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प्रश्न
If the determinant \[\begin{vmatrix}0 & x^2 - a & x^3 - b \\ x^2 + a & 0 & x^2 + c \\ x^4 + b & x - c & 0\end{vmatrix} = 0 \text{ is }\]
पर्याय
a, b, c are in H . P
\[ \alpha\text{ is a root of 4a} x^2 + 12bx + 9c = 0\text{ or a, b, c are in G . P .}\]
a, b, c are in G . P . only
a, b, c are in A . P .
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उत्तर
\[\alpha\text{ is a root of 4a }x^2 + 12bx + 9c = 0\text{ or a, b, c are in G . P .}\]
\[\text{ Let }\Delta = \begin{vmatrix} a & b & 2a\alpha + 3b\\ b & c & 2b\alpha + 3c\\2a\alpha + 3b & 2b\alpha + 3c & 0 \end{vmatrix}\]
\[ = \begin{vmatrix} a - b & b & 2a\alpha + 3b\\ b - c & c & 2b\alpha + 3c\\2a\alpha + 3b - 2b\alpha - 3c & 2b\alpha + 3c & 0 \end{vmatrix} \left[\text{ Applying }C_1 \to C_1 - C_2 \right]\]
\[ = \begin{vmatrix} a - b & b & 2a\alpha + 3b\\ b - c & c & 2b\alpha + 3c\\2\left( a - b \right)\alpha + 3\left( b - c \right) & 2b\alpha + 3c & 0 \end{vmatrix}\]
\[ = \begin{vmatrix} a - b & b & 2a\alpha + 3b\\ b - c & c & 2b\alpha + 3c\\ 0 & 0 & - 2\alpha\left( 2a\alpha + 3b \right) - 3 \left( 2b\alpha + 3c \right) \end{vmatrix} \left[\text{ Applying } R_3 \to R_3 - 2\alpha, R_1 - 3 R_2 \right]\]
\[ = - 2\alpha\left( 2a\alpha + 3b \right) - 3 \left( 2b\alpha + 3c \right)\begin{vmatrix} a - b & b \\ b - c & c \end{vmatrix} \left[\text{ Expanding along }R_3 \right]\]
\[ = - \left( 4a \alpha^2 + 12b\alpha + 9c \right)\left( ac - b^2 \right)\]
\[\text{ But }\Delta = 0 \left[\text{ Given }\right]\]
\[ \Rightarrow - \left( 4a \alpha^2 + 12b\alpha + 9c \right)\left( ac - b^2 \right) = 0\]
\[ \Rightarrow \left( 4a \alpha^2 + 12b\alpha + 9c \right) = 0 \]
\[or \left( ac - b^2 \right) = 0\]
\[ \Rightarrow \alpha\text{ is a root of }4a x^2 + 12bx + 9c = 0\]
\[\text{ or ac }= b^2 \text{ , i . e . a, b, c are in G . P .} \]
