If f ( x ) = | x 3 2 x 2 + 1 1 + 3 x 3 x 2 + 2 2 x x 3 + 6 x 3 - x 4 x 2 - 2 | for all x ? R , then 2 f ( 0 ) + f ' ( 0 ) is equal to [2024]

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Solution

Q.
If $f (x)$ $= \begin{matrix} \end{matrix}$ $| \begin{matrix} x^{3} & 2 x^{2} + 1 & 1 + 3 x \\ 3 x^{2} + 2 & 2 x & x^{3} + 6 \\ x^{3} - x & 4 & x^{2} - 2 \end{matrix} |$ for all $x \in R,$ then $2 f (0) + f' (0)$ is equal to [2024]

1 18

2 42

3 48

4 24

Ans.
(2)

$f (x)$ $=$ $| \begin{matrix} x^{3} & 2 x^{2} + 1 & 1 + 3 x \\ 3 x^{2} + 2 & 2 x & x^{3} + 6 \\ x^{3} - x & 4 & x^{2} - 2 \end{matrix} |$

$f (0)$ $=$ $| \begin{matrix} 0 & 1 & 1 \\ 2 & 0 & 6 \\ 0 & 4 & - 2 \end{matrix} |$ $= 12$

$f' (0)$ $=$ $| \begin{matrix} 0 & 0 & 3 \\ 2 & 0 & 6 \\ 0 & 4 & - 2 \end{matrix} |$ $+$ $| \begin{matrix} 0 & 1 & 1 \\ 0 & 2 & 0 \\ 0 & 4 & - 2 \end{matrix} |$ $+$ $| \begin{matrix} 0 & 1 & 1 \\ 2 & 0 & 6 \\ - 1 & 0 & 0 \end{matrix} |$

$= 24 + 0 - 6 = 18$

$\Rightarrow 2 f (0) + f' (0) = 2 \times 12 + (18) = 24 + 18 \Rightarrow 42$

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