Let f(x) = left|begin{array}{ccc} a & -1 & 0 | vedclass

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(C) Given $f(x) = \left|\begin{array}{ccc} a & -1 & 0 \ ax & a & -1 \ ax^2 & ax & a \end{array}ight|$.Expanding the determinant along the first ro

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$125$

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(C) Given $f(x) = \left|\begin{array}{ccc} a & -1 & 0 \ ax & a & -1 \ ax^2 & ax & a \end{array}ight|$.Expanding the determinant along the first row:$f(x) = a(a^2 + ax) + 1(a^2x + ax^2) + 0$$f(x) = a^3 + a^2x + a^2x + ax^2 = a^3 + 2a^2x + ax^2 = a(a^2 + 2ax + x^2) = a(x + a)^2$.Now,find the derivative $f'(x)$:$f'(x) = \frac{d}{dx}[a(x + a)^2] = 2a(x + a)$.Given the condition $2f'(10) - f'(5) + 100 = 0$:$2[2a(10 + a)] - [2a(5 + a)] + 100 = 0$$4a(10 + a) - 2a(5 + a) + 100 = 0$$40a + 4a^2 - 10a - 2a^2 + 100 = 0$$2a^2 + 30a + 100 = 0$$a^2 + 15a + 50 = 0$$(a + 10)(a + 5) = 0$.So,the values of $a$ are $a = -10$ and $a = -5$.The sum of the squares of these values is $(-10)^2 + (-5)^2 = 100 + 25 = 125$.

Let $f(x) = \left|\begin{array}{ccc} a & -1 & 0 \\ ax & a & -1 \\ ax^2 & ax & a \end{array}\right|$,where $a \in R$. Then the sum of the squares of all the values of $a$ for which $2f'(10) - f'(5) + 100 = 0$ is:

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