(A) Given the determinant equation: $\left|\begin{array}{lll}3 & 5 & x \\ 7 & x & 7 \\ x & 5 & 3\end{array}\right|=0$ Expanding along the first row: $

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(A) Given the determinant equation: $\left|\begin{array}{lll}3 & 5 & x \\ 7 & x & 7 \\ x & 5 & 3\end{array}\right|=0$ Expanding along the first row: $3(3x - 35) - 5(21 - 7x) + x(35 - x^2) = 0$ $9x - 105 - 105 + 35x + 35x - x^3 = 0$ $-x^3 + 79x - 210 = 0$ $x^3 - 79x + 210 = 0$ Since $x = -10$ is a root, $(x + 10)$ is a factor. Performing polynomial division or synthetic division: $(x + 10)(x^2 - 10x + 21) = 0$ $(x + 10)(x - 3)(x - 7) = 0$ The roots are $x = -10, 3, 7$. Thus, the other roots are $3$ and $7$.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Expansion of determinants, Solution of equation in the form of determinants and area of triangle and Equation of Line

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If one of the roots of $\left|\begin{array}{lll}3 & 5 & x \\ 7 & x & 7 \\ x & 5 & 3\end{array}\right|=0$ is $-10$, then the other roots are

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$3, 7$
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$4, 7$
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$3, 9$
D
$3, 4$

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If one of the roots of $\left|\begin{array}{lll}3 & 5 & x \\ 7 & x & 7 \\ x & 5 & 3\end{array}\right|=0$ is $-10$, then the other roots are

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