QJEEM21Ma89fb: JEE Main 2021 Mathematics - Class 12 JEEM21Ma89fb Solution: Let A = [ a i j ] be a 3 × 3 matrix, where a i j = { 1 , i f i…

Question

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Let

A = [a_{i j}]

be a 3

\times

3 matrix, where

a_{i j} = {\begin{matrix} 1 & , & i f i = j \\ - x & , & i f | i - j | = 1 \\ 2 x + 1 & , & o t h e r w i s e . \end{matrix}

Let a function f : R

\to

R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to:

Accepted Answer

A = [\begin{matrix} 1 & - x & 2 x + 1 \\ - x & 1 & - x \\ 2 x + 1 & - x & 1 \end{matrix}]

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

f^{'} (x) = 4 (3 x^{2} - 2 x - 1) = 0

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

∴

f (1) = - 4; f (- \frac{1}{3}) = \frac{20}{27}

Sum

= - 4 + \frac{20}{7} = - \frac{88}{27}

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