Question

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

• A. $-4$
• B. $9$
• C. $-9$
• D. $4$

Answer 1

Answer: (B), (C)

The correct options are
B $9$
C $-9$

$\text{Let}\Delta =\left|\begin{array}{ccc}(1+\alpha {)}^2& (1+2\alpha {)}^2& (1+3\alpha {)}^2\\ (2+\alpha {)}^2& (2+2\alpha {)}^2& (2+3\alpha {)}^2\\ (3+\alpha {)}^2& (3+2\alpha {)}^2& (3+3\alpha {)}^2\end{array}\right|$

$C_2\to C_2-C_1,C_3\to C_3-C_1$
$\Delta =2{\alpha }^2\left|\begin{array}{ccc}(1+\alpha {)}^2& 3\alpha +2& 4\alpha +2\\ (2+\alpha {)}^2& 3\alpha +4& 4\alpha +4\\ (3+\alpha {)}^2& 3\alpha +6& 4\alpha +6\end{array}\right|$

$R_2\to R_2-R_1,R_3\to R_3-R_1$
$\Delta =8{\alpha }^2\left|\begin{array}{ccc}(1+\alpha {)}^2& 3\alpha +2& 2\alpha +1\\ 2\alpha +3& 2& 1\\ 2\alpha +4& 2& 1\end{array}\right|$

$R_3\to R_3-R_2$
$\Delta =8{\alpha }^2\left|\begin{array}{ccc}(1+\alpha {)}^2& 3\alpha +2& 2\alpha +1\\ 2\alpha +3& 2& 1\\ 1& 0& 0\end{array}\right|$

$\Delta =-8{\alpha }^3$
$\Rightarrow -8{\alpha }^3=-648\alpha$
$\Rightarrow \alpha =\pm 9,\text{or}\alpha =0$