A) p
B) \[p+{{p}^{2}}\]
C) \[p+{{p}^{3}}\]
D) Independent of p
Correct Answer: D
Solution :
\[{f}'''(x)=\left| \,\begin{matrix} \frac{{{d}^{3}}}{d{{x}^{3}}}{{x}^{3}} & \frac{{{d}^{3}}}{d{{x}^{3}}}\sin x & \frac{{{d}^{3}}}{d{{x}^{3}}}\cos x \\ 6 & -1 & 0 \\ p & {{p}^{2}} & {{p}^{3}} \\ \end{matrix}\, \right|=\left| \,\begin{matrix} 6 & -\cos x & \sin x \\ 6 & -1 & 0 \\ p & {{p}^{2}} & {{p}^{3}} \\ \end{matrix}\, \right|\] \[\therefore {f}'''(0)=\left| \,\begin{matrix} 6 & -1 & 0 \\ 6 & -1 & 0 \\ p & {{p}^{2}} & {{p}^{3}} \\ \end{matrix}\, \right|=0\], which is independent of p.
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