Question

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

• A. 117
• B. 106
• C. 125
• D. 136

Answer 1

Answer: (C)

$f(x)=\left|\begin{array}{ccc}a& -1& 0\\ ax& a& -1\\ a^2& ax& a\end{array}\right|$
$f(x)=a\left|\begin{array}{ccc}1& -1& 0\\ x& a& -1\\ x^2& ax& a\end{array}\right|$
$=a\left[1\left(a^2+ax\right)+1\left(ax+x^2\right)\right]$
$\Rightarrow f(x)=a(x+a{)}^2$
so, $f^{\prime }(x)=2a(x+a)$
as, $2f^{\prime }(10)-f^{\prime }(5)+100=0$
$\Rightarrow 2\times 2a(10+a)-2a(5+a)+100=0$
$\Rightarrow 40a+4a^2-10a-2a^2+100=0$
$2a^2+30a+100=0$
$\Rightarrow a^2+15a+50=0$
$(a+10)(a+5)=0$
$a=-10$ or $a=-5$
Required $=(-10{)}^2+(-5{)}^2=125$