"Let $$f(x) = \left| {\matrix{ a & { - 1} & 0 \cr {ax} & a & { - 1} \cr {a{x^2}} & {ax} & a \cr } } \right|,\,a \in R$$. 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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$$f(x) = \left| {\matrix{ a & { - 1} & 0 \cr {ax} & a & { - 1} \cr {a{x^2}} & {ax} & a \cr } } \right|,\,a \in R$$

$f(x) = a({a^2} + ax) + 1({a^2}x + a{x^2})$

$= a{(x + a)^2}$

$f'(x) = 2a(x + a)$

Now, $2f'(10) - f'(5) + 100 = 0$

$\Rightarrow 2.\,2a(10 + a) - 2a(5 + a) + 100 = 0$

$\Rightarrow 2a(a + 15) + 100 = 0$

$\Rightarrow {a^2} + 15a + 50 = 0$

$\Rightarrow a = - 10,\, - 5$

$\therefore$ Sum of squares of values of a = 125.

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Let $$f(x) = \left| {\matrix{ a & { - 1} & 0 \cr {ax} & a & { - 1} \cr {a{x^2}} & {ax} & a \cr } } \right|,\,a \in R$$. 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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Let $$f(x) = \left| {\matrix{ a & { - 1} & 0 \cr {ax} & a & { - 1} \cr {a{x^2}} & {ax} & a \cr } } \right|,\,a \in R$$. 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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