A) - 1
B) 0
C) 1
D) Dependent of a
Correct Answer: B
Solution :
\[\frac{{{d}^{n}}}{d{{x}^{n}}}[\Delta (x)]=\left| \begin{matrix} \frac{{{d}^{n}}}{d{{x}^{n}}}{{x}^{n}} & \frac{{{d}^{n}}}{d{{x}^{n}}}\sin x & \frac{{{d}^{n}}}{d{{x}^{n}}}\cos x \\ n! & \sin \left( \frac{n\pi }{2} \right) & \cos \left( \frac{n\pi }{2} \right) \\ a & {{a}^{2}} & {{a}^{3}} \\ \end{matrix} \right|\] \[=\left| \,\,\begin{matrix} n! & \sin \left( x+\frac{n\pi }{2} \right) & \cos \left( x+\frac{n\pi }{2} \right) \\ n! & \sin \left( \frac{n\pi }{2} \right) & \cos \left( \frac{n\pi }{2} \right) \\ a & {{a}^{2}} & {{a}^{3}} \\ \end{matrix}\, \right|\] \[\Rightarrow \] \[{{[{{\Delta }^{n}}(x)]}_{x=0}}=\left| \,\begin{matrix} n\,! & \sin \,\left( 0+\frac{n\pi }{2} \right) & \cos \,\left( 0+\frac{n\pi }{2} \right) \\ n\,! & \sin \,\left( \frac{n\pi }{2} \right) & \cos \,\left( \frac{n\pi }{2} \right) \\ a & {{a}^{2}} & {{a}^{3}} \\ \end{matrix}\, \right|\,=\,0\] {Since \[{{R}_{1}}\equiv {{R}_{2}}\]}.
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