If \(f(x)=\left|\begin{array}{ccc}x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & …
If \(f(x)=\left|\begin{array}{ccc}x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2\end{array}\right|\) for all \(x \in R\), then \(2 f(0)+f^{\prime}(0)\) is equal to
[JEE Main 2024, 31 Jan (Shift 2)]
\(42\)
\(f(0)=\left|\begin{array}{ccc}0 & 1 & 1 \\ 2 & 0 & 6 \\ 0 & 4 & -2\end{array}\right|\)
\(=-2(-2-4)=12\)
Now
\(f^{\prime}(x)=\left|\begin{array}{ccc}3 x^2 & 4 x & 3 \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2\end{array}\right|\)\(+\left|\begin{array}{ccc}x^3 & 2 x^2+1 & 3 x+1 \\ 6 x & 2 & 3 x^2 \\ x^3-x & 4 & x^2-2\end{array}\right|\)\(+\left|\begin{array}{ccc}x^3 & 2 x^2+1 & 3 x+1 \\ 3 x^2+2 & 2 x & x^3+6 \\ 3 x^2-1 & 0 & 2 x\end{array}\right|\)
\({f}^{'}\left(0\right)=\left|\begin{matrix}0 & 0 & 3 \\ 2 & 0 & 6 \\ 0 & 4 & -2\end{matrix}\right|+\left|\begin{matrix}0 & 1 & 1 \\ 0 & 2 & 0 \\ 0 & 4 & -2\end{matrix}\right|+\left|\begin{matrix}0 & 1 & 1 \\ 2 & 0 & 6 \\ -1 & 0 & 0\end{matrix}\right|\)
\(=-2(0-12)+0+0-1(0+6)+1(0-0)\)
\(=24-6=18\)
Now
\(2 f(0)+f^{\prime}(0)= 2.12+18\)
\(= 24+18=42\)
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