Question

If $\Delta(x) = \left| \begin{matrix} x^{n} & \sin x & \cos x \\ n! & \sin\frac{n\pi}{2} & \cos\frac{n\pi}{2} \\ a & a^{2} & a^{3} \end{matrix} \right|,$ then the value of $\frac{d^{n}}{dx^{n}}\lbrack\Delta(x)\rbrack$ at $x = 0$is.

• A. – 1
• B. 0
• C. 1
• D. Dependent of a

Answer 1

Answer: (B)

0

Answer 2

$\frac{\mathbf{d}^{\mathbf{n}}}{\mathbf{d}\mathbf{x}^{\mathbf{n}}}\mathbf{\lbrack}\mathbf{\Delta}\mathbf{(x)\rbrack =}\left| \begin{matrix} \frac{\mathbf{d}^{\mathbf{n}}}{\mathbf{d}\mathbf{x}^{\mathbf{n}}}\mathbf{x}^{\mathbf{n}} & \frac{\mathbf{d}^{\mathbf{n}}}{\mathbf{d}\mathbf{x}^{\mathbf{n}}}\mathbf{\sin}\mathbf{x} & \frac{\mathbf{d}^{\mathbf{n}}}{\mathbf{d}\mathbf{x}^{\mathbf{n}}}\mathbf{\cos}\mathbf{x} \\ \mathbf{n!} & \mathbf{\sin}\left( \frac{\mathbf{n\pi}}{\mathbf{2}} \right) & \mathbf{\cos}\left( \frac{\mathbf{n\pi}}{\mathbf{2}} \right) \\ \mathbf{a} & \mathbf{a}^{\mathbf{2}} & \mathbf{a}^{\mathbf{3}} \end{matrix} \right|$

\mathbf{n!} & \mathbf{\sin}\left( \mathbf{x +}\frac{\mathbf{n\pi}}{\mathbf{2}} \right) & \mathbf{\cos}\left( \mathbf{x +}\frac{\mathbf{n\pi}}{\mathbf{2}} \right) \\ \mathbf{n!} & \mathbf{\sin}\left( \frac{\mathbf{n\pi}}{\mathbf{2}} \right) & \mathbf{\cos}\left( \frac{\mathbf{n\pi}}{\mathbf{2}} \right) \\ \mathbf{a} & \mathbf{a}^{\mathbf{2}} & \mathbf{a}^{\mathbf{3}} \end{matrix} \right|$$ $\mathbf{\Rightarrow}$ $\mathbf{\lbrack}\mathbf{\Delta}^{\mathbf{n}}\mathbf{(x)}\mathbf{\rbrack}_{\mathbf{x = 0}}\mathbf{=}\left| \begin{matrix} \mathbf{n!} & \mathbf{\sin}\left( \mathbf{0 +}\frac{\mathbf{n\pi}}{\mathbf{2}} \right) & \mathbf{\cos}\left( \mathbf{0 +}\frac{\mathbf{n\pi}}{\mathbf{2}} \right) \\ \mathbf{n!} & \mathbf{\sin}\left( \frac{\mathbf{n\pi}}{\mathbf{2}} \right) & \mathbf{\cos}\left( \frac{\mathbf{n\pi}}{\mathbf{2}} \right) \\ \mathbf{a} & \mathbf{a}^{\mathbf{2}} & \mathbf{a}^{\mathbf{3}} \end{matrix} \right|\mathbf{= 0}$ {Since $R_{1} \equiv R_{2}$}.