If one of the roots of 35x7x7x53=0 is – 10, then the other roots are

35x7x7x53=0⇒3(3x−35)−5(21−7x)+x35−x2=0⇒9x−105−105+35x+35x−x3=0⇒x3−79x+210=0⇒(x+10)(x−3)(x−7)=0⇒x=−10,3,7

If one of the roots of 35x7x7x53=0 is – 10, then the other roots are

35x7x7x53=0⇒3(3x−35)−5(21−7x)+x35−x2=0⇒9x−105−105+35x+35x−x3=0⇒x3−79x+210=0⇒(x+10)(x−3)(x−7)=0⇒x=−10,3,7

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If one of the roots of $|\begin{matrix} 3 & 5 & x \\ 7 & x & 7 \\ x & 5 & 3 \end{matrix}| = 0$ is – 10, then the other roots are

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$3, 7$

$4, 7$

$3, 9$

$3, 4$

answer is A.

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Detailed Solution

$|\begin{matrix} 3 & 5 & x \\ 7 & x & 7 \\ x & 5 & 3 \end{matrix}| = 0$

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