Question

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

• A. 48
• B. 24
• C. 42
• D. 18

Answer 1

The Correct Option is C

\[ f(0) = \begin{vmatrix} 0 & 1 & 1 \\ 2 & 0 & 6 \\ 0 & 4 & -2 \end{vmatrix} = 12 \] \[ f'(x) = \begin{vmatrix} 3x^2 & 4x & 3 \\ x^3 & 2x^2 + 1 & 1 + 3x \\ x^3 - x & 4 & x^2 - 2 \end{vmatrix} + \begin{vmatrix} x^3 & 2x^2 + 1 & 1 + 3x \\ 6x & 2x & 3x^2 \\ x^3 - x & 4 & x^2 - 2 \end{vmatrix} + \begin{vmatrix} 3x^2 + 2 & 2x & x^3 + 6 \\ 3x^2 - 1 & 0 & 2x \\ x^3 - x & 4 & x^2 - 2 \end{vmatrix} \]

therefore

\[ \begin{vmatrix} 0 & 0 & 3 \\ 2 & 0 & 6 \\ 0 & 4 & -2 \end{vmatrix} + \begin{vmatrix} 0 & 1 & 1 \\ 0 & 2 & 0 \\ 0 & 4 & -2 \end{vmatrix} + \begin{vmatrix} 0 & 1 & 1 \\ 2 & 0 & 6 \\ -1 & 0 & 0 \end{vmatrix} \] \[ = 24 - 6 = 18 \]

therefore \( 2f(0) + f'(0) = 42 \)