{2} (1 + 3 α)^{2} \\ (2 + α)^{2} (2 + 2 α)^{2} (2 + 3 α)^{2} \\ (3 + α)^{2} (3 + 2 α)^{2} (3 + 3 α)^{2} \end{array}| = - 648 α ?$

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Detailed Solution

We have $|\begin{array}{lll} (1 + α)^{2} (1 + 2 α)^{2} (1 + 3 α)^{2} \\ (2 + α)^{2} (2 + 2 α)^{2} (2 + 3 α)^{2} \\ (3 + α)^{2} (3 + 2 α)^{2} (3 + 3 α)^{2} \end{array}| = - 648 α$

$C_{3} \to C_{3} - C_{2}, C_{2} \to C_{2} - C_{1}$

$\Rightarrow α |\begin{array}{lll} (1 + α)^{2} 2 + 3 α 2 + 5 α \\ (2 + α)^{2} 4 + 3 α 4 + 5 α \\ (3 + α)^{2} 6 + 3 α 6 + 5 α \end{array}|$

$C_{3} \to C_{3} - C_{2}$

$\Rightarrow α |\begin{array}{lll} (1 + α)^{2} 2 + 3 α 2 α \\ (2 + α)^{2} 4 + 3 α 2 α \\ (3 + α)^{2} 6 + 3 α 2 α \end{array}| \Rightarrow 2 α^{2} |\begin{array}{lll} (1 + α)^{2} 2 + 3 α 1 \\ (2 + α)^{2} 4 + 3 α 1 \\ (3 + α)^{2} 6 + 3 α 1 \end{array}|$
Applying $R_{1} \to R_{1} - R_{2} and R_{2} \to R_{2} - R_{3}$
$\Rightarrow 2 α^{2} |\begin{array}{lll} (1 + α)^{2} 2 + 3 α 0 \\ (2 + α)^{2} 4 + 3 α 0 \\ (3 + α)^{2} 6 + 3 α 1 \end{array}| = - 648 α$

$- 8 α^{3} = - 648 α$

$α = \pm 9$

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