Let A=left[a_{i j}right] be a 3 × 3 matrix, where a_{i j}= 1 , quadquadtext { if } i=j quadquad-x ,

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3 and 4 .Determinants and Matrices

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Let $A=\left[a_{i j}\right]$ be a $3 \times 3$ matrix, where

$a_{i j}= 1 , \quad\quad\text { if } i=j$

$\quad\quad-x ,\quad \text { if }|i-j|=1$

$\quad\quad2 x+1, \text { otherwise }$

Let a function f: $\mathrm{R} \rightarrow \mathrm{R}$ be defined as $\mathrm{f}(\mathrm{x})=\operatorname{det}(\mathrm{A})$. Then the sum of maximum and minimum values of $f$ on $R$ is equal to:

A

$\frac{20}{27}$

B

$-\frac{88}{27}$

C

$-\frac{20}{27}$

D

$\frac{88}{27}$

(JEE MAIN-2021)

Solution

$\left[\begin{array}{ccc}1 & -x & 2 x+1 \\ -x & 1 & -x \\ 2 x+1 & -x & 1\end{array}\right]$

$|A|=4 x^{3}-4 x^{2}-4 x=f(x)$

$f(x)=4\left(3 x^{2}-2 x-1\right)=0$

$\Rightarrow x=1 ; x=\frac{-1}{3}$

$\therefore \underbrace{f(1)=-4}_{\text {min }} ; f ; \underbrace{f\left(-\frac{1}{3}\right)=\frac{20}{27}}_{\text {max }}$

$\text { Sum }=-4+\frac{20}{27}=-\frac{88}{27}$

Standard 12

Mathematics

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