Question

Let $a_2, a_3 \in R$ be such that $\left|a_2-a_3\right|=6$ and $f(x)=\begin{vmatrix}1 & a_3 & a_2 \\ 1 & a_3 & 2 a_2-x \\ 1 & 2 a_3-x & a_2\end{vmatrix}, x \in R$. Then the greatest value of $f(x)$ is

• A. 6
• B. 9
• C. 12
• D. 36

Answer 1

Answer: (B)

$ R_2 \rightarrow R_2-R_1 \text { and } R_3 \rightarrow R_3-R_1 $
$f(x)=-\left(a_2-x\right)\left(a_3-x\right)=-x^2+\left(a_2+a_3\right) x-a_2 a_3$
$\left|a_2-a_3\right|=\frac{\sqrt{D}}{|a|}=6 \Rightarrow \sqrt{D}=6 $
$\text { Max. }=\frac{-D}{4 a}=\frac{36}{4}=9$