A) \[1\]
B) \[-1\]
C) \[0\]
D) None of these
Correct Answer: C
Solution :
[c] We have, \[\frac{{{d}^{n}}}{d{{x}^{n}}}[f(x)]=\left| \begin{matrix} 0 & \cos \left( x+\frac{n\pi }{2} \right) & \frac{{{(-1)}^{n}}n!}{{{(x+3)}^{n+1}}} \\ 0 & \cos \frac{n\pi }{2} & \frac{{{(-1)}^{n}}n!}{{{3}^{n+1}}} \\ \alpha & {{\alpha }^{3}} & {{\alpha }^{5}} \\ \end{matrix} \right|\] \[\therefore \,\,\frac{{{d}^{n}}}{d{{x}^{n}}}{{[f(x)]}_{x=0}}=\left| \begin{matrix} 0 & \cos \frac{n\pi }{2} & \frac{{{(-1)}^{n}}n!}{{{3}^{n+1}}} \\ 0 & \cos \frac{n\pi }{2} & \frac{{{(-1)}^{n}}n!}{{{3}^{n+1}}} \\ \alpha & {{\alpha }^{3}} & {{\alpha }^{5}} \\ \end{matrix} \right|=0\] \[(\because {{R}_{1}}and\,\,{{R}_{2}}are\,\,identical)\]You need to login to perform this action.
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