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. $-4$
• D. $-9$

Answer 1

Answer: (B), (D)

$-9$

$\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|$$\begin{array}{cc}& R_2\to R_2-R_1\\ & R_3\to R_3-R_1\end{array}$

(Take common)

$2\left|\begin{array}{ccc}{(1+\alpha )}^2& {(1+2\alpha )}^2& {(1+3\alpha )}^2\\ 3+2\alpha & (3+4\alpha )& (3+6\alpha )\\ (4+2\alpha )& (4+4\alpha )& (4+6\alpha )\end{array}\right|$

$R_3\to R_3–R2$

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

$\therefore 2\alpha .2.\alpha \left|\begin{array}{ccc}(1+\alpha {)}^2& (2+3\alpha )& (2+4\alpha )\\ 3+2\alpha & 2& 2\\ 1& 0& 0\end{array}\right|$

$4{\alpha }^2|-2\alpha |=-8{\alpha }^3=-648\alpha$
$\alpha =0,{\alpha }^2=81,\alpha =\pm 9$