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  4. If one of the roots of | 3 &5 x 7 x 7 x 5 3 | = 0 is -10,then the other roots are

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Q. If one of the roots of $\begin{vmatrix} 3 &5 & x \\ 7 & x & 7 \\ x & 5 & 3 \end{vmatrix} = 0 $ is -10,then the other roots are

BITSATBITSAT 2009

Solution:

Given, $\begin{vmatrix} 3 &5 & x \\ 7 & x & 7 \\ x & 5 & 3 \end{vmatrix} = 0 $
$\Rightarrow \ \ \ 3(3x - 35) - 5(21 - 7x) + x(35 - x^2) = 0 $
$\Rightarrow \ \ \ 9x - 105 - 105 + 35x + 35x - x^3 = 0 $
$\Rightarrow \ \ \ x^3 - 79x + 210 = 0 $
$\Rightarrow \ \ \ (x + 10) (x - 3) (x -7) = 0 $
$\Rightarrow \ \ \ x = -10, 3, 7 $