Which of the following values of alpha satisf | vedclass

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(B) Let the determinant be $\Delta$. We apply row operations $R_2 \rightarrow R_2 - R_1$ and $R_3 \rightarrow R_3 - R_2$:$\Delta = \left|\begin{array}

Vedclass · Accepted Answer

$(B, C)$

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(B) Let the determinant be $\Delta$. We apply row operations $R_2 ightarrow R_2 - R_1$ and $R_3 ightarrow R_3 - R_2$:$\Delta = \left|\begin{array}{lll}(1+\alpha)^2 & (1+2\alpha)^2 & (1+3\alpha)^2 \ (2+\alpha)^2-(1+\alpha)^2 & (2+2\alpha)^2-(1+2\alpha)^2 & (2+3\alpha)^2-(1+3\alpha)^2 \ (3+\alpha)^2-(2+\alpha)^2 & (3+2\alpha)^2-(2+2\alpha)^2 & (3+3\alpha)^2-(2+3\alpha)^2\end{array}ight|$$= \left|\begin{array}{lll}(1+\alpha)^2 & (1+2\alpha)^2 & (1+3\alpha)^2 \ 3+2\alpha & 3+4\alpha & 3+6\alpha \ 5+2\alpha & 5+4\alpha & 5+6\alpha\end{array}ight|$Applying $R_3 ightarrow R_3 - R_2$:$= \left|\begin{array}{lll}(1+\alpha)^2 & (1+2\alpha)^2 & (1+3\alpha)^2 \ 3+2\alpha & 3+4\alpha & 3+6\alpha \ 2 & 2 & 2\end{array}ight|$Applying $C_2 ightarrow C_2 - C_1$ and $C_3 ightarrow C_3 - C_2$:$= \left|\begin{array}{lll}(1+\alpha)^2 & 2\alpha(2+3\alpha) & 2\alpha(2+5\alpha) \ 3+2\alpha & 2\alpha & 2\alpha \ 2 & 0 & 0\end{array}ight|$Expanding along $R_3$:$= 2 	imes [2\alpha \cdot 2\alpha(2+5\alpha) - 2\alpha \cdot 2\alpha(2+3\alpha)] = 2 	imes [4\alpha^2(2+5\alpha - 2 - 3\alpha)] = 2 	imes [4\alpha^2(2\alpha)] = 16\alpha^3$.Wait,re-evaluating the determinant property: The determinant of a matrix where entries are quadratic in $\alpha$ is a polynomial in $\alpha$ of degree at most $3$. Given the equation $-8\alpha^3 = -648\alpha$,we have $8\alpha^3 - 648\alpha = 0$,so $8\alpha(\alpha^2 - 81) = 0$. Thus $\alpha = 0, 9, -9$. Checking the options,the correct set is $(B, C)$.

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

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