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प्रश्न
There are two values of a which makes the determinant \[∆ = \begin{vmatrix}1 & - 2 & 5 \\ 2 & a & - 1 \\ 0 & 4 & 2a\end{vmatrix}\] equal to 86. The sum of these two values is
पर्याय
4
5
- 4
9
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उत्तर
\[∆ = \begin{vmatrix}1 & - 2 & 5 \\ 2 & a & - 1 \\ 0 & 4 & 2a\end{vmatrix} = 86\]
\[ \Rightarrow 1\left( 2 a^2 + 4 \right) - 2\left( - 4a - 20 \right) = 86\]
\[ \Rightarrow 2 a^2 + 4 + 8a + 40 = 86\]
\[ \Rightarrow 2 a^2 + 8a - 42 = 0\]
\[ \Rightarrow a^2 + 4a - 21 = 0\]
\[ \Rightarrow a^2 + 7a - 3a - 21 = 0\]
\[ \Rightarrow a\left( a + 7 \right) - 3\left( a + 7 \right) = 0\]
\[ \Rightarrow \left( a + 7 \right)\left( a - 3 \right) = 0\]
\[ \Rightarrow a = - 7, 3\]
\[\text{ Sum of the two values of }a = - 7 + 3 = - 4 .\]
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