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  • Let <img alt="\mathrm{f(x)=\left|\begin{array}{ccc}a & -1 & 0 \\ a x & a & -1 \\ a x^{2} & a x & a\end{array}\right|, a \in \mathbb{R}}" src="https://entrancecorner.onc

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

Option: 1

117


Option: 2

106


Option: 3

125


Option: 4

136


Answers (1)

best_answer

\begin{aligned} 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] \\ &=a x^{2}+2 a^{2} x+a^{3}=a(x+a)^{2} . \\ &f^{\prime}(x)=2 a(x+a) \\ &f^{\prime}(10)=2 a(10+a)=2 a^{2}+20 a \\ & f^{\prime}(5)=2 a(5+a)=2 a^{2}+10 a . \\ &\Rightarrow f^{\prime}(10)-f^{\prime}(5)+100=0 \\ &\Rightarrow \left(2 a^{2}+20 a\right)-2 a^{2}-10 a+100=0 \\ &\Rightarrow 2 a^{2}+30 a+100=0\\ & \Rightarrow a^{2}+15 a+50=0 \\ &\Rightarrow a=-5,-10 . \\ &\text { sum of squares } =5^{2}+10^{2}=125 \end{aligned}

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Kuldeep Maurya

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