MCQ
Let $f(x)=\left|\begin{array}{ccc}a & -1 & 0 \\ a x & a & -1 \\ a x^{2} & a x & a\end{array}\right|, a \in R$. Then the sum of which the squares of all the values of a for $2 f^{\prime}(10)-f^{\prime}(5)+100=0$ is
- A$117$
- B$106$
- ✓$125$
- D$136$
$f(x)=a\left|\begin{array}{ccc}1 & -1 & 0 \\ x & a & -1 \\ x^{2} & a x & a\end{array}\right|$
$=a\left[1\left(a^{2}+a x\right)+1\left(a x+x^{2}\right)\right]$
$\Rightarrow f ( x )= a ( x + a )^{2}$
$\operatorname{so}, f^{\prime}(x)=2 a(x+a)$
as, $2 f ^{\prime}(10)- f ^{\prime}(5)+100=0$
$\Rightarrow 2 \times 2 a (10+ a )-2 a (5+ a )+100=0$
$\Rightarrow 40 a+4 a^{2}-10 a-2 a^{2}+100=0$
$2 a ^{2}+30 a +100=0$
$\Rightarrow a^{2}+15 a+50=0$
$(a+10)(a+5)=0$
$a=-10$ or $a=-5$
Required $=(-10)^{2}+(-5)^{2}=125$
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